The generalized M. Riesz theorem and transference

Earl Robert Berkson and Thomas Alastair Gillespie

Vol. 120 (1985), No. 2, 279–288

DOI: 10.2140/pjm.1985.120.279

Abstract

We use transference methods to give a new proof for Bochner’s abstract generalization of the M. Riesz Theorem on conjugate harmonic functions. The proof makes direct use of the classical Hilbert transform, providing it with a structural role in the abstract setting. The norm of the classical Hilbert transform is thereby shown to be the least admissible bound in the abstract theorem. Analogous results are obtained for certain classical multiplier transforms related to the Hilbert transform.

Mathematical Subject Classification 2000

Primary: 43A17

Secondary: 42B15

Milestones

Received: 12 April 1984

Published: 1 December 1985