Mathematical Research Letters

Volume 19 (2012)

Number 1

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

Pages: 1 – 7



Maria Carmen Reguera (School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA)

Christoph Thiele (Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA)


We disprove the following a priori estimatefor the Hilbert transform $H$ and the Hardy–Littlewood maximaloperator $M$:\[\sup_{t>0}t w\{x\in \R:|Hf(x)|>t\}\le C\int |f(x)| Mw(x) \,dx.\]This is a sequel to paper \cite{reguera} by the first author,which shows the existence of a weight $w$ and a Haar multiplier operator for which the inequality fails.

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