2 Musielak–Orlicz spaces

We consider a function

$$ \begin{align*}\Phi(x,t) : X \times [0, \infty) \to [0, \infty)\end{align*} $$

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

1. (Φ1) $\Phi (\cdot , t)$ is measurable on X for each $t \ge 0$ and $\Phi (x,\cdot )$ is continuous on $[0, \infty )$ for each $x \in X$ ;

2. (Φ2) there exists a constant $A_1 \ge 1$ such that

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

3. (Φ3) $t \mapsto \Phi (x,t)/t$ is uniformly almost increasing on $(0, \infty )$ , namely there exists a constant $A_2 \ge 1$ such that

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

( $\Phi 2$ ) and ( $\Phi 3$ ) yield

(2.1) $$ \begin{align} \Phi (x,t) \le A_1A_2\, t \quad \text{for} \ 0 \le t \le 1 \quad \text{and} \quad \Phi (x,t) \ge (A_1A_2)^{-1}\, t \quad \text{for} \ t \ge 1. \end{align} $$

Set $\bar {\phi }(x,t) = \sup _{0 < s \le t} \{\Phi (x,s)/s\}$ and $\overline {\Phi }(x,t) = \int _0^t \bar {\phi }(x,r)\, dr$ for $x \in X$ and $t \ge 0$ . Then $\overline {\Phi }(x, \cdot )$ is convex and

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

for all $x \in X$ and $t \ge 0$ . Note that ( $\Phi 3$ ) is used to check the second inequality.

Recall that f is a locally integrable function on X if f is an integrable function on all balls B in X. Given $\Phi (x,t)$ satisfying ( $\Phi $ 1)–( $\Phi $ 3), the Musielak–Orlicz space $L^{\Phi }(X)$ , also called the generalized Orlicz spaces [Reference Harjulehto and Hästö5], is defined as

$$ \begin{align*}L^{\Phi}(X) = \left\{ f \in L^1_{loc}(X)\,; \, \int_{X} \Phi \big(y, |f(y)| \big)\, d\mu(y) < \infty \right\}.\end{align*} $$

The space $L^{\Phi }(X)$ is a Banach space with respect to the norm

$$ \begin{align*}\| f \|_{L^{\Phi}(X)} = \inf \left\{ \lambda> 0\,; \, \int_{X} \overline{\Phi} \big( y, |f(y)|/\lambda \big)\, d\mu(y) \le 1 \right\}.\end{align*} $$

Further, consider the following conditions for $\Phi (x,t)$ : Let $p \ge 1$ be given.

1. (Φ3; p) $t \mapsto t^{-p}\Phi (x,t)$ is uniformly almost increasing on $(0,\infty )$ , namely there exists a constant $A_{2,p} \ge 1$ such that

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

2. (Φ5; Q) for every $\gamma>0$ , there exists a constant $B_{\gamma , Q} \ge 1$ such that

   $$ \begin{align*}\Phi(x,t) \le \Phi(y,B_{\gamma, Q}t)\end{align*} $$
   whenever $x\in X$ , $0<r\le d_X$ , $y\in B(x,r)$ , $\Phi (x,t) \le \gamma r^{-Q(x)}$ and $t \ge 1$ .

Here we will remark on $(\Phi 5;Q)$ .

Remark 2.1 In [Reference Ohno and Shimomura18], we considered the following: Let $q \ge 1$ and $\nu>0$ be given.

1. (Φ3; ∞; 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

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

2. (Φ5; ν) for every $\gamma>0$ , there exists a constant $\tilde {B}_{\gamma , \nu } \ge 1$ such that

   $$ \begin{align*}\Phi(x, t) \le \tilde{B}_{\gamma, \nu} \Phi(y,t)\end{align*} $$
   whenever $x, y \in X$ , $d(x,y) \le \gamma t^{-\nu }$ and $t \ge 1$ .

If $\Phi (x,t)$ satisfies $(\Phi 5;\nu )$ with $\nu>0$ satisfying $\nu \le q/Q^+$ , then it satisfies $(\Phi 5;Q)$ .

We will show this. For $\gamma>0$ , let $x\in X$ , $0<r\le d_X$ , $y \in B(x,r)$ , $\Phi (x,t) \le \gamma r^{-Q(x)}$ and $t \ge 1$ . By $(\Phi 3;\infty ;q)$ , ( $\Phi $ 2) and (1.1)

$$ \begin{align*}1 \le t\le (A_1 A_{2,\infty,q}\Phi(x,t))^{1/q} \le (A_1 A_{2,\infty,q}\gamma r^{-Q(x)})^{1/q} \le C r^{-Q^+/q}. \end{align*} $$

Since $\nu \le q/Q^+$ , $1 \le t\le Cr^{-1/\nu } \le C d(x,y)^{-1/\nu }$ , so that $ d(x,y) \le \gamma ' t^{-\nu } $ and $t \ge 1$ . Hence, by $(\Phi 5;\nu )$ , there exists a constant $B_{\gamma ',\nu } \ge 1$ such that

(2.3) $$ \begin{align} \Phi(x,t) \le B_{\gamma',\nu} \Phi(y,t). \end{align} $$

Taking $B_{\gamma } \ge A_2 B_{\gamma ',\nu } (\ge 1)$ , it follows from (2.3) and $(\Phi 3)$ that

$$\begin{align*}\Phi(x,t) \le A_2 B_{\gamma',\nu} B_{\gamma}^{-1} \Phi(y,B_{\gamma} t) \le \Phi(y,B_{\gamma} t). \end{align*}$$

Thus $\Phi (x,t)$ satisfies $(\Phi 5;Q)$ .

We define $L_{e}(t) = \log (e + t)$ , $L_{c}^{(1)}(t) = L_c(t)$ and $L_{c}^{(j+1)}(t) = L_c( L_c^{(j)}(t))$ . As in [Reference Ohno and Shimomura18], typical examples to apply our theorems are for example

$$ \begin{align*}\Phi_{1}(x,t) = t^{p(x)} \prod_{j=1}^k \big( L_{e}^{(j)}(t) \big)^{q_j(x)} \ \text{and} \ \Phi_2(x,t) = t^{p(x)} + a(x) t^{q(x)} \end{align*} $$

under suitable $p(\cdot ), q(\cdot ), q_j(\cdot )$ and $a(\cdot )$ . For $\Phi _2$ , see [Reference Colombo and Mingione3, Reference Maeda, Mizuta, Ohno and Shimomura11, Reference Ragusa and Tachikawa19].

