Sobolev-Type Inequalities on Musielak–Orlicz–Morrey Spaces of an Integral Form

Abstract

We give Sobolev-type inequalities for variable Riesz potentials $I_{\alpha (\cdot )}f$ of functions in Musielak–Orlicz–Morrey spaces of an integral form ${\mathcal {L}}^{\Phi ,\omega }(G)$. As a corollary, we give Sobolev-type inequalities on ${\mathcal {L}}^{\Phi ,\omega }(G)$ for double phase functions $\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}$.

1 Introduction

Let G be an open bounded set in $\textbf{R}^N$. Let $\alpha (\cdot )$ be a measurable function on G such that

$$\begin{aligned} 0<\inf _{x \in G} \alpha (x)\le \sup _{x \in G} \alpha (x)<N. \end{aligned}$$

We define the Riesz potential of variable order $\alpha (\cdot )$ for a locally integrable function f on G by

$$\begin{aligned} I_{\alpha (\cdot )}f(x) = \int _{G} |x-y|^{\alpha (x)-N} f(y)\, dy; \end{aligned}$$

when $\alpha (\cdot )$ is a constant $\alpha$, this is simply written as $I_{\alpha }f$.

Sobolev-type inequalities for $I_{\alpha }f$ have been established on various function spaces by many researchers. Sobolev-type inequalities were studied on variable exponent Lebesgue spaces $L^{p(\cdot )}$ in [7, 9, 11], on two variable exponent Lebesgue spaces $L^{p(\cdot )}(\log L)^{q(\cdot )}$ in [12, 25], on variable exponent Morrey spaces $L^{p(\cdot ), \nu }$ in [2, 13, 14, 22, 23, 28], on Musielak–Orlicz–Morrey spaces $L^{\Phi ,\kappa }$ in [19, 20].

In the previous paper [32], we gave Sobolev-type inequalities for $I_{\alpha (\cdot )}f$ of functions in variable exponent Morrey spaces of an integral form ${\mathcal {L}}^{p(\cdot ),\omega }(G)$, as an extension of [29, Theorem 5.4] from Morrey spaces of an integral form.

In this paper, we establish a Sobolev-type inequality for $I_{\alpha (\cdot )}f$ of functions in Musielak–Orlicz–Morrey spaces of an integral form ${\mathcal {L}}^{\Phi ,\omega }(G)$ defined by general functions $\Phi (x,t)$ and $\omega (x,r)$ satisfying certain conditions (Theorem 4.5), as an extension of [32, Theorem 4.4]. To do this, we apply Hedberg’s method ([17]) and the boundedness of the maximal operator M in ${\mathcal {L}}^{\Phi ,\omega }(G)$ (Theorem 3.4) which is an extension of [32, Theorem 3.5].

As an application of our general theory, we give Sobolev-type inequalities (Theorem 5.3) in the framework of double phase functions $\Phi (x,t)$ with variable exponents given by

$$\begin{aligned} \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \end{aligned}$$

where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions, $p(x)<q(x)$ for $x\in G$ and $a(\cdot )$ is nonnegative, bounded and Hölder continuous of order $\theta \in (0,1]$. For the studies by Mingione and collaborators, see [3, 4, 8]. We refer to [20, 27] for Sobolev’s inequality and to, e.g., [6, 10, 16, 24, 30, 33] for the recent results.

Throughout the paper, we let C denote various constants independent of the variables in question and $C(a,b,\cdots )$ be a constant that depends on $a,b,\cdots$ only. The symbol $g\sim h$ means that $C^{-1}h\le g\le Ch$ for some constant $C>0$.

2 Musielak–Orlicz–Morrey Spaces of an Integral Form

To define the norm of Musielak–Orlicz–Morrey spaces of an integral form, let us consider a function

$$\begin{aligned} \Phi (x,t) : G \times [0, \infty ) \rightarrow [0, \infty ) \end{aligned}$$

satisfying the following conditions ($\Phi$1) – ($\Phi$3):

($\Phi$1):

$\Phi (\, \cdot \, , t)$ is measurable on G for each $t \ge 0$ and $\Phi (x,\, \cdot \, )$ is continuous on $[0, \infty )$ for each $x \in G$;

($\Phi$2):

there exists a constant $A_1 \ge 1$ such that

$$\begin{aligned} A_1^{-1} \le \Phi (x,1) \le A_1 \quad \text {for all} \ x \in G; \end{aligned}$$

($\Phi 3$):

$t \mapsto \Phi (x,t)/t$ is uniformly almost increasing on $(0, \infty )$, namely there exists a constant $A_2 \ge 1$ such that

$$\begin{aligned} \Phi (x,t_1)/t_1 \le A_2 \Phi (x,t_2)/t_2 \quad \text {for all} \ x \in G \ \text {whenever} \ 0< t_1 < t_2. \end{aligned}$$

We write

$$\begin{aligned} {\bar{\phi }}(x,t) = \sup _{0 < s \le t} (\Phi (x,s)/s) \end{aligned}$$

and

$$\begin{aligned} {\overline{\Phi }}(x,t) = \int _0^t {\bar{\phi }}(x,r)\, dr \end{aligned}$$

for $x \in G$ and $t \ge 0$. Then ${\overline{\Phi }}(x, \cdot )$ is convex and

$$\begin{aligned} \Phi (x,t/2) \le {\overline{\Phi }}(x,t) \le A_2 \Phi (x,t) \end{aligned}$$

(2.1)

for all $x \in G$ and $t \ge 0$ since ${\bar{\phi }}(x,\cdot )$ is increasing on $(0,\infty )$ for each $x\in G$.

