2. Definitions and preliminaries

Let $\mathcal {M}$ denote the class of Lebesgue measurable functions. Let ${{\mathcal {S}}}$ and ${{\mathcal {S}}'}$ be the class of Schwartz functions and tempered distributions, respectively. For any $x\in {\mathbb {R}}^{n}$ and $r>0$, write $B(x,r)=\{y:|y-x|< r\}$. Define ${\mathbb {B}}=\{B(x,r):x\in {\mathbb {R}}^{n},r>0\}$.

Let $p\in (0,\infty )$ and $\omega :{\mathbb {R}}^{n}\to (0,\infty )$, the weighted Lebesgue space $L^{p}(\omega )$ consists of all $f\in {\mathcal {M}}$ satisfying

\[ \|f\|_{L^{p}(\omega)}=\left(\int_{{\mathbb{R}}^{n}}|f(x)|^{p}\omega(x)dx\right)^{\frac{1}{p}}<\infty. \]

For any Lebesgue measurable function $p(\cdot ):{\mathbb {R}}^{n}\to (0,\infty ]$, define

\[ p_{+}=\text{esssup}_{x\in{\mathbb{R}}^{n}}p(x)\quad\mathrm{and}\quad p_{-}=\text{essinf}_{x\in{\mathbb{R}}^{n}}p(x) \]

and

\[ {\mathbb{R}}^{p({\cdot})}_{\infty}=\{x\in{\mathbb{R}}^{n}: p(x)=\infty\}. \]

When $p(\cdot ):{\mathbb {R}}^{n}\to [1,\infty ]$, define the conjugate function $p'(\cdot )$ by $\frac {1}{p(\cdot )}+\frac {1}{p'(\cdot )}=1$.

We recall the definitions of the Lebesgue spaces with variable exponents and the weighted Lebesgue spaces with variable exponents from [Reference Diening, Harjulehto, Hästö and Ružička11, Definitions 3.1.2, 3.2.1 and (5.8.1)].

Definition 2.1 Let $p(\cdot ):{\mathbb {R}}^{n}\to (0,\infty ]$ and $\omega :{\mathbb {R}}^{n}\to (0,\infty )$ be Lebesgue measurable functions. The Lebesgue space with variable exponent consists of all Lebesgue measurable functions $f$ satisfying

\[ \|f\|_{{L^{p({\cdot})}}}=\inf\left\{\lambda>0:\int_{{\mathbb{R}}^{n}}\left|\frac{f(x)}{\lambda}\right|^{p(x)}dx+\left\|\frac{f\chi_{{\mathbb{R}}^{p({\cdot})}_{\infty}}}{\lambda}\right\|_{L^{\infty}}\le 1\right\}<\infty. \]

The weighted Lebesgue space with variable exponent consists of all Lebesgue measurable functions $f$ satisfying

\[ \|f\|_{{L^{p({\cdot})}_{\omega}}}=\|f\omega\|_{{L^{p({\cdot})}}}<\infty. \]

In particular, when $p(\cdot )=p$, $p\in (0,\infty )$, is a constant function, we have $L^{p}_{\omega }=L^{p}(\omega ^{p})$. Notice that for any unbounded Lebesgue measurable set $E$ with $|E|<\infty$, $\int _{E}\omega (x)^{p}dx$ is not necessarily bounded. Therefore, $L^{p}_{\omega }$ is not a Banach function space with respect to the Lebesgue measure. For the definition of Banach function space, the reader is referred to [Reference Bennett and Sharpley2, Chapter 1, Definitions 1.1 and 1.3] and [Reference Diening, Harjulehto, Hästö and Ružička11, Definition 2.7.7].

Theorem 2.1 Let $p(\cdot ):{\mathbb {R}}^{n}\to [1,\infty ]$ be a Lebesgue measurable function. Then, ${L^{p(\cdot )}}$ is a Banach function space and

\[ \|f\|_{{L^{p'({\cdot})}}}=\sup\left\{\left|\int_{{\mathbb{R}}^{n}}f(x)g(x)dx\right|:g\in{L^{p({\cdot})}},\,\|g\|_{{L^{p({\cdot})}}}\le 1\right\}. \]

That is, the associate space of ${L^{p(\cdot )}}$ is ${L^{p'(\cdot )}}$.

The reader is referred to [Reference Diening, Harjulehto, Hästö and Ružička11, Theorem 3.2.13] for the proof of the above result. Notice that ${L^{p(\cdot )}_{\omega }}$ is not necessarily a Banach function space, see [Reference Diening, Harjulehto, Hästö and Ružička11, p.192-193].

In view of the above theorem, we have the following Hölder inequality for the weighted Lebesgue spaces with variable exponents

(2.1)\begin{equation} \int_{{\mathbb{R}}^{n}}|f(x)g(x)|dx= \int_{{\mathbb{R}}^{n}}|f(x)\omega(x)g(x)\omega(x)^{{-}1}|dx\le 2\|f\|_{{L^{p({\cdot})}_{\omega}}}\|g\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}. \end{equation}

Additionally, Theorem 2.1 gives

\begin{align*} & \|f\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}=\|f\omega^{{-}1}\|_{{L^{p'({\cdot})}}}\\& \le C\sup\left\{\left|\int_{{\mathbb{R}}^{n}}f(x)\omega(x)^{{-}1}g(x)\omega(x)dx\right|:g\omega\in{L^{p({\cdot})}},\,\|g\omega\|_{{L^{p({\cdot})}}}\le 1\right\}\\ & \le C\sup\left\{\left|\int_{{\mathbb{R}}^{n}}f(x)g(x)dx\right|:g\in{L^{p({\cdot})}_{\omega}},\,\|g\|_{{L^{p({\cdot})}_{\omega}}}\le 1\right\} \end{align*}

for some $C>0$. That is, the associate space of ${L^{p(\cdot )}_{\omega }}$ with respect to the Lebesgue measure $dx$ is ${L^{p'(\cdot )}_{\omega ^{-1}}}$.

We recall the class of weight functions associated with ${L^{p(\cdot )}_{\omega }}$ from [Reference Cruz-Uribe, Fiorenza and Neugebauer8, Definition 1.4].

Definition 2.2 Let $p(\cdot ):{\mathbb {R}}^{n}\to [1,\infty )$ and $\omega :{\mathbb {R}}^{n}\to (0,\infty )$ be Lebesgue measurable functions. We write $\omega \in A_{p(\cdot )}$ if there exists a constant $C>0$ such that for any $B\in {\mathbb {B}}$,

\[ \|\chi_{B}\omega\|_{{L^{p({\cdot})}}}\|\chi_{B}\omega^{{-}1}\|_{{L^{p'({\cdot})}}}\le C|B|. \]

We also recall the definition of the Muckenhoupt weight functions ${\mathcal {A}}_{p}$ because they are important ingredient in the extrapolation theory. A positive locally integrable function $\omega$ belongs to ${\mathcal {A}}_{p}$ if it satisfies

\[ [\omega]_{{\mathcal{A}}_{p}}=\sup_{B\in{\mathbb{B}}}\left(\frac{1}{|B|}\int_{B}\omega(x)dx\right)\left(\frac{1}{|B|}\int_{B}\omega(x)^{-\frac{p'}{p}}dx\right)^{\frac{p}{p'}}<\infty \]

where $p'=\frac {p}{p-1}$. A locally integrable function $\omega :{\Bbb R}^{n}\to [0,\infty )$ is said to be an ${\mathcal {A}}_{1}$ weight if there is a constant $C>0$ such that for any $B\in {\mathbb {B}}$,

\[ \frac{1}{|B|}\int_{B}\omega(y)dy\le C\omega(x),\quad a.e.\,x\in B. \]

The infimum of all such $C$ is denoted by $[\omega ]_{{\mathcal {A}}_{1}}$. We define ${\mathcal {A}}_{\infty }=\cup _{p\ge 1}{\mathcal {A}}_{p}$.

When $p(\cdot )=p$, $p\in [1,\infty )$, we have $\omega \in A_{p(\cdot )}\Leftrightarrow \omega ^{p}\in {\mathcal {A}}_{p}$.

The following is an important class of exponent functions used in the function spaces with variable exponents.

Definition 2.3 A continuous function $g$ on ${\Bbb R}^{n}$ is locally log-Hölder continuous if there exists $c_{log}>0$ such that

(2.2)\begin{equation} |g(x)-g(y)|\le \dfrac{c_{log}}{-\log(|x-y|)},\quad \forall x,y\in{\Bbb R}^{n},\ |x-y|<\frac{1}{2}.\end{equation}

We denote the class of locally log-Hölder continuous function by $C^{\log }_{loc}({\Bbb R}^{n})$.

Furthermore, a continuous function $g$ is globally log-Hölder continuous if $g\in C^{\log }_{loc}({\Bbb R}^{n})$ and there exist $g_{\infty }\in {\Bbb R}$ and $c_{\infty }>0$ so that

(2.3)\begin{equation} |g(x)-g_{\infty}|\le \dfrac{c_{\infty}}{\log(e+|x|)},\quad \forall x\in{\Bbb R}^{n}.\end{equation}

The class of globally log-Hölder continuous function is denoted by $C^{\log }({\Bbb R}^{n})$.

