INTRODUCTION

In this paper we consider the global Morrey-type spaces \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )\) with variable exponents \(p(\cdot ) \), \(\theta (\cdot )\) and a general function \( w(x,r)\) defining a Morrey-type norm. The Morrey spaces \( {M}_{p,\lambda }\) are introduced in [1] in the frames of the study of partial differential equations. Many classical operators of harmonic analysis (e. g., maximal, fractional maximal, potential operators) were studied in the Morrey-type spaces with constant exponents \(p \), \(\theta \) [2,3,4]. The Morrey spaces also attracted attention of researchers in the area of variable exponent analysis; see [5,6,7,8,9,10]. The Morrey spaces \({\mathcal {L}}_{p(\cdot ),\lambda (\cdot )}\) with variable exponent \(p(\cdot ) \), \(\lambda (\cdot )\) were introduced and studied in [5]. The general versions \( {M}_{p(\cdot ),w(\cdot )}(\Omega )\), \(\Omega \subset {\mathbb {R}}^{n}\), were introduced and studied in [11, 12]. The boundedness of maximal and potential type operators in the generalized Morrey-type spaces with a variable exponent were considered in [11] in the case of bounded sets \(\Omega \subset {\mathbb {R}}^{n}\), in [12] in the case of unbounded sets \(\Omega \subset {\mathbb {R}}^{n}\).

Let \(f\in {L}_{\operatorname {loc}}({\mathbb {R}}^{n}) \).The Hardy–Littlewood maximal operator is defined as

$$ Mf(x)=\mathop {\sup }_{r>0}\frac {1}{|B(x,r)|}\int _{\tilde {B}(x,r)}|f(y)|\thinspace dy, $$

where \(B(x,r)\) is a ball in \({\mathbb R}^{n} \) centered at a point \(x\in {\mathbb {R}}^{n} \) and of a radius \(r \), \(\tilde {B}(x,r)=B(x,r)\cap {\Omega } \), \(\Omega \subset {\mathbb {R}}^{n} \).

The fractional maximal operator of variable order \(\alpha (x) \) is defined as

$$ {M}^{\alpha (\cdot )}f(x)=\mathop {\sup }_{r>0}{|B(x,r)|}^{-1+\frac {\alpha (x)}{n}}\int _{\tilde {B}(x,r)}|f(y)|\thinspace dy, \quad 0\leqslant \alpha (x)<n. $$

In the case \(\alpha (x)=\alpha =\operatorname {const}\), this operator coincides with the classical fractional maximal operator \({M}^{\alpha }\). If \(\alpha (x)=0 \) then \({M}^{\alpha (\cdot )} \) coincides with the operator \(M \).

The Riesz potential \({I}^{\alpha (x)}\) of variable order \( \alpha (x)\) is defined by the following equality:

$$ {I}^{\alpha (x)}f(x)=\int _{{\mathbb {R}}^{n}}\frac {f(y)}{\mathop {\left |x-y\right |} \nolimits ^{n-\alpha (x)}}\thinspace dy, \quad 0<\alpha (x)<n. $$

In the case \(\alpha (x)=\alpha =\operatorname {const}\), this operator coincides with the classical Riesz potential \({I}^{\alpha }\).

1. LEBESGUE SPACES WITH VARIABLE EXPONENT. GENERALIZED MORREY-TYPE SPACES WITH VARIABLE EXPONENTS

Let \(p(x) \) be a measurable function on an open set \(\Omega \subset {\mathbb {R}}^{n}\) with values in \((1,\infty ) \). Put

$$ 1<p_{-}\leqslant p(x)\leqslant p_{+}<\infty , $$
(1)

where \({p}_{-}={p}_{-}(\Omega )=\operatorname {ess\thinspace inf}_{x\in \Omega }p(x)\) and \({p}_{+} ={p}_{+}(\Omega )=\operatorname {ess\thinspace sup}_{x\in \Omega }p\left (x\right ) \). We denote by \(\mathop {L}\nolimits _{p(\cdot )}(\Omega )\) the space of all measurable functions \(f(x) \) on \(\Omega \) such that

$$ {J}_{p(\cdot )} (f)=\int _{\Omega }\mathop {\left |f(x)\right |}\nolimits ^{p(x)} dx<\infty , $$

where the norm is defined as

$$ \|f\|_{p(\cdot )} =\inf \Biggl \{\eta >0:\ J_{p(\cdot )} \left (\frac {f}{\eta } \right )\leqslant 1\Biggr \}. $$

