Discrete hilbert transforms and rearrangement invariant sequence spaces

Abstract

A pair of rearrangement inequalities are obtained for a discrete analogue of the Hilbert transform which lead to necessary and sufficient conditions for certain discrete analogues of the Hilbert transform to be bouonded as linear operators between rearrangement invariant sequence spaces. In particular, if X is a rearrangement invariant space with indices α and β, then 0<β≤α<1 is both necessary and sufficient for these transforms to be bounded from X into itself, which generalizes a well known result of M. Riesz. Applications are made to discerete Hilbert transforms in higher dimensions, in particular, the discrete Riesz transforms are bounded from X into itself if and only if 0<β≤α<1.