According to [Reference Cruz-Uribe, Fiorenza and Neugebauer8, Theorem 1.5], we have the following boundedness result of the Hardy–Littlewood maximal operator on the weighted Lebesgue spaces with variable exponents.

The results given in [Reference Cruz-Uribe, Fiorenza and Neugebauer8, Theorem 1.5] are presented for the weighted Lebesgue space with variable exponent ${L^{p(\cdot )}_{\omega }}$. It is easy to see that $\omega \in A_{p(\cdot )}\Leftrightarrow \omega ^{-1}\in A_{p'(\cdot )}$. Moreover, for any $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$, we have $p'(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}'\le p_{+}'<\infty$. Consequently, [Reference Cruz-Uribe, Fiorenza and Neugebauer8, Theorem 1.5] also yields the boundedness for the Hardy–Littlewood maximal operator on ${L^{p'(\cdot )}_{\omega ^{-1}}}$.

3. Weighted Morrey spaces with variable exponents

The main results for the weighted Morrey spaces with variable exponents are obtained in this section. We establish the extrapolation theory for the weighted Morrey spaces with variable exponents. This extrapolation theory yields the boundedness of pseudo-differential operators and Fourier integral operators on the weighted Morrey spaces with variable exponents.

We start with the definitions of weighted Morrey spaces with variable exponents.

Definition 3.1 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $0< p_{-}\le p_{+}<\infty$, $\omega :{\mathbb {R}}^{n}\to (0,\infty )$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be Lebesgue measurable functions. The weighted Morrey space with variable exponent ${{M_{p(\cdot ),\omega }^{u}}}$ consists of all $f\in {\mathcal {M}}$ satisfying

(3.1)\begin{equation} \|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}=\sup_{y\in{\mathbb{R}}^{n},r>0}\frac{1}{u(\|\chi_{B(y,r)}\|_{{L^{p({\cdot})}_{\omega}}})}\|\chi_{B(y,r)}f\|_{{L^{p({\cdot})}_{\omega}}}<\infty. \end{equation}

To simplify the presentation, for the rest of this paper, we write $u(y,r)=u(\|\chi _{B(y,r)}\|_{{L^{p(\cdot )}_{\omega }}})$, $y\in {\mathbb {R}}^{n}$, $r>0$.

When $p(\cdot )=p$, $1< p<\infty$, is a constant function, $u(y,r)=|B(y,r)|^{\frac {1}{p}-\frac {1}{q}}$, $1\le q\le p<\infty$ and $\omega \equiv 1$, ${{M_{p(\cdot ),\omega }^{u}}}$ becomes the classical Morrey space $M_{p}^{q}$.

When $\omega \equiv 1$, the weighted Morrey space with variable exponent becomes the Morrey spaces with variable exponents [Reference Almeida, Hasanov and Samko1, Reference Guliyev and Samko18, Reference Guliyev, Hasanov and Samko19, Reference Ho25]. When $p(\cdot )=p$, $p\in (0,\infty )$, ${{M_{p(\cdot ),\omega }^{u}}}$ reduces to the weighted Morrey spaces $M^{u}_{p,\omega }$ [Reference Haroske and Skrzypczak21, Reference Kokilashvili and Samko35, Reference Nakamura and Sawano48].

For the mapping properties of the singular integral operators and the Riesz potentials on the weighted Morrey spaces with variable exponents, the reader is referred to [Reference Guliyev, Hasanov and Badalov20, Reference Mizuta and Shimomura42].

We now introduce the weighted block spaces with variable exponents. We need to use the boundedness of the Hardy–Littlewood maximal operator on the weighted block spaces with variable exponents to establish the extrapolation theory on the weighted Morrey spaces with variable exponents.

Definition 3.2 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$, $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. For any Lebesgue measurable function $b$, we write $b\in \mathfrak {b}_{p(\cdot ),\omega }^{u}$ if $\mathrm {supp}b\,\subseteq B(x_{0},r)$ for some $x_{0}\in {\mathbb {R}}^{n}$ and $r>0$ and

\[ \|b\|_{{L^{p({\cdot})}_{\omega}}}\le \frac{1}{u(x_{0},r)}. \]

The block space $\mathfrak {B}_{p(\cdot ),\omega }^{u}$ is defined as

(3.2)\begin{equation} \mathfrak{B}_{p({\cdot}),\omega}^{u}=\bigg\{\sum_{k=1}^{\infty}\lambda_{k}b_{k}:\sum_{k=1}^{\infty}|\lambda_{k}|<\infty\;\mathrm{and}\; b_{k}\in \mathfrak{b}_{p({\cdot}),u}^{\omega}\bigg\}. \end{equation}

The block space $\mathfrak {B}_{p(\cdot ),\omega }^{u}$ is endowed with the norm

(3.3)\begin{equation} \|f\|_{\mathfrak{B}_{p({\cdot}),\omega}^{u}}=\inf\bigg\{\sum_{k=1}^{\infty}|\lambda_{k}|\mathrm{such\ that}\ f=\sum_{k=1}^{\infty}\lambda_{k}b_{k}\,\mathrm{a.e.}\bigg\}. \end{equation}

We now present some supporting results for ${{M_{p(\cdot ),\omega }^{u}}}$ and $\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$. The reader is referred to [Reference Mastyło, Sawano and Tanaka38] for the corresponding results on the Morrey type spaces. We have the Hölder inequalities for ${{M_{p(\cdot ),\omega }^{u}}}$ and $\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$.

Lemma 3.1 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty,$ $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. We have a constant $C>0$ such that

\[ \int_{{\mathbb{R}}^{n}}|f(x)g(x)|dx\le C\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\|g\|_{\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}}. \]

Proof. Let $f\in {{M_{p(\cdot ),\omega }^{u}}}$ and $g\in \mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$. For any $\epsilon >0$, we have $g=\sum _{k=1}^{\infty }\lambda _{k}b_{k}$ with $\mathrm {supp}\,b_{k}\,\subseteq B(x_{k},r_{k})=B_{k}$, $\|b_{k}\|_{{L^{p(\cdot )}_{\omega }}}\le \frac {1}{u(x_{k},r_{k})}$ and

\[ \sum_{k=1}^{\infty}|\lambda_{k}|\le (1+\epsilon)\|g\|_{\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}}. \]

The Hölder inequality gives

\begin{align*} \int_{{\mathbb{R}}^{n}}|f(x)b_{k}(x)|dx& \le C\|f\chi_{B_{k}}\|_{{L^{p({\cdot})}_{\omega}}}\|b_{k}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\\ & =\frac{C}{u(x_{k},r_{k})}\|f\chi_{B_{k}}\|_{{L^{p({\cdot})}_{\omega}}}u(x_{k},r_{k})\|b_{k}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le C\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \end{align*}

Consequently, we have

\begin{align*} \int_{{\mathbb{R}}^{n}}|f(x)g(x)|dx & \le \sum_{k=1}^{\infty}\int_{{\mathbb{R}}^{n}}|f(x)b_{k}(x)|dx\le C\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}} \sum_{k=1}^{n}|\lambda_{k}|\\ & \le C\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\|g\|_{\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}}. \end{align*}

The following proposition gives criteria for the membership $f\in {{M_{p(\cdot ),\omega }^{u}}}$.

Proposition 3.2 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$, $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. If $f\in {\mathcal {M}}$ satisfies

\[ \sup_{b\in \mathfrak{b}_{p'({\cdot}),\omega^{{-}1}}^{u}}\int_{{\mathbb{R}}^{n}}|f(x)b(x)|dx<\infty, \]

then $f\in {{M_{p(\cdot ),\omega }^{u}}}$.

Proof. For any $g\in {L^{p'(\cdot )}_{\omega ^{-1}}}$ with $\|g\|_{{L^{p'(\cdot )}_{\omega ^{-1}}}}\le 1$ and $B=B(x_{0},r)$, $x_{0}\in {\mathbb {R}}^{n}$ and $r>0$, define $b_{g,B}=\frac {1}{u(x_{0},r)}g\chi _{B}$. We find that $b_{g,B}\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$.

Since the associate space of ${L^{p'(\cdot )}_{\omega ^{-1}}}$ with respect to the Lebesgue measure is ${L^{p(\cdot )}_{\omega }}$, we have

\begin{align*} \sup_{g\in {L^{p'({\cdot})}_{\omega^{{-}1}}}\atop \|g\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le 1}\int_{{\mathbb{R}}^{n}}|f(x)b_{g,B}(x)|dx& =\frac{1}{u(x_{0},r)}\sup_{g\in {L^{p'({\cdot})}_{\omega^{{-}1}}}\atop \|g\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le 1}\int_{{\mathbb{R}}^{n}}|f(x)g(x)\chi_{B}(x)|dx\\ & =\frac{1}{u(x_{0},r)}\|\chi_{B(x_{0},r)}f\|_{{L^{p({\cdot})}_{\omega}}}. \end{align*}

By taking the supremum over $B(x_{0},r)\in {\mathbb {B}}$, we obtain

\[ \|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}=\sup_{B(x_{0},r)\in{\mathbb{B}}}\frac{1}{u(x_{0},r)}\|\chi_{B(x_{0},r)}f\|_{{L^{p({\cdot})}_{\omega}}}\le \sup_{b\in \mathfrak{b}_{X',u}}\int_{{\mathbb{R}}^{n}}|f(x)b(x)|dx<\infty. \]

Therefore, $f\in {{M_{p(\cdot ),\omega }^{u}}}$.

