On the boundedness of singular integrals in Morrey spaces and its preduals

TL;DR

This paper establishes a reduction of the boundedness problem for singular integrals in Morrey spaces and their preduals to Lebesgue space boundedness, simplifying analysis for many classical harmonic analysis operators.

Contribution

It introduces a method to reduce boundedness in Morrey spaces and their preduals to Lebesgue space boundedness, applicable to a wide class of harmonic analysis operators.

Findings

Boundedness in Morrey spaces reduces to Lebesgue space boundedness.

Applicable to singular integral operators and the Hardy-Littlewood maximal function.

Uses vector-valued analysis for broader applications.

Abstract

We reduce the boundedness of operators in Morrey spaces $L_p^r({\mathbb R}^n)$, its preduals, $H^{\varrho}L_p ({\mathbb R}^n)$, and their preduals $\overset{\circ}{L}{}^r_p({\mathbb R}^n)$ to the boundedness of the appropriate operators in Lebesgue spaces, $L_p({\mathbb R}^n)$. Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.