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

We reduce the boundedness of operators in Morrey spaces $L_p^r({\mathbb R}^n)$,itspreduals,$H^{\varrho}L_p ({\mathbb R}^n)$,andtheirpreduals$\overset{\circ}{L}{}^r_p({\mathbb R}^n)$ 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.