3 The maximal operator

The following lemma is from [Reference Maeda, Mizuta, Ohno and Shimomura10, Lemma 5.1].

Notice that $F(x,t)=\Phi (x,t)$ satisfies (F1)–(F3) with $K_1=A_1, K_2=A_2, \hat {t}=1$ and $\varepsilon =1$ .

Given a locally integrable function f on X and $\lambda \ge 1$ , the Hardy–Littlewood maximal function $M_{\lambda }f$ is defined by

$$\begin{align*}M_{\lambda}f(x)=\sup_{r>0}\frac{1}{\mu(B(x,\lambda r))}\int_{B(x,r)} |f(y)|\, d\mu(y). \end{align*}$$

Given $\lambda \ge 1$ , we say that X satisfies ( $M \lambda $ ) if there exists a constant $C>0$ such that

(3.1) $$ \begin{align} \mu(\{x\in X : M_{\lambda}f(x)>k\}) \le \frac{C}{k}\int_{X}|f(y)|\, d\mu(y) \end{align} $$

for all measurable functions $f\in L^{1}(X)$ and $k>0$ .

The following lemma is known (see [Reference Sawano and Shimomura22, Lemma 2.1]).

Lemma 3.3 Let $1<p_0<\infty $ and let $\lambda \ge 1$ . Suppose that X satisfies ( $M \lambda $ ). Then there exists a constant $C>0$ such that

$$\begin{align*}\int_{X} \{M_{\lambda}f(x)\}^{p_0}\, d\mu(x) \le C \end{align*}$$

for all measurable functions f on X with $\| f\|_{L^{p_0}(X)} \le 1$ .

For $p\ge 1$ and $\lambda \ge 1$ , set

$$ \begin{align*}I(f;x,r,\lambda) = \frac{1}{\mu(B(x,\lambda r))} \int_{B(x,r)} f(y)\, d\mu(y)\end{align*} $$

and

$$ \begin{align*}J(f;x,r,p, \lambda) = \frac{1}{\mu(B(x,\lambda r))} \int_{B(x,r)} \Phi \left( y, f(y) \right)^{1/p}\, d\mu(y).\end{align*} $$

We show the following lemma instead of [Reference Ohno and Shimomura16, Lemma 3.3].

Lemma 3.5 Let $\lambda \ge 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p \ge 1$ . Then, given $L \ge 1$ , there exist constants $C_1 = C(L) \ge 2$ and $C_2>0$ such that

$$\begin{align*}\Phi \left( x, I(f;x,r,\lambda)/C_1 \right)^{1/p} \le C_2 J(f;x,r,p,\lambda) \end{align*}$$

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

(3.2) $$ \begin{align} \int_{X} \Phi (y,f(y))\, d\mu(y) \le L. \end{align} $$

Proof Let f be as in the statement of this lemma and let $x \in X$ and $0<r\le d_X$ . For simplicity, we set $I = I(f;x,r,\lambda )$ and $J = J(f;x,r,p,\lambda )$ . Hölder’s inequality gives

(3.3) $$ \begin{align} J \le \mu (B(x,\lambda r))^{-1/p}L^{1/p}\le A^{-1/p}L^{1/p}r^{-Q(x)/p} \end{align} $$

by (3.2) and (1.2).

Since $\Phi \left ( y, f(y) \right )^{1/p} \ge (A_1 A_{2,p})^{-1/p} f(y)$ for all $y \in X$ by $(\Phi 3;p)$ , $I \le (A_1 A_{2,p})^{1/p} J$ . Hence, if $J \le 1$ , then in view of $(\Phi 3;p)$ and $(\Phi 2)$

$$ \begin{align*}\Phi (x, I/C_1)^{1/p} \le J \left(A_{2,p}\Phi (x, 1)\right)^{1/p} \le \left(A_1A_{2,p}\right)^{1/p}J\end{align*} $$

whenever $C_1 \ge (A_1 A_{2,p})^{1/p}$ .

Let us consider the case $J>1$ . Noting from $(\Phi 3;p)$ that $\Phi (x,t)^{1/p} \to \infty $ as $t \to \infty $ , we see that there exists $K> 1$ such that

(3.4) $$ \begin{align} \Phi (x, K)^{1/p} = \Phi (x,1)^{1/p} J. \end{align} $$

Using this K, it follows from $(\Phi 3;p)$ that

$$ \begin{align*} \int_{B(x,r)} f(y)\, d\mu(y) &= \int_{B(x,r) \cap \{y\in X: f(y)\le \beta K\}} f(y)\, d\mu(y)\\& \quad +\int_{B(x,r) \cap \{y\in X : f(y)>\beta K\}} f(y)\, d\mu(y) \\&\le \beta K \mu(B(x,r)) + A_{2,p}^{1/p}\beta K \int_{B(x,r)} \frac{\Phi \left( y, f(y) \right)^{1/p} }{\Phi (y,\beta K)^{1/p} }\, d\mu(y). \end{align*} $$

Let $\gamma = A_1A^{-1} L$ . Owing to ( $\Phi 5;Q$ ), there is $\beta =\beta (\gamma ) \ge 1$ , independent of f, x, r, such that

$$ \begin{align*}\Phi(x,K)^{1/p} \le \Phi(y,\beta K)^{1/p} \end{align*} $$

for $y \in B(x,r)$ , since $K>1$ and

$$ \begin{align*} \Phi (x, K) \le A_1 J^p \le A_1A^{-1} L r^{-Q(x)}=\gamma r^{-Q(x)} \end{align*} $$

by (3.4), $(\Phi 2)$ , and (3.3). Therefore

$$\begin{align*}\begin{aligned} \int_{B(x,r)} f(y)\, d\mu(y) & \le \beta K \mu(B(x,r)) + A_{2,p}^{1/p}\beta K \mu(B(x,\lambda r)) \frac{J}{\Phi (x, K)^{1/p}}. \end{aligned} \end{align*}$$

Since $J \Phi (x, K)^{-1/p} = \Phi (x,1)^{-1/p} \le A_1^{1/p}$ by (3.4) and $(\Phi 2)$ , we obtain

$$\begin{align*}\begin{aligned} \int_{B(x,r)} f(y)\, d\mu(y) &\le \beta K\mu(B(x,\lambda r)) \left\{ 1 + \left(A_1A_{2,p}\right)^{1/p} \right\}. \end{aligned} \end{align*}$$

The rest is proved in line with the proof of [Reference Ohno and Shimomura16, Lemma 3.3].

