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Multipliers for spherical harmonic expansions

Jacob Denson

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.

Multipliers for spherical harmonic expansions

Abstract

For any bounded, regulated function , consider the family of operators on the sphere such that for any spherical harmonic of degree . We completely characterize the compactly supported functions for which the operators are uniformly bounded on , in the range . We obtain analogous results in the more general setting of multiplier operators for eigenfunction expansions of an elliptic pseudodifferential operator on a compact manifold , under curvature assumptions on the principal symbol of , and assuming the eigenvalues of are contained in an arithmetic progression. One consequence of our result are new transference principles controlling the boundedness of the multiplier operators associated with a function , in terms of the operator norm of the radial Fourier multiplier operator with symbol . In order to prove these results, we obtain new quasi-orthogonality estimates for averages of solutions to the half-wave equation , via a connection between pseudodifferential operators satisfying an appropriate curvature condition and Finsler geometry.

Paper Structure

This paper contains 16 sections, 16 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. Applications to Eigenfunction Expansions
  4. A Resulting Transference Theorem
  5. Applications To Other Pseudodifferential Operators
  6. Related Results
  7. Summary of Proof
  8. Preliminary Setup
  9. Quasi-Orthogonality Estimates For Solutions to the Half-Wave Equation Obtained From High-Frequency Wave Packets
  10. Some Facts About Finsler geometry
  11. Analysis of Critical Points
  12. Analysis of Regime I via Density Methods
  13. $L^2$ Estimates For Bounded Density Inputs
  14. $L^p$ Estimates Via Density Decompositions
  15. Analysis of Regime II via Local Smoothing
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose $1 < p < 2(d-1)/(d+1)$. For any bounded, regulated function $m$ with compact support in $(0,\infty)$, define operators $T_R$ on $S^d$ by setting where $f_k$ is the projection of $f$ onto the space of degree $k$ spherical harmonics. Then, with implicit constants depending on the support of $m$.

Theorems & Definitions (36)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['zonalconvolutionapplication']}
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • ...and 26 more