Characterizations for S-Convex-Averaging Domains via Two-Dimensional Diffusion-Wave Equations

Abstract

In this paper, we introduce the concept of s-convex-averaging domains, which are extensions of circular and irregular convex domains, by using s-convex functions and generalized Orlicz norms. Based on the quasi-hyperbolic metric and L^p-averaging domains, several fundamental properties of s-convex-averaging domains are characterized. These properties are applied to the domains of a class of two-dimensional diffusion-wave equations. Furthermore, we establish intrinsic relationships between the considered partial differential equations and the geometric structure of s-convex-averaging domains. Finally, the embedding inequality for the solutions of these kinds of partial differential equations is derived.

1. Introduction

As is well-known, the circular and irregular convex domains are generally used to apply to fractional partial differential equations (FPDE) [1,2,3]. Most recently, researchers focus on the regional structures and properties of domains for partial differential equations, such as the two-dimensional distributed order space diffusion equation [4], reaction-diffusion equations with time delay [5] and so on. But in practical life and production, there are significant limitations to the two-dimensional distributed order space diffusion equation on these two kinds of domains. Consider the following two-dimensional distributed order space-fractional diffusion Equation (2D-DO-SFDE for short) [6]:

$${\displaystyle \frac{\partial u}{\partial t}}={\int }_1^{2}P\left(\alpha \right){\displaystyle \frac{{\partial }^{\alpha }u}{{\partial \left|x\right|}^{\alpha }}}+Q\left(\alpha \right){\displaystyle \frac{{\partial }^{\alpha }u}{{\partial \left|y\right|}^{\alpha }}}d\alpha +f(x,y,t),(x,y,t)\in \mathsf{\Omega }\times [0,T],$$

(1)

the initial condition is as follows

$$u(x,y,0)=\psi (x,y),(x,y)\in \mathsf{\Omega },$$

and $\mathsf{\Omega }$ is a certain domain for 2D-DO-SFDE which satisfies zero Dirichlet boundary condition

$$u(x,y,t)=0,(x,y)\in \partial \mathsf{\Omega },$$

Moreover, $\mathsf{\Omega }$ is defined as follows:

$$\mathsf{\Omega }=\left\{(x,y)\right|a_1\left(y\right)\le x\le a_2\left(y\right),b_1\left(x\right)\le y\le b_2\left(x\right)\},$$

where $a_i$ and $b_i$ are boundaries of $\mathsf{\Omega }$ for $i=1,2$. $P\left(\alpha \right)$ and $Q\left(\alpha \right)$ are two non-negative weight functions which satisfy the following conditions for all $\alpha$

$$0<{\int }_{E}P\left(\alpha \right)d\alpha <\infty ,0<{\int }_{E}Q\left(\alpha \right)d\alpha <\infty$$

where E is a suitable domain for $\alpha \in E$.

In the above equation, it is clearly known that $\mathsf{\Omega }$ is an irregular convex domain. This type of domain is suitable for solving the problem for a class of specific reaction–diffusion models or for analyzing the existence of solutions to two-dimensional fractional partial differential equations. However, in more general case, the properties of the solutions to these differential equations will be insufficient to meet the requirements of practical computations.

In this paper, we study s-convex-averaging domains, which are generalizations of irregular convex domains, by using s-convex functions and generalized Orlicz norms. We first characterize s-convex-averaging domains, which are denoted by $L({\phi }_s,\mu )$, and discuss the invariance of $L({\phi }_s,\mu )$ under K-quasiconformal mapping and K-quasiisometric mapping, where ${\phi }_s$ is an s-convex function and $\mu$ is a weight measure satisfying $A_r$-conditions. Next, some particular characteristics for generalized Orlicz spaces are given, and embedding inequalities for the solution of the two-dimensional distributed order space-fractional diffusion equation are derived.

A closed set $E\in {\mathbb{R}}^n$, with nonempty interior, is said to be a convexity domain if there exists a convex local mapping f: $E\to \mathbb{R}$ that does not admit any convex finite-valued extension outside E, see [7,8] for more details on convex functions and convexity domains. The definitions and properties of s-convex functions were introduced in [9]. The generalized Orlicz spaces are given while the young functions are replaced by s-convex functions; for more details on Orlicz spaces, see [10,11]. Then, we extend the convexity domain to the s-convex domain. We further use generalized Orlicz norms and quasi-hyperbolic distance to characterize this kind of domain. The theory of quasi-hyperbolic distance was introduced in [12]. The quasi-conformal mappings and quasi-isometric mappings were given in [13,14,15]. The Lebesgue measure is replaced by a weight function that satisfies the $A_r$-conditions introduced in [16]. Averaging domains are involved in many research fields such as partial differential equations, potential theory, and operator theory; see [17,18,19]. The $L^p$-averaging domains were first introduced by Staples in [20], and then were extended in [21,22]. Combining with the concept of averaging domains and the s-convex domain, we establish the s-convex-averaging domains, which are extensions of convexity domains and irregular convex domains.

Throughout this paper, we assume that $\mathsf{\Omega }$ is a bounded domain. B and $\sigma B$ are the balls with the same center, and diam$\left(\sigma B\right)=\sigma$diam$\left(B\right)$. We use $\left|E\right|$ to denote the Lebesgue measure of a set E. $\omega \left(x\right)$ is the weight function that satisfies $d\mu \left(x\right)=\omega \left(x\right)dx$ and (7) later.

2. Preliminaries

In this section, we need to recall some theories and properties of two types of s-convexity. Then we give the definition of an s-convex-averaging domain. The following definition appeared in [9].

Definition 1

([9]). A function ${\phi }_s:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is said to be s-convex ($0<s\le 1$) in the first sense if the following is met:

$${\phi }_s({\lambda }_{1}u+{\lambda }_{2}v)\le {\lambda }_1^s{\phi }_s\left(u\right)+{\lambda }_2^s{\phi }_s\left(v\right)$$

(2)

for all $u,v\in {\mathbb{R}}_{+}$ and all ${\lambda }_1,{\lambda }_2\ge 0$ with ${\lambda }_1^s+{\lambda }_2^s=1$. We denote this by ${\phi }_1\in K_s^1$.

A function ${\phi }_s:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is said to be s-convex in the second sense if inequality (2) holds for all $u,v\in {\mathbb{R}}_{+}$ and all ${\lambda }_1,{\lambda }_2\ge 0$ that satisfy the following:

$${\lambda }_1+{\lambda }_2=1.$$

(3)

We denote this by ${\phi }_2\in K_s^2$.

Remark 1.

Both s-convexities mean just the convexity when $s=1$.

The product rule and composite property for s-convex functions are as follows:

Lemma 1

([9]). Let $f\in K_{s_1}^1$, and $g\in K_{s_2}^1$, $0<s_1,s_2,s_3<1$.

• If f and g are non-negative functions such that either $f\left(0\right)=0$ and $g\left(0^{+}\right)=g\left(0\right)$ or $g\left(0\right)=0$ and $f\left(0^{+}\right)=f\left(0\right)$, then the product $fg$ of f and g belongs to $K_s^1$, where $s=min\{s_1,s_2\}$.
• If f is a non-decreasing function in $K_{s_3}^2$ and g is a non-negative convex function on $[0,\infty )$, then the composition $f\circ g$ of f with g belongs to $K_{s_3}^2$.

