L ²-average Decay of the Fourier Transform of a Characteristic Function of a Convex Set

Abstract

Let B be a bounded open set in ℝ^d. As we note in the introduction, it is a consequence of the classical method of stationary phase that if $\partial{B}$ is sufficiently smooth and has everywhere non-vanishing Gaussian curvature, then

$$ |\hat{X}B(Rw)| \lesssim R ^{-{\frac{d+1}{2}}}$$

with constants independent of ω