Next, we have the norm conjugate formula for the weighted Morrey spaces with variable exponents.

Proposition 3.3 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$, $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. There exist constants $C_{0},C_{1}>0$ such that for any $f\in {{M_{p(\cdot ),\omega }^{u}}}$,

(3.4)\begin{equation} C_{0}\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le \sup_{b\in \mathfrak{b}_{p'({\cdot}),\omega^{{-}1}}^{u}}\int_{{\mathbb{R}}^{n}}|f(x)b(x)|dx\le C_{1}\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \end{equation}

Proof. In view of the definition of ${{M_{p(\cdot ),\omega }^{u}}}$, we have a $B(x_{0},r)\in {\mathbb {B}}$ such that

(3.5)\begin{equation} \|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le 2\frac{1}{u(x_{0},r)}\|f\chi_{B(x_{0},r)}\|_{{L^{p({\cdot})}_{\omega}}}. \end{equation}

As the associate space of ${L^{p(\cdot )}_{\omega }}$ with respect to the Lebesgue measure is ${L^{p'(\cdot )}_{\omega ^{-1}}}$, we have a $g\in {L^{p'(\cdot )}_{\omega ^{-1}}}$ with $\|g\|_{{L^{p'(\cdot )}_{\omega ^{-1}}}}\le 1$ such that

(3.6)\begin{equation} \|f\chi_{B(x_{0},r)}\|_{{L^{p({\cdot})}_{\omega}}}\le 2\int_{B(x_{0},r)}|f(x)g(x)|dx. \end{equation}

Define $b(x)=\frac {1}{u(x_{0},r)}\chi _{B(x_{0},r)}(x)g(x)$. We have $\text {supp}\,b\subseteq B(x_{0},r)$ and $\|b\|_{{L^{p'(\cdot )}_{\omega ^{-1}}}}\le \frac {1}{u(x_{0},r)}$. Therefore, $b\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$. Consequently, (3.5) and (3.6) yield

\[ C_{0}\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le \int_{{\mathbb{R}}^{n}}|f(x)b(x)|dx\le C_{1}\sup_{b^{{\ast}}\in \mathfrak{b}_{p'({\cdot}),\omega^{{-}1}}^{u}}\int_{{\mathbb{R}}^{n}}|f(x)b^{{\ast}}(x)|dx \]

for some $C_{0},C_{1}>0$. The above inequality gives the first inequality in (3.4). The second inequality in (3.4) follows from Lemma 3.1 and the fact that $\|b\|_{\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}}\le 1$ for any $b\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$.

We now introduce a class of weight function for the study of the boundedness of the Hardy–Littlewood maximal operator on the weighted block spaces with variable exponents.

Definition 3.3 Let $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$ and $\omega \in A_{p(\cdot )}$. We say that a Lebesgue measurable function, $u(\cdot ):(0,\infty )\to (0,\infty )$, belongs to $u\in {\Bbb W}_{p(\cdot ),\omega }$ if there exists a constant $C>0$ such that for any $x\in {\Bbb R}^{n}$ and $r>0$, $u$ fulfils

(3.7)\begin{align} & C\le u(r),\quad r\ge 1, \end{align}

(3.8)\begin{align} & r\le Cu(r),\quad r<1, \end{align}

(3.9)\begin{align} & \sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}}{\|\chi_{B(x,2^{j+1}r)}\|_{{L^{p({\cdot})}_{\omega}}}}u(\|\chi_{B(x,2^{j+1}r)}\|_{{L^{p({\cdot})}_{\omega}}}) \le Cu(\|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}). \end{align}

The reader may consult [Reference Nakamura, Noi and Sawano47, Proposition 2.7] and [Reference Nakamura46, Definition 1.1] for some similar classes of weight functions used to study the generalized Morrey spaces and the generalized weighted Morrey spaces, respectively.

When $p(\cdot )=p$, $p\in (1,\infty )$ is a constant function, we write ${\Bbb W}_{p(\cdot ),\omega }={\mathbb {W}}_{p,\omega }$.

For instance, when $\theta \in (0,1)$, $u_{\theta }(r)=r^{\theta }$ fulfils (3.7) and (3.8). Notice that Definition 2.2 assures that for any $B\in {\mathbb {B}}$, $\|\chi _{B}\|_{{L^{p(\cdot )}_{\omega }}}=\|\chi _{B}\omega \|_{{L^{p(\cdot )}}}<\infty$. Therefore, the inequality in (3.9) is well defined.

Furthermore, (3.7) assures that

(3.10)\begin{equation} 1\le \|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\Rightarrow C\le u(x,r), \quad x\in{\mathbb{R}}^{n},\,r>0. \end{equation}

Similarly, (3.8) guarantees that

(3.11)\begin{equation} \|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\le 1\Rightarrow \|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\le u(x,r), \quad x\in{\mathbb{R}}^{n},\,r>0. \end{equation}

Let $w\in {\mathcal {A}}_{\infty }$, $p(\cdot )=p$, $p\in (1,\infty )$ and $\omega =w^{1/p}$. Recall that for any $w\in {\mathcal {A}}_{\infty }$, there exist constants $\epsilon _{0},C>0$ such that for any $B\in {\mathbb {B}}$ and measurable subset $A$ of $B$, we have

(3.12)\begin{equation} \frac{w(A)}{w(B)}\le C\left(\frac{|A|}{|B|}\right)^{\epsilon_{0}}, \end{equation}

see [Reference Grafakos17, Theorem 9.3.3 (d)]. We find that (3.12) gives

\begin{align*} \sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{{L^{p}_{\omega}}}}{\|\chi_{B(x,2^{j+1}r)}\|_{{L^{p}_{\omega}}}}\frac{u_{\theta}(x,2^{j+1}r)}{u_{\theta}(x,r)}& =\sum_{j=0}^{\infty}\left(\dfrac{\|\chi_{B(x,r)}\|_{{L^{p}_{\omega}}}}{\|\chi_{B(x,2^{j+1}r)}\|_{{L^{p}_{\omega}}}}\right)^{1-\theta}\\ & \le C\sum_{j=0}^{\infty} 2^{-(j+1)n\epsilon_{0}}<\infty. \end{align*}

Thus, $u_{\theta }\in {\mathbb {W}}_{p,\omega }$.

The condition (3.9) is used to obtain the boundedness of the Hardy–Littlewood maximal operator on $\mathfrak {B}_{p(\cdot ),\omega }^{u}$.

Proof. Let $B\in {\Bbb B}$, $x\in {\Bbb R}^{n}$ and $r>0$. Whenever $\|\chi _{B(x,r)}\|_{{L^{p(\cdot )}_{\omega }}}\ge 1$, (3.10) gives

(3.13)\begin{equation} \frac{1}{u(x,r)}\|\chi_{B}\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\le\frac{1}{u(x,r)} \|\chi_{B}\|_{{L^{p({\cdot})}_{\omega}}}\le C\|\chi_{B}\|_{{L^{p({\cdot})}_{\omega}}} \end{equation}

for some $C>0$. When $\|\chi _{B(x,r)}\|_{{L^{p(\cdot )}_{\omega }}}<1$, (3.11) assures that

(3.14)\begin{equation} \frac{1}{u(x,r)}\|\chi_{B}\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\le\frac{1}{u(x,r)} \|\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}\le C. \end{equation}

Consequently, (3.13) and (3.14) yield

\[ \|\chi_{B}\|_{{{M_{p({\cdot}),\omega}^{u}}}}=\sup_{B(x,r)\in{\Bbb B}}\frac{1}{u(x,r)}\|\chi_{B}\chi_{B(x,r)}\|_{{L^{p({\cdot})}_{\omega}}}< C+C\|\chi_{B}\|_{{L^{p({\cdot})}_{\omega}}}. \]

Therefore, $\chi _{B}\in {{M_{p(\cdot ),\omega }^{u}}}$.

The preceding theorem shows that for any $u$ satisfies (3.7) and (3.8), ${{M_{p(\cdot ),\omega }^{u}}}$ is non-trivial. Furthermore, Definition 3.2 and Lemma 3.1 guarantee that for any $f\in {{M_{p(\cdot ),\omega }^{u}}}$ and $B\in {\mathbb {B}}$, we have $\int _{B}|f(x)|dx\le C\|f\|_{{{M_{p(\cdot ),\omega }^{u}}}}$. Therefore, this inequality and Proposition 3.4 assure that ${{M_{p(\cdot ),\omega }^{u}}}$ is a ball Banach function space. For the definition of ball Banach function space, see [Reference Sawano, Ho, Yang and Yang59, Definition 2.2].