This is a Banach space. The conjugate exponent \({p}^{\prime } \) is defined by the formula

$$ {p}^{\prime }(x)=\frac {p(x)}{p(x)-1}.$$

Hölder’s inequality for the variable exponents \(p(\cdot )\) and \({p}^{\prime }(\cdot )\) is of the form

$$ \int _{\Omega }f(x)g(x)dx\leqslant {C(p)}{\|f\|}_{{L}_{p(\cdot )}(\Omega )}{\|g\|}_{{L}_{{p}^{\prime }(\cdot )}(\Omega )},$$

where \(C(p)=\frac {1}{{p}_{-}}+\frac {1}{{p}_{-}^{\prime }}\). The Lebesgue spaces \({L}_{p(\cdot )}\) with variable exponents \(p(\cdot )\) were introduced in [13] and studied in [14, 15].

Define \(\mathcal {P}(\Omega ) \) as the set of measurable functions \(p:\Omega \rightarrow [1,\infty )\). Denote by \({\mathcal {P}}^{\log }(\Omega )\) the set of measurable functions \(p(x) \) satisfying the local log-condition

$$ \left |p(x)-p(y)\right |\leqslant \frac {{A}_{p}}{-\ln \left |x-y\right |}, \thinspace \thinspace \thinspace \mbox {with}\thinspace \thinspace \left | x-y\right |\leqslant \frac {1}{2}\thinspace \thinspace \forall {x},y\in \Omega , $$

where \({A}_{p}\) is independent of \(x \) and \(y \). Next, put \({\mathbb {P}}^{\log }(\Omega ) \) for the set of measurable functions \(p(x) \) meeting both (1) and the log-condition. In the case of \(\Omega \) is an unbounded set, we denote by \({\mathbb {P}}_{\infty }^{\log }(\Omega )\) the set of exponents which is a subset of the set of \({\mathbb {P}}^{\log }(\Omega ) \) and satisfying the decay condition

$$ |p(x)-p(\infty )|\leqslant {A}_{\infty }\ln (2+|x|), \quad x\in {\mathbb {R}}^{n}. $$

Put \({\mathbb {A}}^{\log }(\Omega ) \) for the set of bounded exponents \(\alpha :\Omega \rightarrow {\mathbb {R}}\) satisfying the log-condition.

Let \(\Omega \) be an open bounded set, \(p\in {\mathbb {P}}^{\log }(\Omega )\), and \(\lambda (x) \) be a measurable function on \(\Omega \) with values in \([0,n] \). The variable Morrey space \({\mathcal {L}}_{p(\cdot ),\lambda (\cdot )}(\Omega )\) with the norm

$$ {\|f\|}_{{\mathcal {L}}_{p(\cdot ),\lambda (\cdot )}(\Omega )}=\sup _{x\in \Omega ,t>0}{t}^{-\frac {\lambda (x)}{p(x)}}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,t))}. $$

was introduced in [3]. Let \(w(x,r)\) be nonnegative measurable function on \(\Omega \), where \(\Omega \subset {\mathbb {R}}^{n}\) is an open bounded set. The generalized Morrey-type space \({M}_{p(\cdot ),w(\cdot )}(\Omega ) \) with variable exponent and the norm

$$ {\|f\|}_{{M}_{p(\cdot ),w(\cdot )}(\Omega )}=\sup _{x\in \Omega ,r>0}\frac {{r}^{-\frac {n}{p(x)}}}{w(x,r)}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}. $$

is defined in [11]. Let \(w(x,r)\) be nonnegative measurable function on \(\Omega \), where \(\Omega \subset {\mathbb {R}}^{n}\) is an open unbounded set. The generalized Morrey-type space \({M}_{p(\cdot ),w(\cdot )}(\Omega ) \) with variable exponent with the norm

$$ {\|f\|}_{{M}_{p(\cdot ),w(\cdot )}(\Omega )}=\sup _{x\in \Omega ,r>0}\frac {{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}}{w(x,r)}.$$

is defined in [12]. Put

$$ {\eta }_{p}(x,r)=\begin {cases} \frac {n}{p(x)}&\text {if }r\leqslant 1;\\ \frac {n}{p(\infty )}&\text {if }r>1. \end {cases}$$

Definition 1.1\(. \) Let \(p\in {P}^{\log }(\Omega ) \), \(w(x,r)\) be a positive function on \( \Omega \times [0,\infty ]\), where \(\Omega \in {\mathbb R}^{n}\). The global Morrey-type space \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )\) with variable exponents is defined as the set of functions \(f\in {L}_{p(\cdot )}^{\operatorname {loc}}(\Omega )\) with the finite norm

$$ {\|f\|}_{{GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )}=\sup _{x\in \Omega }{\Bigl \|w(x,r){r}^{-{\eta }_{p}(x,r)}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\Bigr \|}_{{L}_{\theta (\cdot )}(0,\infty )}. $$

