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A sharp estimate for the weighted Hilbert transform via Bellman functions. (English) Zbl 1040.42008
Let $$\omega$$ be a positive $$L^1$$-function on the unit circle $$\mathbb{T},m$$ be the normalized Lebesgue measure on $$\mathbb{T}$$, and $$L^2_\mathbb{T} (\omega)$$ the space with the norm $\| f\|_\omega= \left( \int_\mathbb{T} | f|^2 \omega dm\right)^{1/2}.$ Furthermore, let $Q_2^{inv} (\omega)= \sup_{z=\mathbb{D}} \omega(z) \omega^{-1}(z).$ Here $$\mathbb{D}$$ is the unit disc, $$\omega(z)$$ is the harmonic extension of $$\omega$$: $\omega(z)= \int\omega (t)P_z(t) dm(t),$ where $$P_z(t)= (1-| z|^2)/ | 1-\overline zt|^2$$ and $$\omega^{-1}(z)$$ denotes the extension of $$1/\omega$$. The authors prove: Theorem: The Hilbert transform $$H:L^2_\mathbb{T} (\omega)\to L^2_\mathbb{T} (\omega)$$ has operator norm $$\| H\| \leq C Q^{inv}_2 (\omega)$$, where $$C$$ does not depend on $$Q_2^{inr} (\omega)$$.

##### MSC:
 42A50 Conjugate functions, conjugate series, singular integrals 44A15 Special integral transforms (Legendre, Hilbert, etc.) 47B38 Linear operators on function spaces (general)
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