Abstract
We consider the global Morrey-type spaces \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )\) with variable exponents \(p(x) \), \(\theta (x)\), and \(w(x,r) \) defining these spaces. In the case of unbounded sets \(\Omega \subset {\mathbb {R}}^{n}\), we prove the boundedness of the Hardy–Littlewood maximal operator and potential-type operator in these spaces. We prove Spanne-type results on the boundedness of the Riesz potential \({I}^{\alpha } \) in global Morrey-type spaces with variable exponents \( {GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega ) \).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
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
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:
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
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
where the norm is defined as
This is a Banach space. The conjugate exponent \({p}^{\prime } \) is defined by the formula
Hölder’s inequality for the variable exponents \(p(\cdot )\) and \({p}^{\prime }(\cdot )\) is of the form
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
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
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
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
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
is defined in [12]. Put
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
We assume that the positive measurable function \(w(x,r) \) satisfies the condition
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 \({GM}_{p(\cdot ),\theta (\cdot )}^{\lambda (\cdot )}\):
and
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
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
Theorem 1.3 \(. \) Suppose that \( p\in {\mathbb {P}}_{\infty }^{\log }(\Omega ) \) . Then for each \( f\in {L}_{p(\cdot )}(\Omega )\) we have
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
where \(C \) is independent of \( f,x,t\) .
Proof. Using Theorem 1.3 and Lemma 1.2, we deduce
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 \(t>0\) ,
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
From the results obtained in [16] it follows that
Lemma 1.2 implies
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
Choosing \(\beta >\frac {n}{{q}_{-}} \), we obtain
Therefore,
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
Suppose that \(a \) is a positive fixed number. Put \({\theta }_{1,a}(x)=\operatorname {ess\thinspace inf}_{y\in [x,a)}{\theta }_{1}(y) \),
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
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
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
Denote
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
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
and
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
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
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
Denote
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
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
and
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
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
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 \({I}^{\alpha }f(x)\). \(\quad \square \)
REFERENCES
C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Trans. Am. Math. Soc. 43, 126 (1938).
V. Burenkov and H. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces,” Studia Math. 163, 157 (2004).
V. Burenkov and V. Guliyev, “Necessary and sufficient conditions for boundedness of the Riesz potential in the local Morrey-type spaces,” Potential Analys. 30, 1 (2009).
V. Burenkov, H. Guliyev, and V. Guliyev, “Necessary and sufficient conditions for boundedness of the Riesz potential in the local Morrey-type spaces,” Doklady Math. 75, 103 (2007).
A. Almeida, J. Hasanov, and S. Samko, “Maximal and potential operators in variable exponent Morrey spaces,” Georgian Math. J. 15, 195 (2008).
A. Almeida and P. Hasto, “Besov spaces with variable smoothness and integrability,” J. Funct. Analys. 258, 1628 (2010).
A. Almeida and S. Samko, “Characterization of Riesz and Bessel potentials on variable Lebesgue spaces,” J. Funct. Spaces and Appl. 4, 113 (2006).
A. Almeida and S. Samko, “Embeddings of variable Hajlasz-Sobolev spaces into Holder spaces of variable order,” J. Math. Analys. Appl. 353, 489 (2009).
A. Almeida and S. Samko, “Fractional and hypersingular operators in variable exponent spaces on metric measure spaces,” Mediterr. J. Math. 6, 215 (2009).
J. Alvarez and C. Perez, “Estimates with \({A}_{\infty } \) weights for various singular integral operators,” Boll. Un. Mat. Ital. 7, 123 (1994).
V. Guliyev, J. Hasanov, and S. Samko, “Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey spaces,” J. Math. Sci. 170, 423 (2010).
V. Guliyev and S. Samko, “Maximal, potential, and singular operators in the generalized variable exponent Morrey spaces on unbounded sets,” J. Math. Sci. 193, 228 (2013).
I. Sharapudinov, “The topology of spaces \({L}^{p(t)} \),” Math. Not. 26, 613 (1979).
L. Diening, “Maximal function on generalized Lebesgue spaces \({L}_{p(\cdot )}\),” Math. Inequal. Appl. 7, 245 (2004).
L. Diening, “Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \({L}_{p(\cdot )}\) and \({W}_{k,p(\cdot )}\),” Math. Nachr. 268, 31 (2004).
V. Kokilashvili and S. Samko, “On Sobolev theorem for the Riesz type potentials in Lebesgue spaces with variable exponent,” Z. Analys. Anwend. 22, 899 (2003).
D. E. Edmunds., V. Kokilashvili, and A. Meskhi, “On the boundedness and compactness of weight Hardy operators in \(L_p(x)\) spaces,” Georgian Math. J. 12, 27 (2005).
Author information
Authors and Affiliations
Corresponding authors
About this article
Cite this article
Bokayev, N.A., Onerbek, Z.M. On the Boundedness of Integral Operators in Morrey-Type Spaces with Variable Exponents. Sib. Adv. Math. 32, 79–86 (2022). https://doi.org/10.1134/S1055134422020018
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1055134422020018