Using Lemma 3.5, as in the proof of [Reference Ohno and Shimomura16, Theorem 3.2], we can prove the boundedness of $M_{\lambda }$ on $L^{\Phi }(X)$ , which is an improvement of [Reference Ohno and Shimomura18, Lemma 3.1] ([Reference Ohno and Shimomura16, Theorem 3.2]).

Theorem 3.6 Let $\lambda \ge 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p>1$ . Suppose that X satisfies $(M \lambda )$ . Then there is a constant $C>0$ such that

$$\begin{align*}\| M_{\lambda}f \|_{L^{\Phi}(X)} \le C \| f \|_{L^{\Phi}(X)} \end{align*}$$

for all $f \in L^{\Phi }(X)$ .

Remark 3.7 In [Reference Ohno and Shimomura16, Theorem 3.2], we showed the following boundedness of $M_{\lambda }$ on $L^{\Phi }(X)$ : Let $\lambda \ge 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;\nu )$ for $p>1$ and $0<\nu \le p/Q^+$ . Suppose that X satisfies $(M \lambda )$ . Then there is a constant $C>0$ such that

$$\begin{align*}\| M_{\lambda}f \|_{L^{\Phi}(X)} \le C \| f \|_{L^{\Phi}(X)} \end{align*}$$

for all $f \in L^{\Phi }(X)$ .

Note from Remarks 2.1 and 5.4 that $(\Phi 5;Q)$ is weaker than $(\Phi 5;\nu )$ when $0<\nu \le p/Q^+$ . Hence, Theorem 3.6 is an improvement of [Reference Ohno and Shimomura18, Lemma 3.1] [Reference Ohno and Shimomura16, Theorem 3.2].

Lemma 3.8 Let $\lambda \ge 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p>1$ . Suppose that X satisfies $(M \lambda )$ . Then there is a constant $C>0$ such that

$$\begin{align*}\frac{1}{\mu(B(x,r))}\int_{B(x,r)} \{M_{\lambda}f(y)\}^{\varepsilon}\, d\mu(y) \le C\Phi^{-1}\left(x,r^{-Q(x)}\right)^{\varepsilon} \end{align*}$$

for all $0<\varepsilon \le 1$ , $x\in X$ , $0<r\le d_X$ and nonnegative $f \in L^{\Phi }(X)$ with $\|f\|_{L^{\Phi }(X)}\le 1$ .

Proof By Theorem 3.6, there exists a constant $C_M \ge 1$ such that

$$\begin{align*}\| M_{\lambda}f \|_{L^{\Phi}(X)} \le C_M \| f \|_{L^{\Phi}(X)}. \end{align*}$$

Let f be a nonnegative measurable function with $\|f\|_{L^{\Phi }(X)}\le 1/(2C_M)$ . Then we have by (2.2)

$$\begin{align*}\int_{X} \Phi(y, M_{\lambda}f(y))\, d\mu(y)\le 1. \end{align*}$$

Let $x\in X$ and $0<r\le d_X$ . Set $c_1= A_1A_2 \max \{1,d_X^{Q^{+}}\} \ (\ge 1)$ . Then we have by Lemma 3.1 (5)

$$ \begin{align*} \Phi^{-1}\left(x,c_1r^{-Q(x)}\right) &\ge \min \left\{1, (A_1 A_2)^{-1} c_1r^{-Q(x)}\right\} \\ &\ge \min \left\{1, (A_1 A_2)^{-1} c_1\min\left\{1,d_X^{-Q^{+}}\right\}\right\}= 1 \end{align*} $$

and by Lemma 3.1 (3)

$$ \begin{align*} \Phi\left(x,\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right) &= c_1r^{-Q(x)}, \end{align*} $$

so that, by $(\Phi 5;Q)$ , there exists a constant $\beta \ge 1$ such that

(3.5) $$ \begin{align} c_1r^{-Q(x)} =\Phi\left(x,\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right) \le \Phi\left(y,\beta\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right) \end{align} $$

whenever $y\in B(x,r)$ . We divide the integral into two parts as follows:

$$ \begin{align*} &\frac{1}{\mu(B(x,r))}\int_{B(x,r)} \{M_{\lambda}f(y)\}^{\varepsilon} \, d\mu(y) \\& \quad = \frac{1}{\mu(B(x,r))} \left( \int_{B(x,r) \cap \{y\in X: M_{\lambda}f(y)\le \beta\Phi^{-1}(x,c_1r^{-Q(x)})\}} \{M_{\lambda}f(y)\}^{\varepsilon}\, d\mu(y) \right. \\& \qquad\qquad\qquad\quad \left. +\int_{B(x,r) \cap \{y\in X : M_{\lambda}f(y)>\beta\Phi^{-1}(x,c_1r^{-Q(x)})\}} \{M_{\lambda}f(y)\}^{\varepsilon} \, d\mu(y) \right). \end{align*} $$

Since $t \rightarrow t^{-\varepsilon } \Phi (y,t)$ is uniformly almost increasing on $(0, \infty )$ by $(\Phi 3)$ , we find from (1.2) and (3.5) that

$$ \begin{align*} &\frac{1}{\mu(B(x,r))}\int_{B(x,r)} \{M_{\lambda}f(y)\}^{\varepsilon} \, d\mu(y) \\& \quad \le \left\{\beta\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right\}^{\varepsilon} +A_2\frac{1}{\mu(B(x,r))}\int_{B(x,r)}\{M_{\lambda}f(y)\}^{\varepsilon} \\& \qquad \times \frac{\{M_{\lambda}f(y)\}^{-\varepsilon}\Phi(y, M_{\lambda}f(y))} {\left\{\beta \Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right\}^{-\varepsilon}\Phi\left(y,\beta\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)\right) } \, d\mu(y) \\& \quad \le C\Phi^{-1}\left(x,c_1r^{-Q(x)}\right)^{\varepsilon} \left\{1+\int_{X} \Phi(y, M_{\lambda}f(y))\, d\mu(y)\right\} \\& \quad \le C\Phi^{-1}\left(x,r^{-Q(x)}\right)^{\varepsilon}, \end{align*} $$

where C is independent of $\varepsilon $ . Thus the proof is complete.