We are now ready to show that the Hardy–Littlewood maximal operator is bounded on $\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$.

Proof. Let $b\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$ with $\text {supp}\,b\subseteq B(x_{0},r)$, $x_{0}\in {\mathbb {R}}^{n}$, $r>0$. For any $k\in {\mathbb {N}}\cup \{0\}$, write $B_{k}=B(x_{0}, 2^{k}r)$, $d_{0}=\chi _{B_{1}}\operatorname {M}(b)$ and $d_{k}=\chi _{B_{k+1}\backslash B_{k}}\operatorname {M}(b)$, $k\in {\mathbb {N}}$. Thus, $\text {supp}\, d_{k}\subseteq B_{k+1}\backslash B_{k}$ and $\operatorname {M}(b)=\sum _{k=0}^{\infty }d_{k}$.

Theorem 2.2 assures that

\[ \|d_{0}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le \|\operatorname{M}(b)\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le C\|b\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\le C\frac{1}{u(x_{0},r)} \]

for some $C>0$. Thus, $d_{0}=Ce_{0}$ for some $C>0$ with $e_{0}\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$.

Next, (2.1) gives

\begin{align*} d_{k}& =\chi_{B_{k+1}\backslash B_{k}}\operatorname{M}(b)\le \chi_{B_{k+1}\backslash B_{k}}\frac{1}{2^{kn}r^{n}}\int_{B(x_{0},r)}|b(x)|dx\\ & \le C\chi_{B_{k+1}\backslash B_{k}}\frac{1}{2^{kn}r^{n}}\|b\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}. \end{align*}

Consequently, as $\omega \in A_{p(\cdot )}$, Definition 2.2 guarantees that

\begin{align*} \|d_{k}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}& \le C\|\chi_{B_{k+1}\backslash B_{k}}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\frac{1}{2^{kn}r^{n}}\|b\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}\\ & \le C\|\chi_{B_{k+1}}\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\frac{1}{2^{kn}r^{n}}\|b\|_{{L^{p'({\cdot})}_{\omega^{{-}1}}}}\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}\\ & \le C_{0}\frac{\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}}{\|\chi_{B_{k+1}}\|_{{L^{p({\cdot})}_{\omega}}}}\frac{u(x_{0},2^{k+1}r)}{u(x_{0},r)}\frac{1}{u(x_{0},2^{k+1}r)} \end{align*}

for some $C_{0}>0$.

Define $e_{k}=\frac {1}{\gamma _{k} }d_{k}$ where

(3.15)\begin{equation} \gamma_{k}= C_{0}\frac{\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}}{\|\chi_{B_{k+1}}\|_{{L^{p({\cdot})}_{\omega}}}}\frac{u(x_{0},2^{k+1}r)}{u(x_{0},r)}. \end{equation}

We see that $e_{k}\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$ with $\text {supp}\,d_{k}\subseteq B_{k+1}$. Since (3.9) asserts that

\[ \sum_{k=1}^{\infty}\gamma_{k}=C_{0}\sum_{k=1}^{\infty}\frac{\|\chi_{B_{0}}\|_{{L^{p({\cdot})}_{\omega}}}}{\|\chi_{B_{k+1}}\|_{{L^{p({\cdot})}_{\omega}}}}\frac{u(x_{0},2^{k+1}r)}{u(x_{0},r)}<\infty. \]

Therefore,

(3.16)\begin{equation} \operatorname{M}(b)=\sum_{k=1}^{\infty}\gamma_{k}e_{k}\in\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}\quad\text{with}\quad \|\operatorname{M}(b)\|_{\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}}\le C \end{equation}

for some $C>0$ independent of $b$.

For any $g\in \mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$, we have $g=\sum _{j=1}^{\infty }\lambda _{j}b_{j}$ with $b_{j}\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$ and $\sum _{j=1}^{\infty }|\lambda _{j}|\le 2\|g\|_{\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}}$. Let $\{e_{j,k}\}$ and $\{\gamma _{j,k}\}$ be defined by (3.15) and (3.16) with $b$ replaced by $b_{j}$. We have $\operatorname {M}(b_{j})=\sum _{k=1}^{\infty }\gamma _{j,k}e_{j,k}$. The sub-linearity of $\operatorname {M}$ yields

\[ \operatorname{M}(g)\le \sum_{j=1}^{\infty}|\lambda_{j}|\operatorname{M}(b_{j})=\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}|\lambda_{j}|\gamma_{j,k}e_{j,k}. \]

Since

\[ \sum_{j=1}^{\infty}\sum_{k=1}^{\infty}|\lambda_{j}|\gamma_{j,k}\le C\sum_{j=1}^{\infty}|\lambda_{j}|\le C\|g\|_{\mathfrak{B}_{p'({\cdot}),\omega^{{-}1}}^{u}}, \]

we find that $B=\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }|\lambda _{j}|\gamma _{j,k}e_{j,k}\in \mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}$.

Write

\[ G=\left\{\begin{array}{ll} \operatorname{M}(g)/B, & \, B\not= 0,\\ 0, & \, B=0.\end{array}\right. \]

Obviously,

\[ \operatorname{M}(g)=\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}|\lambda_{j}|\gamma_{j,k}g_{j,k} \]

where $g_{j,k}=Ge_{j,k}$. Since $|G|\le 1$, $g_{j,k}\in \mathfrak {b}_{p'(\cdot ),\omega ^{-1}}^{u}$ with $\text {supp}\, g_{j,k}\subseteq \text {supp}\, e_{j,k}$. Thus, $\|\operatorname {M}(g)\|_{\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}}\le \sum _{j=1}^{\infty }\sum _{k=1}^{\infty }|\lambda _{j}|\gamma _{j,k}\le C\|g\|_{\mathfrak {B}_{p'(\cdot ),\omega ^{-1}}^{u}}$.

A similar result of Theorem 3.5 is obtained in [Reference Sawano and Tanaka58]. We use the preceding theorem to establish the extrapolation theory for ${{M_{p(\cdot ),\omega }^{u}}}$. Let $p_{0}\in (0,\infty )$, $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $1< p_{-}\le p_{+}<\infty$, $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. The operator ${\mathcal {R}}$ is defined by

\[ {\mathcal{R}}h=\sum_{k=0}^{\infty}\frac{\operatorname{M}^{k}(h)}{2^{k}\|\operatorname{M}^{k}\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\to \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}},\quad h\in L^{1}_{loc}, \]

where $\operatorname {M}^{k}$ is the $k$ iterations of the operator $\operatorname {M}$. Notice that the operator ${\mathcal {R}}$ is depending on $p_{0},p(\cdot ),\omega$ and $u$.

Proposition 3.6 Let $p_{0}\in (0,\infty ),$ $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $p_{0}< p_{-}\le p_{+}<\infty$, $\omega \in A_{p(\cdot )}$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be a Lebesgue measurable function. If $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$ and $u^{p_{0}}\in {\mathbb {W}}_{p(\cdot )/p_{0},\omega ^{p_{0}}},$ then ${\mathcal {R}}$ is well defined on $\mathfrak {B}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$ and satisfies

(3.17)\begin{align} |h(x)|& \le {\mathcal{R}}h(x), \end{align}

(3.18)\begin{align} \|{\mathcal{R}}h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}& \le 2\|h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}},\quad h\in \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}, \end{align}

(3.19)\begin{align} [{\mathcal{R}}h]_{{\mathcal{A}}_{1}}& \le 2\|\operatorname{M}\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\to \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}},\quad h\in \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}. \end{align}

Proof. According to Theorem 3.5, the Hardy–Littlewood maximal operator is bounded on $\mathfrak {B}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$. Therefore, ${\mathcal {R}}$ is well defined on $\mathfrak {B}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$.

In view of the definition of ${\mathcal {R}}$, (3.17) and (3.18) are valid. Since $\operatorname {M}$ is a sublinear operator, for any $h\in \mathfrak {B}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$, we get

\begin{align*} \operatorname{M}({\mathcal{R}}h)& \le \sum_{k=0}^{\infty}\frac{\operatorname{M}^{k+1}(h)}{2^{k}\|\operatorname{M}^{k}\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\to \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}}\\ & \le 2\|\operatorname{M}\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\to \mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}{\mathcal{R}}h. \end{align*}

Consequently, ${\mathcal {R}}h\in {\mathcal {A}}_{1}$ and (3.19) is valid.

We now establish the extrapolation theory for the weighted Morrey spaces with variable exponents.

