Attempts to define function spaces, using different approaches, takes its beginning since the appearance of Hardy, Besov and Triebel-Lizorkin spaces, which help to advance the studying nonlinear PDEs. On example, Besov spaces characterized via ball means of differences allowed to establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE in [1]. Another example is Triebel-Lizorkin spaces defined via Fourier transforms, which allowed to explore mild solutions of Navier-Stokes equation in [2].

There are 3 ways to define any function space [3]: Fourier-analytic, derivatives and differences and bounded mean oscillation. Specifying spaces in various ways allows to extend their properties, which can help in the study of PDE solutions, the behavior of pseudo-differential operators, Riemannian manifolds with the corresponding metric, Lie groups and fractals.

The study of quasi-norms, atomic, molecular and wavelet decompositions remains actual since the definition of local Hardy spaces in [4]. Such research was continued on Lorentz spaces for exploring weak and strong solutions of the nonlinear heat equation in [5]. The similar work was realized for Navier-Stokes equations and Keller-Segel system on Besov and Triebe-Lizorkin spaces in [6]. Afterwards, many researchers came up with the idea of combining function spaces into one function spaces, which can be global or hybrid, well described in [7] and [8], respectively.

Examples of hybrid spaces were demonstrated in [9], where the author researched the Caffarelli-Kohn-Nirenberg inequalities on Herz-type Besov spaces ${\dot{K}}_q^{\alpha ,p}{B}_{\beta }^s$ and Triebel-Lizorkin spaces ${\dot{K}}_q^{\alpha ,p}{F}_{\beta }^s$ and presented their quasi-norms, applying Fourier transforms and ball means of differences. Such spaces allowed to combine the properties of Besov, Triebel-Lizorkin and Herz spaces. Another example of hybrid spaces is Besov-weak-Herz spaces $BW{\dot{K}}_{p,q,r}^{\alpha ,s}$ introduced in [10], where authors explored mild solutions of the incompressible Navier-Stokes equations. There were provided $K_{\theta ,r}$-method real interpolations, embeddings and heat semi-group operator inequalities for obtaining estimates of mild solutions on Besov-weak-Herz spaces $BW{\dot{K}}_{p,q,r}^{\alpha ,s}$. The combining of Besov and weak Herz spaces generalized and extended their properties, what allowed to receive valuable results in the studying the Navier-Stokes equations. The unique maximal strong solution and the local strong well-posedness of the Navier–Stokes equations with $f\ne 0$ was constructed on Triebel-Lizorkin-Lorentz spaces in [11], which provided important inequalities containing the Laplace and the Stokes operators.

The main idea of this article arose from the researching quasi-norm characterizations via ball means of differences on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, which are investigated in [12] and [13], the exploring of the atomic decomposition of Herz-type Besov and Triebel-Lizorkin spaces, and the wavelet decomposition for the corresponding spaces in [14]. The definition of Herz-type Besov and Triebel-Lizorkin spaces via differences in [15] impacted to the topic of this research. In this article, we propose Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces, denoted by ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{N}}}_{\mu ,s}^r$ and ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{E}}}_{\mu ,s}^r$, respectively. We establish equivalence between the quasi-norms defined via ball of means differences and Fourier transforms, imply atomic, molecular and wavelet decomposition for corresponding spaces. Taking into account the quasi-norm of Herz spaces, we imply helpful inequalities on Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces, containing Hardy-Littlewood and Peetre maximal operators. The proposed spaces were not present in other papers, then the necessary facts on ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{A}}}_{\mu ,s}^r$, where $\mathcal{A}\in \{\mathcal{N},\mathcal{E}\}$, are not provided. Therefore, there are provided some inequalities, containing maximal functions, embeddings and isomorphisms in this article.

The remaining of the paper is organized as follows. Section 2 is devoted to definition of proposed spaces via Fourier-analytic approach. Section 3 contains necessary theorems, lemmas and propositions for further proofs. In section 4 we characterize Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces, utilizing the ball means of differences. Section 5 and section 6 present atomic, molecular and wavelet decompositions, respectively.

We should provide some helpful notations, which shorten the writing of formulas. Let ${\left\{f_j\right\}}_{j\in {\mathbb{N}}_0}$ be a sequence of functions, then we note where $0<p,q\le \infty$ and ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ is the set of all non-negative integers. For definiteness we adopt the following definition as the Fourier transform and its inverse: for every f in the space of tempered distributions ${\mathcal{S}}^{\prime }$.

Before giving the definition of the Herz spaces, let us introduce the following notations. Let $k\in \mathbb{Z}$. Then For $m\in {\mathbb{N}}_0$, we denote Let us recall the definition of homogeneous Herz spaces [9].

