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

and characterize the optimal rearrangement-invariant space $Y$ for which these inequalities hold.