Theorem 3.7 Let $p_{0}\in (0,\infty ),$ $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $p_{0}< p_{-}\le p_{+}<\infty$, $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$ and $u^{p_{0}}\in {\mathbb {W}}_{p(\cdot )/p_{0},\omega ^{p_{0}}}$. Suppose that ${\mathcal {F}}$ be a family of pairs of non-negative Lebesgue measurable functions such that for every

\[ v\in\{{\mathcal{R}}h:h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\}, \]

we have

(3.20)\begin{equation} \int_{{\Bbb R}^{n}}f(x)^{p_{0}}v(x) dx\le C\int_{{\Bbb R}^{n}}g(x)^{p_{0}}v(x)dx<\infty,\quad (f,g)\in{\mathcal{F}} \end{equation}

where $C$ is independent of $f$ and $g$. For any $(f,g)\in {\mathcal {F}}$ with $g\in {{M_{p(\cdot ),\omega }^{u}}}$, we have

(3.21)\begin{equation} \|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le C\|g\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \end{equation}

Proof. Let $f\in {\mathcal {M}}$ with $(f,g)\in {\mathcal {F}}$ for some $g\in {{M_{p(\cdot ),\omega }^{u}}}$. For any $h\in \mathfrak {b}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$, (3.20) with $v={\mathcal {R}}h$ and (3.17) assure that

\begin{align*} \int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}|h(x)|dx& \le \int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}{\mathcal{R}}h(x)dx\\ & \le \int_{{\mathbb{R}}^{n}}|g(x)|^{p_{0}}{\mathcal{R}}h(x)dx\\ & \le C\||g|^{p_{0}}\|_{{M^{u^{p_{0}}}_{p({\cdot})/p_{0},\omega^{p_{0}}}}}\|{\mathcal{R}}h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}. \end{align*}

Thus, (3.18) gives

\[ \int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}|h(x)|dx\le C\|g\|_{{{M_{p({\cdot}),\omega}^{u}}}}^{p_{0}}\|h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}} \]

for some $C>0$.

Furthermore, we have

\begin{align*} & \sup\bigg\{\int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}|h(x)|dx,\, h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}},\ \bigg\}\\ & \le C\|g\|_{{{M_{p({\cdot}),\omega}^{u}}}}^{p_{0}}. \end{align*}

Proposition 3.3 guarantees that $f\in {{M_{p(\cdot ),\omega }^{u}}}$ and

\begin{align*} \|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}^{p_{0}}& =\||f|^{p_{0}}\|_{{M^{u^{p_{0}}}_{p({\cdot})/p_{0},\omega^{p_{0}}}}}\\ & =\sup\bigg\{\int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}|h(x)|dx,\, h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\bigg\}\\ & \le C\|g\|_{{{M_{p({\cdot}),\omega}^{u}}}}^{p_{0}} \end{align*}

for some $C>0$.

The above theorem is a refined version of the general extrapolation theory [Reference Rubio de Francia51–Reference Rubio de Francia52] by using the idea from [Reference Ho28]. The general extrapolation theory requires the validity of (3.20) for all $\omega \in A_{1}$ while Theorem 3.7 only requires the validity of (3.20) for $\omega \in \{{\mathcal {R}}h:h\in \mathfrak {b}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}\}$. This relaxation gives us extra flexibility to obtain the boundedness of operators without using the density argument.

Theorem 3.7 yields the boundedness of pseudo-differential operators and Fourier integral operators on ${{M_{p(\cdot ),\omega }^{u}}}$.

Definition 3.4 Let $m\in {\mathbb {R}}$. A smooth function $a(x,\xi )\in C^{\infty }({\mathbb {R}}^{n}\times {\mathbb {R}}^{n})$ is a symbol of order $m$ if for any multi-indices $\alpha,\beta$, we have a constant $C>0$ such that

\[ \left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)\right|\le C(1+|\xi|)^{m-\beta}. \]

Let $a$ be a symbol of order $m$. The pseudo-differential operator with symbol $a$ is defined as

\[ T_{a}f(x)=\int_{{\mathbb{R}}^{n}}a(x,\xi)\hat{f}(\xi)e^{2\pi i x\cdot \xi}d\xi,\quad f\in{{\mathcal{S}}} \]

where $\hat {f}$ is the Fourier transform of $f$.

In view of [Reference Miller41, Theorem 2.12], we have the following weighted norm inequality for the pseudo-differential operators of order 0.

We are now ready to present the boundedness of pseudo-differential operators on the weighted Morrey spaces with variable exponents. Notice that in view of [Reference Stein60], $T_{a}$ is bounded on ${{\mathcal {S}}'}$. Thus, $T_{a}$ is well defined on ${{M_{p(\cdot ),\omega }^{u}}}$.

Proof. For any $f\in {{M_{p(\cdot ),\omega }^{u}}}$ and $h\in \mathfrak {b}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}$ Lemma 3.1 and (3.18) guarantee that

\begin{align*} \int_{{\mathbb{R}}^{n}}|f(x)|^{p_{0}}{\mathcal{R}}h(x)dx& \le \||f|^{p_{0}}\|_{{M^{u^{p_{0}}}_{p({\cdot})/p_{0},\omega^{p_{0}}}}}\|{\mathcal{R}}h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}\\ & \le C\|f\|_{{{M_{p({\cdot}),\omega}^{u}}}}^{p_{0}}\|h\|_{\mathfrak{B}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}<\infty. \end{align*}

That is,

(3.22)\begin{equation} {{M_{p({\cdot}),\omega}^{u}}}\hookrightarrow\bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}L^{p_{0}}({\mathcal{R}}h). \end{equation}

Define ${\mathcal {F}}_{0}=\{(|T_{a}f|,|f|): f\in {{M_{p(\cdot ),\omega }^{u}}}\}$. For any

\[ v\in \bigg\{{\mathcal{R}}h:h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\bigg\}, \]

(3.22) ensures that ${{M_{p(\cdot ),\omega }^{u}}}\hookrightarrow L^{p_{0}}(v)$. Theorem 3.8 guarantees that (3.20) is valid for ${\mathcal {F}}_{0}$. Theorem 3.7 asserts that $\|T_{a}f\|_{{{M_{p(\cdot ),\omega }^{u}}}}\le C\|f\|_{{{M_{p(\cdot ),\omega }^{u}}}}$, $\forall f\in {{M_{p(\cdot ),\omega }^{u}}}$, for some $C>0$.

Proof. In view of the left openness property of Muckenhoupt weight functions [Reference Grafakos17, Corollary 9.2.6], we have a $p_{0}\in (1,p)$ such that $\omega ^{p}\in {\mathcal {A}}_{p/p_{0}}$. Whenever $p(\cdot )=p$, $p\in (1,\infty )$ is a constant function, we have $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$ if and only if $\omega ^{p}\in {\mathcal {A}}_{p/p_{0}}$. Thus, we have $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$.

Since $u\in {\mathbb {W}}_{p,\omega }$, (3.9) gives

\[ \sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{L^{p}_{\omega}}}{\|\chi_{B(x,2^{j+1}r)}\|_{L^{p}_{\omega}}}u(x,2^{j+1}r) \le Cu(x,r). \]

As $p_{0}>1$, we find that

\begin{align*} \sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{L^{p/p_{0}}_{\omega^{p_{0}}}}}{\|\chi_{B(x,2^{j+1}r)}\|_{L^{p/p_{0}}_{\omega^{p_{0}}}}}u(x,2^{j+1}r)^{p_{0}}& =\sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{L^{p}_{\omega}}^{p_{0}}}{\|\chi_{B(x,2^{j+1}r)}\|_{L^{p}_{\omega}}^{p_{0}}}u(x,2^{j+1}r)^{p_{0}}\\ & \le \left(\sum_{j=0}^{\infty}\dfrac{\|\chi_{B(x,r)}\|_{L^{p}_{\omega}}}{\|\chi_{B(x,2^{j+1}r)}\|_{L^{p}_{\omega}}}u(x,2^{j+1}r) \right)^{p_{0}} \le Cu(x,r)^{p_{0}}. \end{align*}

Therefore, $u^{p_{0}}\in {\mathbb {W}}_{p/p_{0},\omega ^{p_{0}}}$. Theorem 3.9 yields the boundedness of $T_{a}$ on the weighted Morrey space $M^{u}_{p,\omega }$.

One of the essential components in the proof of [Reference Miller41, Theorem 2.12] is the density of ${{\mathcal {S}}}$ in $L^{p}(\omega )$. As ${{\mathcal {S}}}$ is not necessary dense in ${{M_{p(\cdot ),\omega }^{u}}}$, the argument in [Reference Miller41, Theorem 2.12] cannot directly apply to obtain the boundedness of $T_{a}$ on ${{M_{p(\cdot ),\omega }^{u}}}$. We can overcome this obstacle because we have the embedding (3.22) and Theorem 3.7 requires the validity of 3.20 for all $\omega \in \{{\mathcal {R}}h:h\in \mathfrak {b}_{(p(\cdot )/p_{0})',\omega ^{-p_{0}}}^{u^{p_{0}}}\}$ only.

There are a number of extensions on the study of the boundedness of pseudo-differential operators on weighted Lebesgue spaces [Reference Michalowski, Rule and Staubach40, Reference Nishigiki49, Reference Sato53, Reference Sato54, Reference Yabuta68]. By using Theorem 3.7, we can extend the results in [Reference Michalowski, Rule and Staubach40, Reference Nishigiki49, Reference Sato53, Reference Yabuta68] to the weighted Morrey spaces with variable exponents. For brevity, we omit the details and leave them to the readers.

