On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms

Preprint

Article

On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms

Altmetrics

Downloads

Views

Comments

Stephane Malek^*

This version is not peer-reviewed

Submitted:

05 December 2024

Posted:

06 December 2024

You are already at the latest version

Alerts

Abstract

In this paper, we examine a nonlinear partial differential equation in complex time t and complex space z combined with so-called Mahler transforms acting on time. This equation is endowed with a leading term represented by some infinite order formal differential operator of irregular type which enables the construction of a formal power series solution in t obtained by means of a Borel-Laplace procedure known as k-summability. The so-called k-sums are shown to solve some related differential functional equations involving integral transforms which stem from the analytic deceleration operators appearing in the multisummability theory for formal power series.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 35R10; 35C10; 35C15; 35C20

1. Introduction

This work is dedicated to the study of a nonlinear initial value problem that combines partial derivatives and so-called Mahler transforms with a leading term expressed by means of a formal differential operator of infinite order, with the shape

\begin{matrix} Q (\partial_{z}) u (t, z) = cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) R_{D} (\partial_{z}) u (t, z) + P (t, z, t^{k + 1} \partial_{t}, \partial_{z}, {m_{l_{2}, t}}_{l_{2} \in I}) u (t, z) \\ + Q_{1} (\partial_{z}) u (t, z) \times Q_{2} (\partial_{z}) u (t, z) + f (t, z) \end{matrix}

(1)

for prescribed vanishing initial condition

u (0, z) \equiv 0

. The constituents comprising (1) are described as follows.

The constant $α_{D} > 0$ is a positive real number and $k \geq 1$ is a given natural number.
The elements $Q (X), R_{D} (X), Q_{1} (X)$ and $Q_{2} (X)$ stand for polynomials with complex coefficients.
The expression $P (t, z, V_{1}, V_{2}, {W_{l_{2}}}_{l_{2} \in I})$ represents a polynomial in $t, V_{1}, V_{2}$ , a linear map in the arguments $W_{l_{2}}$ , for $l_{2} \in I$ , where I denotes some finite subset of the positive natural numbers $N ∖ {0}$ and a bounded holomorphic function with respect to z on a horizontal strip in $C$ of the form $H_{β} = {z \in C / | Im (z) | < β}$ , for a given real number $β > 0$ .
The forcing term $f (t, z)$ embodies a polynomial function in t with bounded holomorphic coefficients on $H_{β}$ .
The symbol $m_{l_{2}, t}$ is tagged as the Mahler transform and acts on time t through

$m_{l_{2}, t} u (t, z) = u (t^{l_{2}}, z)$

(2)

for all $l_{2} \in I$ .

The operators (2) arise from the so-called Mahler equations which are linear functional equations of the form

\sum_{k = 0}^{n} a_{k} (z) y (z^{l^{k}}) = 0

(3)

for some given integers

l \geq 2

n \geq 1

and rational coefficients

a_{k} (z) \in C (z)

. The study of these equations is nowadays a very active field of research. Many authors have recently contributed to the understanding of the structure of their solutions and have established bridges with other branches of mathematics such as automata theory or transcendence results in number theory. For the links with automatic sequences and the famous Cobham’s theorem, we refer to the seminal paper [1] by B. Adamczewski and J.P. Bell. For Galoisian aspects and hypertranscendence results related to (3), we refer to the recent work [2] by B. Adamczewski, T. Dreyfus and C. Hardouin. The algebraic structure of the solutions involving so-called Hahn series has been investigated in the series of papers [11,12] by J. Roques and [6] by C. Faverjon and J. Roques.

Mixed type equations comprising Mahler and differential operators have been much less examined, however recent substantial contributions show that they represent a propitious direction for upcoming research. Indeed, the case of coupled systems of linear differential equations and Mahler equations of the form

\{\begin{matrix} x Y^{'} (x) & = A (x) Y (x) \\ Y (x^{q}) & = B (x) Y (x) \end{matrix}

where

A (x), B (x)

are

n \times n

matrices with rational coefficients in

C (x)

and integer

q \geq 2

are considered in the work [13] by R. Schäfke and M. Singer and the general form of their meromorphic solutions on the universal covering

\tilde{C ∖ {0}}

are unveiled. In the paper [10], S. Ōuchi addresses functional equations containing both difference and Mahler operators of the form

u (z) + \sum_{j = 2}^{m} a_{j} u (z + z^{p} φ_{j} (z)) = f (z)

for some integer

p \geq 2

, complex coefficients

a_{j} \in C^{*}

and given holomorphic maps

φ_{j} (z)

and

f (z)

near the origin in

C

. He establishes the existence of a formal power series solution

\hat{u} (z) \in C [[z]]

that is proved to be

p -

summable in suitable directions (see Definition 3 of this work or the textbooks [3] and [4] for the definition of

p -

summability). More recently, in a work in progress [14], H. Yamazawa extends the above statement to more general functional equations with shape

u (z) + L (u (z + z^{p})) = f (z)

where L is a general linear differential operator of finite order with holomorphic coefficients on a disc

D_{r}

, with radius

r > 0

centered at 0, for which formal power series solutions

\hat{u} (z)

are shown to be multisummable in appropriate multidirections in the sense defined in [3], Chapter 6.

The results reached in this paper are holding the line of our previous joint study [9] with A. Lastra where we addressed the next nonlinear problem

\begin{matrix} Q (\partial_{z}) y (t, z, ϵ) = exp (α ϵ^{q} t^{q + 1} \partial_{t}) R (\partial_{z}) y (t, z, ϵ) + H (t, ϵ, {m_{κ, t, ϵ}}_{κ \in J}, ϵ^{q} t^{q + 1} \partial_{t}, \partial_{z}) y (t, z, ϵ) \\ + Q_{1} (\partial_{z}) y (t, z, ϵ) \times Q_{2} (\partial_{z}) y (t, z, ϵ) + f (t, z, ϵ) \end{matrix}

(4)

for given vanishing initial data

y (0, z, ϵ) \equiv 0

. The constant

α > 0

represents some well chosen positive real number and q is taken in the open interval

(1 / 2, 1)

. Here

Q, R, H, Q_{1}

and

Q_{2}

stand for polynomials and the forcing term f is built up in a similar way as above. The symbol

m_{κ, t, ϵ}

, for

κ \in J

(where J stands for a finite subset of the positive real numbers

R_{+}^{*}

), is labeled as the Moebius transform operating on time t by means of

m_{κ, t, ϵ} y (t, z, ϵ) = y (\frac{t}{1 + κ ϵ t}, z, ϵ) .

An additional dependence with respect to a complex parameter

ϵ \in C^{*}

is assumed compared to (1) which gives (4) the quality of a singularly perturbed equation.

For some suitable bounded sector

T

edged at 0 in

C^{*}

and a set

\underset{̲}{E} = {E_{p}}_{0 \leq p \leq ς - 1}

of bounded sectors edged at 0 whose union contains a full neighborhood of 0 in

C^{*}

, we construct genuine bounded holomorphic solutions

y_{p} (t, z, ϵ)

to (4) on the product

T \times H_{β} \times E_{p}

, expressed through a Laplace transform of order q and Fourier inverse transform

y_{p} (t, z, ϵ) = \frac{q}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d_{p}}} ω^{d_{p}} (u, m, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) e^{\sqrt{- 1} z m} \frac{d u}{u} d m

along well chosen halflines

L_{d_{p}} = [0, + \infty) e^{\sqrt{- 1} d_{p}}

with

d_{p} \in R

, where

ω^{d_{p}} (u, m, ϵ)

represents a function called Borel-Fourier map featuring exponential growth of order q on some sector containing

L_{d_{p}}

with respect to u, showing exponential decay relatively to m on

R

and relying analytically on

ϵ

near 0. Furthermore, the partial maps

ϵ \mapsto y_{p} (t, z, ϵ)

are shown to share on the sectors

E_{p}

a common asymptotic expansion

\hat{y} (t, z, ϵ) = \sum_{n \geq 0} y_{n} (t, z) ϵ^{n}

which represents a formal power series in

ϵ

with bounded holomorphic coefficients

y_{n}

n \geq 0

, on the product

T \times H_{β}

. This asymptotic expansion turns out to be (at most) of Gevrey order

1 / q

meaning that constants

C, M > 0

can be singled out with

sup_{t \in T, z \in H_{β}} | y_{p} (t, z, ϵ) - \sum_{n = 0}^{N - 1} y_{n} (t, z) ϵ^{n} | \leq C M^{N} Γ (1 + \frac{N}{q}) {| ϵ |}^{N}

for all integers

N \geq 1

, whenever

ϵ \in E_{p}

The leading term of (4) consists in a formal differential operator of infinite order with respect to t,

exp (α ϵ^{q} t^{q + 1} \partial_{t}) R (\partial_{z}) = \sum_{p \geq 0} \frac{{(α ϵ^{q})}^{p}}{p!} {(t^{q + 1} \partial_{t})}^{(p)} R (\partial_{z})

(5)

where

{(t^{q + 1} \partial_{t})}^{(p)}

stands for the

p -

th iterate of the irregular differential operator

t^{q + 1} \partial_{t}

. The reason for the appearance of such a principal term with infinite order is triggered by the presence of the Moebius transforms

m_{κ, t, ϵ}

κ \in J

, which forbids leading finite order differential operators.

In the present contribution, our aim is to carry out a similar procedure by means of Fourier-Laplace transforms in order to construct solutions to (1) and to related problems to (1). However, the occurence of the Mahler transforms

{m_{l_{2}, t}}_{l_{2} \in I}

in the main term P of (1) modifies utterly the whole picture in comparison with [9].

As a first major change, the choice of a principal term with shape (5) is now insufficient to guarantee the construction of solutions to (1) in our framework. We supplant it by an exponential formal differential operator of higher order

cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) R_{D} (\partial_{z}) = \frac{1}{2} (exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) + exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2})) R_{D} (\partial_{z}) .

The reasons for such an option will be motivated later on in the introduction.

Under fitting conditions on the shape of our main equation (1) itemized in the statement of Theorem 1, Section 5, we can construct a formal power series

\hat{u} (t, z) = \sum_{n \geq 1} u_{n} (z) t^{n}

whose coefficients

u_{n}

n \geq 0

, are bounded holomorphic on

H_{β}

, which solves (1) with vanishing initial data

\hat{u} (0, z) \equiv 0

. This formal series is built up through a Borel-Laplace method similar to the classical

k -

summability approach discussed in [3], that we call

m_{k} -

summability, whose basic results are recalled in Subsection 3.2.1. It means that we can exhibit analytic maps

u^{d} (t, z)

on products

S_{d, ϑ, R} \times H_{β}

where

S_{d, ϑ, R}

stands for a bounded sector edged at 0 with some small radius

R > 0

, well chosen bisecting direction

d \in R

(among a set

Θ_{Q, R_{D}}

explicitely depicted in Lemma 2) and opening

ϑ

slightly larger than

π / k

, such that

the map $u^{d} (t, z)$ is expressed as Fourier-Laplace transform

$u^{d} (t, z) = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} ω^{d} (τ, m) exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m$

where $ω^{d} (τ, m)$ is called the $m_{k} -$ Borel transform of $\hat{u} (t, z)$ with respect to t (see Definition 3) which

–

defines an analytic map with respect to $τ$ with (at most) exponential growth of order k on a union $D_{ρ} \cup S_{d}$ , where $D_{ρ}$ is a disc with small radius $ρ > 0$ and $S_{d}$ is an unbounded sector bisected by d, edged at 0 with small aperture,

–

represents a continuous function relatively to m on $R$ with exponential decay at infinity.
the partial map $t \mapsto u^{d} (t, z)$ is the unique holomorphic map on $S_{d, ϑ, R}$ which has the formal series $\hat{u} (t, z)$ as asymptotic expansion of Gevrey order $1 / k$ , meaning that one can find two constants $C, M > 0$ for which

$sup_{z \in H_{β}} | u^{d} (t, z) - \sum_{n = 1}^{N - 1} u_{n} (z) t^{n} | \leq C M^{N} Γ (1 + \frac{N}{k}) {| t |}^{N}$

holds for all integers $N \geq 2$ , provided that $t \in S_{d, ϑ, R}$ .

Another substantial contrast between the problems (1) and (4) lies in the observation that the holomorphic maps

u^{d} (t, z)

do not (in general) obey the main equation solved by

\hat{u} (t, z)

(see the concluding remark of the work). Instead,

u^{d} (t, z)

is shown to solve two different related functional differential equations (225) or (226) depending on the location of the unbounded sector

S_{d}

C

, see Theorem 2 in Section 5.

In Subsection 3.2.2, we show that the action of the Mahler operator

m_{l_{2}, t}

, for

l_{2} \geq 2

, on the formal series

\hat{u} (t, z)

is described by some integral operator acting on its

m_{k} -

Borel transform

τ \mapsto ω^{d} (τ, m)

{\hat{D}}_{k, k / l_{2}} (ω^{d}) (τ^{l_{2}}) : = - \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} ω^{d} (ξ, m) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(6)

along a closed Hankel path

{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}

confined nearby the origin in

C

. These operators are derived from a version of the analytic deceleration operators introduced by J. Écalle, which turn out to be the inverse for the composition of the so-called analytic acceleration operators which play a central role in the theory of multisummability, see [3], Chapters 5 and 6. As shown in Proposition 7, it comes out that the kernel

τ \mapsto D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}})

appearing in (6) has (at most) an exponential growth rate of order

\frac{k l_{2}}{l_{2} - 1}

on the sector

S_{d}

, which implies that the analytic map

τ \mapsto {\hat{D}}_{k, k / l_{2}} (ω^{d}) (τ^{l_{2}})

itself owns upper bounds of the form

C (m) exp (K | τ |^{\frac{k l_{2}}{l_{2} - 1}})

(7)

provided that

τ \in S_{d}

, for some constant

K > 0

and map

m \mapsto C (m)

with exponential decay on

R

, see Sublemma 1 and 2. As a result, the presence of an infinite order operator with the shape

cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2})

in the leading term of (1) seems mandatory since it acts in the Borel plane on

τ \mapsto ω^{d} (τ, m)

as the multiplication by the map

τ \mapsto cosh (α_{D} {(k τ^{k})}^{2})

whose exponential growth rate on the sector

S_{d}

is of order

2 k

which exceeds

\frac{k l_{2}}{l_{2} - 1}

, for

l_{2} \geq 2

, and compensates the bounds (7). Furthermore, we have favored the cosh function instead of the exponential function exp since it allows a larger choice of sectors

S_{d}

in both the left and right halfplanes

C_{-} = {z \in C / Re (z) < 0}

and

C_{+} = {z \in C / Re (z) > 0}

The exceeding growth rate (7) coming from the action of the Mahler operator

m_{l_{2}, t}

in the Borel plane also compromises the

m_{k} -

sum

u^{d} (t, z)

\hat{u} (t, z)

to become a genuine solution to (1) since only functions with (at most) exponential growth of order k are Laplace transformable. However, we can exhibit some modified functional equations displayed in (225) or (226) involving the analytic transforms (6) that

u^{d}

is shown to obey.

2. Layout of the Main Initial Value Problem

The main problem under study in this work is described as follows

\begin{matrix} Q (\partial_{z}) u (t, z) = cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) R_{D} (\partial_{z}) u (t, z) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A} a_{\underset{̲}{l}} (z) (t^{l_{0}} {(t^{k + 1} \partial_{t})}^{l_{1}} R_{\underset{̲}{l}} (\partial_{z}) u) (t^{l_{2}}, z) \\ + c_{Q_{1} Q_{2}} Q_{1} (\partial_{z}) u (t, z) \times Q_{2} (\partial_{z}) u (t, z) + f (t, z) \end{matrix}

(8)

for given vanishing initial data

u (0, z) \equiv 0

. Below, we display a list of conditions we set on the building blocks of (8). Namely,

–

The constant

k \geq 1

is a natural number. We set

α_{D}

as a positive real number and

c_{Q_{1} Q_{2}} \in C^{*}

stands for a non vanishing complex number. Constraints will be set on these constants that will be disclosed later on in the work, see Proposition 9, Section 4.

–

The set

A

represents a finite subset of

N^{3}

which is asked to fulfill the next record of restrictions.

For all $\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A$ , the next inequality

$l_{2} \leq k$

(9)

holds.
Provided that $\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A$ , the next constraint

$l_{0} + k l_{1} \geq \frac{k}{l_{2}}$

(10)

is required.

–

The maps

Q (X), R_{D} (X), R_{\underset{̲}{l}} (X)

for

\underset{̲}{l} \in A

along with

Q_{1} (X)

and

Q_{2} (X)

are polynomials with complex coefficients. Their degrees are required to obey the next inequalities

\deg (Q) = \deg (R_{D}) \geq \deg (R_{\underset{̲}{l}})

(11)

for all

\underset{̲}{l} \in A

. In addition, we impose that

\deg (R_{D}) \geq max (\deg (Q_{1}), \deg (Q_{2})) .

(12)

Besides, the next technical assumption of geometric nature is set on the polynomials

Q (X)

and

R_{D} (X)

. We ask for the existence of a bounded sectorial annulus

S_{Q, R_{D}} = {z \in C^{*} / r_{Q, R_{D}, 1} \leq | z | \leq r_{Q, R_{D}, 2}, | \arg (z) - d_{Q, R_{D}} | \leq η_{Q, R_{D}}}

(13)

with bisecting direction

d_{Q, R_{D}} \in R

, aperture

η_{Q, R_{D}} > 0

and with inner and outer radii

0 < r_{Q, R_{D}, 1} < r_{Q, R_{D}, 2}

fulfilling the inclusion

{\frac{Q (\sqrt{- 1} m)}{R_{D} (\sqrt{- 1} m)} / m \in R} \subset S_{Q, R_{D}} .

(14)

Precise requirements on the shape of the sectorial annulus

S_{Q, R_{D}}

will be exposed later on in Section 4, see Lemma 2.

–

The differential operator of infinite order

cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2})

is defined as the sum

cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) = \frac{1}{2} (exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) + exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2})),

(15)

where each term is defined as a formal expansion

exp (\pm α_{D} {(t^{k + 1} \partial_{t})}^{2}) = \sum_{p \geq 0} \frac{{(\pm α_{D})}^{p}}{p!} {(t^{k + 1} \partial_{t})}^{(2 p)}

where

{(t^{k + 1} \partial_{t})}^{(2 p)}

stands for the

2 p

-th iterate of the differential operator

t^{k + 1} \partial_{t}

In order to describe the properties of the coefficients

a_{\underset{̲}{l}} (z)

for

\underset{̲}{l} \in A

and forcing term

f (t, z)

, we need to recall the definition of some Banach space of continous functions introduced in the work [5] and the action of the Fourier inverse transform on these spaces. The next two definitions already appear in the work [7].