Next, we introduce the generalized Orlicz spaces with s-convex functions.

Definition 2.

Let ${\phi }_1\in K_s^1$ and ${\phi }_2\in K_s^2$ on $[0,\infty )$ with ${\phi }_i\left(0\right)=0$. Ω be a domain with $\mu \left(\mathsf{\Omega }\right)<\infty$. We denote the generalized Orlicz space $L({\phi }_i,\mathsf{\Omega },\mu )$, the space of all functions $u\in L({\phi }_i,\mathsf{\Omega },\mu )$ for which we obtain the following:

$${\parallel u\parallel }_{L({\phi }_i,\mathsf{\Omega },\mu )}=\inf \{k>0:{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left({\displaystyle \frac{\left|u\right(x\left)\right|}{k}}\right)d\mu \le 1\}$$

(4)

for $i=1,2$. Then, we can easily see the following:

$${\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left({\displaystyle \frac{\left|u\right(x\left)\right|}{{\parallel u\parallel }_{L({\phi }_i,\mathsf{\Omega },\mu )}}}\right)d\mu \le 1$$

(5)

if ${\parallel u\parallel }_{L({\phi }_i,\mathsf{\Omega },\mu )}<\infty$.

The definition of s-convex-averaging domains $L({\phi }_s,\mu )$ is as follows. It is an extension of the $L^p$-averaging domain and irregular convex domains, which were introduced in [4,20].

Definition 3.

Let ${\phi }_1\in K_s^1$ and ${\phi }_2\in K_s^2$ on $[0,\infty )$ with ${\phi }_i\left(0\right)=0$ for $i=1,2$. We say that Ω is a s-convex-averaging domain $L({\phi }_s,\mu )$, if $\mu \left(\mathsf{\Omega }\right)<\infty$ and there exists a constant $C>0$ such that the following is obtained:

$${\displaystyle \parallel u-u_{B_0,\mu }{\parallel }_{L({\phi }_i,\mathsf{\Omega },\mu )}\le C\underset{B\subset \mathsf{\Omega }}{sup}{\parallel u-u_{B,\mu }\parallel }_{L({\phi }_i,B,\mu )}}$$

(6)

for some ball $B_0\subset \mathsf{\Omega }$ and integrable functions u in Ω. $\omega \left(x\right)$ is a weight function and supremum is over all balls B with $4B\subset \mathsf{\Omega }$.

Definition 4

([20]). The quasi-hyperbolic distance between x and y in Ω is given by the following:

$$k(x,y)=k(x,y;\mathsf{\Omega })=inf{\int }_{\gamma }{\displaystyle \frac{1}{d(z,\partial \mathsf{\Omega })}}ds,$$

where γ is any rectifiable curve in Ω joining x to y, $d(z,\partial \mathsf{\Omega })$ is the Euclidean distance between z and the boundary of Ω.

Some properties and applications of the quasi-hyperbolic metric could be found in [20,21]. The following definitions and properties of weight functions appeared in [16,23].

Definition 5

([20,23]). Let a weight function ω satisfy the $A_r$-condition, where $r>1$, and write $\omega \in A_r\left(\mathsf{\Omega }\right)$ when the following is met:

$$\underset{B}{sup}({\displaystyle \frac{1}{\left|B\right|}}{\int }_B\omega dx){({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{\omega }^{1/(1-r)}dx)}^{r-1}<\infty ,$$

(7)

where the supreumum is over all balls $B\subset \mathsf{\Omega }$.

Definition 6

([16]). Let $\sigma >1$. We say that ω satisfies a weak reverse Hölder’s inequality and write $\omega \in WRH\left(\mathsf{\Omega }\right)$ when there exist constants $\theta >1$ and $C>0$ such that the following is met:

$${({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{\omega }^{\theta }dx)}^{1/\theta }\le C{\displaystyle \frac{1}{\left|B\right|}}{\int }_{\sigma B}\omega dx$$

for all balls B with $\sigma B\subset \mathsf{\Omega }$, in fact the set $WRH\left(\mathsf{\Omega }\right)$ is independent of $\sigma >1$.

$\omega$ is a doubling weight function, and satisfies the doubling property: for arbitrary measurable set $\mathsf{\Omega }$, there exist a constant $C>1$, independent of $\mathsf{\Omega }$, such that $\omega \left(2\mathsf{\Omega }\right)\le C\omega \left(\mathsf{\Omega }\right)$. The set of all doubling weight functions denoted by $D\left(\mathsf{\Omega }\right)$. Similarly, for all balls B with $4B\subset \mathsf{\Omega }$, the condition $\omega \left(4B\right)\le C_1\omega \left(B\right)$ is holds, where $C_1$ is a constant, also independent of B.

Lemma 2

([23,24]). Let $\omega \in WD\left(\mathsf{\Omega }\right)$. Suppose that s and q are positive constants and the following is calculated:

$$\mu \left(\right\{x\in B:|u\left(x\right)-u_{B,\mu }|>t\})\le s{e}^{-qt}\mu \left(B\right)$$

for each $t>0$ and each B with $4B\subset \mathsf{\Omega }$. Then there exists a constant $C=C(s,q,n)$ such that

$$|u_{B\left(x\right),\mu }-u_{B\left(y\right),\mu }|\le Ck(x,y)$$

for all x and y in Ω. Here $B\left(x\right)$ is the ball $B(x,d(x,\partial \mathsf{\Omega })/4)$.

3. Properties of $\mathbf{S}$-Convex-Averaging Domain

In this section, we first characterize the s-convex-averaging domain $L({\phi }_s,\mu )$ by using s-convex functions and the quasi-hyperbolic distance. Then we give some properties of s-convex-averaging domain $L({\phi }_s,\mu )$ with Whitney cover.

Theorem 1.

Assume that $\omega \in WRH\left(\mathsf{\Omega }\right)$. Let ${\phi }_i\in K_s^i$ for $i=1,2$ with ${\phi }_i\left(0\right)=0$ and ${\phi }_i\left(t\right)\le e^{bt}$ for some $0\le b<\infty$ and all $t\ge 1$. Then we say that Ω is an s-convex-averaging domain $L({\phi }_s,\mu )$ if and only if the following is met:

$${\int }_{\mathsf{\Omega }}{\phi }_i\left(\alpha k(x,x_0)\right)d\mu <\infty$$

(8)

for each $x_0$ in Ω and some $\alpha >0$.

Theorem 2.

Let ${\phi }_i\in K_s^i$ on $[0,\infty )$ with ${\phi }_i\left(0\right)=0$, ${\phi }_i\left(t\right)\le e^{bt}$, for some $0\le b<\infty$ and all $t\ge 1$, $i=1,2$ and $\mathcal{F}$ be the Whitney cover of Ω consists of cubes $Q_j$ with centers $x_j$. Then we obtain the following:

$${\int }_{\mathsf{\Omega }}{\phi }_i\left({\alpha }_{1}k(x,x_0)\right)d\mu <\infty$$

(9)

if and only if

$$\sum _{Q_j\in \mathcal{F}}{\phi }_i\left({\alpha }_{2}k(x_j,x_0)\right)d\mu <\infty ,$$

(10)

where $x_0$ is a fixed point in Ω, ${\alpha }_1$, and ${\alpha }_2$ are some small positive constants and the measure μ is defined by $d\mu =\omega \left(x\right)dx$, $\omega \in WRH\left(\mathsf{\Omega }\right)$.

