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

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. Copyright © 2012 Academia Scientiarum Fennica.