L^p-boundedness of the multiple Hilbert transform along a surface

James Thomas Vance, Jr.

Vol. 108 (1983), No. 1, 221–241

DOI: 10.2140/pjm.1983.108.221

Abstract

For an appropriate surface σ in Rⁿ, we prove that the multiple Hilbert transform along σ is a bounded operator on L^p(Rⁿ), for p sufficiently close to 2. Our analysis of this singular integral operator proceeds via Fourier transform techniques—that is, on the “multiplier side”—with applications of Stein’s analytic interpolation theorem and the Marcinkiewicz multiplier theorem. At the heart of our argument we have estimates of certain trigonometric integrals.

Mathematical Subject Classification 2000

Primary: 44A15

Secondary: 42B20

Milestones

Received: 1 November 1981

Published: 1 September 1983