Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Convolution estimates related to surfaces of half the ambient dimension

S. W. Drurya1 and Kanghui Guoa2

a1 Department of Pure Mathematics and Mathematical Statistics, McGill University, Montreal H3A 2K6, Canada

a2 Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65809, U.S.A.

Let ƒ be a smooth function of compact support defined in the plane and consider the integral


The estimate


is well-known, see for instance the work of Littman[4]. The operator T amounts to convolution with the measure σ carried by the parabola t → (t, ½t2) and given by dσ = dt. Usually one proves (1) by embedding σ in an analytic family of distributions σz in xs211D2 given by


(Received September 03 1990)

(Revised January 30 1991)