Parameter-dependent pseudodifferential operators of Toeplitz type

Abstract

Grubb's symbol class of parameter-dependent pseudodifferential symbols of finite regularity on R n is shown to split into so-called weakly and strongly parameter-dependent symbols. For weakly parameter-dependent symbols the regularity is shown to have an interpretation as a polynomial weight in the space of homogeneous components. For a suitable sub-class of weakly parameter-dependent symbols we establish a complete pseudodifferential calculus with ellipticity that implies invertibilty of parameter-dependent operators for large values of the parameter. Combined with strongly parameterdependent symbols we obtain a refined calculus for operators of finite regularity, where ellipticity is also defined in case of vanishing regularity. Applications concern resolvents of pseudodifferential operators and parameter-dependent operators of Toeplitz type. PSEUDODIFFERENTIAL OPERATORS OF FINITE REGULARITY 5 with the modulus |y| outside some compact set. If y = (ξ, µ), we write shortly |ξ, µ| := |(ξ, µ)|, ξ, µ = (ξ, µ) , and [ξ, µ] := [(ξ, µ)].A 0-excision function on R m is a smooth function χ(y) that vanishes in a neighborhood of the origin and such that (1 − χ)(y) has compact support. Functions of the form (1 − χ)(y) are called cut-off functions.

Bounded imaginary powers of cone differential operators on higher order Mellin–Sobolev spaces and applications to the Cahn–Hilliard equation

“…For details see e.g. [13,Section 6], [21,Section 2.3] or [26]. Here it suffices to say that ρ varies over the noninvertibility points of the conormal symbol of ∆ 2 in ](n+1)/2−γ−4, (n+1)/2−γ[, and the elements in each F ρ are suitable linear combinations of functions of the form x −ρ+j log k xω(x)e(y), where j ∈ {0, 1, 2, 3}, k ∈ {0, 1} and e ∈ C ∞ (∂B).…”

“…the closure of A considered as an operator on C This result has a long history, see e.g. [2], [18], [21], [26]; the present version is due to Gil, Krainer and Mendoza [13]. A densely defined unbounded operator A on a Banach space X 0 is said to have bounded imaginary powers with angle φ ≥ 0, provided that (i) its resolvent exists in a closed sector Λ θ of angle θ around the negative real axis and decays there like λ −1 as λ → ∞, and (ii) the purely imaginary powers A it , t ∈ R, satisfy the estimate A it L(X0) ≤ M e φ|t| for a suitable constant M .…”

Parametric pseudodifferential operators with point-singularity in the covariable

“…yields the desired parametrix B(μ). However, this result follows from the general theory of abstract pseudodifferential operators and associated Toeplitz operators developed in [14,15]. In fact, in the notation of [15, Section 3.1] let = R + , let G = {g = (M, E) | E vector-bundle over M} be the set of all admissible weights and let H 0 (g) = L 2 (M, E) for g = (M, E).…”

“…Let us conclude with an application to so-called ψdo of Toeplitz type, cf. [14,15]. To this end, for j = 0, 1, let E j be a vector-bundle over M and P j ∈ L 0 (M; E j , E j ) be idempotent, i.e., P 2 j = P j .…”

“…In the present paper, we develop a calculus of parameter-dependent pseudodifferential operators (ψdo), both for operators in Euclidean space R n 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]. The calculus allows to obtain the classical resolvent-trace expansion for elliptic ψdo due to [6] and a systematic treatment of ψdo of Toeplitz type in the sense of [14,15].…”