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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.

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