We consider the multi-dimensional H- integral transform ([1], formula (43)): where (see [1,2,3], ch. 28; [4], ch. 1; [5,6]) $\mathbf{x}=(x_1,x_2,...,x_n)$$\in {\mathbb{R}}^n$; $\mathbf{t}=(t_1,t_2,...,t_n)\in {\mathbb{R}}^n$, ${\mathbb{R}}^n$ be the n-dimensional Euclidean space; $\mathbf{x}·\mathbf{t}=\sum _{n=1}^n{x}_n{t}_n$ denotes their scalar product; in particular, $\mathbf{x}·\mathbf{1}=\sum _{n=1}^n{x}_n$ for 1= (1,1,...,1). The inequality $\mathbf{x}>\mathbf{t}$ means that $x_1>t_1,...,x_n>t_n,$ and inequalities ≥, <, ≤ have similar meanings ; $\underset{\mathbf{0}}{\overset{\infty }{\int }}=\underset{\mathbf{0}}{\overset{\infty }{\int }}\underset{\mathbf{0}}{\overset{\infty }{\int }}···\underset{\mathbf{0}}{\overset{\infty }{\int }}$; by $\mathbb{N}=\{1,2,...\}$ we denote the set of natural numbers, ${\mathbb{N}}_0=\mathbb{N}\bigcup \left\{0\right\}$, ${\mathbb{N}}_0^n={\mathrm{N}}_0\times ...\times {\mathbb{N}}_0$; $k=(k_1,k_2,...,k_n)\in {\mathbb{N}}_0^n$ $(k_i\in {\mathbb{N}}_0,i=1,2,...,n)$ is a multi-index with $k!=k_1!\cdots k_n!$ and $\left|k\right|=k_1+...+k_n$; ${\mathbb{R}}_{+}^n=\{\mathbf{x}\in {\mathbb{R}}^n,\mathbf{x}>0\}$; for $\kappa =({\kappa }_1,{\kappa }_2,...,{\kappa }_n)\in {\mathbb{R}}_{+}^n$ ${\mathbf{D}}^{\kappa }=\frac{{\partial }^{\left|\kappa \right|}}{{(\partial x_1)}^{{\kappa }_1}\cdots {(\partial x_n)}^{{\kappa }_n}}$; $\mathbf{dt}=d{t}_1\cdots d{t}_n;$ ${\mathbf{t}}^{\kappa }=t^{{\kappa }_1}{t}^{{\kappa }_2}\cdots t^{{\kappa }_n};$$f\left(\mathbf{t}\right)=f(t_1,t_2,...,t_n)$; ${\mathbb{C}}^n$$(n\in \mathbb{N})$ be the n-dimensional space of n complex numbers $z=(z_1,z_2,\cdots ,z_n)$$(z_j\in \mathbb{C},$ $j=1,2,\cdots ,n)$; $\overline{\lambda }=({\lambda }_1,{\lambda }_2,...,{\lambda }_n)\in {\mathbb{C}}^n$; $\overline{h}=(h_1,h_2,...,h_n)\in {\mathbb{R}}_{+}^n$; $\frac{\mathrm{d}}{\mathrm{d}\mathbf{x}}=\frac{d}{d{x}_1·d{x}_2···d{x}_n}$;

$\mathbf{m}=(m_1,m_2,...,m_n)\in {\mathbb{N}}_0^n$ and $m_1=m_2=...=m_n$; $\mathbf{n}=({\overline{n}}_1,{\overline{n}}_2,...,{\overline{n}}_n)\in {\mathbb{N}}_0^n$ and ${\overline{n}}_1={\overline{n}}_2=...={\overline{n}}_n$; $\mathbf{p}=(p_1,p_2,...,p_n)\in {\mathbb{N}}_0^n$ and $p_1=p_2=...=p_n$; $\mathbf{q}=(q_1,q_2,...,q_n)\in {\mathbb{N}}_0^n$ and $q_1=q_2=...=q_n$$(0\le \mathbf{m}\le \mathbf{q}$, $0\le \mathbf{n}\le \mathbf{p})$;