Definition 1.

Let

β, μ

be positive real numbers. We set

E_{(β, μ)}

as the vector space of continuous functions

h : R \to C

such that the norm

{| | h (m) | |}_{(β, μ)} = sup_{m \in R} {(1 + | m |)}^{μ} exp (β | m |) | h (m) |

is finite. We observe that the space

E_{(β, μ)}

equipped with the norm

{| | . | |}_{(β, μ)}

represents a Banach space. Furthermore, for given elements

f, g

E_{(β, μ)}

, let us denote

(f * g) (m) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} f (m - m_{1}) g (m_{1}) d m_{1}

(16)

the convolution product of

f, g

. Assume that

μ > 1

. Then,

f * g

belongs to

E_{(β, μ)}

and the next inequality

| | f * {g | |}_{(β, μ)} \leq C_{μ} {| | f | |}_{(β, μ)} {| | g | |}_{(β, μ)}

holds for some constant

C_{μ} > 0

relying on μ. In particular, the Banach space

(E_{(β, μ)}, | | . | |_{(β, μ)})

equipped with the product * turns out to become a Banach algebra.

Definition 2.

Let

f \in E_{(β, μ)}

with

β > 0

μ > 1

. The inverse Fourier transform of f is given by the next integral transform

F^{- 1} (f) (x) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} f (m) exp (\sqrt{- 1} x m) d m

(17)

for all

x \in R

. We observe that the function

F^{- 1} (f)

extends to an analytic bounded function on all strips

H_{β^{'}} = {z \in C / | Im (z) | < β^{'}},

(18)

for given

0 < β^{'} < β

a) We set the function

m \mapsto ϕ (m) = \sqrt{- 1} m f (m)

which belongs to the space

E_{(β, μ - 1)}

. Then, the next differential identity

\partial_{z} F^{- 1} (f) (z) = F^{- 1} (ϕ) (z)

(19)

occurs for all

z \in H_{β^{'}}

with

0 < β^{'} < β

b) Let g be an element of

E_{(β, μ)}

and set ψ as the convolution product of f and g given by the expression (16). Then, the next product formula

F^{- 1} (f) (z) F^{- 1} (g) (z) = F^{- 1} (ψ) (z)

(20)

holds for all

z \in H_{β^{'}}

provided that

0 < β^{'} < β

–

The coefficients

a_{\underset{̲}{l}} (z)

are crafted as follows. For all

\underset{̲}{l} \in A

, we consider maps

m \mapsto A_{\underset{̲}{l}} (m)

that belong to the space

E_{(β, μ)}

, for given

β > 0

and where

μ > 0

is subjected to the conditions

μ > \deg (R_{\underset{̲}{l}}) + 1, μ > max (\deg (Q_{1}) + 1, \deg (Q_{2}) + 1)

(21)

for all

\underset{̲}{l} \in A

. For later use, we denote

A_{\underset{̲}{l}} = | | A_{\underset{̲}{l}} {| |}_{(β, μ)} .

(22)

The upper size of

A_{\underset{̲}{l}}

will be fixed later in due course of the paper, see Proposition 9, Section 4. We set

a_{\underset{̲}{l}} (z) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m) e^{\sqrt{- 1} z m} d m

(23)

as the inverse Fourier transform of

m \mapsto A_{\underset{̲}{l}} (m)

. From Definition 2, it defines a bounded holomorphic function on all the strips

H_{β^{'}}

for any prescribed

0 < β^{'} < β

–

The forcing term

f (t, z)

is built up in the next way. Let

J \in N^{*}

be a subset of the positive natural numbers. For

j \in J

, we mind a map

m \mapsto F_{j} (m)

which belongs to the space

E_{(β, μ)}

for the real numbers

β > 0

and

μ > 1

satisfying (21) given in the previous item. We introduce the notation

F_{j} = | | F_{j} {| |}_{(β, μ)}

(24)

for

j \in J

. The forcing term

f (t, z)

is defined as the next polynomial in t

f (t, z) = \sum_{j \in J} F_{j} (z) Γ (j / k) t^{j}

(25)

where

–: The symbol $Γ (x)$ stands for the classical Gamma function $Γ (x) = \int_{0}^{+ \infty} t^{x - 1} e^{- t} d t$ , for any $x > 0$ .
–: The coefficients $F_{j} (z)$ , $j \in J$ , are the inverse Fourier transforms

$F_{j} (z) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} F_{j} (m) e^{\sqrt{- 1} z m} d m$

(26)

of $F_{j}$ . From Definition 2, $z \mapsto F_{j} (z)$ defines a bounded holomorphic map on any strip $H_{β^{'}}$ with $0 < β^{'} < β$ .

We introduce the next polynomial in the variable

τ

with coefficients in

E_{(β, μ)}

F (τ, m) = \sum_{j \in J} F_{j} (m) τ^{j} .

(27)

According to the definition of the Gamma function, we observe that the forcing term

f (t, z)

has an integral representation as a Laplace transform of order k and inverse Fourier integral

f (t, z) = \frac{k}{{(2 π)}^{1 / 2}} \int_{L_{d_{1}}} \int_{- \infty}^{+ \infty} F (τ, m) exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m

(28)

where

L_{d_{1}} = [0, + \infty) e^{\sqrt{- 1} d_{1}}

stands for any halfline in direction

d_{1} \in R

that depends on the variable t through the restriction

cos (k (d_{1} - \arg (t))) > 0

. Such a representative will be useful in the next section 3.

3. Reduction of the Main Problem to an Integral Equation

In this section, we perform two important reductions of our initial value problem. In the first subsection, we reduce our problem to the study of a differential/convolution equation involving Mahler transforms by means of a Fourier transform. In the second subsection, we further reduce the problem to an integral equation through the application of formal Borel/Laplace transforms of order k. This second reduction is essential in the achievement of our first main result, Theorem 1 in Section 5.

3.1. First Reduction to a Differential/Convolution Equation with Mahler Transforms

We search for solutions to (8) in the form of an inverse Fourier transform

u (t, z) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} U (t, m) e^{\sqrt{- 1} z m} d m

(29)

for some expression

U (t, m)

such that the partial maps

m \mapsto U (t, m)

belong to the space

E_{(β, μ)}

for

β, μ > 0

prescribed in Section 2. The precise shape of

U (t, m)

will be unveiled in the next subsection. With the help of Definition 2, we reach the next

Proposition 1.

The integral expression

u (t, z)

given by (29) formally solves (8)ifthe map

U (t, m)

obeys the next differential/convolution equation comprising Mahler transforms

\begin{matrix} Q (\sqrt{- 1} m) U (t, m) = cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) R_{D} (\sqrt{- 1} m) U (t, m) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (t^{l_{0}} {(t^{k + 1} \partial_{t})}^{l_{1}} U) (t^{l_{2}}, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + c_{Q_{1} Q_{2}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} U (t, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) U (t, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{j \in J} F_{j} (m) Γ (j / k) t^{j} \end{matrix}

(30)

for given initial data

U (0, m) \equiv 0

3.2. Reduction to an Integral Equation

We seek for a solution to the reduced equation (30) expressed as a formal power series

\hat{U} (t, m) = \sum_{n \geq 1} U_{n} (m) t^{n}

(31)

in the time variable t with coefficients

m \mapsto U_{n} (m)

that belong to

E_{(β, μ)}

for

β, μ > 0

assigned in Section 2. In the next two subsections, we provide the required prefatory material for our second step of reduction.

3.2.1. Essentials on Banach Valued $m_{k} -$ Summable Formal Power Series

The objective of this subsection is to remind the reader the notion of

m_{k} -

summability and its basic properties as decribed in our previous work [7] which is a slight adjustment of the concept of

k -

summability discussed in the textbook [4].

Definition 3.

Let

(E, | | . | |_{E})

be a complex Banach space. We select an integer

k \geq 1

and define the sequence

m_{k} (n) = Γ (n / k)

, for all

n \geq 1

. A formal power series

\hat{U} (t) = \sum_{n \geq 1} a_{n} t^{n} \in t E [[t]]

is called

m_{k} -

summable with respect to t in the direction

d \in R

The so-called formal $m_{k} -$ Borel transform of $\hat{U} (t)$ defined by the power series

$B_{m_{k}} (\hat{U}) (τ) = \sum_{n \geq 1} \frac{a_{n}}{Γ (n / k)} τ^{n} \in τ E [[τ]]$

(32)

is convergent on a disc $D_{ρ}$ for some $ρ > 0$ .
The convergent series $B_{m_{k}} (\hat{U}) (τ)$ can be analytically continued with respect to τ (as a function still denoted $B_{m_{k}} (\hat{U}) (τ)$ ) on some unbounded sector

$S_{d, δ} = {τ \in C^{*} / | d - \arg (τ) | < δ}$

with aperture $2 δ$ and bisecting direction $d \in R$ . Moreover, two constants $C > 0$ and $K > 0$ can be found such that

$| | B_{m_{k}} (\hat{U}) (τ) {| |}_{E} \leq C e^{{K | τ |}^{k}}$

(33)

for all $τ \in S_{d, δ}$ .

Let

\hat{U} (t)

be a

m_{k} -

summable formal power series with respect to t in a direction d. We define the Laplace transform of order k in direction d of the

m_{k} -

Borel transform

B_{m_{k}} (\hat{U}) (τ)

by the integral transform

L_{m_{k}}^{d} (B_{m_{k}} (\hat{U})) (t) = k \int_{L_{γ}} B_{m_{k}} (\hat{U}) (τ) exp (- {(\frac{τ}{t})}^{k}) \frac{d τ}{τ}

(34)

along a halfline

L_{γ} = [0, + \infty) e^{\sqrt{- 1} γ} \subset S_{d, δ} \cup {0}

, where γ relies on t and matches the inequality

cos (k (γ - \arg (t))) > Δ_{1}

for some constant

Δ_{1} > 0

The function

t \mapsto L_{m_{k}}^{d} (B_{m_{k}} (\hat{U})) (t)

is bounded holomorphic on any bounded sector

S_{d, θ, R^{1 / k}} = {t \in C^{*} / | t | < R^{1 / k}, | d - \arg (t) | < θ / 2}

(35)

where

\frac{π}{k} < θ < \frac{π}{k} + 2 δ

and

0 < R < \frac{Δ_{1}}{K}

, for K appearing in (33). This function is called the

m_{k} -

sum of

\hat{U} (t)

in the direction d.

The above definition of

m_{k} -

sum of a formal power series is justified by the next proposition (see also Proposition 11 p. 75 from [4] known as Watson’s Lemma)

Proposition 2.

Let

\hat{U} (t)

be a

m_{k} -

summable formal power series with respect to t in some direction d. Then, the Laplace transform

t \mapsto L_{m_{k}}^{d} (B_{m_{k}} (\hat{U})) (t)

is the unique holomorphic map on

S_{d, θ, R^{1 / k}}

which has the formal power series

\hat{U} (t)

as asymptotic expansion of Gevrey order

1 / k

with respect to t on

S_{d, θ, R^{1 / k}}

for any given opening θ under the condition

\frac{π}{k} < θ < \frac{π}{k} + 2 δ

. It means that one can find two constants

C, M > 0

for which the next inequality

| | L_{m_{k}}^{d} (B_{m_{k}} (\hat{U})) (t) - \sum_{p = 1}^{N - 1} a_{p} t^{p} {| |}_{E} \leq C M^{N} Γ (1 + \frac{N}{k}) {| t |}^{N}

(36)

holds for all integers

N \geq 2

, all

t \in S_{d, θ, R^{1 / k}}

In the next proposition, we recall some crucial identities for the formal

m_{k} -

Borel transform under the action of differential operators of irregular type, multiplication by a monomial and products (see Proposition 6 from [7]).

Proposition 3.

Let

(E, | | . | |_{E})

be a complex Banach algebra whose product is denoted *. Let

k, l \geq 1

be natural numbers. Let

{\hat{U}}_{j} (t)

j = 1, 2

, be elements of

t E [[t]]

. The next formal identities hold

\begin{matrix} B_{m_{k}} (t^{k + 1} \partial_{t} {\hat{U}}_{1} (t)) (τ) = k τ^{k} B_{m_{k}} ({\hat{U}}_{1}) (τ), \\ B_{m_{k}} (t^{l} {\hat{U}}_{1} (t)) (τ) = \frac{τ^{k}}{Γ (l / k)} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l}{k} - 1} B_{m_{k}} ({\hat{U}}_{1}) (s^{1 / k}) \frac{d s}{s}, \\ B_{m_{k}} ({\hat{U}}_{1} (t) {\hat{U}}_{2} (t)) (τ) = τ^{k} \int_{0}^{τ^{k}} B_{m_{k}} ({\hat{U}}_{1}) ({(τ^{k} - s)}^{1 / k}) B_{m_{k}} ({\hat{U}}_{2}) (s^{1 / k}) \frac{1}{(τ^{k} - s) s} d s \end{matrix}

(37)

where in the last formula, the product of formal power series is built up by means of the product * in the Banach algebra

E

In the next proposition, we provide the counterpart of the above proposition for the action of differential operators of irregular type, multiplication by a monomial and products for

m_{k} -

sums of formal power series. Its proof is similar to the one given for Lemma 2 of [9].

Proposition 4.

Let

(E, | | . | |_{E}, *)

be a complex Banach algebra. Let

k, l \geq 1

be natural numbers. We consider

{\hat{U}}_{j} (t)

j = 1, 2

, two elements of

t E [[t]]

that are assumed to be

m_{k} -

summable in some direction

d \in R

. For

j = 1, 2

, we set

U_{j}^{d} (t) = k \int_{L_{d}} ω_{j} (τ) exp (- {(\frac{τ}{t})}^{k}) \frac{d τ}{τ}

the

m_{k} -

sum of

{\hat{U}}_{j} (t)

in direction d, where

ω_{j} (τ)

denotes the

m_{k} -

Borel transform of

{\hat{U}}_{j} (t)

. The next identities hold whenever

t \in S_{d, θ, R^{1 / k}}

provided that

θ > π / k

and is taken close enough to

π / k

and the radius

R > 0

is chosen in the vicinity of the origin,

\begin{matrix} t^{k + 1} \partial_{t} U_{j}^{d} (t) = k \int_{L_{d}} {k τ^{k} ω_{j} (τ)} exp (- {(\frac{τ}{t})}^{k}) \frac{d τ}{τ}, \\ t^{l} U_{j}^{d} (t) = k \int_{L_{d}} \{\frac{τ^{k}}{Γ (l / k)} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l}{k} - 1} ω_{j} (s^{1 / k}) \frac{d s}{s}\} exp (- {(\frac{τ}{t})}^{k}) \frac{d τ}{τ}, \\ U_{1}^{d} (t) U_{2}^{d} (t) = k \int_{L_{d}} \{τ^{k} \int_{0}^{τ^{k}} ω_{1} ({(τ^{k} - s)}^{1 / k}) ω_{2} (s^{1 / k}) \frac{1}{(τ^{k} - s) s} d s\} exp (- {(\frac{τ}{t})}^{k}) \frac{d τ}{τ}, \end{matrix}

(38)

where in the closing formula, the product of

E -

valued functions is built up by means of the product * of the algebra

E

3.2.2. Action of the Mahler operators on formal $m_{k} -$ Borel transforms

The aim of this subsection is twofold. We first derive a formula describing the action of the Mahler operators on formal

m_{k} -

Borel transforms for formal power series through so-called formal deceleration operators. Then, for later use in Section 3.2.3, under some additional assumptions, we provide an integral representation of these deceleration operators constructed with the help with some kernel function. At last, we provide some important analytic features of this kernel function.

The next definition is a slightly modified version of the deceleration operators defined in the textbook [3], p. 46.

Definition 4.

Let

(E, | | . | |_{E})

be a complex Banach space. Let

1 \leq k < k^{'}

be two rational numbers. We define the

(k^{'}, k) -

deceleration operator

{\hat{D}}_{k^{'}, k}

from

τ E [[τ]]

into

h E [[h]]

by the formula

{\hat{D}}_{k^{'}, k} (\hat{f}) (h) : = \sum_{n \geq 1} f_{n} \frac{Γ (n / k^{'})}{Γ (n / k)} h^{n}

(39)

for all elements

\hat{f} (τ) = \sum_{n \geq 1} f_{n} τ^{n}

τ E [[τ]]

Remark: This formal deceleration operator

{\hat{D}}_{k^{'}, k}

turns out to be the inverse for the composition of the so-called acceleration operator

{\hat{A}}_{k^{'}, k}

acting on formal series

\hat{f} (τ) = \sum_{n \geq 1} f_{n} τ^{n}

through

{\hat{A}}_{k^{'}, k} (\hat{f}) (h) : = \sum_{n \geq 1} f_{n} \frac{Γ (n / k)}{Γ (n / k^{'})} h^{n}

introduced in the paper [8], Section 4.3, by A. Lastra and the author and which stands for an adjusted version of the classical acceleration operators as defined in [3], Chapter 5.

The next proposition discloses a formula for the formal

m_{k} -

Borel transform under the action of a Mahler operator.

Proposition 5.

Let

(E, | | . | |_{E})

be a complex Banach space. Let

p \geq 2

and

k \geq 1

be natural numbers with

k \geq p

. Let

\hat{U} (t)

be an element of

t E [[t]]

. We set

\hat{V} (t) = \hat{U} (t^{p})

the element of

t E [[t]]

obtained by applying the Mahler operator

t \mapsto t^{p}

\hat{U} (t)

. The next identity holds

B_{m_{k}} (\hat{V}) (τ) = {\hat{D}}_{k, k / p} (B_{m_{k}} (\hat{U})) (τ^{p})

(40)

Proof.

Let us expand

\hat{U}

\hat{U} (t) = \sum_{n \geq 1} u_{n} t^{n}

. Hence,

\hat{V} (t) = \hat{U} (t^{p}) = \sum_{n \geq 1} u_{n} t^{p n}

. According to the first item of Definition 3, we observe that

B_{m_{k}} (\hat{V}) (τ) = \sum_{n \geq 1} u_{n} \frac{τ^{p n}}{Γ (\frac{p n}{k})} .