We assume that the positive measurable function \(w(x,r) \) satisfies the condition

$$ \sup _{x\in \Omega }{\|w(x,r)\|}_{{L}_{\theta (\cdot )}(0,\infty )}<\infty . $$

Then this space contains at least any bounded functions and thereby is nonempty. In the case \(w(x,r)={r}^{-\frac {\lambda (x)}{p(x)}+{\eta }_{p}(x,r)} \), the corresponding space is denoted by \({GM}_{p(\cdot ),\theta (\cdot )}^{\lambda (\cdot )}\):

$$ {GM}_{p(\cdot ),\theta (\cdot )}^{\lambda (\cdot )}(\Omega )={{GM}_{p(\cdot ),w(\cdot ),\theta }|}_{w(x,r)={r}^{-\frac {\lambda (x)}{p(x)}+{\eta }_{p}(x,r)}}, $$

and

$$ {\|f\|}_{{GM}_{p(\cdot ),\theta (\cdot )}^{\lambda (\cdot )}(\Omega )}= \sup _{x\in \Omega }{\Bigl \|w(x,r){r}^{-\frac {\lambda (x)}{p(x)}}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\Bigr \|}_{{L}_{\theta (\cdot )}(0,\infty )}.$$

In the case \(\theta =\infty \), the space \( {GM}_{p(\cdot ),\infty ,w(\cdot )}(\Omega )\) coincides with the generalized Morrey space with variable exponent \({M}_{p(\cdot ),w(\cdot )}(\Omega )\) with the finite quasi-norm

$$ {\|f\|}_{{M}_{p(\cdot ),w(\cdot )}(\Omega )}=\sup _{x\in \Omega }\Bigl \{w(x,r){r}^{-{\eta }_{p}(x,r)}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\Bigr \}. $$

If \(p(\cdot )=p=\operatorname {const} \) and \(\theta (x)=\theta =\operatorname {const}\) then the space \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )\) coincides with the ordinary global Morrey space \({GM}_{p,\theta ,w}(\Omega ) \), considered by V.I. Burenkov and others [2, 3, 4]. Some Spanne- and Adams-type theorems were proved in [5] for bounded sets \(\Omega \). Also, various results on the boundedness of operators are obtained in [11, 12, 16].

We need the following results from [12] for our arguments.

Lemma 1.2 \(. \) Assume that \( p\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) and \( f\in {L}_{\operatorname {loc}}^{p(\cdot )}({\mathbb R}^{n}) \) .Then

$$ {\|f\|}_{{L}_{p(\cdot )}(B(x,t))}\leqslant {C}{t}^{{\eta }_{p}(x,t)}\int _{t}^{\infty }{r}^{-{\eta }_{p}(x,r)-1}{\|f\|}_{{L}_{p(\cdot )}(B(x,r))}\thinspace dr.$$

Theorem 1.3 \(. \) Suppose that \( p\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) . Then for each \( f\in {L}_{p(\cdot )}(\Omega )\) we have

$$ {\|Mf\|}_{{L}_{p(\cdot )}(\tilde {B}(x,t))}\leqslant {C}{t}^{{\eta }_{p}(x,t)}\sup _{r>2t}\Bigl \{{r}^{-{\eta }_{p}(x,r)}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\Bigr \}, $$

where \(C \) is independent of \( f,x\in \Omega \) and \(t>0 \) .

Let us prove the next necessary inequality.

Theorem 1.4 \(. \) Assume that \( p\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) . Then

$$ {\|Mf\|}_{{L}_{p(\cdot )}(\tilde {B}(x,t))}\leqslant {C}{t}^{{\eta }_{p}(x,t)}\int _{t}^{\infty }{s}^{-{\eta }_{p}(x,s)-1}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,s))}\thinspace ds, $$
(2)

where \(C \) is independent of \( f,x,t\) .