In view of Lemmas 3.8 and 3.1 (5), we obtain the following result.

Corollary 3.9 Let $\lambda \ge 1$ and $0<\varepsilon _1\le 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p>1$ . Suppose that X satisfies $(M \lambda )$ . Then there exist constants $0<\varepsilon _2<\varepsilon _1$ and $C>0$ independent of $\varepsilon _1$ and $\varepsilon _2$ such that

(3.6) $$ \begin{align} \frac{1}{\mu(B(x,r))}\int_{B(x,r)} \{M_{\lambda}f(y)\}^{\varepsilon_2}\, d\mu(y) \le Cr^{-\varepsilon_1} \end{align} $$

for all $x\in X$ , $0<r\le d_X$ and nonnegative $f \in L^{\Phi }(X)$ with $\|f\|_{L^{\Phi }(X)}\le 1$ .

Proof Let $0<\varepsilon _1\le 1$ , $x\in X$ and $0<r\le d_X$ . Taking $\varepsilon _2=\varepsilon _1/(1+Q^+)$ , by Lemma 3.1 (5), we find

$$ \begin{align*}\Phi^{-1}\left(x,r^{-Q(x)}\right)^{\varepsilon_2} \le \max\left\{1, A_1A_2r^{-Q(x)}\right\}^{\varepsilon_2} \le Cr^{-Q^+\varepsilon_2} \le Cr^{-\varepsilon_1}, \end{align*} $$

which implies (3.6) by Lemma 3.8.

4 Trudinger-type inequalities

Let E be a measurable subset of X. We set $s_0 = \min \{1, 1/d_X\}$ . As in [Reference Ohno and Shimomura18], we consider a function

(4.1) $$ \begin{align} \Gamma(x,s) : E \times [s_0, \infty) \to (0, \infty) \end{align} $$

satisfying

1. (Γ1) $s \mapsto \Gamma (x,s)$ is uniformly almost increasing on $[s_0, \infty )$ , that is, there exists a constant $c_{\Gamma 1}\ge 1$ such that

   $$\begin{align*}\Gamma(x,s_1)\le c_{\Gamma 1}\Gamma(x,s_2) \end{align*}$$
   for all $x\in E$ and $s_0 \le s_1<s_2$ ;

2. (Γ2) there exists a constant $c_{\Gamma 2}\ge 1$ such that

   $$\begin{align*}\Gamma(x,2)\le c_{\Gamma 2}\Gamma(x, s_0) \quad \text{for all} \ x \in E; \end{align*}$$

3. (Γ_log) there exists a constant $c_{\Gamma \ell }\ge 1$ such that

   $$ \begin{align*} \Gamma(x,s^2) \le c_{\Gamma \ell}\Gamma (x,s) \end{align*} $$
   for all $x\in E$ and $s \ge 1$ .

We borrow the following lemma from [Reference Maeda, Mizuta, Ohno and Shimomura12, Lemma 3.1] to collect properties of the function $\Gamma $ from (4.1).

Further, we assume the following condition:

1. (ΓΦαQ) there exists a constant $c^{**}_1 \ge 1$ such that

   $$\begin{align*}\int_{\delta}^{d_X}\rho^{\alpha(x)}\Phi^{-1}\left(x,\rho^{-Q(x)}\right)\, \frac{d\rho}{\rho} \le c^{**}_{1} \Gamma(x,1/\delta) \end{align*}$$
   for all $x \in E$ whenever $0 < \delta \le d_X/2$ .

Notice that there exists a constant $c_{\Gamma 3}>0$ such that

(4.2) $$ \begin{align} \Gamma(x,2/d_X)\ge c_{\Gamma 3} \end{align} $$

by $(\Gamma \Phi \alpha Q)$ , (1.1) and Lemma 3.1 (5).

Here we prepare the following lemma using $\Gamma $ .

Lemma 4.2 Let $\tau \ge 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 5;Q)$ . Suppose that $(\Gamma \Phi \alpha Q)$ holds. Then there exists a constant $C>0$ such that

(4.3) $$ \begin{align} \sum_{2\delta<2^i\le 2 d_X} \frac{2^{i\alpha(x)}}{\mu(B(x,\tau 2^i))} \int_{B(x,2^i)} f(y)\, d\mu(y) \le C \Gamma(x,1/\delta) \end{align} $$

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

Proof Let f be a nonnegative measurable function with $\|f\|_{L^{\Phi }(X)}\le 1/2$ . Let $x\in E$ and $0<\delta <d_{X}/2$ . Set $c_1= A_1A_2 \max \{1,d_X^{Q^{+}}\} \ (\ge 1)$ . Just as we obtained (3.5), by $(\Phi 5;Q)$ , we have

(4.4) $$ \begin{align} c_1t^{-Q(x)} =\Phi\left(x,\Phi^{-1}\left(x,c_1t^{-Q(x)}\right)\right) \le \Phi\left(y,\beta\Phi^{-1}\left(x,c_1t^{-Q(x)}\right)\right) \end{align} $$

whenever $0<t\le d_X$ and $y\in B(x,t)$ .