${\mathbf{a}}_i=(a_{i_1},a_{i_2},...,a_{i_n}),$$1\le i\le \mathbf{p}$, $a_{i_1},a_{i_2},...,a_{i_n}\in \mathbb{C}$$(i_1=1,2,...,p_1;...;i_n=1,2,...,p_n)$;

${\mathbf{b}}_j=(b_{j_1},b_{j_2},...,b_{j_n}),$$1\le j\le \mathbf{q}$, $b_{j_1},b_{j_2},...,b_{j_n}\in \mathbb{C}$$(j_1=1,2,...,q_1;...;j_n=1,2,...,q_n)$;

${\overline{\alpha }}_i=({\alpha }_{i_1},{\alpha }_{i_2},...,{\alpha }_{i_n}),$$1\le i\le \mathbf{p}$, ${\alpha }_{i_1},{\alpha }_{i_2},...,{\alpha }_{i_n}\in {\mathbb{R}}_1^{+}$$(i_1=1,2,...,p_1;...;i_n=1,2,...,p_n)$;

${\overline{\beta }}_j=({\beta }_{j_1},{\beta }_{j_2},...,{\beta }_{j_n}),$$1\le j\le \mathbf{q}$, ${\beta }_{j_1},{\beta }_{j_2},...,{\beta }_{j_n}\in {\mathbb{R}}_1^{+}$$(j_1=1,2,...,q_1;...;j_n=1,2,...,q_n)$.

The function in the kernel of (1) is the product of H-functions $H_{p,q}^{m,n}\left[z\right]$: where

In the representation (3) L is a specially chosen infinite contour, and the empty products, if any, are taken to be one.

The H-function (3) is the most general of the known special functions and includes as special cases elementary functions, special functions of hypergeometric and bessel type, as well as the Meyer G-function. One may find its properties, for example, in the books by Mathai and Saxena ([7], Ch. 2), Srivastava, Gupta and Goyal ([8], ch. 1), Prudnikov, Brychkov and Marichev ([9], Section 8.3), Kiryakova [10] and Kilbas and Saigo ([11], Ch.1 – Ch.4).

Our paper is devoted to the study of $\mathrm{H}$- transform (1) on Lebesgue-type weighted spaces ${\mathfrak{L}}_{\overline{\nu },\overline{2}}$ of functions $f\left(\mathbf{x}\right)=f(x_1,x_2,...,x_n)$ on ${\mathbb{R}}_{+}^{\mathbf{n}}$, such that

$\overline{\nu }=({\nu }_1,{\nu }_2,...,{\nu }_n)\in {\mathbb{R}}^n$, ${\nu }_1={\nu }_2=...={\nu }_n$, and $\overline{2}=(2,2,...,2)$.

In this paper we apply the results from [2] to obtain mapping properties such as boundedness, the rang and representations for the $\mathrm{H}$- transform (1).

Research results for transformation (1) generalize those obtained earlier for the corresponding one-dimensional transformation ( see [11], Ch. 3): in the space ${\mathfrak{L}}_{\nu ,2}$ of Lebesgue measurable functions f on ${\mathbb{R}}_{+}^1=(0,\infty )$, such that

The $\mathrm{H}$- transform (5) generalizing many integral transforms: transforms with the Meijer G-function, Laplace and Hankel transforms, transforms with Gauss hypergeometric function, transforms with other hypergeometric and Bessel functions in the kernels. One may find a survey of results and bibliography in this field for one-dimensional case in the monograph ([11],Section 6–8). Note that a very important class of transforms under consideration is a class of Buschman–Erdélyi operators, they have many important properties and applications, cf. [1,12,13,14,15,16]. And topic of this paper is also in a very tight connection with transmutation theory, cf. [17,18,19,20,21].

