Abstract
Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb–Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential operators originally proved by Grubb–Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.
Highlights
We develop a calculus of parameter-dependent pseudodifferential operators, both for operators in Euclidean space Rn and for operators on sections of vector-bundles over closed Riemannian manifolds, which is closely related to Grubb’s calculus of operators with finite regularity number [3] and to the Grubb–Seeley calculus introduced in [6]
Definition 5.1 We denote by Sd1,0ν, d, ν ∈ R, the subspace of S1d,0ν consisting of all symbols a(x, ξ ; μ) for which exists a sequence of symbols a[∞ν+ j] ∈ S1ν,+0 j (Rn), j ∈ N0, such that ra,N (x, ξ ; μ) := a(x, ξ ; μ) −
Using the expansions of a# j ∈ S01,−0∞,0−∞ and noting that (a# j )∞ [0] = 0 for every j due to the multiplicativity of the principal limit-symbol, we find a sequence of symbols b[∞j] ∈ S−∞(Rn) such that, for every N ∈ N0, (1 − a)−# = 1 + b[∞j][ξ, μ]− j + rN, j =1
Summary
We develop a calculus of parameter-dependent pseudodifferential operators (ψdo), both for operators in Euclidean space Rn and for operators on sections of vector-bundles over closed Riemannian manifolds, which is closely related to Grubb’s calculus of operators with finite regularity number [3] (for a recent application to fractional heat equations see [5]) and to the Grubb–Seeley calculus introduced in [6]. 4 we show that a(D; μ) is an operator of order d and with regularity number ν ∈ R in the sense of [3] if a admits a decomposition a = a + p with p ∈ Sd and where a admits an expansion of form (1.1), with components satisfying (1.2) but with singular functions a j : introducing polar coordinates (r , φ) on Sn+, centered in the “north-pole” (ξ, μ) = (0, 1), they belong to the weighted space r ν− j CB∞(Sn+), where Sn+ = Sn+ \{(0, 1)} and CB∞ means smooth functions which remain bounded on Sn+ after arbitrary applications of totally characteristic derivatives r ∂r and usual derivatives in φ This observation leads us to consider symbols a = a + p with p ∈ Sd but where the homogeneous components of a originate from the weighted spaces r ν− j CT∞(Sn+), ν ∈ Z, where CT∞(Sn+) is the space of all functions on Sn+ that, in coordinates (r , φ), extend smoothly up to and including r = 0 (the subscript T stands for Taylor expansion). Hoping to help the reader in reading this paper, we finish this introduction by listing the most important spaces of pseudodifferential symbols used in the sequel: S1d,0(Rn), Sd (Rn) : S1d,,0ν , Sd,ν , Shdo,νm : S1d,0, Sd , Shdom : S1d,,0ν , Sd,ν , Shdo,νm : Sd1,,0ν , Sd,ν , Sdho,νm : Sd,ν , Sdho,νm : Section 2.2 Definitions 3.1, 3.3, and 3.4 Definitions 3.7, 3.9, and 3.10 Definitions 3.12, 3.15, and 3.14 Definitions 5.1, 5.14 and 5.12 Definition 5.18
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