Journal of Inequalities and Applications
Volume 1 (1997), Issue 3, Pages 239-252

Weighted inequalities for Hilbert transforms and multiplicators of Fourier transforms

V. Kokilashvili and A. Meskhi

A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Tbilisi 380093, Georgia

Received 12 August 1996

Copyright © 1997 V. Kokilashvili and A. Meskhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


As is well known, invariant operators with a shift can be bounded from Lp into Lq only if 1<pq<. 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<).

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.