The Hilbert transform does not map L^1(Mw) to L^{1,\infty}(w)

TL;DR

This paper demonstrates that a classical weighted inequality for the Hilbert transform, analogous to the Fefferman-Stein inequality for the maximal operator, does not hold, highlighting a fundamental limitation in weighted harmonic analysis.

Contribution

It provides a counterexample showing the failure of a weighted inequality for the Hilbert transform, extending previous results on Haar multipliers.

Findings

The Hilbert transform does not map L^1(Mw) to L^{1, } (w) in general.

The classical weighted inequality fails for the Hilbert transform.

This extends known failures from Haar multipliers to the Hilbert transform.

Abstract

We prove that the analogue for the Hilbert transform of a classical weighted inequality by Fefferman and Stein for the Hardy Littlewood maximal operator does not hold. This is a sequel to paper arXiv:1008.3943 by the first author, which shows the same for the general Haar multiplier operator.