On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals

Abstract

We reduce the boundedness of operators in Morrey spaces $$L_p^r\left( {\mathbb R}^n\right) $$ , its preduals, $$H^{\varrho }L_p ({\mathbb R}^n)$$ , and their preduals $$\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) $$ 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.