Weighted inequalities for Hilbert transforms and multiplicators of Fourier transforms

Abstract

As is well known, invariant operators with a shift can be bounded from Lp into Lq only if 1<p≤q<∞. We show that the case q<p might also hold for weighted spaces. We derive the sufficient conditions for the validity of strong (weak) (p,q) type inequalities for the Hilbert transform when 1<q<p<∞ (q=1,1<p<∞).

Keywords

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