Corollary 1.

Let $\omega \in A_r\left(E_j\right)$ for $j=1,2,\cdots ,n$ and the measure μ be defined by $d\mu =\omega \left(x\right)dx$. Each $E_j$ is an s-convex-averaging domain $L({\phi }_s,\mu )$ for the fixed ${\phi }_s$ and the fixed measure μ. Then we have the following:

i. 

If ${\displaystyle E=\bigcap _{j=1}^n{E}_j}$, then E is an s-convex-averaging domain.

ii. 

If ${\displaystyle E=\bigcup _{j=1}^n{E}_j}$ and $E_i\bigcap E_j\ne ⌀$. Then E is an s-convex-averaging domain.

In order to prove the results, we need the following Lemma:

Lemma 3

([20]). Let $\omega \left(x\right)\in A_r\left(\mathsf{\Omega }\right)$ for some $r>1$ and ${\phi }_i\in K_s^i$ on $[0,\infty )$ with ${\phi }_i\left(0\right)=0,{\phi }_i\left(t\right)\le e^{bt}$, for some $0\le b<\infty$ and $t\ge 1$, $i=1,2$. Let B be any ball in Ω with center $x_B$ and radius r. Then we obtain the following:

$${\int }_B{\phi }_i\left(\alpha k(x,x_B)\right)d\mu <\infty ,$$

(11)

where $d\mu =\omega \left(x\right)dx$, C is a constant independent of B and the constant $\alpha >0$ is small enough.

The following conclusion for s-convex-averaging domain is a key tool for the main results:

Lemma 4.

Let ${\phi }_i\in K_s^i$ for $i=1,2$ and Ω is a domain. Then we obtain the following:

$${\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left(a\right|u-u_{\mathsf{\Omega },\mu }\left|\right)d\mu \le C{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i(b_i|u-c|)d\mu$$

(12)

for any positive constant, $a,b_i,c$, and C are a certain constant, independent of u.

Proof. 

For any constant c, we have the following:

$$\begin{array}{c}\hfill |u_{\mathsf{\Omega },\mu }-c|\le {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}|u-c|d\mu \end{array}$$

(13)

and

$${\phi }_i\left(2a\right|u_{\mathsf{\Omega },\mu }-c\left|\right)\le {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left(2a\right|u-c\left|\right)d\mu ,$$

Hence, we obtain the following:

$${\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left(2a\right|u_{\mathsf{\Omega },\mu }-c\left|\right)d\mu \le {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left(2a\right|u-c\left|\right)d\mu .$$

By the fact that ${\phi }_1\in K_s^1$ for $i=1$, we have the following:

$$\begin{array}{cc}& {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1\left(a\right|u-u_{\mathsf{\Omega },\mu }\left|\right)d\mu \hfill \\ \hfill \le & {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1\left(a\right|u-c|+a|c-u_{\mathsf{\Omega },\mu }\left|\right)d\mu \hfill \\ \hfill \le & {\displaystyle \frac{1-\lambda }{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1({\displaystyle \frac{a}{{(1-\lambda )}^{1/s}}}|u-c|)d\mu +{\displaystyle \frac{\lambda }{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1({\displaystyle \frac{a}{{\lambda }^{1/s}}}|u-c|)d\mu \hfill \\ \hfill \le & {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1(b_1|u-c|)d\mu ,\hfill \end{array}$$

(14)

where $0<\lambda <1$ and $b_1=max\{{\displaystyle \frac{a}{{(1-\lambda )}^{1/s}}},{\displaystyle \frac{a}{{\lambda }^{1/s}}}\}$ is any positive constant. Similar to (14), we can easily obtain the following:

$${\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_2\left(a\right|u-u_{\mathsf{\Omega },\mu }\left|\right)d\mu \le C{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_2(b_2|u-c|)d\mu ,$$

where $b_2=max\{{\displaystyle \frac{a}{(1-\lambda )}},{\displaystyle \frac{a}{\lambda }}\}$ and C is a certain constant, independent of u. □

Now, we give the proof of Theorem 1.

Proof. 

Assume that $\mathsf{\Omega }$ is an s-convex-averaging domain. By Lemma 4, we have the following:

$${\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i\left(a\right|u-u_{\mathsf{\Omega },\mu }\left|\right)d\mu \le {\displaystyle \frac{C}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_i(b_i|u-c|)d\mu .$$

Let B be any ball in $\mathsf{\Omega }$ with center $x_B$ and radius r. Choose $c=u\left(x_B\right)$ and set $u\left(x\right)=k(x,x_0)$. By Lemma 3, we obtain the following:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B{\phi }_i\left(a\right|u-u_{B,\mu }\left|\right)d\mu & \le {\displaystyle \frac{C}{\mu \left(B\right)}}{\int }_B{\phi }_i(b_i|u-u\left(x_B\right)\left|\right)d\mu \hfill \\ \hfill & \le {\displaystyle \frac{C}{\mu \left(B\right)}}{\int }_B{\phi }_i\left(b_{i}k(x,x_B)\right)d\mu \le M\hfill \end{array}$$

for M is a positive constant. Hence, we obtain the following:

$$\begin{array}{c}\hfill \underset{\beta \to 0}{lim}{\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B{\phi }_i\left(\beta \right|u-u_{B,\mu }\left|\right)d\mu ={\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B\underset{\beta \to 0}{lim}{\phi }_i\left(\beta \right|u-u_{B,\mu }\left|\right)d\mu ={\phi }_i\left(0\right)=0,\end{array}$$

where $\beta$ is independent of B. If $\beta$ is small enough, we have the following:

$${\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B{\phi }_i\left(\beta \right|u-u_{B,\mu }\left|\right)d\mu \le 1,$$

which shows that

$$\underset{4B\subset \mathsf{\Omega }}{sup}{\parallel u-u_{B,\mu }\parallel }_{L({\phi }_i,B,\mu )}\le {\beta }^{-1}<\infty .$$

(15)

Combining (6) and (15), we have the following:

$$\parallel u-u_{B_0,\mu }{\parallel }_{L({\phi }_i,B,\mu )}\le C_1{\beta }^{-1}.$$

Choosing $\alpha ={(1-\lambda )}^{1/s}(\parallel u\left(x\right)-u_{B_0,\mu }{{\parallel }_{L({\phi }_1,\mathsf{\Omega },\mu )})}^{-1}$ and by using (5) for $i=1$, we obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha k(x,x_0)\right)d\mu ={\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha u\left(x\right)\right)d\mu \hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha \right|u\left(x\right)-u_{B_0,\mu }|+\alpha |u_{B_0,\mu }\left|\right)d\mu \hfill \\ \hfill & \le (1-\lambda ){\int }_{\mathsf{\Omega }}{\phi }_1\left({\displaystyle \frac{|u\left(x\right)-u_{B_0,\mu }|}{\parallel u\left(x\right)-u_{B_0,\mu }{\parallel }_{L({\phi }_1,\mathsf{\Omega },\mu )}}}\right)d\mu +\lambda {\int }_{\mathsf{\Omega }}{\phi }_1({\displaystyle \frac{\alpha }{{\lambda }^{1/s}}}|u_{B_0,\mu }\left|\right)d\mu \hfill \\ \hfill & \le (1-\lambda )\mu \left(\mathsf{\Omega }\right)+\lambda {\phi }_1(\alpha {\lambda }^{-1/s}|u_{B_0,\mu }\left|\right)<\infty .\hfill \end{array}$$