For $x \in \textbf{R}^N$ and $r>0$, we denote by B(x, r) the open ball centered at x with radius r and $d_{G}=\sup \{|x-y| : x, y \in G \}$. For a set $E\subset \textbf{R}^N$, |E| denotes the Lebesgue measure of E.

We also consider a weight function $\omega (x,r) : G \times (0,\infty ) \rightarrow (0, \infty )$ satisfying the following conditions:

($\omega 0$):

$\omega (\, \cdot \, , r)$ is measurable on G for each $r>0$ and $\omega (x,\, \cdot \, )$ is continuous on $(0, \infty )$ for each $x \in G$;

($\omega 1$):

$r \mapsto \omega (x,r)$ is uniformly almost increasing on $(0, \infty )$, namely there exists a constant ${\tilde{c}}_1 \ge 1$ such that

$$\begin{aligned} \omega (x,r_1) \le {\tilde{c}}_1 \omega (x,r_2) \end{aligned}$$

for all $x \in G$ whenever $0< r_1< r_2< \infty$;

($\omega 2$):

there exists a constant ${\tilde{c}}_2>1$ such that

$$\begin{aligned} {\tilde{c}}_2^{-1} \omega (x, r) \le \omega (x, 2r) \le {\tilde{c}}_2 \omega (x, r) \end{aligned}$$

for all $x \in G$ whenever $r>0$;

($\omega 3; \omega _0$):

there exist constants $\omega _0>0$ and ${\tilde{c}}_3 \ge 1$ such that

$$\begin{aligned} {\tilde{c}}_3^{-1} r^{\omega _0} \le \omega (x,r) \le {\tilde{c}}_3 \end{aligned}$$

for all $x \in G$ and $0<r \le 2d_G$.

Let $f^- := \inf _{x\in G} f(x)$ and $f^+ := \sup _{x\in G} f(x)$ for a measurable function f on G. Let us write that $L_{c}(t) = \log (c + t)$ for $c >1$ and $t \ge 0$, $L_{c}^{(1)}(t) = L_c(t)$, $L_{c}^{(j+1)}(t) = L_c( L_c^{(j)}(t))$.

Example 2.1

Let $\sigma (\cdot )$ and $\beta (\cdot )$ be measurable functions on G such that $0<\sigma ^- \le \sigma ^+ \le \omega _0$ and $-c(\omega _0-\sigma (x))\le \beta (x)\le c$ for all $x\in G$ and some constant $c >0$. Then

$$\begin{aligned} \omega (x,r) = r^{\sigma (x)}L_e(1/r)^{\beta (x)} \end{aligned}$$

satisfies ($\omega 0$), ($\omega 1$), ($\omega 2$) and ($\omega 3; \omega _0$).

Given $\Phi (x,t)$ and $\omega (x,r)$ as above, we define the ${\mathcal {L}}^{\Phi ,\omega }$ norm by

$$\begin{aligned} \Vert f \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)}&= \inf \biggl \{ \lambda > 0\, ; \\&\sup _{x\in G}\left( \int _0^{2d_G} \frac{\omega (x,r)}{|B(x,r)|} \left( \int _{G\cap B(x, r)}{\overline{\Phi }} \left( y,|f(y)|/\lambda \right) \,dy \right) \, \frac{dr}{r} \right) \le 1 \biggl \}, \end{aligned}$$

which is the Luxemburg norm ([18]). The space of all measurable functions f on G with $\Vert f \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} < \infty$ is denoted by ${\mathcal {L}}^{\Phi ,\omega }(G)$. The space ${\mathcal {L}}^{\Phi ,\omega }(G)$ is called a Musielak–Orlicz–Morrey space of an integral form. Here note that $2d_G$ can be replaced by $\kappa d_G$ with $\kappa >1$. In case $\Phi (x,t)=t^{p(x)}$, ${\mathcal {L}}^{\Phi ,\omega }(G)$ is denoted by ${\mathcal {L}}^{p(\cdot ),\omega }(G)$ for simplicity. If $p(\cdot ) \equiv p$, then we write ${\mathcal {L}}^{p(\cdot ),\omega }(G)={\mathcal {L}}^{p,\omega }(G)$.

Remark 2.2

If there exists a constant $C_0>0$ such that

$$\begin{aligned} \int _{0}^{2d_G}\omega (x,r)\, \frac{dr}{r}\le C_0 \end{aligned}$$

for all $x\in G$, then we see that ${\mathcal {L}}^{\Phi ,\omega }(G)\ne \{0\}$ since

$$\begin{aligned} \int _0^{2d_G} \frac{\omega (x,r)}{|B(x,r)|} \left( \int _{G\cap B(x, r)}{\overline{\Phi }} \left( y,1\right) \,dy \right) \, \frac{dr}{r}&\le A_1A_2 \int _{0}^{2d_G}\omega (x,r)\, \frac{dr}{r} \le A_1A_2C_0 \end{aligned}$$

for all $x\in G$ by (2.1) and ($\Phi$2). See also [5, Lemma 1].

We shall also consider the following conditions for $\Phi (x,t)$: Let $p \ge 1$, $q \ge 1$ and $\nu >0$ be given.