(41)

On the other hand, the

m_{k} -

Borel transform of

\hat{U}

writes

B_{m_{k}} (\hat{U}) (τ) = \sum_{n \geq 1} u_{n} τ^{n} / Γ (n / k)

and by Definition 4, we deduce that

{\hat{D}}_{k, k / p} (B_{m_{k}} (\hat{U})) (h) = \sum_{n \geq 1} \frac{u_{n}}{Γ (n / k)} \frac{Γ (n / k)}{Γ (\frac{n}{k / p})} h^{n} = \sum_{n \geq 1} \frac{u_{n}}{Γ (\frac{p n}{k})} h^{n}

(42)

At last, the combination of (41) and (42) yields (40). □

In the next proposition, we display an integral representation for the

(k^{'}, k) -

deceleration operator

{\hat{D}}_{k^{'}, k}

under further assumptions on its source space.

Proposition 6.

Let

(E, | | . | |_{E})

be a complex Banach space. Let

1 \leq k < k^{'}

be two rational numbers. We consider an element

\hat{f} (τ)

τ E [[τ]]

which is assumed to be convergent on some disc

D_{ρ}

with

ρ > 0

. For a given

h \in C^{*}

, we attach the next two items.

We choose a direction $γ_{h} \in R$ and a positive real number $Δ_{1} > 0$ such that

$cos (k^{'} (γ_{h} - \arg (h))) > Δ_{1} .$

(43)
We consider the so-called closed Hankel path denoted ${\tilde{γ}}_{k, h}$ depicted as the union of

–

the oriented segment ${\tilde{γ}}_{k, h, 1} = [0, (ρ / 2) e^{\sqrt{- 1} (γ_{h} + \frac{π}{2 k} + \frac{δ^{'}}{2})}]$

–

the oriented arc of circle

${\tilde{γ}}_{k, h, 3} = {(ρ / 2) e^{\sqrt{- 1} θ} / θ \in [γ_{h} + \frac{π}{2 k} + \frac{δ^{'}}{2}, γ_{h} - \frac{π}{2 k} - \frac{δ^{'}}{2}]}$

–

the oriented segment ${\tilde{γ}}_{k, h, 2} = [(ρ / 2) e^{\sqrt{- 1} (γ_{h} - \frac{π}{2 k} - \frac{δ^{'}}{2})}, 0]$

where $δ^{'} > 0$ is a positive real number taken close to 0.

Then, the

(k^{'}, k) -

deceleration operator

{\hat{D}}_{k^{'}, k}

has the following integral representation

{\hat{D}}_{k^{'}, k} (\hat{f}) (h) = - \frac{k^{'} k}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{k, h}} \hat{f} (ξ) D_{k^{'}, k} (ξ, h) d ξ

(44)

for all

h \in C^{*}

, where the kernel

D_{k^{'}, k} (ξ, h)

is expressed by means of the integral

D_{k^{'}, k} (ξ, h) = \frac{1}{ξ^{k + 1}} \int_{L_{γ_{h}}} u^{k - 1} exp ({(\frac{u}{ξ})}^{k} - {(\frac{u}{h})}^{k^{'}}) d u

(45)

along a the halfline

L_{γ_{h}} = [0, + \infty) e^{\sqrt{- 1} γ_{h}}

Proof.

Let

h \in C^{*}

. At first, from the very definition of the Gamma function, we observe that for all natural numbers

n \geq 1

, we have the next integral form for the monomial

Γ (n / k^{'}) h^{n} = k^{'} \int_{L_{γ_{h}}} u^{n} exp (- {(\frac{u}{h})}^{k^{'}}) \frac{d u}{u}

(46)

along the halfline

L_{γ_{h}}

given in the statement of the proposition 6.

Owing to Proposition 12 from [8] and based on the so-called Hankel formula (see [4], Appendix B.3), we can rewrite the next monomial in an integral form

\frac{u^{n}}{Γ (n / k)} = - \frac{k}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{k, h}} ξ^{n} exp ({(\frac{u}{ξ})}^{k}) \frac{u^{k}}{ξ^{k + 1}} d ξ

(47)

along the Hankel path

{\tilde{γ}}_{k, h}

detailed in the second item of Proposition 6, for all

u \in L_{γ_{h}}

As a result of (46) together with (47), an application of the Fubini theorem yields

\begin{matrix} \frac{Γ (n / k^{'})}{Γ (n / k)} h^{n} = k^{'} \int_{L_{γ_{h}}} \frac{u^{n}}{Γ (n / k)} exp (- {(\frac{u}{h})}^{k^{'}}) \frac{d u}{u} \\ = k^{'} \int_{L_{γ_{h}}} (- \frac{k}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{k, h}} ξ^{n} exp ({(\frac{u}{ξ})}^{k}) \frac{u^{k}}{ξ^{k + 1}} d ξ) exp (- {(\frac{u}{h})}^{k^{'}}) \frac{d u}{u} \\ = - \frac{k^{'} k}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{k, h}} ξ^{n} \frac{1}{ξ^{k + 1}} (\int_{L_{γ_{h}}} u^{k} exp ({(\frac{u}{ξ})}^{k} - {(\frac{u}{h})}^{k^{'}}) \frac{d u}{u}) d ξ . \end{matrix}

(48)

At last, the definition of

{\hat{D}}_{k^{'}, k} (\hat{f}) (h)

given in (39) combined with the integral representation (48) and the uniform convergence of

\hat{f} (τ)

on the disc

D_{ρ / 2}

gives rise to the formula (44) and (45). □

In the forthcoming proposition, we provide crucial technical upper bounds for the kernel

D_{k^{'}, k}

that will be used in the next Section 4, Proposition 9, Lemma 5 and Lemma 6.

Proposition 7.

1) There exists a constant

M_{k^{'}, k} > 0

(depending on

k, k^{'}

) such that the next upper bounds

| D_{k^{'}, k} (ξ, h) | \leq \frac{M_{k^{'}, k}}{{k | ξ |}^{k + 1}} {| h |}^{k} | \frac{h}{ξ} |^{\frac{k^{2}}{k^{'} - k}} exp (| \frac{h}{ξ} |^{κ})

(49)

hold for all

h \in C^{*}

, all

ξ \in {\tilde{γ}}_{k, h}

provided that

| ξ | \leq | h |

, where

κ = \frac{k k^{'}}{k^{'} - k} .

(50)

Observe that

κ > k

since

k^{'} > k

2) There exists a constant

M_{k^{'}, k, 1, 2} > 0

(depending on

k, k^{'}

) such that the upper estimates

| D_{k^{'}, k} (ξ, h) | \leq M_{k^{'}, k, 1, 2} \frac{{| h |}^{k}}{{k | ξ |}^{k + 1}}

(51)

hold for all

h \in C^{*}

, all

ξ \in {\tilde{γ}}_{k, h, 1}

ξ \in {\tilde{γ}}_{k, h, 2}

Proof.

We first express

D_{k^{'}, k}

as a Laplace transform in the combined variable

{(h / ξ)}^{k}

. Indeed, we make the change of variable

u = h t^{1 / k}

in the integral (45) for the variable t belonging to the halfline

L_{γ_{h}^{'}} = [0, + \infty) e^{\sqrt{- 1} γ_{h}^{'}}

with

γ_{h}^{'} = k (γ_{h} - \arg (h))

(52)

which yields

D_{k^{'}, k} (ξ, h) = \frac{1}{ξ^{k + 1}} (\int_{L_{γ_{h}^{'}}} h^{k - 1} t^{\frac{k - 1}{k}} exp ({(\frac{h}{ξ})}^{k} t - t^{k^{'} / k}) \frac{h}{k} t^{\frac{1}{k} - 1} d t = \frac{1}{k ξ^{k + 1}} h^{k} D_{k^{'} / k} ({(\frac{h}{ξ})}^{k})

(53)

where

D_{k^{'} / k} (z) = \int_{L_{γ_{h}^{'}}} exp (- t^{k^{'} / k}) exp (z t) d t .

(54)

In the next lemma, we derive some analytic features and bounds estimates for the Laplace integral

D_{k^{'} / k}

Lemma 1.

a) The map

z \mapsto D_{k^{'} / k} (z)

is an entire function in

C

. Moreover, one can single out some constant

M_{k^{'}, k} > 0

such that

| D_{k^{'} / k} (z) | \leq M_{k^{'}, k} {| z |}^{\frac{k}{k^{'} - k}} exp ({| z |}^{\frac{k^{'}}{k^{'} - k}})

(55)

for all

| z | \geq 1

b) For some fixed constant

Δ_{2} > 0

and the direction

γ_{h}^{'}

given in (52), we consider the sector

S_{γ_{h}^{'}, Δ_{2}} = {z \in C^{*} / cos (\arg (z) + γ_{h}^{'}) < - Δ_{2}} .

(56)

Then, there exists a constant

M_{k^{'}, k, 1, 2} > 0

(relying on

k, k^{'}, Δ_{1}, Δ_{2}

, where

Δ_{1}

stems from (43)) such that

| D_{k^{'} / k} (z) | \leq M_{k^{'}, k, 1, 2}

(57)

for all

z \in S_{γ_{h}^{'}, Δ_{2}}

Proof.

We discuss the first point a). From the Taylor expansion

e^{z t} = \sum_{n \geq 0} {(z t)}^{n} / n!

which converges uniformly on any compact subset of

C

, we deduce that

D_{k^{'} / k} (z) = \sum_{n \geq 0} \frac{a_{n}}{n!} z^{n}

(58)

for any

z \in C

, with

a_{n} = \int_{L_{γ_{h}^{'}}} t^{n} exp (- t^{k^{'} / k}) d t, n \geq 0 .

We make the change of variable

s = t^{k^{'} / k}

in the above integrals defining

a_{n}

and get by definition of the Gamma function that

a_{n} = \frac{k}{k^{'}} \int_{L_{γ_{h}^{''}}} s^{\frac{k}{k^{'}} (n + 1) - 1} e^{- s} d s = \frac{k}{k^{'}} Γ (\frac{k}{k^{'}} (n + 1))

(59)

for all

n \geq 0

, where

γ_{h}^{''} = k^{'} (γ_{h} - \arg (h))

. Combining (58) with (59) yields the expansion

D_{k^{'} / k} (z) = \sum_{n \geq 0} \frac{k}{k^{'}} \frac{Γ (\frac{k}{k^{'}} (n + 1))}{Γ (n + 1)} z^{n}

(60)

for all

z \in C

. On the other hand, from the Beta integral formula (see [4], Appendix B.3), we remind that

\frac{Γ (α) Γ (β)}{Γ (α + β)} = \int_{0}^{1} {(1 - t)}^{α - 1} t^{β - 1} d t \leq 1

(61)

for all real numbers

α, β \geq 1

. From (61) we deduce that

\frac{Γ (\frac{k}{k^{'}} (n + 1))}{Γ (n + 1)} \leq \frac{1}{Γ ((1 - \frac{k}{k^{'}}) n + (1 - \frac{k}{k^{'}}))}

(62)

for all

n \geq n_{k^{'}, k}

, for some integer

n_{k^{'}, k} \geq 1

depending on

k, k^{'}

. As a result of (60) and (62), we obtain a constant

C_{k^{'}, k} > 0

such that

| D_{k^{'} / k} (z) | \leq C_{k^{'}, k} \sum_{n \geq 0} \frac{1}{Γ ((1 - \frac{k}{k^{'}}) n + (1 - \frac{k}{k^{'}}))} {| z |}^{n}

(63)

for all

z \in C

. Now, we call to mind some upper bounds for the so-called Wiman function

E_{α, β} (x) = \sum_{n \geq 0} \frac{x^{n}}{Γ (α n + β)}

for prescribed

α \in (0, 2)

and

β > 0

mentioned in our previous work [8], Proposition 1. Namely, some constant

K_{α, β} > 0

can be found such that

E_{α, β} (x) \leq C_{α, β} x^{\frac{1 - β}{α}} e^{x^{1 / α}}

(64)

for all

x \geq 1

. At last, from (63) and (64), we deduce that awaited bounds (55).

We focus on the second point b). According to the lower bounds (43) and the definition (56) of the sector

S_{γ_{h}^{'}, Δ_{2}}

, we reach a constant

M_{k^{'}, k, 1, 2}

(depending on

k^{'}, k

Δ_{1}

and

Δ_{2}

) with

\begin{matrix} | D_{k^{'} / k} (z) | \leq \int_{0}^{+ \infty} exp (- r^{k^{'} / k} cos (k^{'} (γ_{h} - \arg (h)))) \times exp (| z | r cos (\arg (z) + γ_{h}^{'})) d r \\ \leq \int_{0}^{+ \infty} exp (- r^{\frac{k^{'}}{k}} Δ_{1}) exp (- | z | r Δ_{2}) d r \leq M_{k^{'}, k, 1, 2} \end{matrix}

(65)

for all

z \in S_{γ_{h}^{'}, Δ_{2}}

. □

We turn to the first point 1) of Proposition 7. We observe that the inequality (49) is a straight consequence of the factorization (53) and the upper bounds (55).

We address the second point 2) of Proposition 7. We first observe by construction of

{\tilde{γ}}_{k, h, 1}

and

{\tilde{γ}}_{k, h, 2}

in the second item of Proposition 6, one can find a constant

Δ_{2} > 0

such that

cos (k (γ_{h} - \arg (ξ))) = cos (\arg ({(u / ξ)}^{k})) < - Δ_{2}

(66)

for all

u \in L_{γ_{h}}

, provided that

ξ \in {\tilde{γ}}_{k, h, 1}

ξ \in {\tilde{γ}}_{k, h, 2}

. Besides, for

Δ_{2} > 0

chosen as in (66), we remark that

{(h / ξ)}^{k} \in S_{γ_{h}^{'}, Δ_{2}}

(67)

for all

ξ \in {\tilde{γ}}_{k, h, 1} \cup {\tilde{γ}}_{k, h, 2}

. Indeed, (67) is equivalent to

cos (arg ({(h / ξ)}^{k}) + k (γ_{h} - \arg (h))) < - Δ_{2}

which can be rewritten as (66).

At last, we conclude that the bounds (51) can be derived from the factorization (53) and the upper bounds (57) taking for granted the inclusion (67). □

3.2.3. Statement of the Integral Equation

In this subsection, we denote

\hat{W} (τ, m) = B_{m_{k}} (t \mapsto \hat{U} (t, m)) (τ)

(68)

the formal

m_{k} -

Borel transform of the formal power series expansion (31). The object of this subsection is the derivation of some integral equation fulfilled by the formal power series (68) seen as a series with coefficients in the Banach space

E = E_{(β, μ)}

endowed with the norm

{| | . | |}_{E} = {| | . | |}_{(β, μ)}

In this subsection, we make the assumption that

\hat{W} \in E_{(β, μ)} {τ}

(69)

meaning that

τ \mapsto \hat{W} (τ, m)

is convergent on some disc

D_{ρ}

with

ρ > 0

as a

E_{(β, μ)} -

valued series. We will see in Section 4, where a solution of the integral equation (72) will be constructed in some function space, that this assumption will be satisfied.

In order to improve the legibility of the equation that

\hat{W} (τ, m)

is asked to solve

We introduce the notation

$C_{k, l_{0}, l_{1}} (\hat{W}) (τ, m) : = \frac{τ^{k}}{Γ (l_{0} / k)} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l_{0}}{k} - 1} {(k s)}^{l_{1}} \hat{W} (s^{1 / k}, m) \frac{d s}{s}$

(70)

for all integers $l_{0} \geq 1$ , $l_{1} \geq 0$ .
We define the map

$P_{m} (τ) = Q (\sqrt{- 1} m) - cosh (α_{D} {(k τ^{k})}^{2}) R_{D} (\sqrt{- 1} m)$

(71)

for all $τ \in C$ , all $m \in R$ .

Based upon the transformations formula (37) harked back in Proposition 3 and the formal identity for the Mahler transforms reached in (40) of Proposition 5 together with the integral formula (44) derived in Proposition 6 under our assumption (69), we arrive at the next proposition.

Proposition 8.

The formal power series

\hat{U} (t, m)

given by (31) solves the equation (30) for vanishing initial data

\hat{U} (0, m) \equiv 0

ifthe convergent formal series

\hat{W} (τ, m)

given by (68) obeys the next integral equation

\begin{matrix} P_{m} (τ) \hat{W} (τ, m) = \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) {(k τ^{k})}^{l_{1}} \hat{W} (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) C_{k, l_{0}, l_{1}} (\hat{W}) (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} {(k ξ^{k})}^{l_{1}} \hat{W} (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (\hat{W}) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + c_{Q_{1} Q_{2}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} \hat{W} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times \hat{W} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} + \sum_{j \in J} F_{j} (m) τ^{j} \end{matrix}

(72)

4. Solving the Integral Equation in a Banach Space of Functions with Exponential Growth on Sectors and Decay on the Real Line

In this section, we investigate the existence and unicity of a genuine solution to the above integral equation (72) in the Banach space of functions described in the next definition

Definition 5.

Let

S_{d}

be an unbounded sector edged at 0 with bisecting direction

d \in R

. Let

ν, ρ > 0

be positive real numbers. We consider the natural number

k \geq 1

and

β, μ > 0

the real numbers prescribed in Section 2. We denote

F_{(ν, β, μ, k, ρ)}^{d}

the vector space of continuous functions

(τ, m) \mapsto h (τ, m)

on the product

(S_{d} \cup D_{ρ}) \times R

, which are holomorphic with respect to τ on the union

S_{d} \cup D_{ρ}

for which the norm

{| | h (τ, m) | |}_{(ν, β, μ, k, ρ)} = sup_{τ \in S_{d} \cup D_{ρ}, m \in R} {(1 + | m |)}^{μ} e^{β | m |} \frac{1 + {| τ |}^{2 k}}{| τ |} {exp (- ν | τ |}^{k}) | h (τ, m) |

(73)

is finite. The space

F_{(ν, β, μ, k, ρ)}^{d}

equipped with the norm

{| | . | |}_{(ν, β, μ, k, ρ)}

turns out to be a complex Banach space.

These Banach spaces appear for the first time in the previous paper [7] by A. Lastra and the author.

Our strategy consists in rewriting our main integral equation (72) as a fixed point equation (see (195) below) for which a solution can be constructed in the above Banach space given in Definition 5 for well adjusted parameters

ν

and

ρ

. In order to recast (72) into (195), we need to divide both sides of (72) by the map

P_{m} (τ)

given in (71) provided that the Borel variable

τ

is taken in the vicinity of the origin and along a well chosen unbounded sector, given that the fourier mode m is ranged over

R

In the next lemma, we provide some crucial lower bounds for

P_{m} (τ)

on fitting unbounded domains.

Lemma 2.