Choosing $\alpha =(1-\lambda )(\parallel u\left(x\right)-u_{B_0,\mu }{{\parallel }_{L({\phi }_1,\mathsf{\Omega },\mu )})}^{-1}$ and for $i=2$, similarly we obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\phi }_2\left(\alpha k(x,x_0)\right)d\mu \hfill \\ \hfill & \le {(1-\lambda )}^s{\int }_{\mathsf{\Omega }}{\phi }_2\left({\displaystyle \frac{|u\left(x\right)-u_{B_0,\mu }|}{\parallel u\left(x\right)-u_{B_0,\mu }{\parallel }_{L({\phi }_2,\mathsf{\Omega },\mu )}}}\right)d\mu +{\lambda }^s{\int }_{\mathsf{\Omega }}{\phi }_2({\displaystyle \frac{\alpha }{\lambda }}|u_{B_0,\mu }\left|\right)d\mu \hfill \\ \hfill & \le {(1-\lambda )}^s\mu \left(\mathsf{\Omega }\right)+{\lambda }^s{\phi }_1(\alpha {\lambda }^{-1}|u_{B_0,\mu }\left|\right)<\infty .\hfill \end{array}$$

Next, we assume that (8) holds, then, we only have to show that (6) holds. By (5) we have the following:

$$\begin{array}{cc}\hfill & {\phi }_i\left({\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B{\displaystyle \frac{|u\left(x\right)-u_{B_0,\mu }|}{\parallel u\left(x\right)-u_{B_0,\mu }{\parallel }_{L({\phi }_2,\mathsf{\Omega },\mu )}}}d\mu \right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B{\phi }_i\left({\displaystyle \frac{|u\left(x\right)-u_{B_0,\mu }|}{\parallel u\left(x\right)-u_{B_0,\mu }{\parallel }_{L({\phi }_2,\mathsf{\Omega },\mu )}}}\right)d\mu \le 1.\hfill \end{array}$$

Then we obtain the following:

$${\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B|u\left(x\right)-u_{B_0,\mu }|d\mu \le {\phi }_i^{-1}\left(1\right){\parallel u\left(x\right)-u_{B_0,\mu }\parallel }_{L({\phi }_2,\mathsf{\Omega },\mu )}.$$

Hence, we know that $u\in \mathrm{BMO}(\mathsf{\Omega },\mu )$. By using Lemma 2, we have the following:

$$|u_{B_{Q_j}\left(x_0\right),\mu }-u_{B_{Q_j}\left(x\right),\mu }|\le C_{1}k(x,x_0).$$

(16)

Let $W=\left\{Q_j\right\}$ be a Whitney decomposition of $\mathsf{\Omega }$. Each $Q_j\in W$ is contained in a ball $B_{Q_j}\left(x\right)$ and $\mu B_{Q_j}\left(x\right)\le C\mu \left(Q_j\right)$. Choosing $\alpha ={(1-\lambda )}^{1/s}{\parallel u-u_{B_{Q_j}\left(x\right)}\parallel }_{L\left({\phi }_1\right)}$ and by using (16) and (5) for $i=1$, we obtain the following:

$$\begin{array}{cc}& {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha \right|u-u_{B_0,\mu }\left|\right)d\mu \hfill \\ \le \hfill & \sum _{Q_j}{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{Q_j}{\phi }_1\left(\alpha \right|u-u_{B_{Q_j}\left(x\right),\mu }|+\alpha |u_{B_{Q_j}\left(x\right),\mu }-u_{B_0,\mu }\left|\right)d\mu \hfill \\ \le \hfill & \sum _{Q_j}{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}(1-\lambda ){\int }_{Q_j}{\phi }_1{(\alpha (1-\lambda )}^{-1/s}|u-u_{B_{Q_j}\left(x\right),\mu }|d\mu \hfill \\ & +\sum _{Q_j}{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}\lambda {\int }_{Q_j}{\phi }_1\left(\alpha {\lambda }^{-1/s}{C}_{1}k(x,x_0)\right)d\mu \hfill \\ \le \hfill & \sum _{Q_j}{\displaystyle \frac{\mu \left(B_{Q_j}\left(x\right)\right)}{\mu \left(\mathsf{\Omega }\right)}}{\displaystyle \frac{(1-\lambda )}{\mu \left(B_{Q_j}\left(x\right)\right)}}{\int }_{B_{Q_j}\left(x\right)}{\phi }_1\left({\displaystyle \frac{|u-u_{B_{Q_j}\left(x\right)}|}{\parallel u-u_{B_{Q_j}\left(x\right)}{\parallel }_{L\left({\phi }_1\right)}}}\right)d\mu \hfill \\ & +\sum _{Q_j}{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}\lambda {\int }_{Q_j}{\phi }_1\left(\alpha {\lambda }^{-1/s}{C}_{1}k(x,x_0)\right)d\mu \hfill \\ \le \hfill & (1-\lambda )C_2+{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}\lambda {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha {\lambda }^{-1/s}{C}_{1}k(x,x_0)\right)d\mu \le 1.\hfill \end{array}$$

(17)

If $(1-\lambda )$ is small enough. Choosing $\alpha =(1-\lambda )\parallel u-u_{B_{Q_j}\left(x\right)}{\parallel }_{L\left({\phi }_2\right)}$ for $i=2$, similar to (17), we obtain the following:

$$\begin{array}{cc}& {\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{\phi }_2\left(\alpha \right|u-u_{B_0,\mu }\left|\right)d\mu \hfill \\ \le \hfill & {(1-\lambda )}^s{C}_4+{\displaystyle \frac{1}{\mu \left(\mathsf{\Omega }\right)}}{\lambda }^s{\int }_{\mathsf{\Omega }}{\phi }_2\left(\alpha \lambda C_{3}k(x,x_0)\right)d\mu \le 1.\hfill \end{array}$$

(18)

From (17) and (18), we obtain the following:

$$\parallel u-u_{B_0,\mu }{\parallel }_{L({\phi }_i,\mathsf{\Omega },\mu )}\le \underset{4B\subset \mathsf{\Omega }}{sup}{\parallel u-u_{B,\mu }\parallel }_{L({\phi }_i,B,\mu )}.$$

Then, $\mathsf{\Omega }$ is an s-convex-averaging domain. □

The Lemma of Whitney cover is as follows, which appeared in [25,26]. Obviously, compared with the Besicovitch cover (see [27] for an introduction to the Besicovitch cover), the Whitney cover is more elementary and more suitable for the study of the s-convex-averaging domain.