The properties of the H-function $H_{p,q}^{m,n}\left[z\right]$ (3) depend on the numbers ([11], formulas 1.1.7–1.1.15):

An empty sum in (6), (8), (9), (10) and an empty product in (7), if they occur, are taken to be zero and one, respectively.

There holds the following assertions.

Lemma 1.([11], Lemma 1.2) For $\sigma ,t\in \mathbb{R}$, there holds the estimate

uniformly in σ on any bounded interval in $\mathbb{R}$, where

and ξ and $c^{*}$ are defined in (10) and (11).

Theorem 1.([11],Theorem 3.4) Let $\alpha <\zeta <\beta$ and either of the conditions $a^{*}>0$ or $a^{*}=0$ and $\Delta \zeta +\mathrm{Re}\left(\mu \right)<-1$ are hold. Then for $x>0$, except for $x=\delta$ when $a^{*}=0$ and $\Delta =0$, the relation

holds and the estimate

A set of bounded linear operators acting from a Banach space X into a Banach space Y denote by $[X,Y]$.

Multidimensional Mellin integral transform $\left(\mathfrak{M}f\right)\left(\mathbf{x}\right)$ of function $f\left(\mathbf{x}\right)=f(x_1,x_2,...,x_n),$$\mathbf{x}=(x_1,x_2,...,x_n)\in {\mathbb{R}}_{+}^n,$ is determined by the formula

$\mathbf{s}=(s_1,s_2,...,s_n)\in {\mathbb{C}}^n$. The inverse multidimensional Mellin transform has the form

$\mathbf{x}\in {\mathbb{R}}_{+}^n$, ${\gamma }_j=\mathrm{Re}\left(s_j\right)$$(j=1,\cdots ,n)$. The theory of multidimensional integral transformations (16) and (17) can be recognized, for example, in books ([4], Ch. 1; [22,23]).

We will need the following spaces. As usual, by ${\mathrm{L}}_{\overline{p}}\left({\mathbb{R}}^n\right)$ we will understand the space of functions $f\left(\mathbf{x}\right)=f(x_1,x_2,...,x_n)$, for which

If $\overline{p}=\infty$, then the space ${\mathrm{L}}_{\infty }\left({\mathbb{R}}^n\right)$ is defined as the collection of all measurable functions with a finite norm here $esssup\left|f\right(\mathbf{x}\left)\right|$ is the essential supremum of the function $\left|f\right(\mathbf{x}\left)\right|$ [24].

We need the following properties of the Mellin transform (16).

Lemma 2. ([2], Lemma 1) Let $\overline{\nu }=({\nu }_1,{\nu }_2,...,{\nu }_n)\in {\mathbb{R}}^n$, ${\nu }_1={\nu }_2=...={\nu }_n.$ The following properties of the Mellin transform (16) are valid:

(b)For $f\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$ there holds

if $F(\overline{\nu }+i\mathbf{t})=\prod _{i=1}^n{F}_j({\nu }_j+i{t}_j)$, $F_j({\nu }_j+i{t}_j)\in {\mathrm{L}}_1(-\mathrm{R},\mathrm{R})$, $j=1,2,...,n$, then

(c) For functions $f\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$ and $g\in {\mathfrak{L}}_{1-\overline{\nu },\overline{2}}$ the following equality holds

In [2] we consider the general multi-dimensional integral transform ([2], formula (1)): where the function $\mathrm{k}\left[\mathbf{x}\mathbf{t}\right]$ in the kernel of (20) is the product of some one type special functions:

Transformation (20) satisfies the following theorem.

Theorem 2.([2], Theorem 1) Let $\overline{\nu }=({\nu }_1,{\nu }_2,...,{\nu }_n)\in {\mathbb{R}}^n$$({\nu }_1={\nu }_2=...={\nu }_n)$, $\overline{h}=(h_1,h_2,...,h_n)\in {\mathbb{R}}_{+}^n$, and $\overline{\lambda }=({\lambda }_1,{\lambda }_2,...,{\lambda }_n)\in {\mathbb{C}}^n$.

