Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary

TL;DR

This paper investigates the boundedness of spectral multipliers for generalized Laplacians on compact manifolds with boundary, extending classical results using advanced functional calculus and kernel estimates.

Contribution

It establishes new sufficient conditions for spectral multiplier boundedness of elliptic operators on manifolds with boundary, utilizing a modified functional calculus and kernel bounds.

Findings

Proves boundedness of spectral multipliers under new conditions.

Extends classical results to operators with boundary conditions.

Utilizes a variant of the Cheeger-Gromov-Taylor functional calculus.

Abstract

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary condition. Classically known results, and also recent work in \cite{DOS} and \cite{DM} establish sufficient conditions for $L^\infty-\text{BMO}_L$ continuity of $\varphi(\sqrt{A})$, where $\varphi \in S^0_1(\mathbb{R})$, and $A$ is a suitable elliptic operator. Using a variant of the Cheeger-Gromov-Taylor functional calculus due to \cite{MMV}, and short time bounds on the integral kernel of $e^{tL}$ due to \cite{G}, we prove that a variant of such sufficient conditions holds for our operator $L$.