Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces

Abstract

Abstract. A criteria on the vector-valued Banach function spaces X (B) is obtained so that whenever a vector-valued singular integral operator is bounded on X (B), it can be extended to be a bounded linear operator on the corresponding Morrey type spaces. Using this result, we define the generalized Triebel-Lizorkin-Morrey spaces and obtain the atomic and molecular decompositions. As a particular example of the generalized Triebel-Lizorkin-Morrey spaces, we introduce and study the variable Triebel-Lizorkin-Morrey spaces.

Intrinsic Square Functions on Morrey and Block Spaces with Variable Exponents

“…The boundedness results of some vector-valued singular integral operators on M p(·),u are obtained in [15,22]. These results are also used to study the variable Triebel-Lizorkin-Morrey spaces in [15].…”

“…On the other hand, there are several versions of Morrey spaces with variable exponents, see [1,13,15,20,24]. Therefore, we begin with the definition of the family of Morrey spaces with variable exponents used in this paper.…”

“…The reader is referred to [15,20] for some examples of weight function u(x, t) that satisfies (4.1).…”

Boundedness of Hardy-Littlewood maximal operator on block spaces with variable exponent

“…For the boundedness of the Hardy-Littlewood maximal operator on a Morrey space with variable exponents, the reader is referred to [11]. In addition, the extension of the boundedness of the maximal operator on a vector-valued Morrey space with variable exponents is obtained in [17], [12]. Proposition 3.1.…”

“…For instance, (3.1) is valid for rearrangement-invariant quasi-Banach function spaces, the reader is referred to [14,Lemma 5]. For the general Banach function space, the reader may consult [17].…”