We turn to the study of Fourier integral operators. We recall some definitions and notions from [Reference Ferreira and Staubach12].

Definition 3.5 Let $m\in {\mathbb {R}}$ and $0\le \rho \le 1$. A Lebesgue measurable function $a(x,\xi )$ which is smooth in the frequency variable $\xi$ and bounded in the spatial variable $x$, is said to belong to $L^{\infty }S^{m}_{\rho }$ if for all multi-indices $\alpha$, $a$ satisfies

\[ \sup_{\xi\in{\mathbb{R}}^{n}}(1+|\xi|^{2})^{\frac{-m+\rho|\alpha|}{2}}\|\partial^{\alpha}_{\xi}a({\cdot},\xi)\|_{L^{\infty}({\mathbb{R}}^{n})}<\infty. \]

Comparing to the classical symbol class $S_{\rho,\delta }^{m}$ for the pseudo-differential operators [Reference Stein60, Chapter VII, Section 1.1 (3)], Definition 3.5 gives a class of symbols where smoothness of the $x$ variable is not required.

Let $\varphi (x,\xi )$ be a smooth function and homogeneous of degree 1 in the frequency variable and the amplitude $a(x,\xi )\in L^{\infty }S^{m}_{\rho }$ where $m\in {\mathbb {R}}$ and $0\le \rho \le 1$. The Fourier integral operator associated with $a$ and $\varphi$ is defined as

\[ T_{a,\varphi}f(x)=(2\pi)^{{-}n}\int_{{\mathbb{R}}^{n}}e^{i\varphi(x,\xi)}a(x,\xi)\hat{f}(\xi)d\xi. \]

We now state some conditions for the study of Fourier integral operators.

Definition 3.6 Let $k\in {\mathbb {N}}\cup \{0\}$. A real-valued function $\varphi (x,\xi )$ belongs to the class $L^{\infty }\Phi ^{k}$ if it is homogeneous of degree 1 and smooth on ${\mathbb {R}}^{n}\backslash \{0\}$ in the frequency variable $\xi$, bounded measurable in the spatial variable $x$ and for all multi-indices $|\alpha |\ge k$, it satisfies

\[ \sup_{\xi\in{\mathbb{R}}^{n}\backslash\{0\}}|\xi|^{{-}1+|\alpha|}\|\partial^{\alpha}_{\xi}\varphi({\cdot},\xi)\|_{L^{\infty}}<\infty. \]

Definition 3.7 A real-valued function $\varphi$ satisfies the rough non-degeneracy condition if it is $C^{1}$ on ${\mathbb {R}}^{n}\backslash \{0\}$ in the frequency variable $\xi$, bounded measurable in the spatial variable $x$ and there exists a constant $c>0$ such that for all $x,y\in {\mathbb {R}}^{n}$ and $\xi \in {\mathbb {R}}^{n}\backslash \{0\}$, it satisfies

\[ |\partial_{\xi}\varphi(x,\xi)-\partial_{\xi}\varphi(y,\xi)|\ge c|x-y|. \]

We have the following boundedness result for $T_{a,\varphi }$ on weighted Lebesgue spaces [Reference Ferreira and Staubach12, Theorem 3.11].

We now present the boundedness of the Fourier integral operator $T_{a,\varphi }$ on ${{M_{p(\cdot ),\omega }^{u}}}$.

By using Theorem 3.11 instead of Theorem 3.8, the proof of the preceding theorem follows from the proof of Theorem 3.9. Thus, for brevity, we omit the details.

4. Weighted Triebel–Lizorkin–Morrey spaces with variable exponents

In this section, we extend the study in the previous section to the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. For any $f\in {{\mathcal {S}}'}$, let us denote the Fourier transform of $f$ by $\hat {f}$.

Definition 4.1 Let $\alpha \in {\Bbb R}$, $1\le q<\infty$, $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $0< p_{-}\le p_{+}<\infty$, $\omega :{\mathbb {R}}^{n}\to (0,\infty )$ and $u(\cdot ):(0,\infty )\to (0,\infty )$ be Lebesgue measurable functions. The weighted Triebel–Lizorkin–Morrey space with variable exponent ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$ consists of those $f\in {{\mathcal {S}}'}$ satisfying

(4.1)\begin{equation} \|f\|_{{F^{\alpha,q,u}_{p({\cdot}),\omega}}}= \bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}<\infty, \end{equation}

where $\varphi _{0}\in {{\mathcal {S}}}$ satisfies

(4.2)\begin{equation} \mathrm{supp}\:\hat{\varphi_{0}}\subseteq \{x\in{\mathbb{R}}^{n}\: : |x|\le 2\}\quad and\quad |\hat{\varphi_{0}}(\xi)|\ge C,\, |x|\le 3/2 \end{equation}

and $\varphi _{j}(x)=2^{jn}\varphi (2^{j}x)$, $j\ge 1$ where $\varphi \in {{\mathcal {S}}}$ satisfying

(4.3)\begin{align} & \mathrm{supp}\,\hat{\varphi}\subseteq \{x\in{\mathbb{R}}^{n} : 1/2\le|x|\le 2\} \end{align}

(4.4)\begin{align} & \quad|\hat{\varphi}(\xi)|\ge C,\quad 3/5\le|x|\le 5/3 \end{align}

for some $C>0$.

The above definition is a special case of the Triebel–Lizorkin type spaces defined in [Reference Liang, Sawano, Ullrich, Yang and Yuan37]. In particular, when $\omega \equiv 1$, the weighted Triebel–Lizorkin–Morrey space with variable exponent becomes the inhomogeneous version of the Triebel–Lizorkin–Morrey space with variable exponent [Reference Ho23, Definition 6.6] with $q(\cdot )$ being a constant function. When $\omega \equiv 1$ and $p(\cdot )$ is a constant function, then ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$ reduces to the inhomogeneous version of the Triebel–Lizorkin–Morrey spaces studied in [Reference Strömberg and Torchinsky61, Reference Tang and Xu62]. Moreover, when $\omega \equiv 1$ and $u\equiv 1$, ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$ is the inhomogeneous version of the variable Triebel–Lizorkin spaces introduced in [Reference Xu67, Definition 2].

Let $p,q\in (0,\infty )$, $\alpha \in {\mathbb {R}}$ and $\omega :{\mathbb {R}}^{n}\to (0,\infty )$, the weighted Triebel–Lizorkin space $F^{\alpha,q}_{p}(\omega )$ consists if all $f\in {{\mathcal {S}}'}$ satisfying

\[ \|f\|_{F^{\alpha,q}_{p}(\omega)}= \bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{L^{p}(\omega)}<\infty \]

where $\varphi _{0}$ and $\varphi$ satisfy (4.2) and (4.3), see [Reference Bui5, Reference Bui, Palusyński and Taibleson6] and [Reference Frazier and Jawerth13, p.124]. We see that whenever $p(\cdot )=p$, $p\in (0,\infty )$ is a constant function, ${F^{\alpha,q,u}_{p(\cdot ),\omega }}=F^{\alpha,q}_{p}(\omega ^{p})$.

We first show that the definition of the weighted Triebel–Lizorkin–Morrey space with variable exponent is independent of the choice of functions $\varphi _{0},\varphi$ satisfying (4.2)–(4.4).

Theorem 4.1 Let $\alpha \in {\Bbb R}$, $1\le q<\infty$, $p_{0}\in (0,\infty ),$ $p(\cdot )\in C^{\log }({\Bbb R}^{n})$ with $p_{0}< p_{-}\le p_{+}<\infty$. If $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$ and $u^{p_{0}}\in {\mathbb {W}}_{p(\cdot )/p_{0},\omega ^{p_{0}}}$. Let $\varphi _{0},\psi _{0}\in {{\mathcal {S}}}$ satisfy (4.2) and $\varphi,\psi \in {{\mathcal {S}}}$ satisfy (4.3) and (4.4). There exist constants $C_{0},C_{1}>0$ such that for any $f\in {{\mathcal {S}}'},$ we have

(4.5)\begin{align} C_{0}\bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}& \le \bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\psi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}\nonumber\\ & \le C_{1}\bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \end{align}

Proof. In view of [Reference Frazier and Jawerth13, Remark 2.6 and Proposition 10.14], for any $\omega \in {\mathcal {A}}_{1}$, we have

(4.6)\begin{align} C_{0}\bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{L^{p}(\omega)}& \le \bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\psi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{L^{p}(\omega)}\nonumber\\ & \le C_{1}\bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{L^{p}(\omega)}. \end{align}

Notice that the results given in [Reference Frazier and Jawerth13, Remark 2.6 and Proposition 10.14] are for the homogeneous Triebel–Lizorkin spaces, as stated at [Reference Frazier and Jawerth13, Section 12], the results also apply to the inhomogeneous Triebel–Lizorkin spaces.

We denote the weighted Triebel–Lizorkin–Morrey space with variable exponent generated by $(\varphi _{0},\varphi )$ and $(\psi _{0},\psi )$ by ${F^{\alpha,q,u}_{p(\cdot ),\omega }}(\varphi )$ and ${F^{\alpha,q,u}_{p(\cdot ),\omega }}(\psi )$, respectively.

