Resolvents and complex powers of semiclassical cone operators

Abstract

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter h tends to 0. An example of such an operator is the shifted semiclassical Laplacian h2Δg+1$h^2\Delta _g+1$ on a manifold false(X,gfalse)$(X,g)$ of dimension n≥3$n\ge 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space false[0,1false)h×X×X$[0,1)_h\times X\times X$ of h‐dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of (h2normalΔg+1)w/2${\big (h^2\Delta _g+1\big )}^{w/2}$ for prefixRew∈()−n2,n2$\operatorname{Re}w\in \left(-\tfrac{n}{2},\tfrac{n}{2}\right)$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.

Linear waves on asymptotically flat spacetimes. I

“…(1.17)). The Mellin-transformed normal operator in T lies in the algebra of b-operators near R = 0, and in the high frequency regime in the algebra of semiclassical cone operators [Hin22b,Hin21d]. This is discussed in the general 3b-setting in [Hin23b, § §1.1, 3.3].…”

“…Many of these algebras are well-known; accounts for b-analysis are given in [Mel93,Gri01,Mel96], and the 0-, edge, semiclassical scattering, b-edge, and scattering-b-transition algebras are discussed in the original papers [MM87, Maz91, VZ00, MVW08, GH08]. The 3b-and semiclassical cone algebras were defined in [Hin23b,Hin22b,Hin21d]. For the present paper, the detailed presentations in [HV23b, §2] and [Hin23b, §2] are particularly relevant.…”