Optimal range theorems for operators with p-th power factorable adjoints

Abstract

Consider an operator T : E ! X(µ) from a Banach space E to a Banach function space X(µ) over a finite measure µ such that its dual map is p-th power factorable. We compute the optimal range of T that is defined to be the smallest Banach function space such that the range of T lies in it and the restricted operator has p-th power factorable adjoint. For the case p = 1, the requirement on T is just continuity, so our results give in this case the optimal range for a continuous operator. We give examples from classical and harmonic analysis, as convolution operators, Hardy type operators and the Volterra operator.