On the sharp upper bound related to the weak Muckenhoupt-Wheeden conjecture

TL;DR

This paper demonstrates that a specific upper bound for the Hilbert transform's weak-type norm, involving the A_1 characteristic of weights, cannot be improved, confirming its sharpness.

Contribution

The paper provides a constructed example showing the sharpness of the known upper bound related to the weak Muckenhoupt-Wheeden conjecture.

Findings

The upper bound $[w]_{A_1}\log({ m{e}}+[w]_{A_1})$ for the Hilbert transform's norm is sharp.

No general improvement on this upper bound is possible.

The result confirms the conjecture's limitations in the weighted setting.

Abstract

We construct an example showing that the upper bound $[w]_{A_1}\log({\rm{e}}+[w]_{A_1})$ for the $L^1(w)\to L^{1,\infty}(w)$ norm of the Hilbert transform cannot be improved in general.