Multipliers for spherical harmonic expansions

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.