Proof. Using Theorem 1.3 and Lemma 1.2, we deduce

$$ \begin {aligned} \|Mf\|_{L_p(\cdot )(\tilde {B}(x,t))}&\leqslant Ct^{\eta _p(x,t)}\sup _{r>2t}\Bigl \{r^{-\eta _p(x,r)}\|f\|_{L_p(\cdot )(\tilde {B}(x,r))}\Bigr \} \\ &\leqslant {C}{t}^{{\eta }_{p}(x,t)}\sup _{r>t}\Bigl \{{r}^{-{\eta }_{p}(x,r)}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\Bigr \}\\ &\leqslant {C}{t}^{{\eta }_{p}(x,t)}\sup _{r>t}\Bigl \{\int _{r}^{\infty }{s}^{-{\eta }_{p}(x,s)-1}{\|f\|}_{{L}_{p(\cdot )}(B(x,s))}\thinspace ds\Bigr \}\\ &= Ct^{\eta _p(x,t)} \int _t^\infty s^{-\eta _p(x,s)-1}||f||_{L_p(\cdot )(\tilde {B}(x,s))}\thinspace ds. \end {aligned}$$

The theorem is proved. \(\quad \square \)

The next result generalizes an inequality proved in [12].

Theorem 1.5 \(. \) Let \( p\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) and let the function \( \alpha (x)\) and \( q(x)\) satisfy the condition \(\frac {1}{q(x)}=\frac {1}{p(x)}-\frac {\alpha (x)}{n} \) . Then, for each \( {x}\in {\mathbb {R}}^{n}\) and \(t>0\) ,

$$ {\Biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}f\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))}\leqslant {C}{t}^{{\eta }_{q}(x,t)} \int _{t}^{\infty }{r}^{-{\eta }_{q}(x,r)-1}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\thinspace dr.$$
(3)

Proof. We represent the function \(f \) as \(f(x)={f}_{1}(x)+{f}_{2}(x) \), where \({f}_{1}(x)=f(x){\chi }_{\tilde {B}(x,2t)}\) and \({f}_{2}(x)=f(x){\chi }_{{\Omega }\backslash {\tilde {B}(x,2t)}}\). Then

$$ \frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}f(y)=\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{1}(y)+\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{2}(y).$$

From the results obtained in [16] it follows that

$$ \begin {aligned} {\Biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{1}\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))} &\leqslant {\Biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{1}\Biggr \|}_{{L}_{q(\cdot )}({\mathbb R}^{n})}\\ &\leqslant {C}{\|{f}_{1}\|}_{{L}_{p(\cdot )}({\mathbb R}^{n})}=C{\|f\|}_{{L}_{p(\cdot )}(B(x,2t))}. \end {aligned} $$

Lemma 1.2 implies

$$ {\biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{1}\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))}\leqslant {C}{t}^{{\eta }_{q}(x,t)}\int _{2t}^{\infty }{r}^{-{\eta }_{q}(x,r)-1}{\|f\|}_{{L}_{p(\cdot )}(\tilde {B}(x,r))}\thinspace dr. $$
(4)

If \(|x-z|\leqslant {t}\) and \( |z-y|\geqslant 2t\), we have \(\frac {1}{2}|z-y|\leqslant |x-y|\leqslant \frac {3}{2}|z-y|\). Since \(\frac {1}{{(1+|y|)}^{\gamma (y)}}\leqslant 1\), we infer

$$ \begin {aligned} {\Biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{2}\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))}&\leqslant {\Biggl \|\int _{{\mathbb R}^{n}\backslash {B(x,2t)}}{|z-y|}^{\alpha -n}f(y)\thinspace dy\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))} \\ &\leqslant {C}\int _{{\mathbb R}^{n}\backslash {B(x,2t)}}{|x-y|}^{\alpha -n}|f(y)|\thinspace dy\cdot {\|{\chi }_{B(x,t)}\|}_{{L}_{q(\cdot )}({\mathbb R}^{n})}. \end {aligned} $$

Choosing \(\beta >\frac {n}{{q}_{-}} \), we obtain

$$ \begin {aligned} \int _{{\mathbb R}^{n}\backslash {B(x,2t)}}{|x-y|}^{\alpha -n}|f(y)|\thinspace dy&=\beta \int _{{\mathbb R}^{n}\backslash {B(x,2t)}}{|x-y|}^{\alpha -n+\beta }|f(y)|\Bigl (\int _{|x-y|}^{\infty }{s}^{-\beta -1}\thinspace ds\Bigr )\thinspace dy\\ &=\beta \int _{2t}^{\infty }{s}^{-\beta -1}\Bigl (\int _{y\in {\mathbb R}^{n}:2t\leqslant |x-y|\leqslant {s}}{|x-y|}^{\alpha -n+\beta }|f(y)|\thinspace dy\Bigr )\thinspace ds\\ &\leqslant {C}\int _{2t}^{\infty }{s}^{-\beta -1}{\|f\|}_{{L}_{p(\cdot )}(B(x,s))}\cdot {\|{|x-y|}^{\alpha -n+\beta }\|}_{{L}_{{p}^{\prime }(\cdot )}(B(x,s))}\thinspace ds\\ &\leqslant {C}\int _{2t}^{\infty }{s}^{\alpha -{\eta }_{p}(x,s)-1}{\|f\|}_{{L}_{p(\cdot )}(B(x,s))}\thinspace ds. \end {aligned}$$

