Radial Fourier multipliers in high dimensions

Abstract

Given a fixed p = 2, we prove a simple and effective characterization of all radial multipliers of FL p (R d ), provided that the dimension d is sufficiently large. The method also yields new L q space-time regularity results for solutions of the wave equation in high dimensions.

Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation

“…Based on standard calculations for oscillatory integrals ( [10], [18], [1], [11], [14]) we prove some estimates for scalar products S z , S z ′ ; these results are closely related to the scalar product estimates in [6]. For the Fourier transforms we have…”

“…In the constant coefficient case one can recover from Theorem 1.1 the space time estimates of [6] which correspond to an endpoint version of Sogge's conjecture in the range given for ℓ = d − 1, see also §4 for other generalizations. For previous partial results on Sogge's conjecture, also in lower dimensions, see the groundbreaking paper of Wolff [20] and the subsequent papers [12], [5].…”

“…In this note we prove a variable coefficient version of a recent result in [6] on the local L q space-time regularity results for solutions of wave equations. The solution operators are Fourier integral operators satisfying the 'cinematic curvature' hypothesis introduced in [17] (see also [14]).…”

“…The argument is a variant of one in [6]; it is based on a Calderón-Zygmund type decomposition at height u k (α) where volume is replaced by diameter. By the usual Vitali procedure there is a finite (possibly empty) family B k of disjoint balls so that…”

“…For the case r ≥ 1 one could argue by duality and follow [6] but we shall rely on a result in [15] which gives an estimate for all r > 0.…”