Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary
Mayukh Mukherjee · Mathematische Zeitschrift · 2017
Consider a second order, strongly elliptic negative semidefinite differential operator L (may be 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 Duong et al. (J Funct Anal 196:443–485, 2002) and Duong and McIntosh (Rev Math Iberoam 15:233–265, 1999) establish sufficient conditions for $$L^\infty -\text {BMO}_L$$ continuity of $$\varphi (\sqrt{-L})$$ , where $$\varphi \in S^0_1({\mathbb {R}})$$ , and L is a suitable elliptic operator. Using a variant of the Cheeger–Gromov–Taylor functional calculus due to Mauceri et al. (Math Res Lett 16:861–879, 2009), and short time upper bounds on the integral kernel of $$e^{tL}$$ due to Greiner (Arch Ration Mech Anal 41:168–218, 1971), we prove that a variant of such sufficient conditions holds for our operator L.