Therefore,

$$ {\Biggl \|\frac {1}{{(1+|y|)}^{\gamma (y)}}{I}^{\alpha (\cdot )}{f}_{2}\Biggr \|}_{{L}_{q(\cdot )}(\tilde {B}(x,t))}\leqslant {C}{t}^{{\eta }_{p}(x,t)}\int _{2t}^{\infty }{s}^{-{\eta }_{q}(x,s)-1}{\|f\|}_{{L}_{p(\cdot )}(B(x,s))}\thinspace ds. $$

The latter together with (4) yields (3). The theorem is proved. \(\quad \square \)

Let \(u\) and \(v \) be positive measurable functions. The dual Hardy operator is defined as

$$ {\tilde {H}}_{v,u}f(x)=v(x)\int _{x}^{\infty }f(t)u(t)\thinspace dt, \ x\in {\mathbb R}^{n}.$$

Suppose that \(a \) is a positive fixed number. Put \({\theta }_{1,a}(x)=\operatorname {ess\thinspace inf}_{y\in [x,a)}{\theta }_{1}(y) \),

$$ {\tilde {\theta }}_{1}(x)=\begin {cases} {\theta }_{1,a}(x) &\text { if }x\in [0,a],\\ \overline {\theta }_{1}=\operatorname {const} &\text { if }x\in [a,\infty ). \end {cases} $$

Moreover, we denote \({\mathbb {\theta }}_{1}=\operatorname {ess\thinspace inf}_{x\in {\mathbb R}_{+}}{\theta }_{1}(x) \) and \({\Theta }_{2}=\operatorname {ess\thinspace sup}_{x\in {\mathbb R}_{+}}{\theta }_{2}(x) \) for \(\theta _1(x) \) and \(\theta _2(x) \).

The next theorem is proved in [17].

Theorem 1.6 \(. \) Let \( {\theta }_{1}(x)\) and \( {\theta }_{2}(x)\) be measurable functions on \({\mathbb R}_{+} \) . Suppose that there exists a positive number \(a\) such that \({\theta }_{1}(x)=\overline {\theta }_{1}=\operatorname {const}\) , \( {\theta }_{2}(x)=\overline {\theta }_{2}=\operatorname {const} \) holds for all \(x>a \) , and \( 1<{\theta }_{1}\leqslant {\tilde {\theta }}_{1}(x)\leqslant {\theta }_{2}(x)\leqslant {{\Theta }_{2}}<\infty \) almost everywhere. If

$$ G=\sup _{t>0}\int _{0}^{t}{[v(x)]}^{{\theta }_{2}(x)}{\Bigl (\int _{t}^{\infty }{u}^{{\tilde {\theta }_{1}}^{\prime }(x)}(\tau ),d\tau \Bigr )}^{\frac {{\theta }_{2}(x)}{{({\theta }_{1})}^{\prime }(x)}}dx<\infty $$

then the operator \( {\tilde {H}}_{v,u}\) from \({L}_{{\theta }_{1}(\cdot )}({\mathbb R}^{+}) \) to \( {L}_{{\theta }_{2}(\cdot )}({\mathbb R}^{+}) \) is bounded.

2. MAIN RESULTS

Theorem 2.1 \(. \) Assume that \( p(\cdot )\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) , and \( {\theta }_{1}(x)\) and \( {\theta }_{2}(x)\) are measurable functions on \({\mathbb R}_{+} \) . Suppose that there exists a positive number \(a\) such that we have \({\theta }_{1}(x)=\overline {\theta }_{1}=\operatorname {const} \) , \( {\theta }_{2}(x)=\overline {\theta }_{2}=\operatorname {const} \) for all \(t>a \) , and \( 1<{\theta }_{1}\leqslant {\tilde {\theta }}_{1}(x)\leqslant {\theta }_{2}(x)\leqslant {{\Theta }_{2}}<\infty \) a. e. Let the positive measurable functions \({w}_{1}\) и \({w}_{2}\) satisfy the condition

$$ A=\sup _{x\in \Omega ,t>0}\int _{0}^{t}{({w}_{2}(x,r))}^{{\theta }_{2}(r)}{\Bigl (\int _{t}^{\infty }{\Bigl (\frac {1}{{w}_{1}(x,s)s}\Bigr )}^{{[\tilde {\theta }_{1}(r)]}^{\prime }}\thinspace ds\Bigr )}^{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}\thinspace dr<\infty . $$
(5)

Then the maximal operator \(M \) from \( {GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot )}(\Omega ) \) to \( {GM}_{p(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega ) \) is bounded.