Lemma 5

([25,26]). Each domain Ω has a modified Whitney cover of cubes $W=\left\{Q_j\right\}$, which satisfy ${\bigcup }_j{Q}_j=\mathsf{\Omega }$, ${\sum }_{Q_j\in W}{\chi }_{\sqrt{5/4}{Q}_j}\le N_{{\chi }_{\mathsf{\Omega }}}$ for all $x\in \mathbb{R}$, and some $N>1$, and if $Q_i\bigcap Q_j\ne \varphi$, then there exists a cube $R(\notin W)$ in $Q_i\bigcap Q_j$ such that $Q_i\bigcup Q_j\subset NR$. Moreover, if Ω is a δ-John, then there is a distinguished cube $Q_0\in W$ that can be connected with every cube $Q_j\in W$ by a chain of cubes $Q_0,Q_1,\dots ,Q_k=Q$ from W and such that $Q\subset \rho Q_j$, $j=1,2,\dots ,k$, for some $\rho =\rho (n,\delta )$.

Now, we give the proof of Theorem 2.

Proof. 

By Definition 4, for any $x,x_0$, $k(x,x_0)\le k(x,x_1)+k(x_1,x_0)$. Let $\mathcal{F}$ be a Whitney cover of $\mathsf{\Omega }$ consisting of cubes $Q_j$ with centers $x_j$. Assuming (10) holds, then we have the following:

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\phi \left({\alpha }_{1}k(x,x_0)\right)d\mu & \le {\int }_{{\cup }_{Q_j\in \mathcal{F}}}{\phi }_i\left({\alpha }_{1}k(x,x_0)\right)d\mu \hfill \\ \hfill & \le {\int }_{{\cup }_{Q_j\in \mathcal{F}}}{\phi }_i({\alpha }_{1}k(x,x_j)+{\alpha }_{1}k(x_j,x_0))d\mu .\hfill \end{array}$$

Now let $i=1$, by Definition 1, we have the following:

$$\begin{array}{cc}& {\int }_{{\cup }_{Q_j\in \mathcal{F}}}{\phi }_1({\alpha }_{1}k(x,x_j)+{\alpha }_{1}k(x_j,x_0))d\mu \hfill \\ \le \hfill & {\int }_{{\cup }_{Q_j\in \mathcal{F}}}{\phi }_1({(1-\lambda )}^{1/s}{C}_{1}k(x,x_j)+{\lambda }^{1/s}{C}_{2}k(x_j,x_0))d\mu \hfill \\ \le \hfill & {\displaystyle (1-\lambda )\sum _{Q_j\in \mathcal{F}}{\int }_{Q_j}{\phi }_1\left(C_{1}k(x,x_j)\right)d\mu {\chi }_{\sqrt{\frac{5}{4}}{Q}_j}+\lambda \sum _{Q_j\in \mathcal{F}}{\phi }_1\left(C_{2}k(x_j,x_0)\right)\mu \left(Q_j\right)}\hfill \\ \le \hfill & {\displaystyle (1-\lambda )C_1\mu \left(\mathsf{\Omega }\right)·N_{{\chi }_{\mathsf{\Omega }}}+\lambda \sum _{Q_j\in \mathcal{F}}{\phi }_1\left(C_{2}k(x_j,x_0)\right)\mu \left(Q_j\right),}\hfill \end{array}$$

(19)

where $\lambda \in (0,1)$. And by (10), we know (19) is finite. Next, let $i=2$. Similar to the proof of ${\phi }_1$, we have the following:

$$\begin{array}{cc}\hfill & {\int }_{{\cup }_{Q_j\in \mathcal{F}}}{\phi }_2({\alpha }_{1}k(x,x_j)+{\alpha }_{1}k(x_j,x_0))d\mu \hfill \\ \hfill & {\displaystyle \le \mu \left(\mathsf{\Omega }\right)C_3\sum _{Q_j\in \mathcal{F}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_j}+\sum _{Q_j\in \mathcal{F}}{\phi }_1\left(C_{4}k(x_j,x_0)\right)\mu \left(Q_j\right).}\hfill \end{array}$$

Next we assume (9) holds and prove (10). Let $i=1$, written as follows:

$$\begin{array}{cc}\hfill {\phi }_1\left({\alpha }_{2}k(x_j,x_0)\right)& \le {\phi }_1\left({\alpha }_{2}k(x_j,x)\right)+{\alpha }_{2}k(x,x_0))\hfill \\ & \le {\phi }_1({(1-\lambda )}^{1/s}{C}_{5}k(x_j,x)+{\lambda }^{1/s}{C}_{6}k(x,x_0))\hfill \\ & \le (1-\lambda ){\phi }_1\left(C_{5}k(x_j,x)\right)+\lambda {\phi }_1\left(C_{6}k(x,x_0)\right).\hfill \end{array}$$

(20)

Integrating (20) over $Q_j$, the following is assumed:

$$\begin{array}{cc}\hfill {\phi }_1\left({\alpha }_{2}k(x_j,x_0)\right)\mu \left(Q_j\right)& ={\int }_{Q_j}{\phi }_1\left({\alpha }_{2}k(x_j,x_0)\right)d\mu \hfill \\ & \le (1-\lambda ){\int }_{Q_j}{\phi }_1\left(C_{5}k\left((x_j,x)\right)\right)d\mu \hfill \\ & +\lambda {\int }_{Q_j}{\phi }_1\left(C_{6}k(x,x_0)\right)d\mu .\hfill \end{array}$$

(21)

Using (11) and Lemma 3, we sum (21), written as follows:

$$\begin{array}{cc}& {\displaystyle \sum _{Q_j\in \mathcal{F}}{\phi }_1\left({\alpha }_{2}k(x_j,x_0)\right)\mu \left(Q_j\right)}\hfill \\ \hfill \le & {\displaystyle (1-\lambda )\sum _{Q_j\in \mathcal{F}}{\int }_{Q_j}{\phi }_1(C_{5}k\left((x_j,x)\right)d\mu +\lambda \sum _{Q_j\in \mathcal{F}}{\int }_{Q_j}{\phi }_1\left(C_{6}k(x,x_0)\right)d\mu }\hfill \\ \hfill \le & {\displaystyle (1-\lambda )\sum _{Q_j\in \mathcal{F}}C\mu \left(Q_j\right){\chi }_{\sqrt{\frac{5}{4}}{Q}_j}+\lambda {\int }_{\mathsf{\Omega }}{\phi }_1\left(C_{6}k(x,x_0)\right)d\mu \sum _{Q_j\in \mathcal{F}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_j}}\hfill \\ \hfill \le & (1-\lambda )C\mu \left(\mathsf{\Omega }\right)·N_{\mathsf{\Omega }}+\lambda {\int }_{\mathsf{\Omega }}{\phi }_1\left(C_{6}k(x,x_0)\right)d\mu ·N,\hfill \end{array}$$

