Publication Cover
Applicable Analysis
An International Journal
Volume 5, 1976 - Issue 3
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Original Articles

Discrete hilbert transforms and rearrangement invariant sequence spaces

Pages 193-200 | Received 28 Jan 1972, Published online: 04 Mar 2011
 

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.

The result contained in this paper are taken from the author's Ph. D. thesis written at he University of Toronto under the direction of Professor P.G. Rooney.

The result contained in this paper are taken from the author's Ph. D. thesis written at he University of Toronto under the direction of Professor P.G. Rooney.

Notes

The result contained in this paper are taken from the author's Ph. D. thesis written at he University of Toronto under the direction of Professor P.G. Rooney.

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