A fractional Muckenhoupt-Wheeden theorem and its consequences

TL;DR

This paper proves new conjectures connecting fractional maximal operators and Riesz potentials, leading to sharp weighted inequalities, by leveraging recent techniques from the proof of the $A_2$ conjecture.

Contribution

It establishes a fractional Muckenhoupt-Wheeden theorem and derives sharp weighted inequalities for Riesz potentials using modern harmonic analysis methods.

Findings

Proved conjectures linking Riesz potential and fractional maximal operator.

Established sharp one and two weight norm inequalities.

Extended classical results with modern proof techniques.

Abstract

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the $A_2$ conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.