Provided that the aperture

η_{Q, R_{D}} > 0

of the sector

S_{Q, R_{D}}

diplayed in (13) and that the difference

| r_{Q, R_{D}, 1} - r_{Q, R_{D}, 2} |

of the inner and outer radius of

S_{Q, R_{D}}

are taken small enough, there exists a non empty subset

Θ_{Q, R_{D}}

[- π, π)

and a small radius

ρ > 0

with the next features:

For all $d \in Θ_{Q, R_{D}}$ , one can select an unbounded sector $S_{d}$ edged at 0 with bisecting direction d.
To the above chosen sector $S_{d}$ , one can attach two constants $δ_{S_{d}, k, α_{D}} > 0$ (relying on $S_{d}, k$ and $α_{D}$ ), $Δ_{S_{d}, k} > 0$ (depending on $S_{d}$ and k) with the following lower bounds

$| P_{m} (τ) | \geq | R_{D} (\sqrt{- 1} m) | δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})$

(74)

for all $τ \in S_{d} \cup D_{ρ}$ , all $m \in R$ .

Proof.

For all

m \in R

, we set

H (m) = Q (\sqrt{- 1} m) / R_{D} (\sqrt{- 1} m)

. In a first step, we need to find the complex solutions of the equation

cosh (X) : = \frac{e^{X} + e^{- X}}{2} = H (m) .

(75)

We notice that this equation (75) is equivalent to

{(e^{X})}^{2} - 2 e^{X} H (m) + 1 = 0 .

(76)

If one sets the quantity

δ (m) = | H^{2} {(m) - 1 |}^{1 / 2} exp (\sqrt{- 1} \frac{\arg (H^{2} (m) - 1)}{2})

(77)

for all

m \in R

, then (76) has two infinite sets of solutions

{a_{l} (m)}_{l \in Z}

and

{b_{l} (m)}_{l \in Z}

given by explicit expressions

a_{l} (m) = log | H (m) + δ (m) | + \sqrt{- 1} (\arg (H (m) + δ (m)) + 2 l π)

(78)

for all

l \in Z

and

m \in R

with

b_{l} (m) = log | H (m) - δ (m) | + \sqrt{- 1} (\arg (H (m) - δ (m)) + 2 l π)

(79)

for all

l \in Z

and

m \in R

. Namely, owing to the relation

(H (m) - δ (m)) (H (m) + δ (m)) = 1

for all

m \in R

, we observe that both expressions (78) and (79) are well defined since

H (m) - δ (m)

and

H (m) + δ (m)

are not vanishing quantities and furthermore that the next symmetry occurs

b_{- l} (m) = - a_{l} (m)

(80)

for all integers

l \in Z

and

m \in R

At the next stage, we describe the complex solutions of the equation

cosh (α_{D} k^{2} τ^{2 k}) = H (m) .

(81)

From the above discussion, we deduce that the complex zeros of (81) are given by the union of the roots of the next algebraic equations

α_{D} k^{2} τ^{2 k} = a_{l} (m)

(82)

with

α_{D} k^{2} τ^{2 k} = b_{l} (m)

(83)

for all

l \in Z

. For each

l \in Z

, the

2 k

distinct roots of (82) are given by

τ_{h, l} (m) = | \frac{a_{l} (m)}{α_{D} k^{2}} |^{\frac{1}{2 k}} exp (\sqrt{- 1} (\frac{\arg (a_{l} (m))}{2 k} + \frac{π h}{k}))

(84)

and the

2 k

distinct roots of (83) are expressed through

υ_{h, l} (m) = | \frac{b_{l} (m)}{α_{D} k^{2}} |^{\frac{1}{2 k}} exp (\sqrt{- 1} (\frac{\arg (b_{l} (m))}{2 k} + \frac{π h}{k}))

(85)

for all

0 \leq h \leq 2 k - 1

, all

m \in R

. Furthermore, we notice the symmetry relations

- τ_{h, l} (m) = τ_{h + k, l} (m)

with

- υ_{h, l} (m) = υ_{h + k, l} (m)

provided that

0 \leq h \leq k

, for any given

l \in Z

and

m \in R

Bearing in mind from (14), that

H (m)

belongs to the sector

S_{Q, R_{D}}

for all

m \in R

, provided that the aperture

η_{Q, R_{D}} > 0

S_{Q, R_{D}}

and the difference

| r_{Q, R_{D}, 1} - r_{Q, R_{D}, 2} |

are chosen small enough, there exist directions

d \in R

for which an unbounded sector

S_{d}

edged at 0 with bisecting direction d can be singled out in a way that

S_{d} \cap ({τ_{h, l} (m) / 0 \leq h \leq 2 k - 1, l \in Z, m \in R} \cup {υ_{h, l} (m) / 0 \leq h \leq 2 k - 1, l \in Z, m \in R}) = \emptyset .

(86)

For later use, we choose the sector

S_{d}

with the further assumption that for all

θ \in R

such that

e^{\sqrt{- 1} θ} \in S_{d}

, the next condition

cos (2 k θ) \neq 0

(87)

holds. We denote

Θ_{Q, R_{D}}

the set of all directions d in

(- π, π)

for which sectors

S_{d}

can be selected fulfilling the above two features (86) and (87).

Now, we explain which constraints are set on the radius

ρ > 0

. Provided that

ρ > 0

is chosen close enough to 0, one can choose a small radius

η_{1} > 0

such that

cosh (α_{D} k^{2} τ^{2 k}) \in D (1, η_{1})

(88)

for all

τ \in D_{ρ}

, where

D (1, η_{1})

stands for the disc centered at 1 with radius

η_{1}

. Then, we select the sector

S_{Q, R_{D}}

in a way that

S_{Q, R_{D}} \cap D (1, η_{1}) = \emptyset .

(89)

For the rest of the proof, we choose a sector

S_{d}

for

d \in Θ_{Q, R_{D}}

and a disc

D_{ρ}

as above.

We first come up with lower bounds for the map

P_{m} (τ)

on the domains

(S_{d} \cup D_{ρ}) \cap D_{R}

, for any prescribed large radius

R > 0

. Namely, we factorize

P_{m} (τ)

in the form

P_{m} (τ) = R_{D} (\sqrt{- 1} m) \times [H (m) - cosh (α_{D} k^{2} τ^{2 k})]

(90)

for all

τ \in S_{d} \cup D_{ρ}

, all

m \in R

. Owing to the constraints (86), (88) along with (89) and according to (14), for each given radius

R > 0

(as large as we want), we can find a constant

δ_{1} > 0

such that

| H (m) - cosh (α_{D} k^{2} τ^{2 k}) | \geq δ_{1}

(91)

for all

S_{d} \cup D_{ρ}

with

| τ | \leq R

. Combining (90) and (91) yields lower bounds of the form

| P_{m} (τ) | \geq | R_{D} (\sqrt{- 1} m) | δ_{1}

(92)

for all

τ \in (S_{d} \cup D_{ρ}) \cap D_{R}

and

m \in R

In the last part of the proof, lower bounds for large values of

| τ |

S_{d}

are exhibited. We write

τ \in S_{d}

in the form

τ = r e^{\sqrt{- 1} θ}

, for radius

r \geq 0

and angle

θ \in R

. Then,

Re (α_{D} k^{2} τ^{2 k}) = α_{D} k^{2} r^{2 k} cos (2 k θ)

(93)

According to our choice of

S_{d}

subjected to (87), two cases arise.

Case 1. There exists a constant

Δ_{S_{d}, k, 1} > 0

(depending on

S_{d}

and k) such that

cos (2 k θ) > Δ_{S_{d}, k, 1}

(94)

for all

θ \in R

with

e^{\sqrt{- 1} θ} \in S_{d}

. We perform the next factorization

cosh (α_{D} k^{2} τ^{2 k}) = exp (α_{D} k^{2} τ^{2 k}) \times [\frac{1}{2} + \frac{1}{2} exp (- 2 α_{D} k^{2} τ^{2 k})]

(95)

which allows us to rephrase the next difference as a product

cosh (α_{D} k^{2} τ^{2 k}) - H (m) = exp (α_{D} k^{2} τ^{2 k}) A (τ, m)

(96)

where

\begin{matrix} A (τ, m) = [\frac{1}{2} + \frac{1}{2} exp (- 2 α_{D} k^{2} τ^{2 k})] \\ \times [1 - H (m) exp (- α_{D} k^{2} τ^{2 k}) \times {[\frac{1}{2} + \frac{1}{2} exp (- 2 α_{D} k^{2} τ^{2 k})]}^{- 1}] \end{matrix}

(97)

for all

τ \in S_{d}

m \in R

. According to (94), we note that

| exp (α_{D} k^{2} τ^{2 k}) | = exp (α_{D} k^{2} r^{2 k} cos (2 k θ)) \geq exp (α_{D} k^{2} Δ_{S_{d}, k, 1} {| τ |}^{2 k})

(98)

whenever

τ = r e^{\sqrt{- 1} θ} \in S_{d}

. Besides, observing that

{lim}_{| τ | \to + \infty} | exp (- α_{D} k^{2} τ^{2 k}) | = 0

and keeping in mind that

r_{Q, R_{D}, 1} \leq | H (m) | \leq r_{Q, R_{D}, 2}

, for all

m \in R

, we get some constant

A_{S_{d}, k, α_{D}} > 0

and a radius

R_{1} > 0

(large enough) for which

| A (τ, m) | \geq A_{S_{d}, k, α_{D}}

(99)

for all

τ \in S_{d}

with

| τ | \geq R_{1}

and all

m \in R

Eventually, in gathering the factorizations (90) and (96) together with the lower bounds (98) and (99), we arrive at the lower bounds

| P_{m} (τ) | \geq | R_{D} (\sqrt{- 1} m) | A_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k, 1} {| τ |}^{2 k})

(100)

provided that

τ \in S_{d}

with

| τ | \geq R_{1}

and

m \in R

. At last, combining these last bounds (100) and (92) for

R = R_{1}

, we arrive at the awaited lower bounds (74) for some small constant

δ_{S_{d}, k, α_{D}} > 0

and

Δ_{S_{d}, k} : = Δ_{S_{d}, k, 1}

Case 2. A constant

Δ_{S_{d}, k, 2} > 0

(depending on

S_{d}

and k) can be singled out for which

cos (2 k θ) < - Δ_{S_{d}, k, 2}

(101)

holds for all

θ \in R

with

e^{\sqrt{- 1} θ} \in S_{d}

. In that case, we do factorize

cosh (α_{D} k^{2} τ^{2 k}) = exp (- α_{D} k^{2} τ^{2 k}) \times [\frac{1}{2} + \frac{1}{2} exp (2 α_{D} k^{2} τ^{2 k})]

(102)

and recast the next difference at a product in the form

cosh (α_{D} k^{2} τ^{2 k}) - H (m) = exp (- α_{D} k^{2} τ^{2 k}) B (τ, m)

(103)

where

\begin{matrix} B (τ, m) = [\frac{1}{2} + \frac{1}{2} exp (2 α_{D} k^{2} τ^{2 k})] \\ \times [1 - H (m) exp (α_{D} k^{2} τ^{2 k}) \times {[\frac{1}{2} + \frac{1}{2} exp (2 α_{D} k^{2} τ^{2 k})]}^{- 1}] \end{matrix}

(104)

for all

τ \in S_{d}

m \in R

. Based on our hypothesis (101), we remark that

| exp (- α_{D} k^{2} τ^{2 k}) | = exp (- α_{D} k^{2} r^{2 k} cos (2 k θ)) \geq exp (α_{D} k^{2} Δ_{S_{d}, k, 2} {| τ |}^{2 k})

(105)

as long as

τ = r e^{\sqrt{- 1} θ} \in S_{d}

. On the other hand, since

{lim}_{| τ | \to + \infty} | exp (α_{D} k^{2} τ^{2 k}) | = 0

and granting that

| H (m) | \in [r_{Q, R_{D}, 1}, r_{Q, R_{D}, 2}]

, for all

m \in R

, some constant

B_{S_{d}, k, α_{D}} > 0

and a radius

R_{2} > 0

(large enough) can be deduced for which

| B (τ, m) | \geq B_{S_{d}, k, α_{D}}

(106)

for all

τ \in S_{d}

with

| τ | \geq R_{2}

and all

m \in R

In conclusion, the factorizations (90) and (103) combined with the lower bounds (105) and (106) yield the next lower bounds

| P_{m} (τ) | \geq | R_{D} (\sqrt{- 1} m) | B_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k, 2} {| τ |}^{2 k})

(107)

whenever

τ \in S_{d}

with

| τ | \geq R_{2}

and

m \in R

. Finally, these latter bounds (107) gathered with (92) for

R = R_{2}

give rise to the expected lower bounds (74) for some small constant

δ_{S_{d}, k, α_{D}} > 0

and

Δ_{S_{d}, k} : = Δ_{S_{d}, k, 2}

. □

We now introduce the map

H

acting on the Banach spaces of Definition 5 for which the fixed point theorem will be applied, namely

\begin{matrix} H (ω (τ, m)) : = \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2} P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) {(k τ^{k})}^{l_{1}} ω (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2} P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) C_{k, l_{0}, l_{1}} (ω) (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2} P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} {(k ξ^{k})}^{l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2} P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + c_{Q_{1} Q_{2}} \frac{1}{{(2 π)}^{1 / 2} P_{m} (τ)} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} ω ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} + \sum_{j \in J} \frac{F_{j} (m)}{P_{m} (τ)} τ^{j} . \end{matrix}

(108)

In the next proposition, we show that

H

acts on the Banach space

F_{(ν, β, μ, k, ρ)}^{d}

for well chosen parameters and directions d as a shrinking map. This result is central in our work.

Proposition 9.

Let

ν > 0

be a fixed real number and let

β, μ, k

be prescribed as in Section 2. Let us assume that the sector

S_{Q, R_{D}}

introduced in (13) is selected as in Lemma 2. We choose an unbounded sector

S_{d}

for some direction

d \in Θ_{Q, R_{D}}

and a disc

D_{ρ}

for a suitably small

ρ > 0

fixed as in Lemma 2. We make the following additional restriction on the constant

α_{D}

which is asked to obey the next inequality

α_{D} k^{2} Δ_{S_{d}, k} \geq {(2 / ρ)}^{k}

(109)

where

Δ_{S_{d}, k} > 0

is the constant appearing in (74) from Lemma 2.

Under the assumption that the constants

A_{\underset{̲}{l}} > 0

for

\underset{̲}{l} \in A

set up in (22) and the quantity

| c_{Q_{1} Q_{2}} |

are taken adequately small, one can sort a constant

ϖ > 0

such that the map

H

enjoys the next properties

The next inclusion

$H (B_{ϖ}) \subset B_{ϖ}$

(110)

holds where $B_{ϖ}$ stands for the closed ball of radius $ϖ > 0$ centered at 0 in the space $F_{(ν, β, μ, k, ρ)}^{d}$ .
The shrinking condition

$| | H (ω_{1}) - H (ω_{2}) {| |}_{(ν, β, μ, k, ρ)} \leq \frac{1}{2} | | ω_{1} - ω_{2} {| |}_{(ν, β, μ, k, ρ)}$

(111)

occurs whenever $ω_{1}, ω_{2} \in B_{ϖ}$ .

Proof.

We focus on the first item 1. Let

ω

be an element of

F_{(ν, β, μ, k, ρ)}^{d}

. By definition, the next inequality

| ω (τ, m) | \leq {| | ω | |}_{(ν, β, μ, k, ρ)} {(1 + | m |)}^{- μ} e^{- β | m |} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k})

(112)

holds for all

τ \in S_{d} \cup D_{ρ}

, all

m \in R

In the next six lemma, we provide upper norm bounds for each piece composing the map

H

. The elements of the first sum of operators is minded in the next

Lemma 3.

Let

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} = 0

and

l_{2} = 1

. We can find some constant

C_{1}

(relying on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{1}

) with

| | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) τ^{k l_{1}} ω (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{1} A_{\underset{̲}{l}} {| | ω | |}_{(ν, β, μ, k, ρ)}

(113)

for all

ω \in F_{(ν, β, μ, k, ρ)}^{d}

Proof.

We remind from Section 2 that

R_{\underset{̲}{l}} (X)

is a polynomial of degree

\deg (R_{\underset{̲}{l}})

and

R_{D} (X)

is a polynomial of degree

\deg (R_{D})

not vanishing on

X = \sqrt{- 1} m

, for all

m \in R

. As a result, two constants

R_{\underset{̲}{l}}, R_{D} > 0

can be found with

| R_{\underset{̲}{l}} (\sqrt{- 1} m) | \leq R_{\underset{̲}{l}} {(1 + | m |)}^{\deg (R_{\underset{̲}{l}})}, | R_{D} (\sqrt{- 1} m) | \geq R_{D} {(1 + | m |)}^{\deg (R_{D})}

(114)

for all

m \in R

Let

ω \in F_{(ν, β, μ, k, ρ)}^{d}

. Based on the definition of

A_{\underset{̲}{l}}

given in (22), the lower bounds (74) reached in Lemma 2 together with the upper bounds (112) and the above inequalities (114), we obtain

\begin{matrix} A_{τ, m} = | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) τ^{k l_{1}} ω (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} | \\ \leq \frac{{| τ |}^{k l_{1}} {| | ω | |}_{(ν, β, μ, k, ρ)}}{R_{D} {(1 + | m |)}^{\deg (R_{D})} δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \times \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) \\ \times \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (1 + | m - m_{1} {|)}^{- μ} e^{- β | m - m_{1} |} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} R_{\underset{̲}{l}} (1 + | m_{1} {|)}^{\deg (R_{\underset{̲}{l}})} d m_{1} \end{matrix}

(115)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

. Besides, the triangular inequality

| m | \leq | m - m_{1} | + | m_{1} |

(116)

holds for all

m, m_{1} \in R

and we can choose a constant

M_{S_{d}, k, l_{1}, α_{D}} > 0

such that

sup_{τ \in S_{d} \cup D_{ρ}} \frac{{| τ |}^{k l_{1}}}{exp (α_{D} k^{2} Δ_{S_{d}, k} | τ |^{2 k})} = M_{S_{d}, k, l_{1}, α_{D}} .

