Optimality of function spaces for kernel integral operators

TL;DR

This paper characterizes the optimal target rearrangement-invariant space for kernel integral operators acting on a given space, providing a precise understanding of their boundedness properties, especially in Lorentz spaces.

Contribution

It introduces a general framework for identifying the minimal rearrangement-invariant space ensuring boundedness of kernel integral operators.

Findings

Characterization of the optimal range space for kernel integral operators.

Application of results to Lorentz spaces demonstrating their effectiveness.

Establishment of boundedness criteria for these operators on r.i. spaces.

Abstract

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space $X$ we characterize its optimal range partner, that is, the smallest r.i. space $Y$ such that the operator is bounded from $X$ to $Y$. We apply the general results to Lorentz spaces to illustrate their strength.