Proof. According to the definition and to Theorem 1.6, Hölder’s inequality with variable exponents \(\theta \), \({\theta }^{\prime } \) infers

$$ \begin {aligned} {\|Mf\|}_{{GM}_{p(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega )}&=\sup _{x\in \Omega }{\Biggl \|\frac {{w}_{2}(x,r)}{{r}^{{\eta }_{p}(x,r)}}{\|Mf\|}_{{L}_{p(\cdot )}(B(x,r))}\Biggr \|}_{{L}_{{\theta }_{2}(\cdot )}(0,\infty )}\\ &\leqslant {C}\sup _{x\in \Omega }{\Biggl \|{{w}_{2}(x,r)}\int _{r}^{\infty }{t}^{-{\eta }_{p}(x,t)-1}{\|f\|}_{{L}_{p(\cdot )}(B(x,t))}\thinspace dt\Biggr \|}_{{L}_{{\theta }_{2}(\cdot )}(0,\infty )}. \end {aligned} $$

Denote

$$ {\tilde {H}}_{v,u}f(r)=v(r)\int _{r}^{\infty }g(t)u(t)\thinspace dt,$$

where \(v(r)={w}_{2}(x,r) \), \(g(t)=\frac {{w}_{1}(x,t)}{{t}^{{\eta }_{p}(x,t)}}{\|f\|}_{{L}_{p(\cdot )}(B(x,t))} \), and \(u(t)=\frac {1}{{w}_{1}(x,t)t} \) for every fixed \(x\in \Omega \). Then condition (2) has the form (5), from which the boundedness of the operator \({\tilde {H}}_{v,w}f(r) \) from \({L}_{{\theta }_{1}(\cdot )}(0,\infty )\) to \({L}_{{\theta }_{2}(\cdot )}(0,\infty )\) follows. Consequently, we have

$$ \begin {aligned} {\|Mf\|}_{{GM}_{q(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega )}&\leqslant {A}\cdot \sup _{x\in \Omega }{\Bigl \|{w}_{1}(x,t){t}^{-{\eta }_{p}(x,t)}{\|f\|}_{{L}_{p(\cdot )}(B(x,t))}\Bigr \|}_{{L}_{{\theta }_{1}(\cdot )}(0,\infty )}\\ &=A\cdot {\|f\|}_{{GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot )}}. \end {aligned}$$

The latter infers the boundedness of the operator \(M\) from \({GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot )}\) to \({GM}_{q(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}\). \(\quad \square \)

Corollary 2.2 \(. \) Let \( p(\cdot )\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) , \( {w}_{1}(x,r)={w}_{2}(x,r)={r}^{\beta (x)} \) . If

$$ \inf _{x\in \Omega ,r>0}(\beta (x)+1){[\tilde {\theta }_{1}(r)]}^{\prime }>1$$
(6)

and

$$ \sup _{x\in \Omega ,t>0}\int _{0}^{t}{r}^{{\theta }_{2}(r)\beta (x)}\frac {{t}^{[-(\beta (x)+1){[\tilde {\theta }_{1}(r)]}^{\prime }+1]{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}}}{{[(\beta (x)+1){[\tilde {\theta }_{1}(r)]-1}]}^{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}}\thinspace dr<\infty $$
(7)

then the maximal operator \(M \) from \( {GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{r}^{\beta (\cdot )}}(\Omega ) \) to \( {GM}_{p(\cdot ),{\theta }_{2}(\cdot ),{r}^{\beta (\cdot )}}(\Omega ) \) is bounded.

Proof. Condition (5) has the form

$$ \sup _{x\in \Omega ,t>0}\int _{0}^{t}{r}^{{\theta }_{2}(r)\beta (x)}{\Bigl (\int _{t}^{\infty }{s}^{-[\beta (x)+1]{[\tilde {\theta }_{1}(r)]}^{\prime }}\thinspace ds\Bigr )}^{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}\thinspace dr<\infty . $$

By the convergence of the inner integral, we obtain conditions (6) and (7). \(\quad \square \)

The next theorems give Spanne-type results on the boundedness of the Riesz potential \( {I}^{\alpha }\) in global Morrey-type spaces with variable exponent \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega ) \). In the following theorem \(\alpha =\operatorname {const} \).

