Sharp weighted bounds for fractional integral operators

TL;DR

This paper establishes sharp weighted bounds for fractional integral operators and their maximal functions, leading to improved Sobolev inequalities through advanced extrapolation techniques and two-weight norm characterizations.

Contribution

It provides the first sharp bounds for fractional integral operators and maximal functions in weighted spaces, enhancing understanding of their operator norms.

Findings

Sharp bounds for fractional integral operators derived

Enhanced Sobolev inequalities obtained

Advanced extrapolation techniques applied

Abstract

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.