Multipliers for spherical harmonic expansions

TL;DR

This paper characterizes when certain spherical harmonic multipliers are uniformly bounded on L^p spaces, extends results to elliptic operators on manifolds, and introduces new transference principles and quasi-orthogonality estimates.

Contribution

It provides a complete characterization of bounded multipliers on spheres and extends these results to general elliptic pseudodifferential operators on manifolds, with new transference principles.

Findings

Characterization of compactly supported multipliers for spherical harmonics

Extension of boundedness results to elliptic operators on manifolds

New quasi-orthogonality estimates for solutions to the half-wave equation

Abstract

For any bounded, regulated function $m: [0,\infty) \to \mathbb{C}$, consider the family of operators $\{ T_R \}$ on the sphere $S^d$ such that $T_R f = m(k/R) f$ for any spherical harmonic $f$ of degree $k$. We completely characterize the compactly supported functions $m$ for which the operators $\{ T_R \}$ are uniformly bounded on $L^p(S^d)$, in the range $1/(d-1) < |1/p - 1/2| < 1/2$. We obtain analogous results in the more general setting of multiplier operators for eigenfunction expansions of an elliptic pseudodifferential operator $P$ on a compact manifold $M$, under curvature assumptions on the principal symbol of $P$, and assuming the eigenvalues of $P$ are contained in an arithmetic progression. One consequence of our result are new transference principles controlling the $L^p$ boundedness of the multiplier operators associated with a function $m$, in terms of the $L^p$ operator norm of the radial Fourier multiplier operator with symbol $m(|\cdot|): \mathbb{R}^d \to \mathbb{C}$. In order to prove these results, we obtain new quasi-orthogonality estimates for averages of solutions to the half-wave equation $\partial_t - i P = 0$, via a connection between pseudodifferential operators satisfying an appropriate curvature condition and Finsler geometry.