Truncation in Besov-Morrey and Triebel-Lizorkin-Morrey spaces

TL;DR

This paper investigates the boundedness of truncation operators on Besov-Morrey and Triebel-Lizorkin-Morrey spaces, establishing conditions for boundedness, necessity, and properties like the Fubini property.

Contribution

It provides new conditions under which truncation operators are bounded on these advanced function spaces and explores their structural properties.

Findings

Operators $T^{+}$ and $T$ are bounded under certain parameter conditions.

Some conditions for boundedness are also shown to be necessary.

Triebel-Lizorkin-Morrey spaces often lack the Fubini property when $p < u$.

Abstract

We will prove that under certain conditions on the parameters the operators $T^{+}f = \max(f,0)$ and $ Tf = |f| $ are bounded mappings on the Triebel-Lizorkin-Morrey and Besov-Morrey spaces. Moreover we will show that some of the conditions we mentioned before are also necessary. Furthermore we prove that for $p < u $ in many cases the Triebel-Lizorkin-Morrey spaces do not have the Fubini property.