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

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\bar{M}$ 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 $L^{\mathrm{\infty }}-{\text{BMO}}_L$ continuity of $\phi (\sqrt{A})$, where $\phi \in S_1^0(\mathbb{R})$, 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 $L$.