(117)

Gathering (115), (116) and (117), we come up with

A_{τ, m} \leq \frac{M_{S_{d}, k, l_{1}, α_{D}}}{δ_{S_{d}, k, α_{D}}} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{R_{\underset{̲}{l}}}{R_{D}} \times \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |} A_{1} (m)

(118)

where

A_{1} (m) = {(1 + | m |)}^{μ - \deg (R_{D})} \int_{- \infty}^{+ \infty} \frac{1}{(1 + | m - m_{1} {|)}^{μ} (1 + | m_{1} {|)}^{μ - \deg (R_{\underset{̲}{l}})}} d m_{1}

(119)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

. Owing to Lemma 2.2 of [5], under the assumptions made in (11) and (21), a constant

C_{1.1} > 0

can be deduced with

A_{1} (m) \leq C_{1.1}

(120)

for all

m \in R

. At last, joining (118) and (120) yields a constant

C_{1} > 0

for which

A_{τ, m} \leq C_{1} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}

(121)

as long as

τ \in S_{d} \cup D_{ρ}

m \in R

, which is tantamount to (113). □

The elements of the second sum of operators are considered in the next

Lemma 4.

We set

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} \geq 1

and

l_{2} = 1

. Then, a constant

C_{2} > 0

(depending upon

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{0}, l_{1}

) can be picked out such that

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) [τ^{k} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l_{0}}{k} - 1} s^{l_{1}} ω (s^{1 / k}, m_{1}) \frac{d s}{s}] \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{2} A_{\underset{̲}{l}} {| | ω | |}_{(ν, β, μ, k, ρ)} \end{matrix}

(122)

for all

ω \in F_{(ν, β, μ, k, ρ)}^{d}

Proof.

Take

ω

an element of

F_{(ν, β, μ, k, ρ)}^{d}

. On the ground of the definition

A_{\underset{̲}{l}}

displayed in (22), the lower bounds (74) stated in Lemma 2 together with the upper bounds (112) and the polynomial inequalities (114), we reach

\begin{matrix} B_{τ, m} = | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) [τ^{k} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l_{0}}{k} - 1} s^{l_{1}} ω (s^{1 / k}, m_{1}) \frac{d s}{s}] \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} | \\ \leq \frac{{| τ |}^{k} \int_{0}^{{| τ |}^{k}} {(| τ |}^{k} {- h)}^{\frac{l_{0}}{k} - 1} h^{l_{1} + \frac{1}{k}} e^{ν h} \frac{d h}{h}}{R_{D} {(1 + | m |)}^{\deg (R_{D})} δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} {| | ω | |}_{(ν, β, μ, k, ρ)} \\ \times \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (1 + | m - m_{1} {|)}^{- μ} exp (- β | m - m_{1} |) (1 + | m_{1} {|)}^{- μ} exp (- β | m_{1} |) R_{\underset{̲}{l}} (1 + | m_{1} {|)}^{\deg (R_{\underset{̲}{l}})} d m_{1} \end{matrix}

(123)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

. By applying the change of variable

h = {| τ |}^{k} u

, for

0 \leq u \leq 1

, a constant

M_{S_{d}, k, l_{0}, l_{1}, α_{D}} > 0

is deduced with

\begin{matrix} \frac{{| τ |}^{k} \int_{0}^{{| τ |}^{k}} {(| τ |}^{k} {- h)}^{\frac{l_{0}}{k} - 1} h^{l_{1} + \frac{1}{k}} \frac{d h}{h}}{exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \times \frac{1 + {| τ |}^{2 k}}{| τ |} \\ = \frac{{| τ |}^{l_{0} + k l_{1}} {(1 + | τ |}^{2 k})}{exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \times (\int_{0}^{1} {(1 - u)}^{\frac{l_{0}}{k} - 1} u^{l_{1} + \frac{1}{k}} \frac{d u}{u}) \leq M_{S_{d}, k, l_{0}, l_{1}, α_{D}} \end{matrix}

(124)

for all

τ \in S_{d} \cup D_{ρ}

Collecting (123), (116) and (124) we land up at

B_{τ, m} \leq \frac{M_{S_{d}, k, l_{0}, l_{1}, α_{D}}}{δ_{S_{d}, k, α_{D}}} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{R_{\underset{̲}{l}}}{R_{D}} \frac{| τ |}{1 + {| τ |}^{2 k}} e^{{ν | τ |}^{k}} {(1 + | m |)}^{- μ} e^{- β | m |} A_{1} (m)

(125)

where

A_{1} (m)

is given by (119), as long as

τ \in S_{d} \cup D_{ρ}

m \in R

. Eventually, gathering (125) and (120) gives rise to a constant

C_{2} > 0

(hinging on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{0}, l_{1}

) with

B_{τ, m} \leq C_{2} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}

(126)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

, which is equivalent to (122). □

The components of the third sum of operators are upper bounded in the following

Lemma 5.

Let

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} = 0

and

l_{2} > 1

. There exists a constant

C_{3} > 0

(relying on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{1}, l_{2}, ρ

) such that

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (\int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} ξ^{k l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{3} A_{\underset{̲}{l}} | | ω (τ, m) {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(127)

for all

ω \in F_{(ν, β, μ, k, ρ)}^{d}

Proof.

In the first part of the proof, we provide upper bounds for the integral map

K (τ, m_{1}) : = \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} ξ^{k l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(128)

for

τ \in S_{d} \cup D_{ρ}

and

m_{1} \in R

. Indeed, the next auxiliary result holds.

Sublemma 1.

1) For any

\tilde{ρ} \geq ρ

, there exists a constant

K_{\tilde{ρ}, k, l_{1}, l_{2}}^{1} > 0

with

| K (τ, m_{1}) | \leq K_{\tilde{ρ}, k, l_{1}, l_{2}}^{1} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |}

(129)

for all

τ \in D_{\tilde{ρ}}

, all

m_{1} \in R

2) One can pinpoint a constant

K_{ρ, k, l_{1}, l_{2}}^{2} > 0

such that

| K (τ, m_{1}) | \leq K_{ρ, k, l_{1}, l_{2}}^{2} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}) (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |}

(130)

for all

τ \in S_{d}

with

| τ | \geq {(ρ / 2)}^{1 / l_{2}}

, all

m_{1} \in R

Proof.

Let us consider

ω \in F_{(ν, β, μ, k, ρ)}^{d}

. We observe that the bounds (112) hold. Owing to our assumption (9) and according to the construction of the integral operator

{\hat{D}}_{k, k / l_{2}}

discussed in Proposition 6, for any fixed

τ \in S_{d} \cup D_{\tilde{ρ}}

, one chooses a direction

γ_{τ^{l_{2}}} \in R

and a positive real number

Δ_{1} > 0

such that

cos (k (γ_{τ^{l_{2}}} - \arg (τ^{l_{2}}))) > Δ_{1} .

(131)

From the definition of the Hankel path

{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}

, for any

m_{1} \in R

, one can split the integral

K (τ, m_{1})

as a sum of three pieces

K (τ, m_{1}) = K_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) - K_{γ_{τ^{l_{2}}}^{-}, γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) - K_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

(132)

where

K_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) = \int_{L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{+}}} ξ^{k l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(133)

whose integration path is the segment

L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{+}} = [0, ρ / 2] e^{\sqrt{- 1} (γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2})}

, for some small

δ^{'} > 0

and

K_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) = \int_{C_{ρ / 2; γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}} ξ^{k l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(134)

along the arc of circle

C_{ρ / 2; γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} = {\frac{ρ}{2} e^{\sqrt{- 1} θ} / θ \in [γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2}, γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2}]},

along with

K_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) = \int_{L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{-}}} ξ^{k l_{1}} ω (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(135)

where

L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{-}} = [0, ρ / 2] e^{\sqrt{- 1} (γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2})}

In the next step of the proof, we display bounds for each piece of the splitting (132).

We first provide bounds for

K_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1})

and

K_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

According to Proposition 7 2), we can find a constant

M_{k, k / l_{2}, 1, 2} > 0

such that

| D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) | \leq M_{k, k / l_{2}, 1, 2} \frac{l_{2}}{k} \frac{{| τ |}^{k}}{{| ξ |}^{\frac{k}{l_{2}} + 1}}

(136)

provided that

ξ \in L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{+}}

ξ \in L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{-}}

. Then, under the assumption (10) and bearing in mind the bounds (112) together with (136), we are given a constant

K_{ρ, k, l_{1}, l_{2}} > 0

such that

\begin{matrix} | K_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) | \leq \int_{0}^{ρ / 2} r^{k l_{1}} {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} r e^{ν r^{k}} M_{k, k / l_{2}, 1, 2} \frac{l_{2}}{k} \frac{{| τ |}^{k}}{r^{\frac{k}{l_{2}} + 1}} d r \\ \leq K_{ρ, k, l_{1}, l_{2}} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} . \end{matrix}

(137)

together with

| K_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq K_{ρ, k, l_{1}, l_{2}} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} .

(138)

At a second stage, we discuss bounds for

K_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

. Two cases arise.

Case a. We assume that

τ \in D_{\tilde{ρ}}

. We need upper bounds for the kernel

D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}})

provided that

ξ \in C_{ρ / 2, γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}

. According to the decomposition (53), we observe the next factorization

D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) = \frac{l_{2}}{k} \frac{1}{ξ^{\frac{k}{l_{2}} + 1}} τ^{k} D_{l_{2}} ({(\frac{τ^{l_{2}}}{ξ})}^{k / l_{2}})

(139)

where

D_{l_{2}} (z)

is an entire function on

C

(as shown in Lemma 1). In particular, the function

D_{l_{2}} (z)

is bounded by some constant

M_{l_{2}, \tilde{ρ}} > 0

on the disc

D_{\frac{{\tilde{ρ}}^{k}}{{(ρ / 2)}^{k / l_{2}}}}

. As a result,

| D_{l_{2}} ({(\frac{τ^{l_{2}}}{ξ})}^{k / l_{2}}) | \leq M_{l_{2}, \tilde{ρ}}

(140)

provided that

ξ \in C_{ρ / 2, γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}

and

τ \in D_{\tilde{ρ}}

, since in particular

| ξ | = ρ / 2

and

| τ | \leq \tilde{ρ}

, which yields the bounds

| D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) | \leq \frac{l_{2}}{k} \frac{1}{{(ρ / 2)}^{\frac{k}{l_{2}} + 1}} {| τ |}^{k} M_{l_{2}, \tilde{ρ}}

(141)

for all

ξ \in C_{ρ / 2, γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}

and

τ \in D_{\tilde{ρ}}

. These latter bounds (141) together with (112) allow us to find some constant

K_{\tilde{ρ}, k, l_{1}, l_{2}}^{+ -} > 0

such that

\begin{matrix} | K_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq \int_{γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2}}^{γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2}} {(ρ / 2)}^{k l_{1}} {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times (ρ / 2) e^{ν {(ρ / 2)}^{k}} \frac{l_{2}}{k} \frac{1}{{(ρ / 2)}^{\frac{k}{l_{2}} + 1}} {| τ |}^{k} M_{l_{2}, \tilde{ρ}} (ρ / 2) d θ \\ \leq K_{\tilde{ρ}, k, l_{1}, l_{2}}^{+ -} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} . \end{matrix}

(142)

for all

m_{1} \in R

At last, from the decomposition (132) and the three estimates (137), (138) and (142), we deduce the awaited bounds (129) from the first point 1).

Case b. We assume that

τ \in S_{d}

with

| τ | \geq {(ρ / 2)}^{1 / l_{2}}

. According to Proposition 7 1), we come up with a constant

M_{k, k / l_{2}} > 0

such that

\begin{matrix} | D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) | \leq \frac{M_{k, k / l_{2}}}{\frac{k}{l_{2}} {| ξ |}^{\frac{k}{l_{2}} + 1}} {| τ^{l_{2}} |}^{\frac{k}{l_{2}}} {(| \frac{τ^{l_{2}}}{ξ} |)}^{\frac{{(k / l_{2})}^{2}}{k - \frac{k}{l_{2}}}} \times exp ((| \frac{τ^{l_{2}}}{ξ} {|)}^{\frac{k^{2} / l_{2}}{k - \frac{k}{l_{2}}}}) \\ = \frac{l_{2}}{k} M_{k, k / l_{2}} {| τ |}^{k + \frac{k}{l_{2} - 1}} \frac{1}{{| ξ |}^{\frac{k}{l_{2}} + 1 + \frac{k}{l_{2} (l_{2} - 1)}}} \times exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{| ξ |}^{\frac{k}{l_{2} - 1}}}) \end{matrix}

(143)

for all

ξ \in C_{ρ / 2, γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}

. These last bounds (143) combined with (112) beget a constant

K_{ρ, k, l_{1}, l_{2}}^{+ -; 2} > 0

such that

\begin{matrix} | K_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq \int_{γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2}}^{γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2}} {(ρ / 2)}^{k l_{1}} {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times (ρ / 2) e^{ν {(ρ / 2)}^{k}} \frac{l_{2}}{k} M_{k, k / l_{2}} {| τ |}^{k + \frac{k}{l_{2} - 1}} \frac{1}{{| ρ / 2 |}^{\frac{k}{l_{2}} + 1 + \frac{k}{l_{2} (l_{2} - 1)}}} \times exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{| ρ / 2 |}^{\frac{k}{l_{2} - 1}}}) (ρ / 2) d θ \\ \leq K_{ρ, k, l_{1}, l_{2}}^{+ -; 2} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{| ρ / 2 |}^{\frac{k}{l_{2} - 1}}}) (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \end{matrix}

(144)

for all

m_{1} \in R

In conclusion, the decomposition (132) together with the three bounds (137), (138) and (144) trigger the expected bounds (130) in the second point 2). □

We set

C_{τ, m} = | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) K (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} |

(145)

for

τ \in S_{d} \cup D_{ρ}

m \in R

. Taking heed of the definition

A_{\underset{̲}{l}}

displayed in (22), the lower bounds (74) stated in Lemma 2 together with the upper bounds (129) for

\tilde{ρ} = max (ρ, {(ρ / 2)}^{1 / l_{2}})

along with (130) and the polynomial inequalities (114), we arrive at

\begin{matrix} C_{τ, m} \leq \frac{1}{R_{D} {(1 + | m |)}^{\deg (R_{D})}} \\ \times \frac{max (K_{\tilde{ρ}, k, l_{1}, l_{2}}^{1} | τ |^{k - 1}, K_{ρ, k, l_{1}, l_{2}}^{2} | τ |^{k - 1 + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}))}{δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} {\times (1 + | τ |}^{2 k}) e^{- {ν | τ |}^{k}} \\ \times [\frac{| τ |}{1 + {| τ |}^{2 k}} e^{{ν | τ |}^{k}} {| | ω | |}_{(ν, β, μ, k, ρ)}] \times \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (1 + | m - m_{1} {|)}^{- μ} e^{- β | m - m_{1} |} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times R_{\underset{̲}{l}} (1 + | m_{1} {|)}^{\deg (R_{\underset{̲}{l}})} d m_{1} \end{matrix}

(146)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

Two situations ensue. In the case

l_{2} = 2

, we observe that

2 k = \frac{k l_{2}}{l_{2} - 1}

and in the situation

l_{2} > 2

, we notice that

2 k > \frac{k l_{2}}{l_{2} - 1}

. In both cases, under the assumption (109), we deduce a constant

M_{S_{d}, k, l_{1}, l_{2}, α_{D}, ρ} > 0

with

\begin{matrix} \frac{max (K_{\tilde{ρ}, k, l_{1}, l_{2}}^{1} | τ |^{k - 1}, K_{ρ, k, l_{1}, l_{2}}^{2} | τ |^{k - 1 + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}))}{δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \\ {\times (1 + | τ |}^{2 k}) e^{- {ν | τ |}^{k}} \leq M_{S_{d}, k, l_{1}, l_{2}, α_{D}, ρ} \end{matrix}

(147)

for all

τ \in S_{d} \cup D_{ρ}

The collection of (146), (116) and (147) spawns

C_{τ, m} \leq M_{S_{d}, k, l_{1}, l_{2}, α_{D}, ρ} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{R_{\underset{̲}{l}}}{R_{D}} [\frac{| τ |}{1 + {| τ |}^{2 k}} e^{{ν | τ |}^{k}} {(1 + | m |)}^{- μ} e^{- β | m |}] A_{1} (m)

(148)

where

A_{1} (m)

is stated in (119), whenever

τ \in S_{d} \cup D_{ρ}

m \in R

. Finally, the last bounds (148) and (120) foster a constant

C_{3} > 0

(depending on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{1}, l_{2}, ρ

) with

C_{τ, m} \leq C_{3} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}

(149)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

, which can be recast as (127).

The constituents of the fourth sum involved in (108) are evaluated in the next

Lemma 6.

We select

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} \geq 1

and

l_{2} > 1

. Then a constant

C_{4} > 0

(relying on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{0}, l_{1}, l_{2}, ρ

) can be found such that

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (\int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{4} A_{\underset{̲}{l}} | | ω (τ, m) {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(150)

provided that

ω \in F_{(ν, β, μ, k, ρ)}^{d}

Proof.

The proof follows closely the one displayed for Lemma 5. The first part of the discussion is devoted to upper bounds for the integral map

\tilde{K} (τ, m_{1}) : = \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(151)

where by definition

C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) : = \frac{ξ^{k}}{Γ (l_{0} / k)} \int_{0}^{ξ^{k}} {(ξ^{k} - s)}^{\frac{l_{0}}{k} - 1} {(k s)}^{l_{1}} ω (s^{1 / k}, m_{1}) \frac{d s}{s}

(152)

for all

τ \in S_{d} \cup D_{ρ}

and

m_{1} \in R

. Namely, the following statement holds.

Sublemma 2.

1) For any prescribed

\tilde{ρ} \geq ρ

, there exists a constant

K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} > 0

with

| \tilde{K} (τ, m_{1}) | \leq K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |}

(153)

provided that

τ \in D_{\tilde{ρ}}

and

m_{1} \in R

2) A constant

K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} > 0

can be singled out such that

| \tilde{K} (τ, m_{1}) | \leq K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}) (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |}

(154)

as long as

τ \in S_{d}

with

| τ | \geq {(ρ / 2)}^{1 / l_{2}}

and

m_{1} \in R

Proof.

We set up

ω \in F_{(ν, β, μ, k, ρ)}^{d}

and we take

τ \in S_{d} \cup D_{\tilde{ρ}}

for some given

\tilde{ρ} \geq ρ

. Keeping the same notations as in Lemma 5, we can break up the integral

\tilde{K} (τ, m_{1})

in three parts

\tilde{K} (τ, m_{1}) = {\tilde{K}}_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) - {\tilde{K}}_{γ_{τ^{l_{2}}}^{-}, γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) - {\tilde{K}}_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

(155)

where

{\tilde{K}}_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) = \int_{L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{+}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(156)

and

{\tilde{K}}_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) = \int_{C_{ρ / 2; γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(157)

together with

{\tilde{K}}_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) = \int_{L_{[0, ρ / 2]; \frac{k}{l_{2}}; γ_{τ^{l_{2}}}^{-}}} C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ

(158)

where the paths of integration are the same as in the integrals (133), (134) and (135).