($\Phi 3;0; p$):

$t \mapsto t^{-p}\Phi (x,t)$ is uniformly almost increasing on (0, 1], namely there exists a constant $A_{2,0,p} \ge 1$ such that

$$\begin{aligned} t_1^{-p} \Phi (x,t_1) \le A_{2,0,p}\, t_2^{-p}\, \Phi (x,t_2) \quad \text {for all} \ x \in G \ \text {whenever} \ 0< t_1 < t_2 \le 1 ; \end{aligned}$$

($\Phi 3;\infty ; q$):

$t \mapsto t^{-q}\Phi (x,t)$ is uniformly almost increasing on $[1, \infty )$, namely there exists a constant $A_{2,\infty ,q} \ge 1$ such that

$$\begin{aligned} t_1^{-q}\Phi (x,t_1) \le A_{2,\infty ,q}\, t_2^{-q}\Phi (x,t_2) \quad \text {for all} \ x \in G \ \text {whenever} \ 1 \le t_1 < t_2 ; \end{aligned}$$

$(\Phi 5;\nu )$:

for every $\gamma >0$, there exists a constant $B_{\gamma , \nu } \ge 1$ such that

$$\begin{aligned} \Phi (x,t) \le B_{\gamma , \nu } \Phi (y,t) \end{aligned}$$

whenever $x, y \in G$, $|x-y| \le \gamma t^{-\nu }$ and $t \ge 1$.

Remark 2.3

We refer to [1, p. 2544] and [15, Section 7.3] for $(\Phi 5;\nu )$. If $\Phi (x,t)$ satisfies $(\Phi 3;\infty ;q)$, then it satisfies $(\Phi 3;\infty ;q')$ for $1 \le q' \le q$. If $\Phi (x,t)$ satisfies $(\Phi 5;\nu )$, then it satisfies $(\Phi 5;\nu ')$ for all $\nu ' \ge \nu$.

We give some examples of $\Phi (x,t)$.

Example 2.4

Let $p(\cdot )$ and $q_j(\cdot )$, $j=1, \ldots , k$ be given measurable functions on G such that $1<p^-\le p^+<\infty$ and $-\infty<q_j^-\le q_j^+<\infty$, $j=1,\, \ldots \, k$. Then,

$$\begin{aligned} \Phi _{p(\cdot ),\{q_j(\cdot )\}}(x,t) = t^{p(x)} \prod _{j=1}^k \bigl ( L_{e}^{(j)}(t) \bigr )^{q_j(x)} \end{aligned}$$

satisfies ($\Phi$1), ($\Phi$2) and ($\Phi 3$). This function satisfies ($\Phi 3;\infty ; q$) for $1\le q < p^-$ in general and for $1 \le q \le p^-$ in case $q_{j}^- \ge 0$ for all $j=1, \ldots , k$.

Moreover, we see that $\Phi _{p(\cdot ),\{q_j(\cdot )\}}(x,t)$ satisfies $(\Phi 5;\nu )$ for every $\nu >0$ if $p(\cdot )$ is log-Hölder continuous, namely

$$\begin{aligned} |p(x) - p(y)| \le \frac{C_{p}}{L_e (1/|x-y|)} \quad (x,\, y \in G) \end{aligned}$$

with a constant $C_{p} \ge 0$ and $q_{j}(\cdot )$ is $(j+1)$-log-Hölder continuous, namely

$$\begin{aligned} |q_{j}(x) - q_{j}(y)| \le \frac{C_{j}}{L_e^{(j+1)} (1/|x-y|)} \quad (x,\, y \in G) \end{aligned}$$

with constants $C_{j} \ge 0$ for each $j=1, \ldots k$.

Example 2.5

Theorem 3.4 applies, e.g., to the following nondoubling functions

$$\begin{aligned} \Phi _1 (t) = e^{p(x)t}-p(x)t-1, \ \Phi _2 (t) = e^{t}t^{p(x)}, \ \Phi _3 (t) = e^{t^{p(x)}}-1 \end{aligned}$$

which satisfy ($\Phi$1), ($\Phi$2) and ($\Phi$3). We refer to [21, Examples 3-5] for the conditions on p and q which ($\Phi 3;0; p$) and ($\Phi 3;\infty ; q$) hold.

Example 2.6

The double phase function with variable exponents

$$\begin{aligned} \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \ x \in G, \ t \ge 0 , \end{aligned}$$

where $p(x)<q(x)$ for $x\in G$, $a(\cdot )$ is a nonnegative, bounded and Hölder continuous function of order $\theta \in (0,1]$, was studied in [20]. We refer to [20, Lemma 5.1] and Section 5 for the conditions on $p(\cdot )$ and $q(\cdot )$ which $(\Phi 1)$, $(\Phi 2)$, $(\Phi 3)$, $(\Phi 3;0;p^-)$, $(\Phi ;\infty ;p^-)$ and $(\Phi 5; \nu )$ hold.

3 Boundedness of the Maximal Operator

For a locally integrable function f on G, the Hardy-Littlewood maximal function Mf is defined by

$$\begin{aligned} Mf(x)=\sup _{r >0}\frac{1}{|B(x,r)|} \int _{G\cap B(x,r)} |f(y)|\, dy. \end{aligned}$$

We know the boundedness of M on ${\mathcal {L}}^{p,\omega }(G)$.

Lemma 3.1

([32, Lemma 3.2]) Suppose

$(\omega 1')$:

$r \mapsto r^{-\varepsilon _{1}}\omega (x, r)$ is uniformly almost increasing in $(0,d_G]$ for some $\varepsilon _1>0$.

If $p>1$, then there is a constant $C >0$ such that

$$\begin{aligned} \Vert Mf \Vert _{{\mathcal {L}}^{p,\omega }(G)} \le C \Vert f \Vert _{{\mathcal {L}}^{p,\omega }(G)} \end{aligned}$$

for all $f \in {\mathcal {L}}^{p,\omega }(G)$.

Remark 3.2

Note that ($\omega 1'$) implies ($\omega 1$).

