Research Article | Open Access
Morrey Meets Herz with Variable Exponent and Applications to Commutators of Homogeneous Fractional Integrals with Rough Kernels
We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous fractional integral operator and its commutator on Morrey-Herz space with variable exponent, where for is a homogeneous function of degree zero, , and is a BMO function.
The Morrey spaces first appeared in 1938 in the work of Morrey  in relation to some problems in partial differential equations. In  the authors introduced central Morrey spaces. The Herz spaces are a class of function spaces introduced by Herz in the study of absolutely convergent Fourier transforms in 1968; see . The complete theory of Herz spaces for the case of general indexes was established by Lu et al. in 2008; see . Recently, Lu and Xu defined the homogeneous Morrey-Herz spaces in .
The theory of function spaces with variable exponent was extensively studied by researchers since the work of Kováčik and Rákosník  appeared in 1991; see [7, 8] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids , in the study of image processing , and in differential equations with nonstandard growth . In 2009, Izuki established the Herz spaces with variable exponent and Morrey-Herz spaces with variable exponent; see [12, 13]. In , the authors introduced the nonhomogeneous central Morrey spaces of variable exponent.
Suppose that denotes the unit sphere in equipped with normalized Lebesgue measure. Let for be a homogeneous function of degree zero. For , the homogeneous fractional integral operator is defined by Denote when . In 1971, Muckenhoupt and Wheeden  gave the weighted boundedness of with power weights. Recently, Tan and Liu  gave some boundedness of on function spaces with variable exponent. For the applications of fractional integral operator in fractional order system, we choose to refer to .
Let be a locally integrable function; the commutator of homogeneous fractional integral operator is defined by
Motivated by the above references, in the present paper we will study the boundedness for the homogeneous fractional integral operator and its commutator on the Morrey-Herz space with variable exponent.
Let us explain the outline of this article. In Section 2, we first briefly recall some standard notations and lemmas in variable Lebesgue spaces. In Section 3, we will give the relation between central Morrey spaces, Herz spaces, and Morrey-Herz spaces with variable exponent. We will establish the boundedness for the homogeneous fractional integral operator and its commutator on the Morrey-Herz space with variable exponent in Sections 4 and 5.
2. Variable Lebesgue Spaces
Given an open set and a measurable function , denotes the set of measurable functions on such that for some This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm These spaces are referred to as variable spaces, since they generalized the standard spaces: if is constant, then is isometrically isomorphic to .
The space is defined by Define to be set of such that Define to be set of such that Denote In addition, we denote the Lebesgue measure and the characteristic function of a measurable set by and , respectively. The notation means that there exist constants such that .
In variable spaces there are some important lemmas as follows.
Lemma 1 (see ). If and satisfies then ; that is, the Hardy-Littlewood maximal operator is bounded on .
Lemma 2 (see ). Let . If and , then is integrable on and whereThis inequality is named the generalized Hölder inequality with respect to the variable spaces.
Lemma 3 (see ). Let . Then there exists a positive constant such that for all balls in and all measurable subsets where depend on .
Throughout this paper will change with .
Lemma 4 (see ). Suppose . Then there exists a constant such that for all balls in
3. The Relation between Central Morrey Spaces, Herz Spaces, and Morrey-Herz Spaces with Variable Exponent
Let and for . Denote and as the sets of all positive and nonnegative integers, for if and .
We can give the following definitions of function spaces with variable exponent.
Definition 5. Let and . The homogeneous central Morrey space with variable exponent is defined by where The nonhomogeneous central Morrey space with variable exponent is defined by where
Definition 6 (see ). Let , , and . The homogeneous Herz space with variable exponent is defined by where The nonhomogeneous Herz space with variable exponent is defined by where
Definition 7. Let , , , and . The homogeneous Morrey-Herz space with variable exponent (see ) is defined by where The nonhomogeneous Morrey-Herz space with variable exponent is defined by where
Remark 8. If , , then and .
Remark 9. If , then and .
Remark 10. If is a constant, then and .
4. Estimate for the Homogeneous Fractional Integral Operator
In this section we will prove the boundedness of the homogeneous fractional integral operators on Morrey-Herz spaces with variable exponent.
To prove Theorem 11, we need the following lemmas.
Lemma 12 (see ). If , , , and , then
Lemma 13 (see ). Define a variable exponent by Then we have for all measurable functions and .
Proof of Theorem 11. We only prove the homogeneous case. Similar to the method of , it is easy to prove that for . So the nonhomogeneous case can be proved in the same way. Let . Denote for each ; then we have . Note that ; we have We first estimate . For each , , and a.e. , using the generalized Hölder inequality, we have Noting , we denote and . By Lemmas 12 and 13 we have When and , by Lemma 14, we have When we have Sinceby Lemmas 3 and 4, we have So we have When , take . Since , by the Hölder inequality, we have When , we have Next we estimate ; by the -boundedness of the commutator we have If , then we have For , we have For , by we have If , noting , we can take a constant so that . By the Hölder inequality we have For , we have