In the ongoing part of the proof, we disclose bounds for each piece of the decomposition (155). As a preliminary, we rearrange the map (152) by means of the parametrization

s = ξ^{k} s_{1}

where

0 \leq s_{1} \leq 1

C_{k, l_{0}, l_{1}} (ω) (ξ, m_{1}) = \frac{ξ^{l_{0} + k l_{1}}}{Γ (l_{0} / k)} \int_{0}^{1} {(1 - s_{1})}^{\frac{l_{0}}{k} - 1} k^{l_{1}} s_{1}^{l_{1}} ω (ξ s_{1}^{1 / k}, m_{1}) \frac{d s_{1}}{s_{1}}

(159)

and from the bounds (112), we observe that

| ω (ξ s_{1}^{1 / k}, m_{1}) {| \leq | | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \frac{| ξ | s_{1}^{1 / k}}{1 + | ξ s_{1}^{1 / k} |^{2 k}} {exp (ν | ξ |}^{k} s_{1})

(160)

for all

ξ \in {\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}

0 \leq s_{1} \leq 1

and

m_{1} \in R

Bounds for the integrals

{\tilde{K}}_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1})

and

{\tilde{K}}_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

along segments are first achieved.

On the ground of the bounds (136), the factorization (159) and the upper estimates (160), under the assumption (10), a constant

K_{ρ, k, l_{0}, l_{1}, l_{2}} > 0

can be reached with

\begin{matrix} | {\tilde{K}}_{γ_{τ^{l_{2}}}^{+}} (τ, m_{1}) | \leq \int_{0}^{ρ / 2} \frac{r^{l_{0} + k l_{1}}}{Γ (l_{0} / k)} \times (\int_{0}^{1} {(1 - s_{1})}^{\frac{l_{0}}{k} - 1} k^{l_{1}} s_{1}^{l_{1}} s_{1}^{1 / k} \frac{d s_{1}}{s_{1}}) \\ \times {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} r e^{ν r^{k}} M_{k, k / l_{2}, 1, 2} \frac{l_{2}}{k} \frac{{| τ |}^{k}}{r^{\frac{k}{l_{2}} + 1}} d r \\ \leq K_{ρ, k, l_{0}, l_{1}, l_{2}} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} . \end{matrix}

(161)

together with

| {\tilde{K}}_{γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq K_{ρ, k, l_{0}, l_{1}, l_{2}} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} .

(162)

In the next phase, bounds for the integral

{\tilde{K}}_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1})

along the arc of circle are devised. Two cases are distinguished.

Case a. The variable

τ

belongs to

D_{\tilde{ρ}}

. Based on the bounds (141), the factorization (159) and the upper estimates (160), we are given a constant

K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{+ -} > 0

such that

\begin{matrix} | {\tilde{K}}_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq \int_{γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2}}^{γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2}} {(ρ / 2)}^{l_{0} + k l_{1}} \frac{1}{Γ (l_{0} / k)} (\int_{0}^{1} {(1 - s_{1})}^{\frac{l_{0}}{k} - 1} k^{l_{1}} s_{1}^{l_{1}} s_{1}^{1 / k} \frac{d s_{1}}{s_{1}}) \\ \times {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \times (ρ / 2) e^{ν {(ρ / 2)}^{k}} \frac{l_{2}}{k} \frac{1}{{(ρ / 2)}^{\frac{k}{l_{2}} + 1}} {| τ |}^{k} M_{l_{2}, \tilde{ρ}} (ρ / 2) d θ \\ \leq K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{+ -} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} . \end{matrix}

(163)

for all

m_{1} \in R

Eventually, departing from the splitting (155) and taking heed of the three upper bounds (161), (162) and (163), the due bounds (153) from the first point 1) are established.

Case b. The variable

τ

is assumed to belong to

S_{d}

under the constraint

| τ | \geq {(ρ / 2)}^{1 / l_{2}}

. Acquired from the bounds (143), the factorization (159) and the upper estimates (160), a constant

K_{ρ, k, l_{0}, l_{1}, l_{2}}^{+ -; 2} > 0

exists such that

\begin{matrix} | {\tilde{K}}_{γ_{τ^{l_{2}}}^{+}, γ_{τ^{l_{2}}}^{-}} (τ, m_{1}) | \leq \int_{γ_{τ^{l_{2}}} - \frac{π l_{2}}{2 k} - \frac{δ^{'}}{2}}^{γ_{τ^{l_{2}}} + \frac{π l_{2}}{2 k} + \frac{δ^{'}}{2}} {(ρ / 2)}^{l_{0} + k l_{1}} \frac{1}{Γ (l_{0} / k)} (\int_{0}^{1} {(1 - s_{1})}^{\frac{l_{0}}{k} - 1} k^{l_{1}} s_{1}^{l_{1}} s_{1}^{1 / k} \frac{d s_{1}}{s_{1}}) \\ \times {| | ω | |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times (ρ / 2) e^{ν {(ρ / 2)}^{k}} \frac{l_{2}}{k} M_{k, k / l_{2}} {| τ |}^{k + \frac{k}{l_{2} - 1}} \frac{1}{{| ρ / 2 |}^{\frac{k}{l_{2}} + 1 + \frac{k}{l_{2} (l_{2} - 1)}}} \times exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{| ρ / 2 |}^{\frac{k}{l_{2} - 1}}}) (ρ / 2) d θ \\ \leq K_{ρ, k, l_{0}, l_{1}, l_{2}}^{+ -; 2} {| | ω | |}_{(ν, β, μ, k, ρ)} {| τ |}^{k + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{| ρ / 2 |}^{\frac{k}{l_{2} - 1}}}) (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \end{matrix}

(164)

for all

m_{1} \in R

In conclusion, from the splitting (155) together with the three upper bounds (161), (162) and (164), the forecast bounds (154) from the second point 2) hold. □

The remaining part of the proof is similar to the one of Lemma 5. Namely, we define

{\tilde{C}}_{τ, m} = | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) \tilde{K} (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} |

(165)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

. The definition

A_{\underset{̲}{l}}

displayed in (22), the lower bounds (74) established in Lemma 2, the upper bounds (153) for

\tilde{ρ} = max (ρ, {(ρ / 2)}^{1 / l_{2}})

along with (154) and the polynomial inequalities (114), beget the next inequality

\begin{matrix} {\tilde{C}}_{τ, m} \leq \frac{1}{R_{D} {(1 + | m |)}^{\deg (R_{D})}} \\ \times \frac{max (K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} | τ |^{k - 1}, K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} | τ |^{k - 1 + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}))}{δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} {\times (1 + | τ |}^{2 k}) e^{- {ν | τ |}^{k}} \\ \times [\frac{| τ |}{1 + {| τ |}^{2 k}} e^{{ν | τ |}^{k}} {| | ω | |}_{(ν, β, μ, k, ρ)}] \times \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (1 + | m - m_{1} {|)}^{- μ} e^{- β | m - m_{1} |} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times R_{\underset{̲}{l}} (1 + | m_{1} {|)}^{\deg (R_{\underset{̲}{l}})} d m_{1} \end{matrix}

(166)

as long as

τ \in S_{d} \cup D_{ρ}

m \in R

Two alternative arise. In the case

l_{2} = 2

, we observe that

2 k = \frac{k l_{2}}{l_{2} - 1}

and in the situation

l_{2} > 2

, we notice that

2 k > \frac{k l_{2}}{l_{2} - 1}

. Under the assumption (109), needed only in the case

l_{2} = 2

, we deduce a constant

M_{S_{d}, k, l_{0}, l_{1}, l_{2}, α_{D}, ρ} > 0

with

\begin{matrix} \frac{max (K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} | τ |^{k - 1}, K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} | τ |^{k - 1 + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}))}{δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \\ {\times (1 + | τ |}^{2 k}) e^{- {ν | τ |}^{k}} \leq M_{S_{d}, k, l_{0}, l_{1}, l_{2}, α_{D}, ρ} \end{matrix}

(167)

for all

τ \in S_{d} \cup D_{ρ}

The gathering of (166), (116) and (167) yields

{\tilde{C}}_{τ, m} \leq M_{S_{d}, k, l_{0}, l_{1}, l_{2}, α_{D}, ρ} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{R_{\underset{̲}{l}}}{R_{D}} [\frac{| τ |}{1 + {| τ |}^{2 k}} e^{{ν | τ |}^{k}} {(1 + | m |)}^{- μ} e^{- β | m |}] A_{1} (m)

(168)

where

A_{1} (m)

is defined in (119), whenever

τ \in S_{d} \cup D_{ρ}

m \in R

. Finally, the last bounds (168) and (120) trigger a constant

C_{4} > 0

(depending on

μ, R_{D}, R_{\underset{̲}{l}}, S_{d}, k, α_{D}, l_{0}, l_{1}, l_{2}, ρ

) with

{\tilde{C}}_{τ, m} \leq C_{4} {| | ω | |}_{(ν, β, μ, k, ρ)} A_{\underset{̲}{l}} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}

(169)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

, which can be rewritten using norms as (150).

An integral expression related to the fifth building block of (108) is assessed in the next

Lemma 7.

One can single out a constant

C_{5} > 0

(relying on

μ, R_{D}, Q_{1}, Q_{2}, S_{d}, k, α_{D}

) with

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω_{2} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{5} | | ω_{1} {| |}_{(ν, β, μ, k, ρ)} | | ω_{2} {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(170)

for all

ω_{1}, ω_{2} \in F_{(ν, β, μ, k, ρ)}^{d}

Proof.

Let us take

ω_{1}, ω_{2} \in F_{(ν, β, μ, k, ρ)}^{d}

. Owing to the bounds (112), we deduce the next upper estimates

\begin{matrix} | ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) | \leq | | ω_{1} {| |}_{(ν, β, μ, k, ρ)} (1 + | m - m_{1} {|)}^{- μ} e^{- β | m - m_{1} |} \\ \times \frac{| {(τ^{k} - s)}^{1 / k} |}{1 + | τ^{k} {- s |}^{2}} exp (ν | τ^{k} - s |) \end{matrix}

(171)

and

| ω_{2} (s^{1 / k}, m_{1}) | \leq | | ω_{2} {| |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \frac{| s^{1 / k} |}{1 + {| s |}^{2}} exp (ν | s |)

(172)

for all

τ \in S_{d} \cup D_{ρ}

, all

s \in [0, τ^{k}]

, with

m, m_{1} \in R

. Besides, since

Q_{1} (X)

and

Q_{2} (X)

are polynomials, two constants

Q_{1}, Q_{2} > 0

can be exhibited such that

| Q_{1} (\sqrt{- 1} (m - m_{1})) | \leq Q_{1} (1 + | m - m_{1} {|)}^{\deg (Q_{1})}, | Q_{2} (\sqrt{- 1} m_{1}) | \leq Q_{2} (1 + | m_{1} {|)}^{\deg (Q_{2})}

(173)

for all

m, m_{1} \in R

. From these latter bounds and bearing in mind the lower estimates (74), we come up to

\begin{matrix} D_{τ, m} = | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω_{2} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} | \\ \leq \frac{1}{R_{D} {(1 + | m |)}^{\deg (R_{D})} δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})} \\ \times \int_{- \infty}^{+ \infty} {| τ |}^{k} \int_{0}^{{| τ |}^{k}} | | ω_{1} {| |}_{(ν, β, μ, k, ρ)} (1 + | m - m_{1} {|)}^{- μ} e^{- β | m - m_{1} |} \frac{{(| τ |}^{k} {- h)}^{1 / k}}{1 + {(| τ |}^{k} {- h)}^{2}} {exp (ν (| τ |}^{k} - h)) \\ \times Q_{1} (1 + | m - m_{1} {|)}^{\deg (Q_{1})} \times | | ω_{2} {| |}_{(ν, β, μ, k, ρ)} (1 + | m_{1} {|)}^{- μ} e^{- β | m_{1} |} \\ \times \frac{h^{1 / k}}{1 + h^{2}} e^{ν h} Q_{2} (1 + | m_{1} {|)}^{\deg (Q_{2})} \frac{1}{(| τ |^{k} - h) h} d h d m_{1} \end{matrix}

(174)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

. Moreover, by using the change of variable

h = {| τ |}^{k} u

, for

0 \leq u \leq 1

, one can select a constant

{\overset{ˇ}{M}}_{S_{d}, k, α_{D}} > 0

such that

\begin{matrix} sup_{τ \in S_{d} \cup D_{ρ}} \frac{{(1 + | τ |}^{2 k} {) | τ |}^{k - 1} \int_{0}^{{| τ |}^{k}} \frac{{(| τ |}^{k} {- h)}^{1 / k} h^{1 / k}}{(| τ |^{k} - h) h} \frac{1}{1 + {(| τ |}^{k} {- h)}^{2}} \frac{1}{1 + h^{2}} d h}{exp (α_{D} k^{2} Δ_{S_{d}, k} | τ |^{2 k})} \\ = sup_{τ \in S_{d} \cup D_{ρ}} \frac{{(1 + | τ |}^{2 k}) | τ | \int_{0}^{1} \frac{{(1 - u)}^{\frac{1}{k} - 1}}{1 + {| τ |}^{2 k} {(1 - u)}^{2}} \frac{u^{\frac{1}{k} - 1}}{1 + {| τ |}^{2 k} u^{2}} d u}{exp (α_{D} k^{2} Δ_{S_{d}, k} | τ |^{2 k})} \leq {\overset{ˇ}{M}}_{S_{d}, k, α_{D}} . \end{matrix}

(175)

Combining (174) and (175) prompts

\begin{matrix} D_{τ, m} \leq \frac{{\overset{ˇ}{M}}_{S_{d}, k, α_{D}}}{δ_{S_{d}, k, α_{D}}} \frac{Q_{1} Q_{2}}{R_{D}} | | ω_{1} {| |}_{(ν, β, μ, k, ρ)} | | ω_{2} {| |}_{(ν, β, μ, k, ρ)} \\ \times \{\frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}\} A_{2} (m) \end{matrix}

(176)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

, where

A_{2} (m) = {(1 + | m |)}^{μ - \deg (R_{D})} \int_{- \infty}^{+ \infty} \frac{1}{(1 + | m - m_{1} {|)}^{μ - \deg (Q_{1})}} \frac{1}{(1 + | m_{1} {|)}^{μ - \deg (Q_{2})}} d m_{1} .

(177)

According to Lemma 2.2 of [5] and under the assumptions (12) and (21), a constant

C_{5.1} > 0

is obtained with

A_{2} (m) \leq C_{5.1}

(178)

for all

m \in R

. Finally, from (176) and (178) we deduce a constant

C_{5} > 0

(depending on

μ, R_{D}, Q_{1}, Q_{2}, S_{d}, k, α_{D}

) with

D_{τ, m} \leq C_{5} | | ω_{1} {| |}_{(ν, β, μ, k, ρ)} | | ω_{2} {| |}_{(ν, β, μ, k, ρ)} \times \{\frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (1 + {| m |)}^{- μ} e^{- β | m |}\}

(179)

for all

τ \in S_{d} \cup D_{ρ}

m \in R

which precisely means that (170) holds true. □

In the next lemma, the tail piece of (108) is investigated.

Lemma. 8.

A constant

F_{J} > 0

(depending on

F_{j}

j \in J

R_{D}, S_{d}, k, α_{D}, ν

) can be singled for which

| | \sum_{j \in J} \frac{F_{j} (m)}{P_{m} (τ)} τ^{j} {| |}_{(ν, β, μ, k, ρ)} \leq F_{J} .

(180)

Proof.

By definition of (24), we notice that

| F_{j} (m) | \leq F_{j} {(1 + | m |)}^{- μ} e^{- β | m |}

(181)

for all

m \in R

. Besides, according to the geometric assumption (14) and the lower bounds (74) reached in Lemma 2, we see that

| P_{m} (τ) | \geq min_{m \in R} | R_{D} (\sqrt{- 1} m) | δ_{S_{d}, k, α_{D}} exp (α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})

(182)

for all

τ \in S_{d} \cup D_{ρ}

, all

m \in R

. As a result, we get

\begin{matrix} E_{τ, m} = | \sum_{j \in J} \frac{F_{j} (m)}{P_{m} (τ)} τ^{j} | \leq \sum_{j \in J} F_{j} {(1 + | m |)}^{- μ} e^{- β | m |} \frac{1}{{min}_{m \in R} | R_{D} (\sqrt{- 1} m) | δ_{S_{d}, k, α_{D}}} \\ \times {| τ |}^{j - 1} exp (- α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k}) (1 + {| τ |}^{2 k}) exp (- {ν | τ |}^{k}) \times [\frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k})] \end{matrix}

(183)

provided that

τ \in S_{d} \cup D_{ρ}

m \in R

. Since

J \subset N^{*}

does not contain the origin, a constant

{\hat{M}}_{j, S_{d}, k, ν, α_{D}} > 0

can be pinpointed such that

sup_{τ \in S_{d} \cup D_{ρ}} {| τ |}^{j - 1} {(1 + | τ |}^{2 k}) exp (- α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k}) exp (- {ν | τ |}^{k}) = {\hat{M}}_{j, S_{d}, k, ν, α_{D}} .

(184)

Lastly, adding up (183) and (184), we arrive at

E_{τ, m} \leq \{\sum_{j \in J} F_{j} \frac{{\hat{M}}_{j, S_{d}, k, ν, α_{D}}}{{min}_{m \in R} | R_{D} (\sqrt{- 1} m) | δ_{S_{d}, k, α_{D}}}\} {(1 + | m |)}^{- μ} e^{- β | m |} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k})

(185)

as long as

τ \in S_{d} \cup D_{ρ}

m \in R

. Lemma 8 follows. □

Now, we choose the constants

A_{\underset{̲}{l}}

for

\underset{̲}{l} \in A

and

c_{Q_{1} Q_{2}} \in C^{*}

close enough to 0 in a manner that one can find some radius

ϖ > 0

fulfilling the next constraint

\begin{matrix} \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} k^{l_{1}} C_{1} A_{\underset{̲}{l}} ϖ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} \frac{k^{l_{1}}}{Γ (l_{0} / k)} C_{2} A_{\underset{̲}{l}} ϖ \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} k^{l_{1}} \frac{k^{2} / l_{2}}{2 π} C_{3} A_{\underset{̲}{l}} ϖ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} \frac{k^{2} / l_{2}}{2 π} C_{4} A_{\underset{̲}{l}} ϖ \\ + | c_{Q_{1} Q_{2}} | \frac{1}{{(2 π)}^{1 / 2}} C_{5} ϖ^{2} + F_{J} \leq ϖ \end{matrix}

(186)

for the constants

C_{j} > 0

1 \leq j \leq 5

and

F_{J} > 0

appearing in the above lemmas. Eventually, the appliance of the bounds recorded in the lemmas 3, 4, 5, 6, 7 and 8 under the condition (186) yields the expected inclusion (110).

