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.