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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 \(Q_2^{inr} (\omega)\).

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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