Let $\omega (x,r) = r^{\sigma (x)} L_{e}(1/r)^{\beta (x)}$ be as in Example 2.1. Then note that ($\omega 1'$) holds for $0<\varepsilon _1<\sigma ^-$.

Lemma 3.3

Suppose $\Phi (x,t)$ satisfies $(\Phi 3;0;p)$, $(\Phi 3;\infty ;q)$ and $(\Phi 5;\nu )$ for $p \ge 1$, $q \ge 1$ and $\nu >0$ satisfying $\nu \le q/\omega _0$. Set

$$\begin{aligned} I(f;x,r) = \frac{1}{|B(x,r)|} \int _{G\cap B(x,r)} f(y)\, dy \end{aligned}$$

and

$$\begin{aligned} J(f;x,r) = \frac{1}{|B(x,r)|} \int _{G\cap B(x,r)} \Phi \bigl ( y, f(y) \bigr )^{1/p_0}\, dy \end{aligned}$$

for $x \in G$ and $0<r \le d_G$, where $1 \le p_0 \le \min (p,\, q)$. Then, given $L \ge 1$, there exist constants $C_1 = C(L) \ge 2$ and $C_2>0$ such that

$$\begin{aligned} \Phi \bigl ( x, I(f;x,r)/C_1 \bigr )^{1/p_0} \le C_2 J(f;x,r) \end{aligned}$$

for all $x \in G$, $0<r \le d_G$ and for all nonnegative measurable functions f on G such that $f(y) \ge 1$ or $f(y)=0$ for each $y \in G$ and

$$\begin{aligned} \sup _{z\in G}\left( \int _0^{2d_G} \frac{\omega (z,t)}{|B(z,t)|} \left( \int _{G\cap B(z, t)}\Phi \left( y,f(y)\right) \,dy \right) \, \frac{dt}{t} \right) \le L. \end{aligned}$$

(3.1)

Proof

Given f as in the statement of the lemma, $x \in G$ and $0<r <d_G$, set $I = I(f;x,r)$ and $J = J(f;x,r)$. Taking f, note that (3.1) implies

$$\begin{aligned}&\frac{\omega (x,r)}{|B(x,r)|} \int _{G\cap B(x,r)}\Phi \left( y,f(y)\right) \,dy \\&\le C_0\int _{r}^{2r} \frac{\omega (x,t)}{|B(x,t)|} \left( \int _{G\cap B(x, t)}\Phi \left( y,f(y)\right) \,dy \right) \, \frac{dt}{t}\le C_0L, \end{aligned}$$

so that

$$\begin{aligned} J \le C_0^{1/p_0}\omega (x,r)^{-1/p_0}L^{1/p_0}. \end{aligned}$$

(3.2)

We treat only the case $J >1$. Since $\Phi (x,t)^{1/p_0} \rightarrow \infty$ as $t \rightarrow \infty$ by $(\Phi 3;\infty ;q)$ and $p_0\le q$, there exists $K > 1$ such that

$$\begin{aligned} \Phi (x,K)^{1/p_0} = \Phi (x,1)^{1/p_0} J. \end{aligned}$$

(3.3)

With this K, we have by $(\Phi 3;\infty ;q)$ and $p_0\le q$

$$\begin{aligned} \int _{G\cap B(x,r)} f(y)\, dy&\le K |B(x,r)| + A_{2,\infty ,p_0}^{1/p_0} K \int _{G\cap B(x,r)} \frac{\Phi \bigl ( y, f(y) \bigr )^{1/p_0} }{\Phi (y,K)^{1/p_0} }\, dy. \end{aligned}$$

Since $K >1$, by $(\Phi 3;\infty ;q)$, we have

$$\begin{aligned} \Phi (x,1)^{1/p_0}J = \Phi (x,K)^{1/p_0} \ge A_{2,\infty ,q}^{-1/p_0}K^{q/p_0}\Phi (x,1)^{1/p_0}, \end{aligned}$$

so that, in view of (3.2) and $(\omega 3;\omega _0)$,

$$\begin{aligned} K^{q} \le A_{2,\infty ,q}J^{p_0} \le C_0A_{2,\infty ,q} \omega (x,r)^{-1}L \le C_0A_{2,\infty ,q}{\tilde{c}}_3 L r^{-\omega _0} \end{aligned}$$

or $r \le \gamma K^{-q/\omega _0}$ with $\gamma = (C_0A_{2,\infty ,q}{\tilde{c}}_3L)^{1/\omega _0}$. Thus, if $|x-y| \le r$, then

$$\begin{aligned} |x-y| \le \gamma K^{-q/\omega _0} \le \gamma K^{-\nu } \end{aligned}$$

since $\nu \le q/\omega _0$. Hence, by ($\Phi 5; \nu$) with $B_{\gamma , \nu }^{1/p_0}=\beta$

$$\begin{aligned} \begin{aligned} \int _{G\cap B(x,r)} f(y)\, dy&\le K|B(x,r)| \left\{ 1 + \left( A_1A_{2,\infty ,p_0}\right) ^{1/p_0} \beta \right\} \end{aligned} \end{aligned}$$

as in the proof of [20, Lemma 3.3]. See [21, Lemma 9] and [20, Lemma 3.3] for details. $\square$

In view of Lemmas 3.1 and 3.3, we show the boundedness of M on ${\mathcal {L}}^{\Phi ,\omega }(G)$ as an extension of [32, Theorem 3.5].

