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Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary. (English) Zbl 1398.53042

Math. Z. 289, No. 3-4, 1011-1031 (2018); erratum ibid. 296, No. 1-2, 881 (2020).
Author’s abstract: 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 [X. T. Duong et al., J. Funct. Anal. 196, No. 2, 443–485 (2002; Zbl 1029.43006); Rev. Mat. Iberoam. 15, No. 2, 233–265 (1999; Zbl 0980.42007)] establish sufficient conditions for \(L^\infty -\mathrm {BMO}_L\) 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 [G. Mauceri et al., Math. Res. Lett. 16, No. 5–6, 861–879 (2009; Zbl 1202.53043)], and short time upper bounds on the integral kernel of \(e^{tL}\) due to [P. Greiner, Arch. Ration. Mech. Anal. 41, 163–218 (1971; Zbl 0238.35038)], we prove that a variant of such sufficient conditions holds for our operator \(L\).

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35L05 Wave equation
47A60 Functional calculus for linear operators
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