Hilbert transform, Toeplitz operators and Hankel operators, and invariant \(A_\infty\) weights
type: Item
instance of: scholarly article
title: Hilbert transform, Toeplitz operators and Hankel operators, and invariant \(A_\infty\) weights
zbMATH Open document ID: 0896.42009
DOI: 10.4171/RMI/223
author: Alexander Volberg
author: Sergei Treil
author: Dechao Zheng
published in: Revista Matemática Iberoamericana
publication date: 1998-10-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39541
review text: Several sufficient conditions for the boundedness of the Hilbert transform \(H\): \(L^p(\mathbb{T},u) \to L^p(\mathbb{T},v)\) are found. For instance, \(H\) is bounded if for some \(r>1\) we have \[ \sup_{z\in\mathbb{D}} v^r(z) \bigl(u^{-r/(p-1)} (z)\bigr)^{p-1} <\infty \] (for a function \(f\) on \(\mathbb{T}\), \(f(z)\) is the Poisson integral of \(f\) at \(z\in \mathbb{D})\). The same condition with \(r=1\) turns out to be necessary and sufficient if both \(v\) and \(u^{-1/(p-1)}\) are invariant \(A_\infty\)-weights -- \(w\) is called an invariant \(A_\infty\)-weight if \[ \sup_{z\in\mathbb{D}} w(z)\exp \bigl(-(\log w)(z) \bigr) <\infty. \] The properties of invariant \(A_\infty\)-weights are studied. Next, \(H:L^p(\mathbb{T},u) \to L^p(\mathbb{T},v)\) is bounded if \[ \sup_{z\in\mathbb{D}} \| v^{1/p} \circ\varphi_z \|_{L^\Phi} \| u^{-1} \circ \varphi_z \|_{L^\Psi} <\infty, \] where \(\varphi_z (\zeta) =(z-\zeta) (1-\overline z \zeta)^{-1}\) and \((\Phi, \Psi)\) is a couple of Young functions subject to certain conditions. The proofs are based on estimates of specific maximal function operators related in a natural way to a given Banach lattice of measurable functions (ideal space) on \(\mathbb{T}\). The same technique is used to obtain new conditions sufficient for the \(L^2\)-boundedness of the product \(T_fT_g\) of \(H_fH_g\) of two Toeplitz or Hankel operators. Again, these conditions are in terms of certain expressions involving \(f,g\), and some Banach lattice (in particular, Orlicz or Lorentz) norms.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 1136585
zbMATH Keywords: weighted spaces
zbMATH Keywords: Hankel operator
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: invariant \(A_\infty\)-weights
zbMATH Keywords: Hilbert transform
zbMATH Keywords: Young functions
zbMATH Keywords: maximal function
reviewed by: S. V. Kislyakov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1136585