Weighted inequalities for Hilbert transforms and multiplicators of Fourier transforms

Abstract

As is well known, invariant operators with a shift can be bounded from into only if . We show that the case might also hold for weighted spaces. We derive the sufficient conditions for the validity of strong (weak) type inequalities for the Hilbert transform when . The examples of couple of weights which guarantee the fulfillness of two-weighted strong (weak) type inequalities for singular integrals are presented. The method of proof of the main results allows us to generalize the results of this paper to the singular integrals which are defined on homogeneous groups. The Fourier multiplier theorem is also proved.

Keywords

• Hilbert transform
• two-weight inequality
• weak and strong type estimates
• Fourier transform
• multiplier