Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces

TL;DR

This paper establishes conditions for the boundedness of kernels of functions of the Laplacian on non-compact symmetric spaces, providing sharp bounds in certain cases and broad implications for spectral analysis.

Contribution

It introduces a decay-based condition on functions ensuring uniform kernel bounds for Laplacian functions on symmetric spaces, extending pointwise estimates.

Findings

Kernel bounds depend only on decay of F, not derivatives

Results apply to a wide class of functions of the Laplace-Beltrami operator

Bounds are sharp for specific oscillatory functions when G has real rank one

Abstract

Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated with a semisimple Lie group $G$. Assume $F$ is a Borel function on $\mathbb{R}^+$. We give a condition on $F$ such that the kernels of the functions $F(L)$ are uniformly bounded. This condition involves the decay of $F$ only and not its derivatives. By a known correspondence, this implies pointwise estimates for a wide range of functions of the Laplace-Beltrami operator on symmetric spaces. In particular, when $G$ is of real rank one and $F(x)={\rm e}^{it\sqrt x}\psi(\sqrt x)$, our bounds are sharp.