(22)

where $(1-\lambda )C\mu \left(\mathsf{\Omega }\right)·N_{\mathsf{\Omega }}$ is a constant. And by (9), we know (22) is finite. Similarly, when $i=2$, we obtain the following:

$$\begin{array}{cc}\hfill & {\phi }_2\left({\alpha }_{2}k(x_j,x_0)\right)\mu \left(Q_j\right)\le {(1-\lambda )}^s{\int }_{Q_j}{\phi }_2\left(C_{7}k\left((x_j,x)\right)\right)d\mu +{\lambda }^s{\phi }_2\left(C_8(x,x_0)\right)d\mu \hfill \end{array}$$

and

$$\begin{array}{cc}& {\displaystyle \sum _{Q_j\in \mathcal{F}}{\phi }_2\left({\alpha }_{2}k(x_j,x_0)\right)\mu \left(Q_j\right)}\hfill \\ \hfill \le & {(1-\lambda )}^{s}C\mu \left(\mathsf{\Omega }\right)·N_{\mathsf{\Omega }}+{\lambda }^s{\int }_{\mathsf{\Omega }}{\phi }_2\left(C_{8}k(x,x_0)\right)d\mu ·N.\hfill \end{array}$$

(23)

The above (23) is finite, then we complete the proof of Theorem 2. □

4. Invariant of $\mathbf{S}$-Convex-Averaging Domain

In this section, we prove one of our main results, the invariant of s-convex-averaging domains under the quasi-conformal mappings and quasi-isometric mappings. The following lemma appeared in [13].

Lemma 6

([13]). Let Ω and ${\mathsf{\Omega }}^{\prime }$ be proper subdomains of ${\mathbb{R}}^n$, and $f:\mathsf{\Omega }\subset {\mathbb{R}}^n\to {\mathsf{\Omega }}^{\prime }$ be a K-quasi-conformal mapping. Then there exists a constant c, depending only on n and K, such that the following is calculated:

$$k(f\left(x_1\right),f\left(x_2\right);{\mathsf{\Omega }}^{\prime })\le c(n,k)max\left(k(x_1,x_2;\mathsf{\Omega }),k^t(x_1,x_2;\mathsf{\Omega })\right)$$

for all $x_1$, $x_2\in \mathsf{\Omega }$, where $t=K^{1/(1-n)}\le 1$.

The Definition of K-quasi-isometry is given in [13,14,15].

Definition 7

([14,15]). A mapping f defined in Ω is said to be a K-quasi-isometry, $K>1$, if the following is met:

$${\displaystyle K^{-1}\le \frac{\left|f\right(x)-f(y\left)\right|}{|x-y|}\le K}$$

for all $x\ne y$.

The properties of K-quasi-isometric mapping is as follows which appeared in [15].

Lemma 7

([15]). Let $f:\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime }$ be a K-quasi-isometric mapping. If $\omega \left(x\right)\in A_r$, then $\omega \left(f\left(x\right)\right)\in A_r$.

Lemma 8

([15]). Let $f:\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime }$ be a K-quasi-isometric mapping. Then we obtain the following:

$$K^{-n}\mu \left(D\right)\le \nu \left(D^{\prime }\right)\le K^n\mu \left(D\right),$$

where $D\subset \mathsf{\Omega }$ and $D^{\prime }=f\left(D\right)\subset {\mathsf{\Omega }}^{\prime }$.

Now, we show the invariant of s-convex-averaging domains.

Theorem 3.

If $f:\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime }$ is a surjective K-quasi-conformal mapping and $J_f$ is the Jacobian determinant of f. Then, we obtain the following:

(a) 

${\mathsf{\Omega }}^{\prime }$ is an s-convex-averaging domain $L({\phi }_s,m)$, where m is Lebesgue measure, then Ω is an s-convex-averaging domain $L({\phi }_s,\mu )$ with $d\mu =J_{f}dm$.

(b) 

Ω is an s-convex-averaging domain $L({\phi }_s,\mu )$, where m is Lebesgue measure, then ${\mathsf{\Omega }}^{\prime }$ is an s-convex-averaging domain $L({\phi }_s,m)$ with $d\mu =J_{f}dx$.

Proof. 

We first consider (a). Let $x_0,x\in \mathsf{\Omega }$ and $y=f\left(x\right),y_0=f\left(x_0\right)$. By Lemma 6, we obtain the following:

$$k(f^{-1}\left(y\right),f^{-1}\left(y_0\right);\mathsf{\Omega })\le C_1\left(k(y,y_0;{\mathsf{\Omega }}^{\prime })+k^t(y,y_0;{\mathsf{\Omega }}^{\prime })\right).$$

(24)

Let $\lambda$ be a constant with $\lambda \in (0,1)$. Using (24) and ${\phi }_1\in K_s^1$, we obtain the following:

$$\begin{array}{cc}& {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha k(x,x_0;\mathsf{\Omega })\right)J_{f}dx\hfill \\ \hfill =& {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha k(f^{-1}\left(y\right),f^{-1}\left(y_0\right);\mathsf{\Omega })\right)J_{f}dx\hfill \\ \hfill \le & {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_1(\alpha C_{1}k(y,y_0;{\mathsf{\Omega }}^{\prime })+\alpha C_1{k}^t(y,y_0;{\mathsf{\Omega }}^{\prime }))dy\hfill \\ \hfill \le & {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_1({(1-\lambda )}^{(1/s)}\alpha C_{2}k(y,y_0;{\mathsf{\Omega }}^{\prime })+{\lambda }^{1/s}\alpha C_3{k}^t(y,y_0;{\mathsf{\Omega }}^{\prime }))dy\hfill \\ \hfill \le & (1-\lambda ){\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_1\left(\alpha C_{2}k(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy+\lambda {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_1\left(C_3{k}^t(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy.\hfill \end{array}$$

(25)

On the other hand, let $i=2$. We then have the following:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\phi }_2\left(\alpha k(x,x_0;\mathsf{\Omega })\right)J_{f}dx\hfill \\ \hfill & \le {(1-\lambda )}^s{\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_2\left(\alpha C_{2}k(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy\hfill \\ \hfill & +{\lambda }^s{\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_2\left(C_3{k}^t(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy.\hfill \end{array}$$

(26)

By Theorem 1, (25) and (26) are finite. So $\mathsf{\Omega }$ is an s-convex-averaging domain $L({\phi }_s,\mu )$. Next, we derive (b). Similar to (a), we have the following:

$$k(f\left(x\right),f\left(x_0\right);{\mathsf{\Omega }}^{\prime })\le C_{1}max\left(k(x,x_0;\mathsf{\Omega }),k^t(x,x_0;\mathsf{\Omega })\right).$$

Since $\mathsf{\Omega }$ is an s-convex-averaging domain $L({\phi }_s,\mu )$, there exists constant ${\beta }_i>0$ for $i=1,2$ such that the following is obtained:

$${\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_{i}k(x,x_0;\mathsf{\Omega })\right)d\mu <\infty .$$

