Weighted Grand Herz-Type Spaces and Its Applications
In this paper, we introduce the weighted grand Herz spaces and weighted grand Herz-type Hardy spaces. The decompositional characterizations of these spaces are established. As its applications, the boundedness of some sublinear operators are established.
In the process of investigating the minimal hypothesis for the integrability of the Jocobian, Iwanniec and Sbordone  introduced grand Lebesgue spaces, denoted by , and being a bounded domain. Greco et al.  generalized the space to with and . During the last decades, grand Lebesgue spaces have attracted many researchers and have been considered in various aspects, such as PDE, interpolation theory, and boundedness of various operators, e.g. Hardy operator, maximal operator, singular integral operator, and fractional integral operator, see [3–7].
Usually, the underlying space of grand Lebesgue spaces would be an open set with finite measure (in or more generally homogeneous spaces) since that the basic idea of their definition is based on the supposition that in grand Lebesgue space, , must be integrable to the power with all . In 2011, Samko and Umarkhardzhiev  introduced grand Lebesgue spaces on an arbitrary set which may has infinite measure in . They also introduced a version of weighted grand Lebesgue spaces and proved that some operators bounded in a Lebesgue space with Muckenhoupt weights were also bounded in the corresponding weighted grand Lebesgue space (for bounded or unbounded ).
Let and be an open set. To introduce the weighted grand Lebesgue spaces in the case , it is quite natural to make use of weighted space; for any nonnegative weight function , any measurable function on , we write
Definition 1 (see ). Let , , and , , be a weight. The grand Lebesgue space with power weights is defined as the set of with the finite norm where and .
According to , the embedding with holds if and only if In the case is unbounded, we call a positive number admissible for the weight , if the condition holds with this . We denote the Muckenhoupt class of weights on by . It is known that Then, all are admissible for all the weights .
Suppose and , with the choice and , we have where is independent of . Since , we have
Herz spaces are arising from Herz’s study on the absolute convergence of Fourier transform , which is an important function space in harmonic analysis. Chen and Lau  and García-Cuerva  studied some Hardy spaces associated with nonhomogeneous Herz spaces with special indexes in and , respectively. Lu et al. studied some Hardy spaces associated with homogeneous Herz spaces with special indexes, and the complete theory for the case of general indexes was established in 1995. We refer to  for more details. The weighted Herz spaces on were established by Lu and Yang in . Let be a sublinear operator satisfying the condition for any integrable function with compact support, in  such a sublinear operator was proved be bounded on , and provided is bounded on , , and . The restriction on the weights has been relaxed to and in . A certain weighted Herz-type Hardy space was introduced, its atomic decomposition was established in , and a boundedness theory of sublinear operators was given, where . The atomic decomposition for the weighted Herz-type Hardy space with more general weights was established in , and local Caldrón-Zygmund operators of nonconvolutional type was proved to be bounded from these weighted Herz-type Hardy spaces into weighted Herz spaces.
Grand variable Herz spaces using grand Lebesgue sequence spaces and were introduced by Nafis et al. in , and the boundedness of sublinear operators on these spaces was established in this paper. The boundedness of multilinear Calderón-Zygmund operators on these spaces was established in .
Inspired by these work, we introduce the weighted grand Herz spaces and weighted grand Herz-type Hardy spaces which are products of weighted grand Lebesgue spaces on and Herz-type spaces. The decompositional characterizations of these spaces are established. As its applications, the boundedness of some sublinear operators is established on these function spaces.
We recall the definition of Muckenhoupt weight as follows. Let , and a weight is a Muckenhoupt weight if for any ball , with a constant independent of the ball . Further, if and only if there is a constant such that for any ball ,
Lemma 2. Let for some and B be any ball. Then, (i)For any function , where is independent of and .(ii)If is a measurable subset of , then there exists a such that where and are independent of and , respectively.
The following lemmas (see ) and notations will be used in this paper.
Lemma 3. For every and , there exists a such that for all , where and .
