Transferring spherical multipliers on compact symmetric spaces

TL;DR

This paper establishes a transference theorem connecting $L^{p}$ spherical multipliers on compact symmetric spaces with those on their tangent vector spaces, generalizing a classical Fourier analysis result.

Contribution

It introduces a two-sided transference theorem for $L^{p}$ spherical multipliers on compact symmetric spaces, extending deLeeuw's classical theorem to a broader geometric setting.

Findings

Proves a two-sided transference theorem for $L^{p}$ spherical multipliers.

Generalizes deLeeuw's theorem from tori and Euclidean spaces to symmetric spaces.

Bridges harmonic analysis on symmetric spaces with classical Fourier analysis.

Abstract

We prove a two-sided transference theorem between $L^{p}$ spherical multipliers on the compact symmetric space $U/K$ and $L^{p}$ multipliers on the vector space $i\mathfrak{p},$ where the Lie algebra of $U$ has Cartan decomposition $\mathfrak{k\oplus }i\mathfrak{p}$. This generalizes the classic theorem transference theorem of deLeeuw relating multipliers on $% L^{p}(\mathbb{T)}$ and $L^{p}(\mathbb{R)}$.