Let $i=1$ and $\alpha \le min\{\left({\beta }_1{(1-\lambda )}^{1/s}\right)/C_1,{\beta }_1{\lambda }^{1/s}/C_1\}$, to obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_1\left(\alpha k(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}{\phi }_1(\alpha C_{1}k(x,x_0;\mathsf{\Omega })+\alpha C_1{k}^t(x,x_0;\mathsf{\Omega }))J_{f}dx\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\phi }_1{(\alpha (1-\lambda )}^{1/s}{(1-\lambda )}^{-1/s}{C}_{1}k(x,x_0;\mathsf{\Omega })\hfill \\ \hfill & +\alpha {\lambda }^{1/s}{\lambda }^{-1/s}{C}_1{k}^t(x,x_0;\mathsf{\Omega }))J_{f}dx\hfill \\ \hfill & \le (1-\lambda ){\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha {(1-\lambda )}^{-1/s}{C}_{1}k(x,x_0;\mathsf{\Omega })\right)J_{f}dx\hfill \\ \hfill & +\lambda {\int }_{\mathsf{\Omega }}{\phi }_1\left(\alpha {\lambda }^{-1/s}{C}_1{k}^t(x,x_0;\mathsf{\Omega })\right)J_{f}dx\hfill \\ \hfill & \le (1-\lambda ){\int }_{\mathsf{\Omega }}{\phi }_1\left({\beta }_{1}k(x,x_0;\mathsf{\Omega })\right)d\mu +\lambda {\int }_{\mathsf{\Omega }}{\phi }_1\left({\beta }_1{k}^t(x,x_0;\mathsf{\Omega })\right)d\mu .\hfill \end{array}$$

(27)

For $i=2$, choose $\alpha \le min\{{\beta }_2(1-\lambda )/C_1,{\beta }_2\lambda /C_1\}$. Similarly, the following is assumed:

$$\begin{array}{cc}\hfill & {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_2\left(\alpha k(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy\hfill \\ \hfill & \le {(1-\lambda )}^s{\int }_{\mathsf{\Omega }}{\phi }_2\left({\beta }_{2}k(x,x_0;\mathsf{\Omega })\right)d\mu +{\lambda }^s{\int }_{\mathsf{\Omega }}{\phi }_2\left({\beta }_2{k}^t(x,x_0;\mathsf{\Omega })\right)d\mu .\hfill \end{array}$$

(28)

By Theorem 1, we obtain the first part of (27) and (28) are finite. For the second part of (27) and (28), the following is assumed:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_i{k}^t(x,x_0;\mathsf{\Omega })\right)d\mu \hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_i{k}^t(x,x_0;\mathsf{\Omega })\right){\chi }_{\{x:k(x,x_0;\mathsf{\Omega })\le 1\}}d\mu \hfill \\ \hfill & +{\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_i{k}^t(x,x_0;\mathsf{\Omega })\right){\chi }_{\{x:k(x,x_0;\mathsf{\Omega })>1\}}d\mu \hfill \\ \hfill & \le {\int }_{\{x:k(x,x_0;\mathsf{\Omega })\le 1\}}{\phi }_i\left({\beta }_i\right)d\mu +{\int }_{\{x:k(x,x_0;\mathsf{\Omega })>1\}}{\phi }_i\left({\beta }_{i}k(x,x_0;\mathsf{\Omega })\right)d\mu \hfill \\ \hfill & \le {\phi }_i\left({\beta }_i\right)\mu \left(\mathsf{\Omega }\right)+{\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_{i}k(x,x_0;\mathsf{\Omega })\right)d\mu .\hfill \end{array}$$

(29)

Combining (27)–(29), we have the following:

$${\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_i\left(\alpha k(y,y_0;{\mathsf{\Omega }}^{\prime })\right)dy\le {\phi }_i\left({\beta }_i\right)\mu \left(\mathsf{\Omega }\right)+{\int }_{\mathsf{\Omega }}{\phi }_i\left({\beta }_{i}k(x,x_0;\mathsf{\Omega })\right)d\mu <\infty .$$

The proof of Theorem 3 has been completed. □

Theorem 4.

If $f:\mathsf{\Omega }\to {\mathsf{\Omega }}^{\prime }$ is a K-quasi-isometric mapping and Ω is an s-convex-averaging domain $L({\phi }_s,\mu )$, then ${\mathsf{\Omega }}^{\prime }$ is an s-convex-averaging domain $L({\phi }_s,\nu )$, where μ and ν are measures defined by $d\nu =\omega \left(f\right(x\left)\right)dx$ and $d\mu =\omega \left(x\right)dx$.

Proof. 

Let $\gamma$ be a quasihyperbolic arc joining x to y in $\mathsf{\Omega }$ and set ${\gamma }^{\prime }=f\left(\gamma \right)$. Then we obtain the following:

$$\begin{array}{c}\hfill d(f\left(x\right),\partial {\mathsf{\Omega }}^{\prime })=K^{-1}d(x,\partial {\mathsf{\Omega }}^{\prime }).\end{array}$$

(30)

By Definition 4 and (30), we have the following:

$$k(f\left(x\right),f\left(y\right);{\mathsf{\Omega }}^{\prime })\le K^{2}k(x,y;\mathsf{\Omega }).$$

(31)

Let $\alpha =\beta /K^2$. By (30) and (31) and Theorem 1, we have the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_s\left(\alpha k(f\left(x\right),f\left(x_0\right);{\mathsf{\Omega }}^{\prime })\right)d\nu & ={\int }_{{\mathsf{\Omega }}^{\prime }}{\phi }_s\left(\alpha k(f\left(x\right),f\left(x_0\right);{\mathsf{\Omega }}^{\prime })\right)\omega \left(f\left(x\right)\right)dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}{\phi }_s\left(\beta k(x,x_0;\mathsf{\Omega })\right)\omega \left(x\right)J_{f}dx\hfill \\ \hfill & \le K^n{\int }_{\mathsf{\Omega }}{\phi }_s\left(\beta k(x,x_0;\mathsf{\Omega })\right)d\mu \hfill \\ \hfill & \le C{\beta }^{-1}<\infty ,\hfill \end{array}$$

where $\beta$ is coincident with that in Theorm 1, and C is a certain constant, independent of x and $x_0$. We have finished the proof of Theorem 4. □

5. Embedding Inequality for Two-Dimensional Equations

In this section, by $A\lesssim B$ we mean that $A\le CB$ with some positive constant C independent of appropriate quantities. If $A\lesssim B$ and $B\lesssim A$, we write $A\sim B$, and then we say that A and B are equivalent. We first show the definition of generalized Orilcz–Morrey spaces and the concept of class $G(p,q,C)$. Generalized Orilcz–Morrey spaces are important tools for the local solutions of partial differential equations; see [28,29,30] for more theories of Orilcz–Morrey spaces. So we establish the embedding inequality for the solutions of two-dimensional diffusion-wave equations on the s-convex-averaging domain with generalized Orilcz–Morrey norms.

Definition 8

([11]). Let ${\mathsf{\Phi }}_1\in K_s^1$ and ${\mathsf{\Phi }}_2\in K_s^2$. We say that ${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2$ lie in the class $G(p,q,C)$, $1\le p<q<\infty$, and $C\ge 1$, if $1/C\le {\mathsf{\Phi }}_i\left(t^{1/p}\right)/g\left(t\right)\le C$ and $C^{-1}\le {\mathsf{\Phi }}_i\left(t^{1/q}\right)/h\left(t\right)\le C$ for all $t>0$ and $i=1,2$, where g is a convex increasing function and h is a concave increasing function on $[0,\infty )$.