Theorem 2.3 \(. \) Assume that \( p(\cdot )\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) , the constant number \( \alpha \) is positive, and \( {(\alpha p(\cdot ))}_{+}=\sup _{x\in \Omega }\alpha p(x)<n \) . Let \( {\theta }_{1}(x)\) and \( {\theta }_{2}(x)\) be measurable functions on \({\mathbb R}_{+} \) . Suppose that there exists a positive number \(a\) such that, for all \(x>a \) , we have \( {\theta }_{1}(x)=\overline {\theta }_{1}=\operatorname {const} \) , \( {\theta }_{2}(x)=\overline {\theta }_{2}=\operatorname {const} \) , and \( 1<{\theta }_{1}\leqslant {\tilde {\theta }}_{1}(x)\leqslant {\theta }_{2}(x)\leqslant {{\Theta }_{2}}<\infty \) almost everywhere. Let the functions \({p}_{1}(x)\) and \({p}_{2}(x)\) satisfy the equality \(\frac {1}{{p}_{2}(x)}=\frac {1}{{p}_{1}(x)}-\frac {\alpha (x)}{n} \) and let the functions \( {w}_{1}\) and \( {w}_{2}\) meet the condition

$$ T=\sup _{x\in \Omega ,t>0}\int _{0}^{t}{({w}_{2}(x,r))}^{{\theta }_{2}(r)}{\Bigl (\int _{t}^{\infty }{\Bigl (\frac {{s}^{\alpha -1}}{{w}_{1}(x,s)}\Bigr )}^{{[\tilde {\theta }_{1}(r)]}^{\prime }}\thinspace ds\Bigr )}^{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}\thinspace dr<\infty . $$
(8)

Then the operators \( {I}_{\alpha }\) and \( {M}_{\alpha }\) from \( {GM}_{{p}_{1}(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot ))}(\Omega ) \) to \( {GM}_{{p}_{2}(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega ) \) are bounded.

Proof. Using the definition and results from [12] (see also Theorem 1.5), we have

$$ \begin {aligned} {\|{I}^{\alpha }\|}_{{GM}_{q(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega )}&=\sup _{x\in \Omega }{\Bigl \|{w}_{2}(x,r){r}^{-{\eta }_{q}(x,r)}{\|{I}_{\alpha }f\|}_{{L}_{q(\cdot )}(B(x,r))}\Bigr \|}_{{L}_{{\theta }_{2}(\cdot )}(0,\infty )}\\ &\leqslant {C}\sup _{x\in \Omega }{\Biggl \|{w}_{2}(x,r)\int _{r}^{\infty }{t}^{-{\eta }_{q}(x,t)-1}{||f||}_{{L}_{p(\cdot )}(B(x,t))}\thinspace dt\Biggr \|}_{{L}_{{\theta }_{2}(\cdot )}(0,\infty )}. \end {aligned} $$

Denote

$$ {\tilde {H}}_{v,u}f(r)=v(r)\int _{r}^{\infty }g(t)u(t)\thinspace dt,$$

where \(v(r)={w}_{2}(x,r) \), \(g(t)=\frac {{w}_{1}(x,t)}{{t}^{{\eta }_{p}(x,t)}}{\|f\|}_{{L}_{p(\cdot )}(B(x,t))} \), and \(u(t)=\frac {{t}^{{\eta }_{p}(x,t)-{\eta }_{q}(x,t)-1}}{{w}_{1}(x,t)} \) for every fixed \(x\in \Omega \). Then condition (2) has the form (8), from which the boundedness of the operator \({\tilde {H}}_{v,w}f(r) \) acting from \({L}_{{\theta }_{1}(\cdot )}(0,\infty )\) to \({L}_{{\theta }_{2}(\cdot )}(0,\infty )\) follows. Consequently, we have

$$ \begin {aligned} {\|{I}^{\alpha }f\|}_{{GM}_{q(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega )}&\leqslant {T}\cdot \sup _{x\in \Omega }{\Bigl \|{w}_{1}(x,t){t}^{-{\eta }_{p}(x,t)}{\|f\|}_{{L}_{p(\cdot )}(B(x,t))}\Bigr \|}_{{L}_{{\theta }_{1}(\cdot )}(0,\infty )}\\ &=T\cdot {\|f\|}_{{GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot )}}. \end {aligned} $$

This means that the operator \({I}^{\alpha } \) from \({GM}_{{p}_{1}(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot )}\) to \( {GM}_{{p}_{2}(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )} \) is bounded. The theorem is proved. \(\quad \square \)

