Oscillatory integrals and Fourier transforms of surface carried measures

Abstract

We suppose that S S is a smooth hypersurface in R n + 1 {{\mathbf {R}}^{n + 1}} with Gaussian curvature κ \kappa and surface measure d S dS , w w is a compactly supported cut-off function, and we let μ α {\mu _\alpha } be the surface measure with d μ α = w κ α d S d{\mu _\alpha } = w{\kappa ^\alpha }\,dS . In this paper we consider the case where S S is the graph of a suitably convex function, homogeneous of degree d d , and estimate the Fourier transform μ ^ α {\hat \mu _\alpha } . We also show that if S S is convex, with no tangent lines of infinite order, then μ ^ α ( ξ ) {\hat \mu _\alpha }(\xi ) decays as | ξ | − n / 2 |\xi {|^{ - n / 2}} provided α ⩾ [ ( n + 3 ) / 2 ] \alpha \geqslant [(n + 3)/2] . The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.