Hilbert transform, Toeplitz operators and Hankel operators, and invariant \(A_\infty\) weights
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.
scientific article; zbMATH DE number 1136585