The embedding (3.22) guarantees that for any $f\in {F^{\alpha,q,u}_{p(\cdot ),\omega }}(\psi )$

\[ \left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\psi}_{j}|)^{q}\right)^{\frac{1}{q}}\in \bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}L^{p_{0}}({\mathcal{R}}h). \]

Define

\[ {\mathcal{F}}=\bigg\{\bigg(\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}, \left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\psi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg):f\in {F^{\alpha,q,u}_{p({\cdot}),\omega}}(\psi)\bigg\}. \]

The first inequality in (4.6) shows that (3.20) is valid for ${\mathcal {F}}$. Theorem 3.7 yields a constant $C_{0}>0$ such that for any $f\in {F^{\alpha,q,u}_{p(\cdot ),\omega }}(\psi )$

\[ C_{0}\bigg\|\left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le \bigg\|\left(\sum_{j=0}^{\infty}(2^{ j\alpha}|f\ast{\psi}_{j}|)^{q}\right)^{\frac{1}{q}}\bigg\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \]

We establish the first inequality in (4.9). The second inequality in (4.9) follows similarly.

We study the boundedness of pseudo-differential operators on ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$. Let $L$ be a non-negative integer, $r\in (0,\infty )\backslash {\mathbb {N}}$ and $\delta \in [0,1]$. Suppose that $a$ satisfies

(4.7)\begin{align} & |\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\le \frac{1}{(1+|\xi|)^{|\alpha|}}, \, |\alpha|\le L,\, \end{align}

(4.8)\begin{align} & |\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x+y,\xi)-\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\le \frac{|y|^{r-[r]}}{(1+|\xi|)^{|\alpha|-\delta r}}, \, |\alpha|\le L,\,|\beta|=[r] \end{align}

where $[r]$ is the integral part of $r$.

According to [Reference Sato54, Theorem 1 and Remaek 3], we have the boundedness result for the pseudo-differential operators on the weighted Triebel–Lizorkin spaces.

We now extend the boundedness of $T_{a}$ to ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$.

Proof. In view of Theorem 4.2, for any $\omega \in {\mathcal {A}}_{1}\subseteq {\mathcal {A}}_{p_{0}/q}$, the pseudo-differential operator $T_{a}$ is bounded on $F^{\alpha,q}_{p_{0}}(\omega )$. In view of (3.19) and the embedding (3.22) guarantees that

(4.9)\begin{equation} {F^{\alpha,q,u}_{p({\cdot}),\omega}}\hookrightarrow\bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}F^{\alpha,u}_{{p_{0}}}({\mathcal{R}}h), \end{equation}

we are allowed to apply Theorem 3.7 with

\[ {\mathcal{F}}=\bigg\{\Bigg(\left(\sum_{j=0}^{\infty}(2^{j\alpha}|T_{a}f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}, \left(\sum_{j=0}^{\infty}(2^{j\alpha}|f\ast{\varphi}_{j}|)^{q}\right)^{\frac{1}{q}}\Bigg):f\in {F^{\alpha,q,u}_{p({\cdot}),\omega}}\bigg\} \]

and obtain the boundedness of $T_{a}$ on ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$.

As a special case of Theorem 4.3, we have the boundedness of the pseudo-differential operator $T_{a}$ on the weighted Triebel–Lizorkin–Morrey spaces.

There are some other results on the boundedness of pseudo-differential operators on the weighted Triebel–Lizorkin spaces such as [Reference Bui4, Section 3.4 (c)]. We can use the method in Theorem 4.2 to extend the result in [Reference Bui4, Section 3.4 (c)] to ${{M_{p(\cdot ),\omega }^{u}}}$. For brevity, we leave the details to the readers.

At the end of this section, we consider the boundedness of Fourier integral operators on ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$.

Theorem 4.5 Let $\alpha \in {\Bbb R},$ $1\le q<\infty,$ $p\in (1,\infty ),$ $\omega \in {\mathcal {A}}_{p},$ $a(\xi )$ is a symbol of order $-\frac {n+1}{2}$ and $\varphi \in C^{\infty }({\mathbb {R}}^{n}\backslash \{0\})$ is a positive homogeneous function of degree 1. If $|\mathrm {det}_{n-1}\partial ^{2}_{\xi \xi }\varphi (\xi )|\ge c>0$ and for any $|\alpha |\ge 1,$ there is a constant $C>0$ such that

(4.10)\begin{equation} |\partial^{\alpha}_{\xi}\varphi(\xi)|\le C|\xi|^{1-|\alpha|}, \quad \xi\in {\mathbb{R}}^{n}\backslash\{0\}, \end{equation}

then the Fourier integral operator

\[ Tf(x)=\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}}^{n}}e^{i\varphi(\xi)+ix\cdot\xi}a(\xi)\hat{f}(\xi)d\xi \]

is bounded on $F^{\alpha,q}_{p}(\omega )$.

The reader is referred to [Reference Ferreira and Staubach12, Theorem 4.1.4] for the proof of the above theorem. In view of the embedding (4.9) and the preceding theorem, $T$ is well defined on ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$.

The above theorem and Theorem 3.7 yield the boundedness of the Fourier integral operator $T$ on ${F^{\alpha,q,u}_{p(\cdot ),\omega }}$.

As the proof of the preceding theorem is similar to the proof of Theorem 4.3, for simplicity, we omit the details.

5. Weighted Hardy–Morrey spaces with variable exponents

In this section, we study the weighted Hardy–Morrey spaces with variable exponents by using the extrapolation theory. This approach had been used in [Reference Ho27, Reference Ho30, Reference Ho31] for the weighted Hardy spaces with variable exponents, the Orlicz-slice Hardy spaces and the Hardy local Morrey spaces with variable exponents.

We begin with the definition of the weighted Hardy–Morrey spaces with variable exponents. Let ${\mathcal {F}}=\{\|\cdot \|_{\alpha _{i},\beta _{i}}\}$ be any finite collection of semi-norms on ${{\mathcal {S}}}$ and

\[ {\mathcal{S}}_{\mathcal{F}}=\{\psi\in{{\mathcal{S}}}:\|\psi\|_{\alpha_{i},\beta_{i}}\le 1, \text{ for all } \|\cdot\|_{\alpha_{i},\beta_{i}}\in{\mathcal{F}}\}. \]

For any $f\in {{\mathcal {S}}'}$, write

\[ {\mathcal{M}}_{\mathcal{F}}f(x)=\sup_{\psi\in{\mathcal{S}}_{\mathcal{F}}}\sup_{t>0}|(f\ast\psi_{t})(x)| \]

where for any $t>0$, write $\psi _{t}(x)=t^{-n}\psi (x/t)$.

Definition 5.1 Let $p(\cdot ):{\mathbb {R}}^{n}\to (0,\infty )$, $\omega :{\mathbb {R}}^{n}\to (0,\infty )$ and $u: (0,\infty )\to (0,\infty )$ be Lebesgue measurable functions. The weighted Hardy–Morrey space with variable exponent ${{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$ consists of all $f\in {{\mathcal {S}}'}$ satisfying

\[ \|f\|_{{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}}=\|{\mathcal{M}}_{\mathcal{F}}f\|_{{{M_{p({\cdot}),\omega}^{u}}}}<\infty. \]

When $\omega \equiv 1$, the weighted Hard–Morrey spaces with variable exponents become the Hardy–Morrey spaces with variable exponents [Reference Ho25, Reference Ho29]. When $u\equiv 1$, the weighted Hardy Morrey spaces with variable exponents reduce to the Hardy spaces with variable exponents [Reference Ho26, Reference Ho27]. When $p(\cdot )=p$, $p\in (0,1]$, is a constant function, ${{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$ becomes the weighted Hardy–Morrey space ${\mathcal {H}}^{u}_{p,\omega }$. For $p\in (0,1]$ and $\omega :{\mathbb {R}}^{n}\to (0,\infty )$, the weighted Hardy space $H^{p}(\omega )$ consists of all $f\in {{\mathcal {S}}'}$ satisfies

\[ \|f\|_{H^{p}(\omega)}=\|{\mathcal{M}}_{\mathcal{F}}f\|_{L^{p}(\omega)}<\infty. \]

We recall the definition of regular Calderón–Zygmund operators from [Reference García-Cuerva and Kazarian15, Definitions 2.1, 2.2 and 2.6].

Definition 5.2 We say that $T$ is a singular integral operator with kernel $K(x,y)$ if for every bounded function $f$ with compact support

\[ Tf(x)=\int_{{\mathbb{R}}^{n}}K(x,y)f(y)dy,\quad x\in{\mathbb{R}}^{n}\backslash\text{supp}f. \]

If $T$ is bounded on $L^{p}$ for some $p\in (1,\infty )$, $T$ is called as a Calderón–Zygmund operator.

