Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function

Abstract

We characterize boundedness of a convolution operator with a fixed kernel between the classes $S^p(v)$, defined in terms of oscillation, and weighted Lorentz spaces $\Gamma^q(w)$, defined in terms of the maximal function, for $0 < p,q \le \infty$. We prove corresponding weighted Young-type inequalities of the form $$ |f\ast g|{\Gamma^q(w)} \le C |f|{\S^p(v)}|g|\_Y $$ and characterize the optimal rearrangement-invariant space $Y$ for which these inequalities hold.