Theorem 3.4

Suppose $\Phi (x,t)$ satisfies $(\Phi 3;0;p)$, $(\Phi 3;\infty ;q)$ and $(\Phi 5;\nu )$ for $p > 1$, $q > 1$ and $\nu >0$ satisfying $\nu \le q/\omega _0$. Assume that $(\omega 1')$ holds. Then there is a constant $C >0$ such that

$$\begin{aligned} \Vert Mf \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \le C \Vert f \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \end{aligned}$$

for all $f \in {\mathcal {L}}^{\Phi ,\omega }(G)$.

Proof

Set $p_0 = \min (p,\, q)$. Then $p_0 >1$. Consider the function

$$\begin{aligned} \Phi _0(x,t) = \Phi (x,t)^{1/p_0}. \end{aligned}$$

Let f be a nonnegative measurable function on G with $\Vert f \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \le 1/2$. Let $f_1 = f \chi _{\{ x\in G:f(x) \ge 1 \}}$, $f_2 = f - f_1$. Applying Lemma 3.3 to $f_1$ and $L = 1$, there exist constants $C_1\ge 2$ and $C_2>0$ such that

$$\begin{aligned} \Phi _0 \bigl ( x, Mf_1(x)/C_1 \bigr ) \le C_2 M [\Phi _0 \bigl ( \cdot , f_1( \cdot ) \bigr )] (x), \end{aligned}$$

so that

$$\begin{aligned} \Phi \bigl ( x, Mf_1(x)/C_1 \bigr ) \le C_2^{p_0} \left[ M [\Phi _0 \left( \cdot , f( \cdot ) \right) ](x) \right] ^{p_0} \end{aligned}$$

(3.4)

for all $x \in G$.

On the other hand, since $Mf_2\le 1$, we have by $(\Phi 2)$ and $(\Phi 3)$

$$\begin{aligned} \Phi \bigl ( x, Mf_2(x)/C_1 \bigr ) \le A_1A_2 \end{aligned}$$

(3.5)

for all $x \in G$.

By (2.1), (3.4), (3.5) and Lemma 3.1, we obtain

$$\begin{aligned}&\int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)}{\overline{\Phi }} \left( x,Mf(x)/(2C_1)\right) \,dx \right) \, \frac{dr}{r} \\&\quad \le \frac{A_2}{2}\biggl \{\int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)}\Phi \left( x,Mf_1(x)/C_1\right) \,dx \right) \, \frac{dr}{r} \\&\qquad +\int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)}\Phi \left( x,Mf_2(x)/C_1\right) \,dx \right) \, \frac{dr}{r} \biggl \} \\&\quad \le C\left\{ \int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)} \left[ M [\Phi _0 \left( \cdot , f( \cdot ) \right) ](x) \right] ^{p_0}\,dx \right) \, \frac{dr}{r} +\int _0^{2d_G} \omega (z,r)\, \frac{dr}{r} \right\} \\&\quad \le C \end{aligned}$$

for all $z\in G$ since there exists a constant $C_3>0$ such that

$$\begin{aligned} \int _0^{2d_G} \omega (z,r)\, \frac{dr}{r} =\int _0^{2d_G} r^{-\varepsilon _1}\omega (z,r)\cdot r^{\varepsilon _1}\, \frac{dr}{r} \le C\int _0^{2d_G} r^{\varepsilon _1}\, \frac{dr}{r} \le C_3 \end{aligned}$$

(3.6)

for all $z\in G$ by ($\omega 1'$) and $(\omega 3;\omega _0)$. Thus, this theorem is proved. $\square$

4 Sobolev-Type Inequality

We recall the following lemma from [19].

Lemma 4.1

([19, Lemma 5.1]) Let F(x, t) be a positive function on $G \times (0, \infty )$ satisfying the following conditions:

(F1):

$F(x,\, \cdot \, )$ is continuous on $(0, \infty )$ for each $x \in G$;

(F2):

there exists a constant $K_1 \ge 1$ such that

$$\begin{aligned} K_1^{-1} \le F(x,1) \le K_1 \quad \text {for all} \ x \in G; \end{aligned}$$

(F3):

$t \mapsto t^{-\varepsilon '}F(x,t)$ is uniformly almost increasing for some $\varepsilon ' >0$; namely there exists a constant $K_2 \ge 1$ such that

$$\begin{aligned} t_1^{-\varepsilon '}F(x,t_1) \le K_2 t_2^{-\varepsilon '}F(x,t_2) \quad \text {for all} \ x \in G \ \text {whenever} \ 0< t_1 < t_2. \end{aligned}$$

Set

$$\begin{aligned} F^{-1}(x,s) = \sup \{t >0\, ;\, F(x,t) < s \} \end{aligned}$$

for $x \in G$ and $s >0$. Then:

(1):

$F^{-1}(x, \cdot )$ is nondecreasing.

(2):

$$\begin{aligned} F^{-1}(x,\lambda t) \le (K_2 \lambda )^{1/\varepsilon '} F^{-1}(x,t) \end{aligned}$$

(4.1)

for all $x \in G$, $t >0$ and $\lambda \ge 1$.

(3):

$$\begin{aligned} F(x,F^{-1}(x,t)) = t \end{aligned}$$

for all $x \in G$ and $t >0$.

(4):

$$\begin{aligned} K_2^{-1/\varepsilon '} t \le F^{-1}(x,F(x,t)) \le K_2^{2/\varepsilon '} t \end{aligned}$$

for all $x \in G$ and $t >0$.

(5):

$$\begin{aligned} \min \left\{ 1, \left( \frac{s}{K_1 K_2} \right) ^{1/\varepsilon '} \right\} \le F^{-1}(x,s) \le \max \{1, (K_1 K_2 s)^{1/\varepsilon '} \} \end{aligned}$$

for all $x \in G$ and $s>0$.