(a) If the transformation operator (20) satisfies the condition $\mathrm{K}\in [{\mathfrak{L}}_{\overline{\nu },\overline{2}},{\mathfrak{L}}_{1-\overline{\nu },\overline{2}}]$, then the kernel on the right side of (20) $\mathrm{k}\in {\mathfrak{L}}_{1-\overline{\nu },\overline{2}}$. If we set for ${\nu }_j\ne 1-(\mathrm{Re}\left({\lambda }_j\right)+1)/h_j,j=1,2,...,n,$

almost everywhere, then function $\theta \in {\mathrm{L}}_{\infty }\left({\mathbb{R}}^n\right)$, and for $f\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$ there holds the relation

(c) Based on statements (a) or (b) with $\theta \ne 0$, $\mathrm{K}$ is one-to-one transformation from the space ${\mathfrak{L}}_{\overline{\nu },\overline{2}}$ into the space ${\mathfrak{L}}_{1-\overline{\nu },\overline{2}}$, and if in addition $1/\theta \in {\mathrm{L}}_{\infty }\left({\mathbb{R}}^n\right)$, then $\mathrm{K}$ maps ${\mathfrak{L}}_{\overline{\nu },\overline{2}}$ onto ${\mathfrak{L}}_{1-\overline{\nu },\overline{2}}$, and for functions $f,g\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$ the relation

To formulate the results for the transform $\mathrm{H}f$ (1) we need the following constants ([1], (57)–(60)), analogical for one-dimensional case defined via the parameters of the H - function (3) ([11], (3.4.1), (3.4.2), (1.1.7), (1.1.8), (1.1.10)):

let $\tilde{\alpha }=({\tilde{\alpha }}_1,{\tilde{\alpha }}_2,...,{\tilde{\alpha }}_n)$ and $\tilde{\beta }=({\tilde{\beta }}_1,{\tilde{\beta }}_2,...,{\tilde{\beta }}_n)$ where and so on let $a^{*}=(a_1^{*},a_2^{*},...,a_n^{*})$, $\Delta =({\Delta }_1,{\Delta }_2,...,{\Delta }_n)$ and and so on let $\mu =({\mu }_1,{\mu }_2,...,{\mu }_n)$ and

The exceptional set ${\mathcal{E}}_{\overline{\mathcal{H}}}$ of a function ${\overline{\mathcal{H}}}_{\mathbf{p},\mathbf{q}}^{\mathbf{m},\mathbf{n}}\left(\mathbf{s}\right)$: is called a set of vectors $\overline{\nu }=({\nu }_1,{\nu }_2,...,{\nu }_n)\in {\mathbb{R}}^n$ $({\nu }_1={\nu }_2=...={\nu }_n),$ such that ${\tilde{\alpha }}_k<1-{\nu }_k<{\tilde{\beta }}_k,$ $k=1,2,...n,$ where the parameters ${\tilde{\alpha }}_k,{\tilde{\beta }}_k(k=1,2,...,n)$ are defined by formulas (24), and functions ${\mathcal{H}}_{p_k,q_k}^{m_k,{\overline{n}}_k}\left(s_k\right)(k=1,2,...,n)$ of the view (4) have zeros on lines $\mathrm{Re}\left(s_k\right)<1-{\nu }_k(k=1,2,...,n)$, respectively (see [1], (61)).

