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

Abstract

Consider an operator $T:E \to X(\mu)$ from a Banach space $E$ to a Banach function space $X(\mu)$ over a finite measure $\mu$ 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.