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On the two weights problem for the Hilbert transform. (English) Zbl 0880.44002

This paper deals with the two weights problem for the Hilbert transform. This means to give some sufficient conditions on pairs of weights functions \((u,v)\) so that the classical Hilbert transform becomes a bounded operator from \(L^2(u)\) to \(L^2(v)\). The authors mention that the weights functions are scalar, matrix or operator valued. In the introduction they describe briefly three conditions under which the Hilbert transform is a bounded operator as above. When \(u=v\) these conditions are equivalent to the classical Muckenhoupt \(A_2\) condition. Before the direct proof of the main theorem the authors give important preliminaries in the two-weighted estimates of integral operators such as lemmas on Cotlar, Schur, Carleson conditions and an operator version of Jensen’s inequality.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A63 Linear operator inequalities
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