Vector-valued operators with singular kernel and Triebel–Lizorkin block spaces with variable exponents

Abstract

We establish two principles which state that, whenever an operator is bounded on a given Banach function space, then under some simple conditions, it is also bounded on the corresponding Morrey spaces and block spaces. By applying these principles on some concrete operators, we generalize the Fefferman-Stein vector-valued inequalities, define and study the Triebel-Lizorkin block spaces with variable exponents, and extend the mapping properties of the fractional integral operators to Morrey-type spaces and block-type 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].…”

“…The following recalls the vector-valued block spaces used in [22] for the studies of the vector-valued operators on block spaces and Morrey spaces.…”

“…The boundedness of some vector-valued operators with singular kernel such as the Calderón-Zygmund operators and fractional integral operators on B q p(·),u is presented in [22].…”

Strong maximal operator and singular integral operators in weighted Morrey spaces on product domains

“…The above conditions are product domain version of the conditions used in [, (5.0.2)] and [, Definition 3.1].…”

Definability of singular integral operators on Morrey–Banach spaces

“…Recently, the studies of Morrey spaces is extending to Morrey space built on some non Lebesgue spaces such as Morrey-Lorentz spaces [3,18,31], Orlicz-Morrey spaces [11,29,28], Morrey spaces with variable exponents [1,13,15,19,22,24,25,26,33,32]. On the other hand, for instance, in [15,19], we are lack of a precise definition of the action of singular integral operators on the above mentioned Morrey type spaces. It is important to give a precise definition on the singular integral operators studied in the above mentioned results as it gives a solid foundation for us to study the boundedness of singular integral operators on Morrey spaces.…”