Remark 4.2

Note that $F(x,t)=\Phi (x,t)$ is a function satisfying (F1), (F2) and (F3) with $K_1=A_1, K_2=A_2$ and $\varepsilon ' =1$.

We consider the following condition:

$(\Phi \omega \alpha )$:

there exist constants $\varepsilon _2 >0$ and $A_4\ge 1$ such that

$$\begin{aligned} r_2^{\varepsilon _2+\alpha (x)} \Phi ^{-1}(x,\omega (x,r_2)^{-1}) \le A_4 r_1 ^{\varepsilon _2+\alpha (x)} \Phi ^{-1}(x,\omega (x,r_1 )^{-1}) \end{aligned}$$

for all $x \in G$ whenever $0< r_1< r_2 < d_G$.

Lemma 4.3

Suppose $\Phi (x,t)$ satisfies $(\Phi 3;\infty ;q)$ and $(\Phi 5;\nu )$ for $q \ge 1$ and $\nu >0$ satisfying $\nu \le q/\omega _0$. Assume that $(\Phi \omega \alpha )$ holds. Then there exists a constant $C >0$ such that

$$\begin{aligned} \int _{G\setminus B(x,\delta )} |x-y|^{\alpha (x)-N}f(y)\, dy \le C \delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1}) \end{aligned}$$

for all $x\in G$, $0<\delta <d_{G}/2$ and nonnegative $f \in {\mathcal {L}}^{\Phi ,\omega }(G)$ with $\Vert f\Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)}\le 1$.

Proof

Let f be a nonnegative measurable function with $\Vert f\Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)}\le 1/2$. Let $x\in G$ and $0<\delta <d_{G}/2$. By $(\Phi 3)$ and $(\Phi 3;\infty ;q)$,

$$\begin{aligned} \min \{1, (A_1 A_2)^{-1}s\} \le F^{-1}(x,s) \le \max \{1, (A_1 A_{2,\infty ,q} s)^{1/q}\}; \end{aligned}$$

cf. Lemma 4.1 (5). Set

$$\begin{aligned} c_1= \max \left\{ A_1A_2 {\tilde{c}}_3,\, (A_1A_{2,\infty ,q}{\tilde{c}}_3)^{-1}d_G^{\omega _0} \right\} . \end{aligned}$$

Then we have by ($\omega 3; \omega _0$), Lemma 4.1 and the condition $\nu \le q/\omega _0$

$$\begin{aligned} \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \ge \min \{1, (A_1 A_2)^{-1} c_1{\tilde{c}}_3^{-1}\} \ge 1 \end{aligned}$$

and

$$\begin{aligned} \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right)&\le \max \{1, (A_1 A_{2,\infty ,q} c_1{\tilde{c}}_3 |x-y|^{-\omega _0} )^{1/q} \} \\&= (A_1 A_{2,\infty ,q} c_1{\tilde{c}}_3 d_G^{-\omega _0} )^{1/q} (|x-y|/d_G)^{-\omega _0/q} \\&\le (A_1 A_{2,\infty ,q} c_1{\tilde{c}}_3 d_G^{-\omega _0} )^{1/q} (|x-y|/d_G)^{-1/\nu } \end{aligned}$$

for all $x, y\in G$. Hence,

$$\begin{aligned} |x-y| \le c_2 \left\{ \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \right\} ^{-\nu } \end{aligned}$$

for all $x, y\in G$, where $c_2=d_G (A_1A_{2,\infty ,q} c_1 {\tilde{c}}_3 d_G^{-\omega _0})^{\nu /q}$. We find by $(\Phi 3)$, $(\Phi 5;\nu )$ and Lemma 4.1 (3)

$$\begin{aligned}&\int _{G\setminus B(x,\delta )} |x-y|^{\alpha (x)-N}f(y)\, dy \\&\quad \le \int _{G \setminus B(x,\delta )} |x-y|^{\alpha (x)-N} \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \, dy \\&\quad +A_2\int _{G \setminus B(x,\delta )} |x-y|^{\alpha (x)-N}f(y) \\&\quad {}\times \frac{f(y)^{-1}\Phi (y,f(y))}{\left\{ \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \right\} ^{-1} \Phi \left( y, \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \right) }\, dy\\&\quad \le \int _{G \setminus B(x,\delta )} |x-y|^{\alpha (x)-N} \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \, dy \\&\quad +C\int _{G \setminus B(x,\delta )} |x-y|^{\alpha (x)-N}\omega (x,|x-y|) \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \Phi (y,f(y))\, dy\\&\quad = I_1+ C I_2. \end{aligned}$$

Let $j_0$ be the smallest integer such that $2^{j_0}\delta \ge d_G$. By $(\omega 1)$, $(\omega 2)$, (4.1) and $(\Phi \omega \alpha )$, we obtain

$$\begin{aligned} I_1&= \sum _{j=1}^{j_0}\int _{G\cap (B(x,2^{j}\delta )\setminus B(x,2^{j-1}\delta ))} |x-y|^{\alpha (x)-N} \Phi ^{-1}\left( x,c_1\omega (x,|x-y|)^{-1}\right) \, dy \\&\le C\sum _{j=1}^{j_0}(2^{j}\delta )^{\alpha (x)} \Phi ^{-1}\left( x,\omega (x,2^{j}\delta )^{-1}\right) \\&\le C \delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1}) \end{aligned}$$

as in the proof of [29, Lemma 4.2].