The spaces ${\dot{K}}_{p,q}^{\alpha }$ are quasi-Banach spaces, and if $min(p,q)\ge 1$ then ${\dot{K}}_{p,q}^{\alpha }$ are also Banach spaces. When $\alpha =0$ and $0<p=q\le \infty$ the space ${\dot{K}}_{p,p}^0$ coincides with the Lebesgue space $L^p$ . In addition ${\dot{K}}_{p,p}^{\alpha }=L^p({\mathbb{R}}^n{,|·|}^{\alpha p})$, where

Let $B_r\left(x_0\right)$ be an open ball in ${\mathbb{R}}^n$ centered at $x_0$ and radius $r>0$. Next we define the Herz-type Morrey space.

Let $\mathcal{S}\left({\mathbb{R}}^n\right)$ and ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right)$ be Schwarz and the tempered distributions spaces, respectively. Suppose, that $\varphi \in \mathcal{S}\left({\mathbb{R}}^n\right)$ is a nonegative radial function such that and where ${\varphi }_k\left(\xi \right)=\varphi \left(2^{-k}\xi \right)$. Using a partition of unity, we can define Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces via Fourier-analytic approach.

Let us introduce the Hardy-Littlewood maximal operator:And we note ${\mathcal{M}}^{\left(\tau \right)}f\left(x\right)={\left(\mathcal{M}\right|f|}^{\tau }{)}^{1/\tau }$, $0<\tau <\infty$. Suppose that ${\left\{{\varphi }_k\right\}}_{k=0}^{\infty }\subset \mathcal{S}\left({\mathbb{R}}^n\right)$, $f\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right)$, $a>0$, $x\in {\mathbb{R}}^n$ and $k\in {\mathbb{N}}_0$, then we introduce the Peetre maximal function defined as These maximal operators are useful for characterization of Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces via ball means differences.

In this section introduced spaces were defined via Fourier analytic approach. This approach was useful in [16], where estimates of mild solutions of the Navier-Stokes equation, containing semi-group heat operator on weak Herz-type Besov-Morrey spaces, were explored. The Fourier-analytic approach on Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces facilitates the studying of mild solutions of nonlinear PDEs, where the heat semi-group operator and the Lerray projection are intensively engaged. For example, a mathematical model of waves on shallow water surfaces described by Korteweg-de Vries equation [17], Keller–Segel System [18] presents a cellular chemotaxis model, or Fokker-Planck equations [19] demonstrates models of anomalous diffusion processes. The developing of atomic, molecular and wavelet decompositions can advance the studying of ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{N}}}_{\mu ,r}^s$ and ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{E}}}_{\mu ,r}^s$ spaces on smooth and singular manifolds, especially observing them not only using the Fourier-analytic approach, but also using the difference approach.

In this section we provide useful inequalities and properties of spaces ${\dot{K}}_{p,q}^{\alpha }{\dot{\mathcal{A}}}_{\mu ,r}^s$, where $\mathcal{A}\in \{\mathcal{N},\mathcal{E}\}$.

According to Lemma 1, it is possible to obtain the general inequality for the arbitrary family of functions.

We also verify the relation between f and ${\left|f\right|}^u$ in the next lemma.

Now it is necessary to provide helpful inequalities, containing Hardy-Littlewood and Peetre maximal functions.

We should to collect some basic properties of Herz-type Besov- Morrey and Triebel-Lizorkin-Morrey spaces.

Let us imply the Hardy-type inequality, containing the norm of ${\dot{K}}_{p,q}^{\alpha }{\mathcal{M}}_{\mu }$, which is a corollary of Lemma 3.9 from [23].

From [23] we provide useful inequality.

Remark, that $\Delta$ is defined in section 4.

For atomic, molecular and wavelet decompositions we need to introduce the following facts. Let $f\in {\dot{K}}_{p,q}^{\alpha }{\mathcal{M}}_{\mu }$ . Then for all $\eta >0$ there holds Now, we consider the lifting properties. If $\sigma \in \mathbb{R}$, then operator $I_{\sigma }$ is defined as It is well-known that $I_{\sigma }$ is an one to one mapping on ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right)$ and $\mathcal{S}\left({\mathbb{R}}^n\right)$, respectively. Moreover, the equality $I_{\sigma }{I}_{\tau }=I_{\sigma +\tau }$ is valid for all $\sigma ,\tau \in \mathbb{R}$.

It is possible to provide the corollary of Theorem 3.

Let us provide three useful statements, which can help to prove atomic and molecular decompositions.

Let us define $k_0=\varphi$ and $k={\Delta }^{2N}\varphi$, where $N>0$ is a large integer depending on the parameters $\alpha ,p,q,\mu ,r,s$. Let $k_j\left(x\right)=2^{jn}k\left(2^{j}x\right)$ for $j\in \mathbb{N}$. Next theorem is a generalization of Theorem 4.1 from [14], where instead of Morrey spaces used Herz-type Morrey spaces.