We turn to the second item 2. Let us fix the radius

ϖ > 0

as above and select

ω_{1}, ω_{2} \in B_{ϖ} \subset F_{(ν, β, μ, k, ρ)}^{d}

. In the next list of lemmas, we discuss bounds for each piece of the difference

H (ω_{1}) - H (ω_{2})

A direct issue of Lemma 3 gives rise to

Lemma. 9.

Take

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} = 0

and

l_{2} = 1

. Then,

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) τ^{k l_{1}} [ω_{1} (τ, m_{1}) - ω_{2} (τ, m_{1})] R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \\ \leq C_{1} A_{\underset{̲}{l}} | | ω_{1} - ω_{2} {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(187)

holds for the constant

C_{1} > 0

disclosed in Lemma 3.

As a consequence of Lemma 4, we obtain

Lemma. 10.

Let

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} \geq 1

and

l_{2} = 1

. Then,

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) [τ^{k} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l_{0}}{k} - 1} s^{l_{1}} [ω_{1} (s^{1 / k}, m_{1}) - ω_{2} (s^{1 / k}, m_{1})] \frac{d s}{s}] \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{2} A_{\underset{̲}{l}} | | ω_{1} - ω_{2} {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(188)

holds for the constant

C_{2} > 0

croping up in Lemma 4.

An application of Lemma 5 yields

Lemma. 11.

Let

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} = 0

and

l_{2} > 1

. The next inequality

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (\int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} ξ^{k l_{1}} [ω_{1} (ξ, m_{1}) - ω_{2} (ξ, m_{1})] D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{3} A_{\underset{̲}{l}} | | ω_{1} (τ, m) - ω_{2} (τ, m) {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(189)

holds for the constant

C_{3} > 0

appearing in Lemma 5.

Lemma 6 enables to set up the next

Lemma. 12.

We choose

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

with

l_{0} \geq 1

and

l_{2} > 1

. Then,

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) (\int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω_{1} - ω_{2}) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \\ \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} {| |}_{(ν, β, μ, k, ρ)} \leq C_{4} A_{\underset{̲}{l}} | | ω_{1} - ω_{2} {| |}_{(ν, β, μ, k, ρ)} \end{matrix}

(190)

holds true for the constant

C_{4} > 0

showing up in Lemma 6.

In order to control the norm of the nonlinear terms of the difference

H (ω_{1}) - H (ω_{2})

, we rewrite the next difference as a sum

\begin{matrix} ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) ω_{1} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \\ - ω_{2} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) ω_{2} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \\ = [ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) - ω_{2} ({(τ^{k} - s)}^{1 / k}, m - m_{1})] Q_{1} (\sqrt{- 1} (m - m_{1})) \times ω_{1} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \\ + ω_{2} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \times [ω_{1} (s^{1 / k}, m_{1}) - ω_{2} (s^{1 / k}, m_{1})] Q_{2} (\sqrt{- 1} m_{1}) . \end{matrix}

(191)

As a result of Lemma 7 and the above reordering (191), we come up with the next

Lemma. 13.

The following inequality

\begin{matrix} | | \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} ω_{1} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω_{1} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} \\ - \frac{1}{P_{m} (τ)} \int_{- \infty}^{+ \infty} (τ^{k} \int_{0}^{τ^{k}} ω_{2} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω_{2} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} | | \leq C_{5} | | ω_{1} - ω_{2} {| |}_{(ν, β, μ, k, ρ)} \\ \times [| | ω_{1} {| |}_{(ν, β, μ, k, ρ)} + | | ω_{2} {| |}_{(ν, β, μ, k, ρ)}] \end{matrix}

(192)

holds where

C_{5} > 0

is the constant arising in Lemma 7.

We adjust the constants

A_{\underset{̲}{l}}

\underset{̲}{l} \in A

and

c_{Q_{1} Q_{2}} \in C^{*}

nearby the origin in a way that the next restriction

\begin{matrix} \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} k^{l_{1}} C_{1} A_{\underset{̲}{l}} + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} = 1}} \frac{1}{{(2 π)}^{1 / 2}} \frac{k^{l_{1}}}{Γ (l_{0} / k)} C_{2} A_{\underset{̲}{l}} \\ + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} = 0, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} k^{l_{1}} \frac{k^{2} / l_{2}}{2 π} C_{3} A_{\underset{̲}{l}} + \sum_{\overset{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A}{l_{0} \geq 1, l_{2} > 1}} \frac{1}{{(2 π)}^{1 / 2}} \frac{k^{2} / l_{2}}{2 π} C_{4} A_{\underset{̲}{l}} \\ + | c_{Q_{1} Q_{2}} | \frac{1}{{(2 π)}^{1 / 2}} C_{5} 2 ϖ \leq 1 / 2 \end{matrix}

(193)

holds. The collection of lemmas 9,10,11,12 and 13, accounting of the above condition (193) yields the contraction property (111).

At the end, we choose the constants

A_{\underset{̲}{l}}

for

\underset{̲}{l} \in A

and

c_{Q_{1} Q_{2}} \in C^{*}

appropriately close to 0, along with a radius

ϖ > 0

in a manner that both conditions (186) and (193) hold at once. It follows that the map

H

obeys both features (110) and (111). Proposition 9 follows.

The next proposition provides sufficient conditions for which the auxiliary equation (72) is endowed with a solution in the Banach space described in Definition 5.

Proposition 10.

Under the assumptions made in the statement of Proposition 9, we can find a constant

ϖ > 0

for which the auxiliary equation (72) hosts a unique solution

ω^{d}

which belongs to the space

F_{(ν, β, μ, k, ρ)}^{d}

and is subjected to the bounds

| | ω^{d} {| |}_{(ν, β, μ, k, ρ)} \leq ϖ .

(194)

Proof.

For

ϖ > 0

suitably chosen as in Proposition 9, we observe that the map

H

induces a contractive application from the metric space

(B_{ϖ}, d)

into itself, where

B_{ϖ}

stands for the closed ball of radius

ϖ > 0

centered at 0 in

F_{(ν, β, μ, k, ρ)}^{d}

and the distance d is induced from the norm

{| | . | |}_{(ν, β, μ, k, ρ)}

by the expression

d (x, y) = | | x - {y | |}_{(ν, β, μ, k, ρ)}

. Since

(F_{(ν, β, μ, k, ρ)}^{d}, | | . | |_{(ν, β, μ, k, ρ)})

is a Banach space, the metric space

(B_{ϖ}, d)

is complete. Then, according to the classical contractive mapping theorem, the map

H

has a fixed point we denote

ω^{d}

B_{ϖ}

, meaning that

H (ω^{d}) = ω^{d},

(195)

which implies in particular that the analytic map

ω^{d}

solves the equation (72). Proposition 10 ensues. □

5. Statement of the Main Results

We are in position to state the first prominent result of our work.

Theorem 1.

Let us take for granted that the assumptions (9), (10), (11), (12), (14), (21), (23), (25), (26) hold for the shape of the main problem (8) with vanishing initial condition

u (0, z) \equiv 0

We assume furthermore that the sector

S_{Q, R_{D}}

defined in (13) obeys the requirements asked in Lemma 2. We select

−: an unbounded sector $S_{d}$ edged at 0, with bisecting direction d belonging to the set $Θ_{Q, R_{D}}$ (introduced in Lemma 2) fulfilling the two conditions (86), (87),
−: a disc $D_{ρ}$ whose radius $ρ > 0$ fits the restrictions (88), (89).

Then, provided that

−: the constant $α_{D}$ appearing in the leading operator (15) of infinite order in (8) is chosen in agreement with (109),
−: the constants $A_{\underset{̲}{l}}$ , for $\underset{̲}{l} \in A$ , set up in (22) and the coefficient $c_{Q_{1} Q_{2}}$ of the nonlinear term of (8) are taken close enough to 0

there exist a formal power series

\hat{u} (t, z) = \sum_{n \geq 1} u_{n} (z) t^{n}

solution to (8) with

\hat{u} (0, z) \equiv 0

whose coefficients $u_{n}$ belong to the Banach space $O_{b} (H_{β^{'}})$ of bounded holomorphic functions on the strip $H_{β^{'}}$ (given in (18)) for any prescribed $0 < β^{'} < β$ endowed with the sup norm ${| | . | |}_{\infty}$ ,
which is $m_{k} -$ summable in any direction d chosen as above in the set $Θ_{Q, R_{D}}$ as a series with coefficients in $(O_{b} (H_{β^{'}}), | | . | |_{\infty})$ (see Definition 3).

Proof.

Under the assumptions made in Theorem 1, we observe that Proposition 10 holds. For the given sector

S_{d}

with

d \in Θ_{Q, R_{D}}

and disc

D_{ρ}

as constructed in Lemma 2, for any given real number

ν > 0

and for the constants

β, μ, k > 0

prescribed in Section 2, we depart from the solution

ω^{d}

of the auxiliary equation (72) that belongs to the Banach space

F_{(ν, β, μ, k, ρ)}^{d}

under the condition

| | ω^{d} {| |}_{(ν, β, μ, k, ρ)} \leq ϖ

(196)

for some well chosen constant

ϖ > 0

. By construction, since the partial map

τ \mapsto ω^{d} (τ, m)

is holomorphic on the disc

D_{ρ}

it has a convergent power series expansion

ω^{d} (τ, m) = \hat{ω} (τ, m) : = \sum_{n \geq 1} ω_{n} (m) τ^{n}

(197)

on the disc

D_{ρ / 2}

, where the coefficients

ω_{n} (m)

can be expressed in integral form

ω_{n} (m) = \frac{1}{2 π \sqrt{- 1}} \int_{C_{ρ / 2}} \frac{ω^{d} (ξ, m)}{ξ^{n + 1}} d ξ

(198)

along the positively oriented circle

C_{ρ / 2}

centered at 0 with radius

ρ / 2

. According to (196), a constant

C_{ϖ, k, ν, ρ} > 0

(relying on

ϖ, k, ν, ρ

) can be deduced with

| ω_{n} (m) | \leq C_{ϖ, k, ν, ρ} {(2 / ρ)}^{n} {(1 + | m |)}^{- μ} e^{- β | m |}

(199)

for all

m \in R

. In particular, each coefficient

m \mapsto ω_{n} (m)

belongs to the Banach space

(E_{(β, μ)}, | | . | |_{(β, μ)})

and

\hat{ω} (τ, m) \in E_{(β, μ)} {τ} .

(200)

Furthermore, owing to (196), we notice that the partial map

τ \mapsto \hat{ω} (τ, m)

, seen as a holomorphic map on

D_{ρ / 2}

in the Banach space

E_{(β, μ)}

can be extended to a holomorphic map denoted

τ \mapsto ω^{d} (τ, m)

on the sector

S_{d}

with bounds

| | ω^{d} {(τ, m) | |}_{(β, μ)} \leq ϖ \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k})

(201)

for all

τ \in S_{d}

Let us define the formal power series

\hat{U} (t, m) = \sum_{n \geq 1} U_{n} (m) t^{n}

(202)

where the coefficients

U_{n} (m)

are defined by

U_{n} (m) = ω_{n} (m) Γ (n / k)

(203)

for all

n \geq 1

. By construction, the series

\hat{ω} (τ, m)

given by (197) represents the

m_{k} -

Borel transform of the formal power series (202). From (200) and (201), we deduce that the formal series

\hat{U} (t, m)

m_{k} -

summable in direction d, viewed as series with coefficients in

(E_{(β, μ)}, | | . | |_{(β, μ)})

, see Definition 3.

According to Proposition 10, the convergent series

\hat{ω} (τ, m)

matches the auxiliary equation (72). Taking heed of Proposition 8, we deduce that the formal series (202) solves the differential/convolution equation (30). Let us introduce the formal power series

\hat{u} (t, z) = \sum_{n \geq 1} u_{n} (z) t^{n}

(204)

where the coefficients

u_{n} (z)

are defined as the inverse Fourier transform

u_{n} (z) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} U_{n} (m) e^{\sqrt{- 1} z m} d m

(205)

for all integers

n \geq 1

z \in H_{β^{'}}

, for any given

0 < β^{'} < β

. As claimed by Proposition 1, it follows that

\hat{u} (t, z)

formally solves the main equation (8).

Bearing in mind (199) and (203), the next upper bounds

| u_{n} (z) | \leq \frac{1}{{(2 π)}^{1 / 2}} C_{ϖ, k, ν, ρ} Γ (n / k) {(2 / ρ)}^{n} \int_{- \infty}^{+ \infty} {(1 + | m |)}^{- μ} e^{- (β - β^{'}) | m |} d m

(206)

hold provided that

n \geq 1

and

z \in H_{β^{'}}

, with prescribed

0 < β^{'} < β

. In particular, we observe that each coefficient

u_{n}

belongs to

(O_{b} (H_{β^{'}}), | | . | |_{\infty})

, for

n \geq 1

. It ensues that the series

\sum_{n \geq 0} \frac{{sup}_{z \in H_{β^{'}}} | u_{n} (z) |}{Γ (n / k)} {(ρ^{'})}^{n}

is convergent for any

0 < ρ^{'} < ρ / 2

. As a result, the

m_{k} -

Borel transform of

\hat{u}

given by

B_{m_{k}} (\hat{u}) (τ) = \sum_{n \geq 1} \frac{u_{n} (z)}{Γ (n / k)} τ^{n}

(207)

is convergent on

D_{ρ^{'}}

as a series in coefficients in the Banach space

(O_{b} (H_{β^{'}}), | | . | |_{\infty})

, meaning that

B_{m_{k}} (\hat{u}) \in τ O_{b} (H_{β^{'}}) {τ} .

(208)

Besides, the expansion (197) allows the

m_{k} -

Borel transform

B_{m_{k}} (\hat{u})

to be expressed in integral form

B_{m_{k}} (\hat{u}) (τ) = \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} ω^{d} (τ, m) e^{\sqrt{- 1} z m} d m

(209)

for all

z \in D_{ρ^{'}}

with

0 < ρ^{'} < ρ / 2

. Based on the bounds (201), it follows that the map

τ \mapsto B_{m_{k}} (\hat{u}) (τ)

viewed as a function from

D_{ρ^{'}}

into

(O_{b} (H_{β^{'}}), | | . | |_{\infty})

can be analytically continued along the unbounded sector

S_{d}

and is subjected to the bounds

sup_{z \in H_{β^{'}}} | B_{m_{k}} (\hat{u}) (τ) | \leq \frac{ϖ}{{(2 π)}^{1 / 2}} \frac{| τ |}{1 + {| τ |}^{2 k}} {exp (ν | τ |}^{k}) (\int_{- \infty}^{+ \infty} {(1 + | m |)}^{- μ} e^{- (β - β^{'}) | m |} d m)

(210)

as long as

τ \in S_{d}

. Finally, bearing in mind Definition 3, on the ground of the two above features (208) and (210), we deduce that the formal solution

\hat{u} (t, z)

to (8) with

\hat{u} (0, z) \equiv 0

given by (204) is

m_{k} -

summable in direction d. □

In the second foremost outcome of the paper (Theorem 2), we disclose some functional equations satisfied by the

m_{k} -

sum of the formal solution

\hat{u} (t, z)

to (8) built up in Theorem 1. Before stating the main result, we need to introduce some integral operators acting on Fourier-Laplace transforms that are described in the next

Proposition 11.

We consider an unbounded sector

S_{d}

edged at 0 with bisecting direction

d \in Θ_{Q, R_{D}}

and a small disc

D_{ρ}

centered at 0 with radius

ρ > 0

chosen as in Lemma 2. We take for granted that the constant

α_{D} > 0

appearing in (15) fulfills the condition (109). We fix some real number

ν > 0

and prescribe the constants

β, μ, k > 0

as in Section 2.

For any given

ω^{d}

F_{(ν, β, μ, k, ρ)}^{d}

, we define the Fourier-Laplace transform

u^{d} (t, z) = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} ω^{d} (τ, m) exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m

(211)

along the halfline

L_{d} = [0, + \infty) e^{\sqrt{- 1} d}

. According to Definitions 2 and 3, we know that the function

u^{d}

represents a bounded holomorphic map on the product

S_{d, ϑ, R} \times H_{β^{'}}

, where

S_{d, ϑ, R}

is a bounded sector of the form (35) for an angle ϑ satisfying

\frac{π}{k} < ϑ < \frac{π}{k} + Op (S_{d})

, with

Op (S_{d})

standing for the opening of

S_{d}

and for a small enough radius

R > 0

, where

H_{β^{'}}

is the strip displayed in (18) for any given

0 < β^{'} < β

We distinguish two different situations.

We assume that the constant $Δ_{S_{d}, k} > 0$ appearing in the lower bounds (74) satisfies the requirement

$cos (2 k θ) > Δ_{S_{d}, k}$

(212)

for all $θ \in R$ with $e^{\sqrt{- 1} θ} \in S_{d}$ . We introduce the next integral operator defined by its action on $u^{d}$ as follows

$\begin{matrix} exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2}) u^{d} (t, z) : = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} exp (- α_{D} {(k τ^{k})}^{2}) ω^{d} (τ, m) \\ \times exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m . \end{matrix}$

(213)

Let $\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A$ , where $A$ is depicted in Section 2. According to the notation (70), we set

$C_{k, l_{0}, l_{1}} (ω^{d}) (τ, m) : = \frac{τ^{k}}{Γ (l_{0} / k)} \int_{0}^{τ^{k}} {(τ^{k} - s)}^{\frac{l_{0}}{k} - 1} {(k s)}^{l_{1}} ω^{d} (s^{1 / k}, m) \frac{d s}{s}$

(214)

for all integers $l_{0} \geq 1$ , $l_{1} \geq 0$ . Besides, when $l_{0} = 0$ , we denote

$C_{k, 0, l_{1}} (ω^{d}) (τ, m) : = {(k τ^{k})}^{l_{1}} ω^{d} (τ, m) .$

(215)

We define the next integral operator through its action on $u^{d}$ by

$\begin{matrix} G_{l_{0}, l_{1}, l_{2}, k, α_{D}}^{-} (u^{d}) (t, z) : = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} exp (- α_{D} {(k τ^{k})}^{2}) \\ \times (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \times exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m . \end{matrix}$

(216)

Then, both functions (213) and (216) are well defined and bounded holomorphic on the product $S_{d, ϑ, R} \times H_{β^{'}}$ .
The constant $Δ_{S_{d}, k} > 0$ that arises in the lower bounds (74) is assumed to obey the condition

$cos (2 k θ) < - Δ_{S_{d}, k}$

(217)

for all $θ \in R$ with $e^{\sqrt{- 1} θ} \in S_{d}$ . We set up the following two integral operators acting on $u^{d}$ by means of

$\begin{matrix} exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) u^{d} (t, z) : = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} exp (α_{D} {(k τ^{k})}^{2}) ω^{d} (τ, m) \\ \times exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m \end{matrix}$

(218)

and for any $\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A$ ,

$\begin{matrix} G_{l_{0}, l_{1}, l_{2}, k, α_{D}}^{+} (u^{d}) (t, z) : = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} exp (α_{D} {(k τ^{k})}^{2}) \\ \times (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \times exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m, \end{matrix}$

(219)

keeping the notations (214) and (215). As a result, the two expressions (218) and (219) represent bounded holomorphic maps on the product $S_{d, ϑ, R} \times H_{β^{'}}$ .