Let $\gamma >0$. We say that $K$ is $\gamma$-regular with respect to $y$ if for every multi-index $\alpha$ with $|\alpha |<\gamma$,

\[ |\partial^{\alpha}_{y}K(x,y)|\le C|x-y|^{{-}n-|\alpha|} \]

and for those multi-index satisfying $\alpha =[\gamma ]$

\[ |\partial^{\alpha}_{y}K(x,y)-\partial^{\alpha}_{y}K(x,z)|\le C\frac{|y-z|^{\gamma-|\alpha|}}{|x-y|^{n+\gamma}} \]

whenever $|y-z|\le \frac {1}{2}|x-y|$.

The following theorem gives the mapping properties of the Calderón–Zygmund operator on the weighted Hardy spaces.

The reader is referred to [Reference García-Cuerva and Kazarian15, Theorem 2.8] for the proof of Theorem 5.1.

Proof. Whenever $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$, according to (3.22), we have

(5.1)\begin{equation} {\mathcal{M}}_{\mathcal{F}}f\in{{M_{p({\cdot}),\omega}^{u}}}\hookrightarrow\bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}L^{p_{0}}({\mathcal{R}}h). \end{equation}

That is,

(5.2)\begin{equation} {{\mathcal{H}}_{p({\cdot}),\omega}^{u}}\hookrightarrow \bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}H^{p_{0}}({\mathcal{R}}h). \end{equation}

Consequently, (3.19) and Theorem 5.1 show that for any

\[ v\in \bigg\{{\mathcal{R}}h:h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\bigg\}, \]

we have a constant $C>0$ such that for any $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$

\[ \int_{{\mathbb{R}}^{n}}|Tf(x)|^{p_{0}}v(x)dx\le C\int_{{\mathbb{R}}^{n}}({\mathcal{M}}_{\mathcal{F}}f(x))^{p_{0}} v(x)dx. \]

Applying Theorem 3.7 with

\[ {\mathcal{F}}=\{(|Tf|,{\mathcal{M}}_{\mathcal{F}}f):f\in{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}\}, \]

we obtain a constant $C>0$ such that for any $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$, we have

\[ \|Tf\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le C\|{\mathcal{M}}_{\mathcal{F}}f\|_{{{M_{p({\cdot}),\omega}^{u}}}}=C\|f\|_{{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}}.\quad\rule{1.5mm}{1.5mm} \]

In particular, we have the mapping properties of the Calderón-Zygmund operators on the weighted Hardy–Morrey spaces.

The mapping properties of Calderón–Zygmund operators on the weighted Morrey spaces with variable exponents were established in [Reference Guliyev, Hasanov and Badalov20, Theorem 4.3].

In [Reference García-Cuerva and Kazarian15], the boundedness of the Calderón–Zygmund operators is used to establish the wavelet characterization of weighted Hardy spaces. We also have the wavelet characterization for the weighted Hardy–Morrey spaces with variable exponents. We obtain this characterization by solely using Theorem 3.7. For the wavelet characterizations of the Besov–Morrey spaces and the Triebel–Lizorkin–Morrey spaces, the reader is referred to [Reference Sawano56].

We recall some definitions from wavelet theory. A function $\psi \in L^{2}({\mathbb {R}})$ is an orthonormal wavelet if the system

\[ \psi_{j,k}(x)=2^{j/2}\psi(2^{j}x-k),\quad j,k\in{\mathbb{Z}} \]

is an orthonormal basis for $L^{2}({\mathbb {R}})$.

For any wavelet $\psi$, write

\[ {\mathcal{W}}_{\psi}f=\left(\sum_{j,k\in{\mathbb{Z}}}2^{j}|\langle f,\psi_{j,k}\rangle |^{2}\chi_{I_{j,k}}\right)^{1/2} \]

where $I_{j,k}=[2^{-j}k,2^{-j}(k+1)]$.

We have the following wavelet characterization of the weighted Hardy spaces [Reference García-Cuerva and Martell16, Theorem 4.2].

Theorem 5.4 Let $\alpha \ge 1,$ $\frac {1}{\alpha }\le p,$ $\omega \in {\mathcal {A}}_{1}$ and $\psi \in C^{[\alpha ]}$. If there exist $C,r,\epsilon >0$ such that

(5.3)\begin{align} & \int_{{\mathbb{R}}}x^{k}\psi(x)dx=0,\quad k=0,1\cdots,[\alpha]-1, \end{align}

(5.4)\begin{align} & |\psi(x)|\le C(1+|x|)^{-[\alpha]-r-1},\quad \forall x\in{\mathbb{R}}, \end{align}

(5.5)\begin{align} & |\psi^{(k)}(x)|\le C(1+|x|)^{-\alpha-\epsilon},\quad \forall x\in{\mathbb{R}},\quad\text{and}\quad k=0,1\cdots,[\alpha], \end{align}

then there exist $C_{0}>C_{1}>0$ such that

\[ C_{0}\|f\|_{H^{p}(\omega)}\le \|{\mathcal{W}}_{\psi}f\|_{L^{p}(\omega)}\le C_{1}\|f\|_{H^{p}(\omega)}. \]

For any $w\in {\mathcal {A}}_{1}$, Theorem 5.4 guarantees that the linear functional $l_{\psi _{j,k}}(f)=\langle f,\psi _{j,k}\rangle$ is bounded on the weighted Hardy space $H^{p}(w)$. In view of the embedding (5.2), we see that the functional $l_{\psi _{j,k}}$ is also bounded on ${{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$. Thus, $\langle f,\psi _{j,k}\rangle$ and ${\mathcal {W}}_{\psi }f$ are well defined on ${{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$. For the dual spaces of the weighted Hardy spaces, the reader is referred to [Reference García-Cuerva14, Theorem II.4.4].

We give the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents in the following theorem.

Theorem 5.5 Let $\alpha \ge 1$ and $\psi \in C^{[\alpha ]}$. Suppose that $\psi$ satisfies (5.3)-(5.5). Let $p_{0}\in (\alpha ^{-1},1),$ $p(\cdot )\in C^{\log }({\Bbb R})$ with $p_{0}< p_{-}\le p_{+}<\infty$. If $\omega ^{p_{0}}\in A_{p(\cdot )/p_{0}}$ and $u^{p_{0}}\in {\mathbb {W}}_{p(\cdot )/p_{0},\omega ^{p_{0}}},$ then there exist $C_{0}>C_{1}>0$ such that for any $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$

(5.6)\begin{equation} C_{0}\|f\|_{{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}}\le \|{\mathcal{W}}_{\psi}f\|_{{{M_{p({\cdot}),\omega}^{u}}}}\le C_{1}\|f\|_{{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}}. \end{equation}

Proof. For any $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$ with ${\mathcal {W}}_{\psi }f\in {{M_{p(\cdot ),\omega }^{u}}}$, (3.22) assures that

\[ {\mathcal{W}}_{\psi}f\in \bigcap_{h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}}L^{p_{0}}({\mathcal{R}}h). \]

Therefore, for any

\[ v\in \bigg\{{\mathcal{R}}h:h\in \mathfrak{b}_{(p({\cdot})/p_{0})',\omega^{{-}p_{0}}}^{u^{p_{0}}}\bigg\}, \]

Theorem 5.4 shows that

\[ \int_{{\mathbb{R}}}({\mathcal{M}}_{\mathcal{F}}f(x))^{p_{0}} v(x)dx\le C\int_{{\mathbb{R}}}({\mathcal{W}}_{\psi}f(x))^{p_{0}}v(x)dx. \]

By applying Theorem 3.7 with

\[ {\mathcal{F}}=\{({\mathcal{M}}_{\mathcal{F}}f,{\mathcal{W}}_{\psi}f):f\in {{\mathcal{H}}_{p({\cdot}),\omega}^{u}}\}, \]

we obtain a constant $C_{0}>0$ such that for any $f\in {{\mathcal {H}}_{p(\cdot ),\omega }^{u}}$,

\[ C_{0}\|f\|_{{{\mathcal{H}}_{p({\cdot}),\omega}^{u}}}\le \|{\mathcal{W}}_{\psi}f\|_{{{M_{p({\cdot}),\omega}^{u}}}}. \]

We establish the first inequality in (5.6). The proof of the second inequality in (5.6) follows similarly.

The following wavelet characterization of the weighted Hardy–Morrey space is a special case of Theorem 5.5.

Corollary 5.6 Let $\alpha \ge 1,$ $\psi \in C^{[\alpha ]},$ $p\in (\alpha ^{-1},1]$ and $\omega ^{p}\in {\mathcal {A}}_{1}$. Suppose that $\psi$ satisfies (5.3) and (5.5). If there exists a $p_{0}\in (\alpha ^{-1},p)$ such that $u^{p_{0}}\in {\mathbb {W}}_{p/p_{0},\omega ^{p_{0}}},$ then there exist constants $C_{0},C_{1}>0$ such that for any $f\in {\mathcal {H}}^{u}_{p,\omega }$

\[ C_{0}\|f\|_{{\mathcal{H}}^{u}_{p,\omega}}\le \|{\mathcal{W}}_{\psi}f\|_{M^{u}_{p,\omega}}\le C_{1}\|f\|_{{\mathcal{H}}^{u}_{p,\omega}}. \]