By $(\Phi 3)$ , (4.4), and (1.2)

(4.5) $$ \begin{align} &\frac{t^{\alpha(x)}}{\mu(B(x,\tau t))}\int_{B(x,t)} f(y)\, d\mu(y) \nonumber \\& \quad \le \frac{t^{\alpha(x)}}{\mu(B(x,\tau t))}\int_{B(x,t)} \beta\Phi^{-1}\left(x,c_1t^{-Q(x)}\right) \, d\mu(y) \nonumber \\& \qquad + A_2\frac{t^{\alpha(x)}}{\mu(B(x,\tau t))}\int_{B(x,t)} f(y) \frac{f(y)^{-1}\Phi(y,f(y))}{\left\{\beta\Phi^{-1}\left(x,c_1t^{-Q(x)}\right)\right\}^{-1} \Phi\left(y, \beta\Phi^{-1}\left(x,c_1t^{-Q(x)}\right)\right)}\, d\mu(y) \nonumber \\& \quad \le C\left\{t^{\alpha(x)}\Phi^{-1}\left(x,c_1t^{-Q(x)}\right) +t^{\alpha(x)}\Phi^{-1}\left(x,c_1t^{-Q(x)}\right)\int_{B(x,t)} \Phi(y,f(y))\, d\mu(y)\right\} \nonumber \\& \quad \le Ct^{\alpha(x)}\Phi^{-1}\left(x, t^{-Q(x)}\right)\nonumber\\ \end{align} $$

for all $0<t\le d_X$ . It follows that

$$ \begin{align*} & \sum_{2\delta<2^i\le 2d_X} \frac{2^{i\alpha(x)}}{\mu(B(x,\tau 2^i))} \int_{B(x,2^i)} f(y)\, d\mu(y) \\& \quad \le \sum_{2\delta<2^i < d_X} \frac{2^{i\alpha(x)}}{\mu(B(x,\tau 2^i))} \int_{B(x,2^i)} f(y)\, d\mu(y) + C \frac{d_X^{\alpha(x)}}{\mu(B(x,\tau d_X))} \int_{B(x,d_X)} f(y)\, d\mu(y). \end{align*} $$

As for the rest of the proof, we obtain (4.3) in view of (4.5), Lemma 3.1 (1) and $(\Gamma \Phi \alpha Q)$ , as in the proof of [Reference Ohno and Shimomura18, Lemma 3.5].

As in [Reference Ohno and Shimomura18], we consider a function

$$\begin{align*}\Psi (x,t) : E \times [0, \infty) \to [0, \infty) \end{align*}$$

satisfying the following properties:

1. (Ψ1) $\Psi (\cdot , t)$ is measurable on E for each $t \in [0, \infty )$ and $\Psi (x, \cdot )$ is continuous on $[0, \infty )$ for each $x \in E$ ;

2. (Ψ2) there is a constant $\widetilde {Q}_1 \ge 1$ such that $\Psi (x,t_1) \le \Psi (x, \widetilde {Q}_1t_2)$ for all $x \in E$ whenever $0 < t_1 < t_2$ ;

3. (ΨΓ) there are constants $\widetilde {Q}_2,\ \widetilde {Q}_3 \ge 1$ and $s_0^* \ge s_0$ such that $\Psi (x, \Gamma (x,s)/\widetilde {Q}_2) \le \widetilde {Q}_3 s$ for all $x \in E$ and $s\ge s_0^*$ .

In line with the proof of [Reference Ohno and Shimomura18, Theorem 3.6], using Corollary 3.9, we prove Trudinger-type inequalities, which is an extension of [Reference Ohno and Shimomura18, Theorem 3.6].

Theorem 4.3 Let $1\le \lambda \le \tau $ and $0<\varepsilon \le 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p>1$ . Suppose that X satisfies $(M \lambda )$ and $(\Gamma \Phi \alpha Q)$ holds. Then there exist constants $c_1=c_1(\varepsilon )>0$ and $c_2>0$ independent of $\varepsilon $ such that

(4.6) $$ \begin{align} \frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E\cap B(z,r)} \Psi \left(x, \frac{|J_{\alpha(\cdot), \tau}f(x)|}{c_1}\right)\, d\mu(x) \le c_2 \end{align} $$

for all $z\in X$ , $0<r\le d_X$ and $f \in L^{\Phi }(X)$ with $\|f\|_{L^{\Phi }(X)}\le 1$ .

Proof Let f be a nonnegative measurable function on X with $\|f\|_{L^{\Phi }(X)}\le 1$ . Then Corollary 3.9 implies that there exist constants $0<\varepsilon _2<\varepsilon $ and $C>0$ independent of $\varepsilon $ and $\varepsilon _2$ such that

(4.7) $$ \begin{align} \frac{1}{\mu(B(z,r))}\int_{B(z,r)} \{M_{\lambda}f(x)\}^{\varepsilon_2}\, d\mu(x) \le Cr^{-\varepsilon} \end{align} $$

for all $z\in X$ and $0<r\le d_X$ . Let $x \in E$ . For $0 < \delta \le d_X/2$ , Lemma 4.2 and $\lambda \le \tau $ imply

$$\begin{align*}\begin{aligned} J_{\alpha(\cdot), \tau}f(x) & \le C\left\{ \delta^{\alpha(x) } M_{\lambda}f(x) + \Gamma \left( x, \frac{1}{\delta} \right)\right\} \end{aligned} \end{align*}$$

with a constant $C>0$ independent of x.

When $M_{\lambda }f(x) \le 2/d_X$ , we take $\delta = d_X/2$ . By (4.2) and Lemma 4.1 (1), we see that there exists a constant $C>0$ independent of x such that

(4.8) $$ \begin{align} J_{\alpha(\cdot), \tau}f(x) \le C \Gamma \left( x, \frac{2}{d_X} \right) \le C \Gamma (x, s_0^*) \qquad \text{if} \ M_{\lambda}f(x) \le 2/d_X. \end{align} $$

Next, consider the case when $2/d_X < M_{\lambda }f(x) < \infty $ . By Lemma 4.1 (3) and $(\Gamma 1)$ , there exists a constant $m>0$ such that $\Gamma (x,s)/s\le m \Gamma (x,2/d_X)$ for $s \ge 2/d_X$ . Let

$$\begin{align*}\delta = (d_X/2) \left[ \frac{\Gamma (x, M_{\lambda}f(x))}{m\Gamma(x,2/d_X)M_{\lambda}f(x)} \right]^{1/\alpha(x)}. \end{align*}$$