Proof.

We focus on the first item. Under the restriction (212), we remind from (98) that the inequality

| exp (- α_{D} k^{2} τ^{2 k}) | \leq exp (- α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})

(220)

holds provided that

τ \in S_{d}

. It follows that the expression (213) is well defined and bounded holomorphic on

S_{d, ϑ, R} \times H_{β^{'}}

. For

\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A

, one sets

K (τ, m) = \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ .

(221)

According to Sublemma 1 and 2, we get that

For any given $\tilde{ρ} \geq ρ$ , there exists a constant $K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} > 0$ with

$| K (τ, m) | \leq K_{\tilde{ρ}, k, l_{0}, l_{1}, l_{2}}^{1} | | ω^{d} {| |}_{(ν, β, μ, k, ρ)} {| τ |}^{k} {(1 + | m |)}^{- μ} e^{- β | m |}$

(222)

provided that $τ \in D_{\tilde{ρ}}$ and $m \in R$ .
A constant $K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} > 0$ can be reached such that

$| K (τ, m) | \leq K_{ρ, k, l_{0}, l_{1}, l_{2}}^{2} | | ω^{d} {| |}_{(ν, β, μ, k, ρ)} {| τ |}^{k + \frac{k}{l_{2} - 1}} exp (\frac{{| τ |}^{\frac{k l_{2}}{l_{2} - 1}}}{{(ρ / 2)}^{\frac{k}{l_{2} - 1}}}) {(1 + | m |)}^{- μ} e^{- β | m |}$

(223)

as long as $τ \in S_{d}$ with $| τ | \geq {(ρ / 2)}^{1 / l_{2}}$ and $m \in R$ .

Since

2 k = \frac{k l_{2}}{l_{2} - 1}

when

l_{2} = 2

and

2 k > \frac{k l_{2}}{l_{2} - 1}

for

l_{2} > 2

, from the bounds (220), (222), (223), under the constraint (109), we deduce that the integral (216) is well defined and represents a bounded holomorphic function on

S_{d, ϑ, R} \times H_{β^{'}}

In the second part of the proof, the second item is discussed. It follows from the condition (217) and the lower bounds (105) showing that

| exp (α_{D} k^{2} τ^{2 k}) | \leq exp (- α_{D} k^{2} Δ_{S_{d}, k} {| τ |}^{2 k})

(224)

for all

τ \in S_{d}

. Hence, the expression (218) turns out to be well defined and bounded holomorphic on

S_{d, ϑ, R} \times H_{β^{'}}

At last, taking heed of the above upper bounds (222), (223), (224) and the restriction (109), we observe that the integral (219) represents a bounded holomorphic function on

S_{d, ϑ, R} \times H_{β^{'}}

. □

The second principal result of this paper is disclosed in the next

Theorem 2.

Let us assume that the hypotheses formulated in Theorem 1 hold. Let

d \in R

be a direction chosen in the set

Θ_{Q, R_{D}}

(discussed in Lemma 2). We consider the formal power series

\hat{u} (t, z)

solution of our main equation (8) with initial vanishing data

\hat{u} (0, z) \equiv 0

. From Theorem 1, we know that

\hat{u} (t, z)

m_{k} -

summable in the given direction d. We denote

u^{d} (t, z)

its

m_{k} -

sum in the direction d. The map

u^{d} (t, z)

defines a bounded holomorphic function on the product

S_{d, ϑ, R} \times H_{β^{'}}

, where

S_{d, ϑ, R}

denotes a bounded sector shaped as (35) for an angle ϑ satisfying

\frac{π}{k} < ϑ < \frac{π}{k} + Op (S_{d})

, with

Op (S_{d})

representing the opening of

S_{d}

and for a small enough radius

R > 0

, where

H_{β^{'}}

is the strip given in (18) for any given

0 < β^{'} < β

Two alternatives arise.

Assume that the unbounded sector $S_{d}$ (displayed in the first item of Theorem 1) and the constant $Δ_{S_{d}, k} > 0$ stemming from the lower bounds (74) conform the condition (212). Then, the $m_{k} -$ sum $u^{d} (t, z)$ solves the next functional equation involving the integral operators (213) and (216) given by

$\begin{matrix} exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2}) Q (\partial_{z}) u^{d} (t, z) = [\frac{1}{2} + \frac{1}{2} exp (- 2 α_{D} {(t^{k + 1} \partial_{t})}^{2})] R_{D} (\partial_{z}) u^{d} (t, z) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} = 1} a_{\underset{̲}{l}} (z) R_{\underset{̲}{l}} (\partial_{z}) exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2}) [t^{l_{0}} {(t^{k + 1} \partial_{t})}^{l_{1}} u^{d} (t, z)] \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} > 1} a_{\underset{̲}{l}} (z) R_{\underset{̲}{l}} (\partial_{z}) G_{l_{0}, l_{1}, l_{2}, k, α_{D}}^{-} (u^{d}) (t, z) \\ + c_{Q_{1} Q_{2}} exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2}) [Q_{1} (\partial_{z}) u^{d} (t, z) Q_{2} (\partial_{z}) u^{d} (t, z)] \\ + exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2}) f (t, z) \end{matrix}$

(225)

on the domain $S_{d, ϑ, R} \times H_{β^{'}}$ provided that the radius $R > 0$ is taken small enough.
Take for granted that the sector $S_{d}$ and the constant $Δ_{S_{d}, k} > 0$ obey the condition (217). Then, the $m_{k}$ -sum $u^{d} (t, z)$ is a solution of the following functional equation which comprises the integral operators (218) and (219) displayed as

$\begin{matrix} exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) Q (\partial_{z}) u^{d} (t, z) = [\frac{1}{2} + \frac{1}{2} exp (2 α_{D} {(t^{k + 1} \partial_{t})}^{2})] R_{D} (\partial_{z}) u^{d} (t, z) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} = 1} a_{\underset{̲}{l}} (z) R_{\underset{̲}{l}} (\partial_{z}) exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) [t^{l_{0}} {(t^{k + 1} \partial_{t})}^{l_{1}} u^{d} (t, z)] \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} > 1} a_{\underset{̲}{l}} (z) R_{\underset{̲}{l}} (\partial_{z}) G_{l_{0}, l_{1}, l_{2}, k, α_{D}}^{+} (u^{d}) (t, z) \\ + c_{Q_{1} Q_{2}} exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) [Q_{1} (\partial_{z}) u^{d} (t, z) Q_{2} (\partial_{z}) u^{d} (t, z)] \\ + exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) f (t, z) \end{matrix}$

(226)

on the product $S_{d, ϑ, R} \times H_{β^{'}}$ as long as the radius $R > 0$ is chosen nearby the origin.

Proof.

The assumptions made in Theorem 1 enable Proposition 10 to be applied. For some prescribed sector

S_{d}

with

d \in Θ_{Q, R_{D}}

and disc

D_{ρ}

as put forward in Lemma 2, for any given real number

ν > 0

and for the constants

β, μ, k > 0

chosen in Section 2, we depart from the solution

ω^{d}

of the auxiliary equation (72) that belongs to the Banach space

F_{(ν, β, μ, k, ρ)}^{d}

under the condition (196) for some well chosen constant

ϖ > 0

. Owing to the integral representation (209), we know from Definition 3 and the expression (34) of the Laplace transform of order k, that the

m_{k} -

sum

u^{d} (t, z)

of the formal solution

\hat{u} (t, z)

of (8) given by (204) is expressed as the next Fourier-Laplace transform

u^{d} (t, z) = \frac{k}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} \int_{L_{d}} ω^{d} (τ, m) exp (- {(\frac{τ}{t})}^{k}) e^{\sqrt{- 1} z m} \frac{d τ}{τ} d m

(227)

which defines a bounded holomorphic map on the product

S_{d, ϑ, R} \times H_{β^{'}}

, where

S_{d, ϑ, R}

stands for a bounded sector shaped as (35) with an angle

ϑ

subjected to

\frac{π}{k} < ϑ < \frac{π}{k} + Op (S_{d})

, where

Op (S_{d})

represents the opening of

S_{d}

and for a small enough radius

R > 0

, where

H_{β^{'}}

represents the strip given in (18) for any given

0 < β^{'} < β

We first assume the condition (212) imposed in the first item of Theorem 2. We multiply the auxiliary equation (72) fulfilled by

ω^{d} (τ, m)

by the function

exp (- α_{D} {(k τ^{k})}^{2})

which yields the next equality

\begin{matrix} Q (\sqrt{- 1} m) exp (- α_{D} {(k τ^{k})}^{2}) ω^{d} (τ, m) = [\frac{1}{2} + \frac{1}{2} exp (- 2 α_{D} {(k τ^{k})}^{2})] R_{D} (\sqrt{- 1} m) ω^{d} (τ, m) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} = 1} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) exp (- α_{D} {(k τ^{k})}^{2}) C_{k, l_{0}, l_{1}} (ω^{d}) (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} > 1} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) exp (- α_{D} {(k τ^{k})}^{2}) \\ \times (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + c_{Q_{1} Q_{2}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} exp (- α_{D} {(k τ^{k})}^{2}) (τ^{k} \int_{0}^{τ^{k}} ω^{d} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω^{d} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} + exp (- α_{D} {(k τ^{k})}^{2}) \times \sum_{j \in J} F_{j} (m) τ^{j} \end{matrix}

(228)

for all

τ \in S_{d} \cup D_{ρ}

and all

m \in R

. We apply the Laplace transform

L_{m_{k}}

of order k in direction d displayed by the formula (34) and the inverse Fourier transform (17) to the left and right handside of the above equality (228). From the first item of Proposition 11, together with the identities (19), (20) in Definition 2 and the formula (38) disclosed in Proposition 4, we deduce that the

m_{k} -

sum

u^{d} (t, z)

given by (227) solves the functional equation (225) on the domain

S_{d, ϑ, R} \times H_{β^{'}}

, provided that the radius

R > 0

is chosen small enough.

Assume the condition (217) holds as asked in the second item of Theorem 2. Each side of the equation (72) satisfied by

ω^{d} (τ, m)

is then multiplied by the function

exp (α_{D} {(k τ^{k})}^{2})

which is recast in the form

\begin{matrix} Q (\sqrt{- 1} m) exp (α_{D} {(k τ^{k})}^{2}) ω^{d} (τ, m) = [\frac{1}{2} + \frac{1}{2} exp (2 α_{D} {(k τ^{k})}^{2})] R_{D} (\sqrt{- 1} m) ω^{d} (τ, m) \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} = 1} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) exp (α_{D} {(k τ^{k})}^{2}) C_{k, l_{0}, l_{1}} (ω^{d}) (τ, m_{1}) R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + \sum_{\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A; l_{2} > 1} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} A_{\underset{̲}{l}} (m - m_{1}) exp (α_{D} {(k τ^{k})}^{2}) \\ \times (- \frac{k^{2} / l_{2}}{2 \sqrt{- 1} π} \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m_{1}) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ) \times R_{\underset{̲}{l}} (\sqrt{- 1} m_{1}) d m_{1} \\ + c_{Q_{1} Q_{2}} \frac{1}{{(2 π)}^{1 / 2}} \int_{- \infty}^{+ \infty} exp (α_{D} {(k τ^{k})}^{2}) (τ^{k} \int_{0}^{τ^{k}} ω^{d} ({(τ^{k} - s)}^{1 / k}, m - m_{1}) Q_{1} (\sqrt{- 1} (m - m_{1})) \\ \times ω^{d} (s^{1 / k}, m_{1}) Q_{2} (\sqrt{- 1} m_{1}) \frac{1}{(τ^{k} - s) s} d s) d m_{1} + exp (α_{D} {(k τ^{k})}^{2}) \times \sum_{j \in J} F_{j} (m) τ^{j} \end{matrix}

(229)

whenever

τ \in S_{d} \cup D_{ρ}

and

m \in R

. The Laplace transform

L_{m_{k}}

of order k in direction d given by the formula (34) and the inverse Fourier transform (17) are applied to the left and right handside of the above equality (229). According to the second item of Proposition 11 and the identities discussed in Definition 2 and Proposition 4, we deduce that the

m_{k} -

sum

u^{d} (t, z)

expressed as a double integral (227) obeys the functional equation (226) on the product

S_{d, ϑ, R} \times H_{β^{'}}

, on the condition that the radius

R > 0

is chosen small enough. □

At last, we justify the statement of Theorem 2 with the next

Remark Observe that the

m_{k} -

sum

u^{d} (t, z)

of the formal solution of (8) given by the expression (227) does not (in general) fulfill the same equation (8). There are two reasons for that.

−: The action of the infinite order differential operator

$cosh (α_{D} {(t^{k + 1} \partial_{t})}^{2}) = \frac{1}{2} exp (α_{D} {(t^{k + 1} \partial_{t})}^{2}) + \frac{1}{2} exp (- α_{D} {(t^{k + 1} \partial_{t})}^{2})$

given in (15) is not well defined on $u^{d} (t, z)$ since the map $τ \mapsto exp (α_{D} {(k τ^{k})}^{2})$ or $τ \mapsto exp (- α_{D} {(k τ^{k})}^{2})$ has an exponential growth of order $2 k$ where a growth rate of at most order k is required on the sector $S_{d}$ .
−: The action of the Mahler operator $t \mapsto t^{l_{2}}$ is not properly settled on $t^{l_{0}} {(t^{k + 1} \partial_{t})}^{l_{1}} u^{d} (t, z)$ for any given $\underset{̲}{l} = (l_{0}, l_{1}, l_{2}) \in A$ since the analytic map

$τ \mapsto \int_{{\tilde{γ}}_{\frac{k}{l_{2}}, τ^{l_{2}}}} C_{k, l_{0}, l_{1}} (ω^{d}) (ξ, m) D_{k, \frac{k}{l_{2}}} (ξ, τ^{l_{2}}) d ξ$

endows (at most) an exponential growth of order $\frac{k l_{2}}{l_{2} - 1}$ which exceeds the admissible order k on the sector $S_{d}$ .

References

B. Adamczewski, J.P. Bell. A problem about Mahler functions. > Ann. Sc. Norm. Super. Pisa Cl. Sci 2017, 17, 1301–1355. [Google Scholar]
B. Adamczewski, T. Dreyfus, C. Hardouin. Hypertranscendence and linear difference equations. J. Amer. Math. Soc. 2021, 34, 475–503. [Google Scholar] [CrossRef]
W. Balser. From divergent power series to analytic functions. Theory and application of multisummable power series; Lecture Notes in Mathematics; Springer-Verlag: Berlin, 1994; p. x+108. [Google Scholar]
W. Balser. Formal power series and linear systems of meromorphic ordinary differential equations; Universitext. Springer-Verlag: New York, 2000; p. xviii+299. [Google Scholar]
O. Costin, S. Tanveer. Short time existence and Borel summability in the Navier-Stokes equation in R³. Comm. Partial Differential Equations 2009, 34, 785–817. [Google Scholar] [CrossRef]
C. Faverjon, J. Roques. Hahn series and Mahler equations: algorithmic aspects. J. Lond. Math. Soc., II Ser. 2024, 110, e12945. [Google Scholar] [CrossRef]
A. Lastra, S. Malek. Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems. J. Differential Equations 2015, 259, 5220–5270. [Google Scholar] [CrossRef]
A. Lastra, S. Malek. On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems. Advances in Difference Equations 2015, 2015, 200. [Google Scholar] [CrossRef]
A. Lastra, S. Malek. On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms. Adv. Difference Equ. 2018, 386, 40. [Google Scholar]
S. Ōuchi. On some functional equations with Borel summable solutions. Funkc. Ekvacioj, Ser. Int. 2015, 58, 223–251. [Google Scholar] [CrossRef]
J. Roques. On the local structure of Mahler systems. Int. Math. Res. Not. 2021, 2021, 9937–9957. [Google Scholar] [CrossRef]
J. Roques. Frobenius method for Mahler equations. J. Math. Soc. Japan 2024, 76, 229–268. [Google Scholar]
R. Schäfke, M. Singer. Consistent systems of linear differential and difference equations. J. Eur. Math. Soc. (JEMS) 2019, 21, 2751–2792. [Google Scholar] [CrossRef]
H. Yamazawa. Multisummability of formal solutions for some infinite order linear differential equations, talk at the workshop FASdiff17 in Alcalá de Henares, Spain, 2017. Available online: http://www3.uah.es/fasdiff17/.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms

Abstract

1. Introduction

2. Layout of the Main Initial Value Problem

3. Reduction of the Main Problem to an Integral Equation

3.1. First Reduction to a Differential/Convolution Equation with Mahler Transforms

3.2. Reduction to an Integral Equation

3.2.1. Essentials on Banach Valued m k − Summable Formal Power Series

3.2.2. Action of the Mahler operators on formal m k − Borel transforms

3.2.3. Statement of the Integral Equation

4. Solving the Integral Equation in a Banach Space of Functions with Exponential Growth on Sectors and Decay on the Real Line

5. Statement of the Main Results

References

MDPI Initiatives

Important Links

Subscribe

3.2.1. Essentials on Banach Valued $m_{k} -$ Summable Formal Power Series

3.2.2. Action of the Mahler operators on formal $m_{k} -$ Borel transforms