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

Abstract

We disprove the following a priori estimate for the Hilbert transform H and the Hardy Littlewood maximal operator M :This is a sequel to paper [5] by the first author, which shows the existence of a Haar multiplier operator for which the inequality holds.

Classical Fourier Analysis

“…The conjecture whether T L 1 (M(w))→L 1,∞ (w) < ∞ holds for a weight w was disproved by Reguera [287] when T is a Haar multiplier and then by Reguera and Thiele [288] for the Hilbert transform. However, the slightly weaker version of this inequality, in which M(w) is replaced by the Orlicz maximal operator M L(log L) ε (w), holds for any ε > 0 and any Calderón-Zygmund operator T , as shown by Pérez [277].…”

The Hardy-Littlewood maximal type operators between Banach function spaces

“…It was conjectured in that paper that this result would be false in the case ε = 0 where M L(log L) ε is replaced by M, but only recently this result was shown to be false by Reguera and Thiele in [21] (see also the previous work by Reguera in [20]). …”

On weak weighted estimates of the martingale transform and a dyadic shift

“…It took many years to disprove (1.2). This was done by Maria Reguera and Christoph Thiele [7] (for the martingale transform), [8] (for the Hilbert transform). The constructions involve a very irregular (almost a sum of delta measures) weight w, so there was a hope that such an effect cannot appear when the weight is regular in the sense that w ∈ A 1 .…”