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<∞).
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.