As in the proof of [Reference Ohno and Shimomura18, Theorem 3.6], by (4.2)

$$ \begin{align*} \delta^{\alpha(x)} M_{\lambda}f(x) \le C \Gamma(x, M_{\lambda}f(x)) \end{align*} $$

and by $(\Gamma 1)$ and Lemma 4.1 (1) and (2),

$$\begin{align*}\Gamma \left( x, \frac{1}{\delta} \right) \le C \Gamma (x,M_{\lambda}f(x)) \le C \Gamma \left(x, \{M_{\lambda}f(x)\}^{\varepsilon_2}\right), \end{align*}$$

where C is dependent of $\varepsilon $ and $\varepsilon _2$ . Hence, there exists a constant $C>0$ independent of x such that

(4.9) $$ \begin{align} J_{\alpha(\cdot), \tau}f(x) \le C \Gamma \left(x, \{M_{\lambda}f(x)\}^{\varepsilon_2}\right) \qquad \text{if} \ 2/d_X < M_{\lambda}f(x) < \infty. \end{align} $$

It follows from (4.8) and (4.9) that there exists a constant $C^*>0$ such that

$$\begin{align*}J_{\alpha(\cdot), \tau}f(x) \le C^* \Gamma \big( x, \max\{s_0^*, \{M_{\lambda}f(x)\}^{\varepsilon_2}\} \big) \end{align*}$$

for a.e. $x \in E$ .

Letting $c_1 = \widetilde {Q}_1 \widetilde {Q}_2 C^*$ , using $(\Psi 2)$ and $(\Psi \Gamma )$ , we have

$$\begin{align*}\begin{aligned} \Psi \left( x, \frac{J_{\alpha(\cdot), \tau}f(x)}{c_1} \right) & \le \Psi \Big( x, \Gamma \big( x, \max\left\{s_0^*, \{M_{\lambda}f(x)\}^{\varepsilon_2}\right\} \big)/\widetilde{Q}_2 \Big) \\ & \le \widetilde{Q}_3 \max \{ s_0^*, \{M_{\lambda}f(x)\}^{\varepsilon_2} \} \le \widetilde{Q}_3(s_0^* + \{M_{\lambda}f(x)\}^{\varepsilon_2}) \end{aligned} \end{align*}$$

for a.e. $x \in E$ , so that by (4.7)

$$ \begin{align*} &\frac{1}{\mu(B(z,r))}\int_{E\cap B(z,r)} \Psi \left(x, \frac{J_{\alpha(\cdot), \tau}f(x)}{c_1}\right)\, d\mu(x) \\ & \le \widetilde{Q}_3 s_0^* + \widetilde{Q}_3 \frac{1}{\mu(B(z,r))} \int_{E\cap B(z,r)} \{M_{\lambda}f(x)\}^{\varepsilon_2}\, d\mu(x)\\ & \le c_2r^{-\varepsilon} \end{align*} $$

for all $z\in X$ and $0<r\le d_X$ , as required.

5 Double phase functionals

Let $p(\cdot )$ and $q(\cdot )$ be real valued measurable functions on X such that

1. (P1) $1 \le p^- \le p^+ < \infty ,$

2. (Q1) $1 \le q^- \le q^+ < \infty .$

Suppose that $p(\cdot )$ and $q(\cdot )$ satisify

1. (P2) $p(\cdot )$ is log-Hölder continuous, that is,

   $$ \begin{align*}|p(x) - p(y)| \le \frac{C_{p}}{\log (e+1/d(x,y))} \quad (x,\, y \in X)\end{align*} $$
   with a constant $C_{p} \ge 0$ , and

2. (Q2) $q(\cdot )$ is log-Hölder continuous, that is,

   $$ \begin{align*}|q(x) - q(y)| \le \frac{C_{q}}{\log (e+1/d(x,y))} \quad (x,\, y \in X)\end{align*} $$
   with a constant $C_{q} \ge 0$ .

In this section, we consider a double phase functional $\Phi (x,t)$ with variable exponents given by

$$\begin{align*}\Phi(x,t) = t^{p(x)} + a(x) t^{q(x)}\left(=t^{p(x)} + (b(x) t)^{q(x)}\right), \ x \in X, \ t \ge 0 ,\end{align*}$$

where $p(x)<q(x)$ for $x\in X$ , $a(\cdot )$ is non-negative, bounded and Hölder continuous of order $\theta \in (0,1]$ and $b(x)=a(x)^{1/q(x)}$ ([Reference Maeda, Mizuta, Ohno and Shimomura11, Reference Ragusa and Tachikawa19], cf. [Reference Baroni, Colombo and Mingione1, Reference Colombo and Mingione3]). Notice that $\Phi (x,t)$ satisfies ( $\Phi $ 1)–( $\Phi $ 3) and that $\Phi (x,t)$ satisfies $(\Phi 5;Q)$ for $\theta \ge \sup _{x\in X_0}\{Q(x)(q(x)/ p(x)-1)\}$ , where $X_0=\left \{x\in X : a(x)>0\right \}$ (see [Reference Ohno and Shimomura17, Lemma 6.1]).

Set

$$\begin{align*}E_1 = \{ x \in X :a(x)=0, Q(x)= \alpha(x)p(x) \}, \end{align*}$$

$$\begin{align*}E_2 = \{ x \in X :a(x)>0, Q(x)= \alpha(x)q(x) \} \end{align*}$$

and $E = E_1 \cup E_2$ . Let us define

$$\begin{align*}\Gamma(x,s)= \left\{ \begin{array}{@{}ll} \log(e+s), & x \in E_1 \\ b(x)^{-1}\log(e+s), & x \in E_2 \end{array} \right. \end{align*}$$

for $s \ge s_0$ . Here note from [Reference Ohno and Shimomura18, Lemma 4.1] that $\Gamma (x,s)$ satisfies $(\Gamma 1)$ , $(\Gamma 2)$ , $(\Gamma _{\mathrm {log}})$ and $(\Gamma \Phi \alpha Q)$ .

Consider the function

$$ \begin{align*} \Psi(x,t)=\left\{ \begin{array}{@{}ll} \exp t, & x \in E_1 \\ \exp\left(b(x)t\right) & x \in E_2 \\ \end{array} \right. \end{align*} $$

for $t>0$ . Notice that $\Psi (x,t)$ satisfies $(\Psi 1)$ , $(\Psi 2)$ , and $(\Psi \Gamma )$ with $s_0^*=2/d_X$ .

By Theorem 4.3, we obtain the following Trudinger’s inequality, which is an extension of [Reference Ohno and Shimomura18, Corollary 4.2].

