Classification of singular integrals over a local field

Charles P. Downey

Vol. 59 (1975), No. 2, 417–426

DOI: 10.2140/pjm.1975.59.417

Abstract

The singular integral operators over a local field K whose kernels are multiplicative characters of the unit sphere of K are shown to be precisely those continuous operators on ℒ₂(K) which commute with translation and dilation, anticommute with an appropriately defined rotation, and whose multipliers satisfy a smoothness condition. The characterization is analogous to that of the Hilbert transform over the real numbers.

Mathematical Subject Classification 2000

Primary: 44A25

Secondary: 43A70

Milestones

Received: 14 October 1974

Revised: 12 April 1975

Published: 1 August 1975