Lemma 9

([11]). Let Φ be an s-convex function in the class $G(p,q,C)$, $1\le p<q<\infty$, $C\ge 1$. Ω be any bounded $L(\mathsf{\Phi },\mu )$-averaging domain. Assume that $\mathsf{\Phi }\left(\right|u\left|\right),\mathsf{\Phi }\left(d\right|u\left|\right)\in L^1(\mathsf{\Omega },\mu )$. Then we obtain the following:

$${\int }_B\mathsf{\Phi }\left({\displaystyle \frac{|u-u_B|}{k}}\right)d\mu \lesssim {\int }_{\sigma B}\mathsf{\Phi }\left({\displaystyle \frac{\left|du\right|}{k}}\right)d\mu$$

(32)

for any constant $k>0$ and all balls B with $\sigma B\subset \mathsf{\Omega }$, where the measure μ is defined by $d\mu =\omega \left(x\right)dx$ and $\omega \left(x\right)$ is a weight function satisfies (7).

If $s=1$, both ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ become convex functions if they are non-decreasing and continuous. More details of the class $G(p,q,C)$ were introduced in [11]. Next, we show the definition of generalized Orilcz–Morrey spaces.

Definition 9.

Let $\phi (x,r)$ be a positive measurable function on ${\mathbb{R}}^n\times (0,\infty )$ and Φ be an s-convex function. We denote by $L^{\phi ,\mathsf{\Phi }}\left({\mathbb{R}}^n\right)$ the generalized Orlicz-Morrey spaces, the space of all functions $u\in L_{loc}^{\mathsf{\Phi }}\left({\mathbb{R}}^n\right)$ for which the following is obtained:

$${\parallel u\parallel }_{L^{\phi ,\mathsf{\Phi }}}:=\underset{x\in R^n,r>0}{sup}\phi {(x,r)}^{-1}{\parallel u\parallel }_{L^{\mathsf{\Phi }}\left(B(x,r)\right)}<\infty ,$$

(33)

where $B(x,r)\subset {\mathbb{R}}^n$, ${\parallel u\parallel }_{L^{\mathsf{\Phi }}\left(B(x,r)\right)}$ satisfies (4).

We give the embedding inequality on the s-convex-averaging domain in the following theorem with the generalized Orlicz-Morrey norms.

Theorem 5.

Let u be a solution of the two-dimensional distributed order space-fractional diffusion Equation (1). Let also Φ be an s-convex function in the class $G(p,q,C)$, and $\phi (x,r)$ is almost decreasing for $r\in (0,\infty )$. $1\le p<q<\infty$, and Ω is an s-convex-averaging domain. Assume that $\mathsf{\Phi }\left(\right|u\left|\right),\mathsf{\Phi }\left(d\right|u\left|\right)\in L^1(\mathsf{\Omega },\mu )$. Then there exists a constant C, such that the following is calculated:

$$\parallel u-u_{B_0}{\parallel }_{L^{\phi ,\mathsf{\Phi }}\left(\mathsf{\Omega }\right)}\le C{\parallel du\parallel }_{L^{\phi ,\mathsf{\Phi }}\left(\mathsf{\Omega }\right)},$$

where $B_0\subset \mathsf{\Omega }$ is a ball, the measure μ is defined by $d\mu =\omega \left(x\right)dx$ and $\omega \left(x\right)$ is a weight function satisfies (7).

Proof. 

By Definition 9, Definition 2, and Lemma 9, we have the following:

$$\begin{array}{cc}\hfill & \parallel u-u_B{\parallel }_{L^{\phi ,\mathsf{\Phi }}\left(B(x,r)\right)}\hfill \\ \hfill & =\underset{x\in \mathsf{\Omega },r>0}{sup}\phi {(x,r)}^{-1}{\int }_B\mathsf{\Phi }\left({\displaystyle \frac{|u-u_B|}{k}}\right)d\mu \hfill \\ \hfill & \lesssim \underset{x\in \mathsf{\Omega },r>0}{sup}\phi {(x,\sigma r)}^{-1}{\int }_{\sigma B}\mathsf{\Phi }\left({\displaystyle \frac{\left|du\right|}{k}}\right)d\mu \hfill \\ \hfill & \lesssim {\parallel du\parallel }_{L^{\phi ,\mathsf{\Phi }}\left(\sigma B\right)}.\hfill \end{array}$$

(34)

Using (6) and (34), we have the following:

$$\begin{array}{cc}\hfill \parallel u-u_{B_0}{\parallel }_{L^{\phi ,\mathsf{\Phi }}(\mathsf{\Omega },\mu )}& \lesssim \underset{B\subset \mathsf{\Omega }}{sup}{\parallel u-u_B\parallel }_{L^{\phi ,\mathsf{\Phi }}(B,\mu )}\hfill \\ \hfill & \lesssim \underset{B\subset \mathsf{\Omega }}{sup}{\parallel du\parallel }_{L^{\phi ,\mathsf{\Phi }}(\sigma B,\mu )}\hfill \\ \hfill & \lesssim {\parallel du\parallel }_{L^{\phi ,\mathsf{\Phi }}(\mathsf{\Omega },\mu )}.\hfill \end{array}$$

We have completed the proof of Theorem 5. □

Remark 2.

Let ${\kappa }_1$ and ${\kappa }_2$ are two classes of s-convex functions in $K_s^1$ and $K_s^2$. By Lemma 1, we obtain the following:

$${\parallel u\parallel }_{L^{\left({\kappa }_1{\kappa }_2\right)}}:=\underset{x\in {\mathbb{R}}^n,r>0}{sup}{\displaystyle \frac{1}{{\kappa }_1{\kappa }_2}}{\parallel u\parallel }_{L^{\left({\kappa }_1{\kappa }_2\right)}}<\infty$$

and

$${\parallel u\parallel }_{L^{({\kappa }_1\circ {\kappa }_2)}}:=\underset{x\in {\mathbb{R}}^n,r>0}{sup}{\displaystyle \frac{1}{{\kappa }_1\circ {\kappa }_2}}{\parallel u\parallel }_{L^{({\kappa }_1\circ {\kappa }_2)}}<\infty .$$

It is easy to know that Theorem 5 holds with the norms ${\parallel ·\parallel }_{L^{\left({\kappa }_1{\kappa }_2\right)}}$ and ${\parallel ·\parallel }_{L^{({\kappa }_1\circ {\kappa }_2)}}$, respectively.

Remark 3.

The s-convex-averaging domains are generalized domains of the $L^p$-averaging domains. By Theorem 3.14 in [20], we know that John domains are special cases of $L^p$-averaging domains. So Theorem 5 holds for the above domains, respectively.

6. Conclusions

In this paper, we extend the circular and irregular convex domains and give a new concept of s-convex-averaging domains. The s-convex-averaging domains can be decomposed into a form covered by countably many small domains by the Whitney covering method. Each small domain is an s-convex-averaging domain for a fixed ${\phi }_s$ and fixed measure $\mu$. We prove the invariance of s-convex-averaging domains under the K-quasi-conformal mappings and K-quasi-isometric mappings. The embedding inequality of solutions of two-dimensional diffusion-wave equations is established on the s-convex-averaging domain with Orilcz–Morrey norms.