L^p estimates for operators associated to flat curves without the Fourier transform

Anthony Carbery, James Thomas Vance, Jr., Stephen Wainger, David K. Watson and James Wright

Vol. 167 (1995), No. 2, 243–262

DOI: 10.2140/pjm.1995.167.243

Abstract

The purpose of this paper is to provide new proofs to known theorems on the L^p boundedness of the maximal function and Hilbert transform corresponding to curves in Rⁿ which are “infinitely flat” at the origin. The old proofs use the Fourier transform in a crucial way. The present proofs avoid the Fourier transform and hence at least have the potential of being used in more general situations.

Mathematical Subject Classification 2000

Primary: 42B25

Milestones

Received: 20 October 1992

Revised: 27 November 1992

Published: 1 February 1995