Corollary 2.4 \(. \) Let \( p(\cdot )\in {P}_{\infty }^{\log }(\Omega ) \) and \( {w}_{1}(x,r)={w}_{2}(x,r)={r}^{\beta (x)} \) . If

$$ \sup _{x\in \Omega ,r>0}(\alpha -\beta (x)-1){[\tilde {\theta }_{1}(r)]}^{\prime }<-1$$
(9)

and

$$ \sup _{x\in \Omega ,t>0}\int _{0}^{t}{r}^{{\theta }_{2}(r)\beta (x)}\frac {{t}^{[[\alpha -\beta (x)-1]{[\tilde {\theta }_{1}(r)]}^{\prime }+1]\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}}{[\beta (x)+1-\alpha ]{[\tilde {\theta }_{1}(r)]}^{\prime }-1}\thinspace dr<\infty$$
(10)

then the operators \({I}_{\alpha } \) and \( {M}_{\alpha }\) from \( {GM}_{p(\cdot ),{\theta }_{1}(\cdot ),{r}^{\beta }}(\Omega ) \) to \( {GM}_{p(\cdot ),{\theta }_{2}(\cdot ),{r}^{\beta }}(\Omega ) \) are bounded.

Proof. Condition (8) takes the form

$$ \sup _{x\in \Omega ,t>0}\int _{0}^{t}{r}^{{\theta }_{2}(r)\beta (x)}{\Bigl (\int _{t}^{\infty }{s}^{(\alpha -1-\beta (x)){[\tilde {\theta }_{1}(r)]}^{\prime }}\thinspace ds\Bigr )}^{{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}}\thinspace dr<\infty . $$

By the convergence of the inner integral, we deduce conditions (9) and (10). \(\quad \square \)

In the following theorem, \(\alpha (x)\) is a variable exponent.

Theorem 2.5 \(. \) Assume that \( p(\cdot )\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) , the function \( \alpha (x)\) satisfies the condition \(\alpha (x)>0 \) and \( {(\alpha (\cdot ) p(\cdot ))}_{+}=\sup _{x\in \Omega }\alpha (x) p(x)<n \) . Let \( {\theta }_{1}(x)\) and \( {\theta }_{2}(x)\) be measurable functions on \({\mathbb R}_{+} \) . Suppose that there exists a positive number \(a\) such that, for all \(x>a \) , we have \( {\theta }_{1}(x)=\overline {\theta }_{1}=\operatorname {const} \) , \( {\theta }_{2}(x)=\overline {\theta }_{2}=\operatorname {const} \) , and \( 1<{\theta }_{1}\leqslant {\tilde {\theta }}_{1}(x)\leqslant {\theta }_{2}(x)\leqslant {{\Theta }_{2}}<\infty \) almost everywhere. Let the functions \({p}_{1}(x)\) and \({p}_{2}(x)\) satisfy the equality \(\frac {1}{{p}_{2}(x)}=\frac {1}{{p}_{1}(x)}-\frac {\alpha (x)}{n} \) and let the functions \( {w}_{1}\) and \( {w}_{2}\) meet the condition

$$ T=\sup _{x\in \Omega ,t>0}\int _{0}^{t}{({w}_{2}(x,r))}^{{\theta }_{2}(r)}{\Bigl (\int _{t}^{\infty }{\Bigl (\frac {{s}^{\alpha (x)-1}}{{w}_{1}(x,s)}\Bigr )}^{{[\tilde {\theta }_{1}(r)]}^{\prime }}\thinspace ds\Bigr )}^{\frac {{\theta }_{2}(r)}{{[\tilde {\theta }_{1}(r)]}^{\prime }}}\thinspace dr<\infty . $$

Then the operators \( \frac {1}{{(1+|x|)}^{\gamma (x)}}{I}^{\alpha (\cdot )} \) and \( \frac {1}{{(1+|x|)}^{\gamma (x)}}{M}^{\alpha (\cdot )} \) acting from the space \( {GM}_{{p}_{1}(\cdot ),{\theta }_{1}(\cdot ),{w}_{1}(\cdot ))}(\Omega ) \) to the space \( {GM}_{{p}_{2}(\cdot ),{\theta }_{2}(\cdot ),{w}_{2}(\cdot )}(\Omega ) \) are bounded.

Proof. The proof of this theorem is the same as that of Theorem 2.3: It is sufficient to substitute \(\frac {1}{{(1+|x|)}^{\gamma (x)}}{I}^{\alpha (\cdot )}f(x)\) for \({I}^{\alpha }f(x)\). \(\quad \square \)