Corollary 5.1 Let $1\le \lambda \le \tau $ and $0<\varepsilon \le 1$ . Assume that $p^->1$ and $\theta \ge \sup _{x\in X_0}\{Q(x) (q(x)/p(x)-1)\}$ . Suppose that X satisfies $(M \lambda )$ . Then there exist constants ${c_1=c_1(\varepsilon )>0}$ and $c_2>0$ independent of $\varepsilon $ such that

(5.1) $$ \begin{align} & \frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E_1\cap B(z,r)} \exp\left(\frac{|J_{\alpha(\cdot), \tau}f(x)|}{c_1}\right)\, d\mu(x) \nonumber \\& \quad + \frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E_2\cap B(z,r)} \exp\left(\frac{b(x)|J_{\alpha(\cdot), \tau}f(x)|}{c_1}\right)\, d\mu(x) \le c_2 \end{align} $$

for all $z\in X$ , $0<r\le d_X$ and $f \in L^{\Phi }(X)$ with $\|f\|_{L^{\Phi }(X)}\le 1$ .

Remark 5.2 In [Reference Ohno and Shimomura18, Corollary 4.2], we showed the following result: Let $1\le \lambda \le \tau $ . Assume that $p^->1$ and $\sup _{x\in X_0}(q(x)-p(x))/\theta \le p^-/Q^+$ . Suppose that X satisfies $(M \lambda )$ . Then there exist constants $c_1, c_2>0$ such that

(5.2) $$ \begin{align} \int_{E_1} \exp\left(\frac{|J_{\alpha(\cdot), \tau}f(x)|}{c_1}\right)\, d\mu(x) + \int_{E_2} \exp\left(\frac{b(x)|J_{\alpha(\cdot), \tau}f(x)|}{c_1}\right)\, d\mu(x) \le c_2 \end{align} $$

for all $f \in L^{\Phi }(X)$ with $\| f \|_{L^{\Phi }(X)} \le 1$ .

Note that the assumption $\theta \ge \sup _{x\in X_0}\{Q(x)(q(x)/p(x)-1)\}$ in Corollary 5.1 is better than the assumption $\sup _{x\in X_0}(q(x)-p(x))/\theta \le p^-/Q^+$ in [Reference Ohno and Shimomura18, Corollary 4.2] since

$$\begin{align*}\theta \ge Q^+(q(x)-p(x))/p^- \ge Q(x)(q(x)-p(x))/p(x)=Q(x)(q(x)/p(x)-1) \end{align*}$$

for all $x\in X_0$ . Furthermore, (5.1) implies (5.2) when $r=d_X$ since $0<d_X^{\varepsilon }/\mu (z,d_X)=d_X^{\varepsilon }/\mu (X)<\infty $ and $E_j\cap B(z,d_X)=E_j$ for all $z\in X$ and $j=1,2$ . Therefore, Corollary 5.1 extends [Reference Ohno and Shimomura18, Corollary 4.2].

Remark 5.3 We give examples of functions $a(\cdot )$ , $\alpha (\cdot )$ , $p(\cdot )$ , and $q(\cdot )$ that satisfy the assumptions in Corollary 5.1 but not the assumption $\sup _{x\in X_0}(q(x)-p(x))/\theta \le p^-/Q^+$ in the previous paper [Reference Ohno and Shimomura18]. This shows the significance of considering variable order $Q(\cdot )$ .

Let X and $Q(\cdot )$ be as in Example 1.1. Then note from [Reference Nazarov, Treil and Volberg13] that X satisfies $(M 3)$ . We assume that $a(\cdot )$ is non-negative, bounded and Hölder continuous of order $\theta \in (0,1]$ such that $a(x)>0$ for all $x\in X_1$ and $a(x)=0$ for all $x\in X_2$ , that is, $X_1=X_0=\{ x \in X: a(x)>0 \}$ . For an examples of $a(\cdot )$ ,

$$\begin{align*}a(x)= \left\{\begin{array}{@{}ll} x_1^{\theta} & \text{for }x=(x_1,x_2)\in X_1 \\ 0 & \text{for }x\in X_2. \end{array} \right. \end{align*}$$

Let $1<p_0<\infty $ and $q_0= (\theta +1)p_0$ . We consider

$$\begin{align*}p(x)=p_0-\frac{c_0}{\log(e+1/|x|)} \end{align*}$$

and

$$\begin{align*}q(x)=q_0-\frac{c_0(\theta+1)}{\log(e+1/|x|)} \end{align*}$$

for $x\in X$ , where $c_0$ is a constant satisfying

$$\begin{align*}0<c_0<\min\left\{p_0-1, (q_0-1)/(\theta+1), (q_0-p_0)/\theta\right\}. \end{align*}$$

Further we consider a function

$$\begin{align*}\alpha(x)= \left\{\begin{array}{@{}ll} 1/q(x) & \text{for }x\in X_1 \\ 2/p(x) & \text{for }x\in X_2. \end{array} \right. \end{align*}$$

Then $p(\cdot )$ satisfies (P1) and (P2) since $0<c_0<p_0-1$ , $q(\cdot )$ satisfies (Q1) and (Q2) since $0<c_0<(q_0-1)/(\theta +1),$ and $p(x)<q(x)$ holds for all $x\in X$ since $0<c_0<(q_0-p_0)/\theta $ . Further, we can show that

$$ \begin{align*} E_1 &= \{ x \in X :a(x)=0, Q(x)= \alpha(x)p(x) \}=X_2, \\ E_2 &= \{ x \in X :a(x)>0, Q(x)= \alpha(x)q(x) \}=X_1 \end{align*} $$

and $\sup _{x\in X_0}\{Q(x)(q(x)/p(x)-1)\} = \theta $ since $q_0= (\theta +1)p_0$ . However, $\sup _{x\in X_0} (q(x)-p(x))/\theta \le p^-/Q^+$ does not hold since

$$ \begin{align*} q(x)-p(x) =\theta\left(p_0-\frac{c_0}{\log(e+1/|x|)}\right) =\frac{\theta p(x)}{Q(x)}>\frac{\theta p^-}{Q^+} \end{align*} $$

for all $x\in X_0=X_1$ since $Q^+=2$ .