Lemma 4. Let ; if is bounded on for every and is bounded on for every for some , then is also bounded on the weighted grand Lebesgue space , where is an arbitrary positive admissible number.
Let be a measurable set, denotes the Lebesgue measure of , and denotes the characteristic function. We write , , and , where and , , where . means the conjugate exponent function, namely, . means where .
2. Weighted Grand Herz Spaces
Now, we give the definition of weighted grand Herz spaces.
Definition 6. Suppose , , , and . Let and be nonnegative weight functions and . The homogeneous weighted grand Herz space on is defined by where The nonhomogeneous weighted grand Herz space on is defined by where
It is obvious that . Therefore, the weighted grand Herz spaces are natural generalization of the weighted grand Lebesgue spaces.
By the embedding between weighted grand Lebesgue space and weighted Lebesgue space, we can get the following proposition immediately.
Proposition 7. Suppose , , , and . Let and and , , and . Then, we have
In order to state our decomposition theorem, we need the following definition.
Definition 8. Suppose , , let and be nonnegative weight functions on . (a)A function on is said to be a central -unit if (i) for some (ii), where (b)A function on is said to be a central -unit of restrict type if (i) for some (ii)
Theorem 10. Suppose , , , and . Let , , and be a nonnegative weight function on . The following two statement are equivalent. (i)(ii) can be represented by where each is a dyadic central -unit with the support and , , , and .
Proof. Firstly, we prove (i) can imply (ii). For , write
where and .
It is obvious that is nonnegative, , and . Hence, for each k, is a dyadic central -unit with the support . Now, we prove (ii) can imply (i). Let be a decomposition of satisfying the hypothesis in (ii). For each , we can get If , we have Since , then there exists and such that Thus, If , by Hölder inequality, Since , then there exists and such that Thus, Hence, we obtain . This completes the proof of Theorem 10.
The following theorem shows the boundedness of some sublinear operators on weighted Herz spaces.
Theorem 11. Suppose , , , and . Let and , and there exists some such that , where is an arbitrary positive admissible number. and satisfy either of the following conditions. (i), , and (ii), , and If a sublinear operator satisfies the condition for any integrable function with compact support and is bounded on , then is bounded on . Similarly, the nonhomogeneous weighted grand Herz space has the same result.
Proof. Without loss of generality, we assume that is nonnegative. Let for , and then we have . By the -boundedness of , it is easy to obtain For , , , , and then we have . where . By Remark 5, Lemma 2, , and for , we have If , we have If , by the inequality, we have Finally, we estimate . In this case, , , and , we have . where . By Remark 5, Lemma 2, and for , we have By the same argument as , we obtain Combining the estimates of , , and , we complete the proof of Theorem 11.
Corollary 12. Let ; by Lemma 4, Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operator are bounded on the weighted grand Lebesgue space , , for every admissible . Then, by Theorem 11, and are bounded on and .
3. Weighted Grand Herz-Type Hardy Spaces
We begin with recalling some definitions. For , let denote the greatest integer when and when . Define where is the space of Schwartz functions, and . Moreover, we define
Let be the space of Schwartz distributions. For , we define where for . is usually called the grand maximal of , see .
Now, we give the definition of the weighted grand Herz-type Hardy spaces.
Definition 13. Suppose , , , and . Let and , and and . The homogeneous weighted grand Herz-type Hardy space on is defined by The nonhomogeneous weighted grand Herz-type Hardy space on is defined by
By the embedding between weighted grand Herz spaces and weighted Herz spaces, we can get the following proposition immediately.
Proposition 14. Suppose , , , and . Let and and , , and . Then, we have
Applying Theorem 11, we can obtain the following relation between the weighted grand Herz spaces and the weighted grand Herz-type Hardy spaces.
Theorem 15. Suppose , , , and . Let and , and there exists some such that , where is an arbitrary positive admissible number. and satisfy either of the following conditions. (i), , and (ii), , and Then, and .
Thus, we are interested in the case for the weighted grand Herz-type Hardy spaces. We can establish their atomic decomposition.
Definition 16. Let and ,