For $I_2$, it follows from $(\Phi \omega \alpha )$, (4.1), $(\omega 1)$ and $(\omega 2)$ that

$$\begin{aligned} I_2&\le C \delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1}) \int _{G \setminus B(x,\delta )} \frac{\omega (x,|x-y|)}{|B(x,|x-y|)|}\Phi (y,f(y))\, dy \\&\le C\delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1})\sum _{j=1}^{j_0} \frac{\omega (x,2^j\delta )}{|B(x,2^j\delta )|} \int _{G\cap B(x,2^j\delta )} \Phi (y,f(y))\, dy \\&\le C\delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1}) \end{aligned}$$

as in the proof of [29, Lemma 4.2]. Thus, the present lemma is proved. $\square$

To state our main theorem, we consider a function

$$\begin{aligned} \Psi (x,t) : G \times [0, \infty ) \rightarrow [0, \infty ) \end{aligned}$$

that satisfies $(\Phi 1)$ – $(\Phi 3)$ and

$(\Psi \Phi )$:

there exists a constant $A' \ge 1$ such that

$$\begin{aligned} \Psi \left( x,t \left( \omega ^{-1} \left( x, \Phi (x, t)^{-1} \right) \right) ^{\alpha (x)} \right) \le A'\Phi (x,t) \end{aligned}$$

for all $x\in G$ and $t\ge 1$.

Remark 4.4

In [26], we considered the condition like $(\Psi \Phi )$ for Musielak–Orlicz spaces.

We give a Sobolev-type inequality for $I_{\alpha (\cdot )}f$ of functions in ${\mathcal {L}}^{\Phi ,\omega }(G)$ by Theorem 3.4, as an extension of [32, Theorem 4.4].

Theorem 4.5

Suppose $\Phi (x,t)$ satisfies $(\Phi 3;0;p)$, $(\Phi 3;\infty ;q)$ and $(\Phi 5;\nu )$ for $p > 1$, $q > 1$ and $\nu >0$ satisfying $\nu \le q/\omega _0$. Assume that $(\omega 1')$ and $(\Phi \omega \alpha )$ hold. Then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert I_{\alpha (\cdot )}f\Vert _{{\mathcal {L}}^{\Psi ,\omega }(G)} \le C\Vert f\Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \end{aligned}$$

for all $f\in {\mathcal {L}}^{\Phi ,\omega }(G)$.

Proof

Let f be a nonnegative measurable function on G such that $\Vert f\Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)}\le 1$. We may assume that

$$\begin{aligned} \sup _{z\in G}\int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)}\Phi \left( x,Mf(x)\right) \,dx \right) \, \frac{dr}{r}\le 1 \end{aligned}$$

(4.2)

by Theorem 3.4. Let $x\in G$ and $0<\delta <d_G/2$. By Lemma 4.3, we find

$$\begin{aligned} I_{\alpha (\cdot )}f(x)&=\int _{G\cap B(x,\delta )} |x-y|^{\alpha (x)-N}f(y)\, dy +\int _{G\setminus B(x,\delta )} |x-y|^{\alpha (x)-N}f(y)\, dy \\&\le C\left\{ \delta ^{\alpha (x)}Mf(x)+\delta ^{\alpha (x)} \Phi ^{-1}(x,\omega (x,\delta )^{-1}) \right\} . \end{aligned}$$

If $\omega ^{-1}\left( x, \Phi (x,Mf(x))^{-1}\right) \ge d_G/2$, then, taking $\delta =d_G/2$, we have $I_{\alpha (\cdot )}f(x)\le C$ by Lemma 4.1, $(\omega 1)$ and $(\omega 3;\omega _0)$. If $\omega ^{-1}\left( x, \Phi (x,Mf(x))^{-1}\right) <d_G/2$, then take $\delta =\omega ^{-1}\left( x, \Phi (x,Mf(x))^{-1}\right)$. Then we have

$$\begin{aligned} I_{\alpha (\cdot )}f(x)\le CMf(x) \left( \omega ^{-1}\left( x, \Phi (x, Mf(x))^{-1} \right) \right) ^{\alpha (x)} \end{aligned}$$

by Lemma 4.1. Therefore, we obtain

$$\begin{aligned} I_{\alpha (\cdot )}f(x) \le C_1' \max \left\{ Mf(x) \left( \omega ^{-1}\left( x, \Phi (x, Mf(x))^{-1} \right) \right) ^{\alpha (x)}, 1 \right\} , \end{aligned}$$

so that by ($\Psi \Phi$), we have

$$\begin{aligned} \Psi \left( x, I_{\alpha (\cdot )}f(x)/C_1' \right)&\le C \left\{ \Psi \left( x, Mf(x) \left( \omega ^{-1}\left( x, \Phi (x, Mf(x))^{-1} \right) \right) ^{\alpha (x)} \right) + 1 \right\} \\&\le C \left\{ \Phi \bigl ( x, Mf(x) \bigr ) + 1 \right\} . \end{aligned}$$

Hence, it follows from (4.2) and (3.6) that

$$\begin{aligned}&\int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)} \Psi \left( x, I_{\alpha (\cdot )}f(x)/C_1' \right) \,dx \right) \, \frac{dr}{r} \\&\quad \le C\left\{ \int _0^{2d_G} \frac{\omega (z,r)}{|B(z,r)|} \left( \int _{G\cap B(z, r)}\Phi \left( x,Mf(x)\right) \,dx \right) \, \frac{dr}{r} +\int _0^{2d_G} \omega (z,r)\, \frac{dr}{r} \right\} \\&\quad \le C \end{aligned}$$

for all $z\in G$. Thus, we complete the proof. $\square$

Remark 4.6

When $\Phi (x,t)=t^{p(x)}$, Theorem 4.5 was proved in [32, Theorem 4.4].