Remark 5.4 In general, if $\Phi (x,t)$ satisfies $(\Phi 5;\nu )$ with $\nu>0$ satisfying $\nu \le p^-/Q^+$ , then it satisfies $(\Phi 5;Q)$ (see Remark 2.1). In this remark, we show that $(\Phi 5;Q)$ is weaker than $(\Phi 5;\nu )$ when $0<\nu \le p^-/Q^+$ . For this purpose, let X, $a(\cdot )$ , $p(\cdot )$ , and $q(\cdot )$ be as in Remark 5.3. First, Remark 5.3 implies that $\Phi (x,t)$ satisfies $(\Phi 5;Q)$ . Next, we show that $\Phi (x,t)$ does not satisfy $(\Phi 5;\nu )$ for $0<\nu \le p^-/Q^+$ . Let $x=(t^{-\nu },0)$ and $y=(0,0)$ for $t\ge 1$ . Then we find that $x,y\in X$ , $d(x,y)=t^{-\nu }$ and

$$ \begin{align*} \lim_{t\to \infty}\frac{\Phi(x,t)}{\Phi(y,t)} &=\lim_{t\to \infty} \frac{t^{p_0-c_0/\log(e+1/|x|)}+x_1^{\theta}t^{q_0-c_0(\theta+1)/\log(e+1/|x|)}}{t^{p_0}} \\ &=\lim_{t\to \infty}\left(t^{-c_0/\log(e+t^{\nu})} +t^{\theta (p_0-\nu)-c_0(\theta+1)/\log(e+t^{\nu})}\right)\\ &=\infty \end{align*} $$

for all $0<\nu <p_0$ since $q_0= (\theta +1)p_0$ . Hence, $\Phi (x,t)$ does not satisfy $(\Phi 5;\nu )$ for all $0<\nu <p_0$ , so that $\Phi (x,t)$ does not satisfy $(\Phi 5;\nu )$ for $0<\nu \le p^-/Q^+$ since $p^-/Q^+=p^-/2<p_0/2<p_0$ .

6 Sobolev functions

In this section, we consider the case when $\mu $ is doubling. Given $u\in L^{1}_{loc}(X)$ and g be a nonnegative measurable function on X, we say that the pair $u, g$ satisfies a Poincaré inequality in X if there exist constants $A_0>0$ and $\sigma \ge 1$ such that

$$\begin{align*}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}\left|u(y)-u_{B(x,r)}\right|\, d\mu(y) \le \frac{A_0 r}{\mu(B(x,\sigma r))}\int_{B(x,\sigma r)}g(y)\, d\mu(y) \end{align*}$$

for all $x\in X$ and $r>0$ , where $u_{B(x,r)}=\int _{B(x,r)} u(y)\, d\mu (y)/\mu (B(x,r))$ .

As in [Reference Ohno and Shimomura14], we consider Musielak–Orlicz–Hajłasz–Sobolev spaces $M^{1,\Phi }(X)$ . We say that a function $u\in L^{\Phi }(X)$ belongs to $M^{1,\Phi }(X)$ if there exists a nonnegative function $g\in L^{\Phi }(X)$ , called a Hajłasz gradient of u, such that

(6.1) $$ \begin{align} |u(x)-u(y)|\le d(x,y)(g(x)+g(y)) \end{align} $$

for $\mu $ -almost every $x,y\in X$ . Notice that the Poincaré inequality holds if we integrate both sides in (6.1) over y and x.

When $\mu $ satisfies the doubling condition and the pair $u, g$ satisfies a Poincaré inequality in X, it follows from [Reference Hajłasz and Koskela4, Theorem 5.2] that

$$ \begin{align*}|u(x)-u_B| \le C J_{1,1}g(x) \end{align*} $$

for a.e. $x \in B$ . Thanks to this estimate, Theorem 4.3 gives the following theorem.

Theorem 6.1 Given $u\in L^{1}_{loc}(X)$ and $g\in L^{\Phi }(X)$ be a nonnegative measurable function on X, suppose that the pair $u, g$ satisfies a Poincaré inequality in X. Let $0<\varepsilon \le 1$ . Let $\Phi (x,t)$ satisfy $(\Phi 3;p)$ and $(\Phi 5;Q)$ for $p>1$ . Assume that $(\Gamma \Phi \alpha Q)$ with $\alpha (\cdot )\equiv 1$ holds. Then there exist constants $c_1=c_1(\varepsilon )>0$ and $c_2>0$ independent of $\varepsilon $ such that

$$\begin{align*}\frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E\cap B \cap B(z,r)} \Psi \left(x, \frac{|u(x)-u_B|}{c_1}\right)\, d\mu(x) \le c_2 \end{align*}$$

for all $z\in X$ , $0<r\le d_X$ and balls B.

Corollary 5.1 gives the following corollary for double phase case, as an extension of [Reference Ohno and Shimomura18, Corollary 5.2].

Corollary 6.3 Let $p(\cdot )$ , $q(\cdot )$ , $a(\cdot )$ , $E_1$ , and $E_2$ with $\alpha (\cdot )\equiv 1$ be as in Section 5 and let

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

for $x \in X$ and $t \ge 0$ . Given $u\in L^{1}_{loc}(X)$ and $g\in L^{\Phi }(X)$ be a nonnegative measurable function on X, suppose that the pair $u, g$ satisfies a Poincaré inequality in X. Let $0<\varepsilon \le 1$ . If $p^->1$ and $\theta \ge \sup _{x\in X_0}\{Q(x)(q(x)/p(x)-1)\}$ , then there exist constants $c_1=c_1(\varepsilon )>0$ and $c_2>0$ independent of $\varepsilon $ such that

(6.2) $$ \begin{align} & \frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E_1\cap B\cap B(z,r)} \exp\left(\frac{|u(x)-u_B|}{c_1}\right)\, d\mu(x) \nonumber \\ &+ \frac{r^{\varepsilon}}{\mu(B(z,r))}\int_{E_2\cap B\cap B(z,r)} \exp\left(\frac{b(x)|u(x)-u_B|}{c_1}\right)\, d\mu(x) \le c_2 \end{align} $$

for all $z\in X$ , $0<r\le d_X$ and balls B.