Applying multidimensional Mellin transformation (16) to (1), formally we obtain:

Theorem 3.Suppose that

and that either of the conditions

or

(b) If $f\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$ and $g\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$, then for H there holds the relation (23):

$\left(\mathrm{H}f\right)\left(\mathbf{x}\right)=\overline{h}{\mathbf{x}}^{1-(\overline{\lambda }+1)/\overline{h}}$

$\left(\mathrm{H}f\right)\left(\mathbf{x}\right)=-\overline{h}{\mathbf{x}}^{1-(\overline{\lambda }+1)/\overline{h}}$

Proof. Let $\overline{\omega }\left(\mathbf{t}\right)=\overline{\mathcal{H}}(1-\overline{\nu }+i\mathbf{t})$$=\prod _{k=1}^n\mathcal{H}(1-{\nu }_k+i{t}_k).$ By virtue of (4), (24), and the conditions (29) the functions ${\mathcal{H}}_{p_1,q_1}^{m_1,{\overline{n}}_1}\left(s_1\right),$${\mathcal{H}}_{p_2,q_2}^{m_2,{\overline{n}}_2}\left(s_2\right),$...,${\mathcal{H}}_{p_n,q_n}^{m_n,{\overline{n}}_n}\left(s_n\right)$ are analytic in the strips ${\tilde{\alpha }}_1<1-{\nu }_1<{\tilde{\beta }}_1,...,{\tilde{\alpha }}_n<1-{\nu }_n<{\tilde{\beta }}_n,{\nu }_1={\nu }_2=...={\nu }_n$, respectively. In accordance with (12) and conditions (30) or (31), $\overline{\omega }\left(\mathbf{t}\right)=O\left(1\right)$ as $\left|\mathbf{t}\right|\to \infty$. Therefore $\overline{\omega }\in {\mathrm{L}}_{\infty }\left({\mathbb{R}}^n\right)$, and hence we obtain from Theorem 2 $\left(b\right)$ that there exists a transform $\mathrm{H}\in$$[{\mathfrak{L}}_{\overline{\nu },\overline{2}},{\mathfrak{L}}_{1-\overline{\nu },\overline{2}}]$ such that for $f\in {\mathfrak{L}}_{\overline{\nu },\overline{2}}$. This means that the equality (28) holds when condition $\mathrm{Re}\left(\mathbf{s}\right)=1-\overline{\nu }$ is met. Since the functions ${\mathcal{H}}_{p_1,q_1}^{m_1,{\overline{n}}_1}\left(s_1\right),$${\mathcal{H}}_{p_2,q_2}^{m_2,{\overline{n}}_2}\left(s_2\right),$...,${\mathcal{H}}_{p_n,q_n}^{m_n,{\overline{n}}_n}\left(s_n\right)$ are analytic in the strips ${\tilde{\alpha }}_1<1-{\nu }_1<{\tilde{\beta }}_1,...,{\tilde{\alpha }}_n<1-{\nu }_n<{\tilde{\beta }}_n,{\nu }_1={\nu }_2=...={\nu }_n$, respectively, and have isolated zeros, then $\overline{\omega }\left(\mathbf{t}\right)\ne 0$ almost everywhere. So it follows from the Theorem 2$\left(c\right)$ that $\mathrm{H}\in [{\mathfrak{L}}_{\overline{\nu },\overline{2}},{\mathfrak{L}}_{1-\overline{\nu },\overline{2}}]$ is a one-to-one transform. If $a_k^{*}=0$, ${\Delta }_k(1-{\nu }_k)+\mathrm{Re}\left({\mu }_k\right)=0$$(k=1,2,...n)$ and $\overline{\nu }$ is not in the exceptional set ${\mathcal{E}}_{\overline{\mathcal{H}}}$ of $\overline{\mathcal{H}}$, then $1/\overline{\omega }\in {\mathrm{L}}_{\infty }\left({\mathbb{R}}^n\right)$, and from Theorem 2 $\left(c\right)$ we have that $\mathrm{H}$ transforms the space ${\mathfrak{L}}_{\overline{\nu },\overline{2}}$ onto ${\mathfrak{L}}_{1-\overline{\nu },\overline{2}}$. This completed the proof of the statement $\left(a\right)$ of the theorem.