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

S0305004100070201_eqnU001

The estimate

S0305004100070201_eqn001

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

S0305004100070201_eqnU002

(Received September 03 1990)

(Revised January 30 1991)