$L^p$-estimates for FIO-cone multipliers

TL;DR

This paper develops $L^p$-estimates for a new class of Fourier integral operator-based cone multipliers in three dimensions, improving existing bounds and applying results to maximal averages on surfaces.

Contribution

It introduces and analyzes FIO-cone multipliers adapted to cone geometry, extending $L^p$-estimates and confirming a conjecture on maximal averages for certain surfaces.

Findings

Established $L^p$-bounds for FIO-cone multipliers in $4/3 \,\leq p \leq 4$ range.

Improved estimates by a factor of $R^{-|1/p-1/2|}$ over previous methods.

Confirmed a conjecture on the critical Lebesgue exponent for specific surfaces.

Abstract

The classical cone multipliers are Fourier multiplier operators which localize to narrow $1/R$-neighborhoods of the truncated light cone in frequency space. By composing such convolution operators with suitable translation invariant Fourier integral operators (FIOs), we obtain what we call FIO-cone multipliers. We introduce and study classes of such FIO-cone multipliers on $\Bbb R^3$, in which the phase functions of the corresponding FIOs are adapted in a natural way to the geometry of the cone and may even admit singularities at the light cone. By building on methods developed by Guth, Wang and Zhang in their proof of the cone multiplier conjecture in $\Bbb R^3,$ we obtain $L^p$-estimates for FIO-cone multipliers in the range $4/3\le p\le 4$ which are stronger by the factor $R^{-|1/p-1/2|}$ than what a direct application of the method of Seeger, Sogge and Stein for estimating FIOs would give. An important application of our theory is to maximal averages along smooth analytic surfaces in $\Bbb R^3.$ It allows to confirm a conjecture on the the critical Lebesgue exponent for a prototypical surface from a small class of ``exceptional'' surfaces, for which this conjecture had remained open.