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The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) characteristic. (English) Zbl 1139.44002
The paper is concerned with the sharp bound for the operator norm of the Hilbert transform in \(L^p(\omega)\). The author shows that the norm of the Hilbert transform as an operator in the weighted space \(L_{\mathbb R}^p(\omega)\) for \(2<p<\infty\) is bounded by a constant multiple of the first power of the classical \(A_p\) characteristic of \(\omega\). This result is sharp.
He also proves a bilinear impeding theorem with simple conditions. One of the theorem is Theorem 1. There exists a constant \(c\) so that for all weights is \(E\in A_2\) the Hilbert transform as an operator in weighted space \(H:L^2_{\mathbb R}(\omega)\rightarrow L_{\mathbb R}^2(\omega)\) has operator norm \(\| H\| \leq cQ_2(\omega)\) and this result is sharp.

MSC:
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47B38 Linear operators on function spaces (general)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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