Hilbert transform, Toeplitz operators and Hankel operators, and invariant $A_{\infty }$ weights

Abstract

In this paper several sufficient conditions for boundedness of the Hilbert transform between two weighted $L^p$-spaces are obtained. Invariant $A_{\infty }$ weights are introduced. Several characterizations of invariant $A_{\infty }$ weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators or Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.