Convolution with measures on hypersurfaces

Abstract

Let S be a smooth hypersurface in ℝⁿ with surface area measure ds and Gaussian curvature κ(s). Define the convolution operator T by

formula here

for suitable functions f on ℝⁿ. We are interested in the L^p − L^q mapping properties of T. Write [Sscr ] for the type set of T, the set

formula here

It is well known (see, e.g. [O1]) that [Sscr ] is contained in the closed triangle [Tscr ] with vertices (0, 0), (1, 1) and (n/(n+1), 1/(n+1)). This paper is concerned with estimates of the form

formula here

The estimate (1) is interesting because it serves as a weak substitute for the L^(n+1)/n − Lⁿ⁺¹ boundedness of T. For example, if S is compact and (1) holds, then well-known arguments show that [Sscr ] differs from the full triangle [Tscr ] by at most the point (n/(n + 1), 1/(n + 1)). Our main result is a condition sufficient to imply (1). Its statement requires the following definition.