Remark 4.7

Let $\Phi _{p(\cdot ),\{q_j(\cdot )\}}(x,t) = t^{p(x)} \prod _{j=1}^k \bigl ( L_{e}^{(j)}(t) \bigr )^{q_j(x)}$ and $\omega (x,r) = r^{\sigma (x)} L_{e}(1/r)^{\beta (x)}$.

Set

$$\begin{aligned} \Psi (x,t)= \bigl [ \Phi _{p(\cdot ),\{q_j(\cdot )\}}(x,t) \bigr ]^{p^*(x)/p(x)} L_e(t)^{p^*(x)\alpha (x) \beta (x)/\sigma (x)} , \end{aligned}$$

where $1/p^*(x)=1/p(x)-\alpha (x)/\sigma (x)$. Then $\Psi (x,t)$ satisfies condition ($\Psi \Phi$) (see [31, Remark 3.14]).

5 Double Phase Functions with Variable Exponents

In this section, let

$$\begin{aligned} \omega (x,r) = r^{\sigma (x)} L_{e}(1/r)^{\beta (x)} \end{aligned}$$

be as in Example 2.1 (Remark 3.2) and let $p(\cdot )$ and $q(\cdot )$ be real valued measurable functions on G such that

(P1):

$1 \le p^- \le p^+ < \infty ,$

(Q1):

$1 \le q^- \le q^+ < \infty .$

We assume that

(P2):

$p(\cdot )$ is log-Hölder continuous, that is,

$$\begin{aligned} |p(x) - p(y)| \le \frac{C_{p}}{L_{e}(1/|x-y|)} \quad (x,\, y \in G) \end{aligned}$$

with a constant $C_{p} \ge 0$, and

(Q2):

$q(\cdot )$ is log-Hölder continuous, that is,

$$\begin{aligned} |q(x) - q(y)| \le \frac{C_{q}}{L_{e}(1/|x-y|)} \quad (x,\, y \in G) \end{aligned}$$

with a constant $C_{q} \ge 0$.

As an example and application, we consider the case where $\Phi (x,t)$ is a double phase function with variable exponents given by

$$\begin{aligned} \Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \ x \in G, \ t \ge 0 ,\end{aligned}$$

where $p(x)<q(x)$ for $x\in G$, $a(\cdot )$ is nonnegative, bounded and Hölder continuous of order $\theta \in (0,1]$ (cf. [20, 33]).

This $\Phi (x,t)$ satisfies $(\Phi 1)$, $(\Phi 2)$, $(\Phi 3;0;p^-)$ and $(\Phi 3;\infty ;p^-)$. $\Phi (x,t)$ also satisfies $(\Phi 5; \nu )$ for $\nu \ge \sup _{x\in G_0}(q(x)-p(x))/\theta$; see [20, Lemma 5.1].

Let $G_0=\{ x \in G: a(x)>0 \}$.

In view of Theorem 3.4, we have the boundedness of the maximal operator on ${\mathcal {L}}^{\Phi ,\omega }(G)$ in the framework of double phase functions $\Phi$.

Theorem 5.1

If $p^->1$ and $\sup _{x\in G_0}(q(x)-p(x))/\theta \le p^-/\omega _0$, then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert Mf \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \le C \Vert f \Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \end{aligned}$$

for all $f \in {\mathcal {L}}^{\Phi ,\omega }(G)$.

Let $p^*(x)$ and $q^*(x)$ be defined by

$$\begin{aligned} \frac{1}{p^*(x)} = \frac{1}{p(x)} - \frac{\alpha (x)}{\sigma (x)} \end{aligned}$$

when $1/p(x) - \alpha (x)/\sigma (x)>0$, and

$$\begin{aligned} \frac{1}{q^*(x)} = \frac{1}{q(x)} - \frac{\alpha (x)}{\sigma (x)} \end{aligned}$$

when $1/q(x) - \alpha (x)/\sigma (x)>0$. In this section, set

$$\begin{aligned} \Psi (x,t)&= t^{p^*(x)} L_{e}(t)^{\alpha (x) p^*(x)\beta (x)/\sigma (x)}\\&\quad + \left( a(x)^{1/q(x)} t\right) ^{q^*(x)} L_{e}\left( a(x)^{1/q(x)}t\right) ^{\alpha (x) q^*(x)\beta (x)/\sigma (x)} \end{aligned}$$

for $x \in G$ and $t \ge 0$.

Lemma 5.2

([20, Lemma 5.6 (1), (3)])

(1):

If $\inf _{x\in G_0}(\sigma (x)/q(x) - \alpha (x))>0$ and $\inf _{x\in G \setminus G_0}(\sigma (x)/p(x) - \alpha (x))>0$, then $(\Phi \omega \alpha )$ holds.

(2):

$\Psi (x,t)$ satisfies $(\Psi \Phi )$.

Finally, by Lemma 5.2 and Theorem 4.5, we obtain a Sobolev inequality in our setting.

Theorem 5.3

If $p^->1$, $\inf _{x\in G_0}(\sigma (x)/q(x) - \alpha (x))>0$, $\inf _{x\in G \setminus G_0}(\sigma (x)/p(x) - \alpha (x))>0$ and $\sup _{x\in G_0}(q(x)-p(x))/\theta \le p^-/\omega _0$, then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert I_{\alpha (\cdot )}f\Vert _{{\mathcal {L}}^{\Psi ,\omega }(G)} \le C\Vert f\Vert _{{\mathcal {L}}^{\Phi ,\omega }(G)} \end{aligned}$$

for all $f\in {\mathcal {L}}^{\Phi ,\omega }(G)$.