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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
##### Keywords:
operator norm; sharp bound; Hilbert transform
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