The Mellin-Edge Quantisation for Corner Operators

Abstract

We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold M with second order singularities. The typical ingredients come from the "most singular" stratum of M which is a second order edge where the infinite transversal cone has a base B that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over B. In this respect our result is formally analogous to a quantisation rule of [9] for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over B that tend to infinity.for any cut-off function ω. Similarly as (1.3) we have K 0,0 (X ∧ ) = r −n/2 L 2 (R + × X). Moreover, for every s, γ, e ∈ R we form the spaces K s,γ;e (X ∧ ) := r −e K s,γ (X ∧ ).For purposes below we introduce a family of isomorphisms κ δ : K s,γ (X ∧ ) → K s,γ (X ∧ ), (κ δ u)(r, x) := δ n+1 2 u(δr, x), where n := dim X. Below we will also employ subspaces K s,γ Θ (X ∧ ) ⊂ K s,γ (X ∧ ) of flatness −ϑ relative to γ for some −∞ ≤ ϑ < 0 and the half-open weight interval Θ = (ϑ, 0]. More precisely, we set K s,γ Θ (X ∧ ) = lim ←− ε>0 K s,γ−ϑ−ε (X ∧ ) in the respective Fréchet topology. In the case Θ = (−∞, 0] the space K s,γ Θ (X ∧ ) = K s,∞ (X ∧ ) is independent of γ.

Existence of multiple solutions for quasi-linear degenerate elliptic equations

“…Remark 2.3. Although the definitions of weighted Sobolev spaces on manifolds with corner singularity are complex (see more in [15]), Definitions 2.1 and 2.2 fit the present problem (1.1). Here since this paper concentrates on M = (0, δ) × (0, δ) × X with small enough positive δ, it sufficient to consider the case in the support of ω and σ in the definition 2.2.…”

“…where σ(y) = σ(e −y ) and the H m,γ l,0 (R + × R × X) (see more in [14] and [15]) denotes the space of all w(x 1 , y, x…”

Recent developments on Pseudo-Differential Operators (I)

“…It turns out that the details require new structures, not only new classes of weighted Sobolev spaces and edge spaces, as established in [17], but also new techniques of proving continuity of operators in those spaces, see the paper [23] of Seiler. Elements of the calculus for higher singularities are developed in [15], [16], [19], [3], [4], [20], [21], [22], and in many other papers. The corner theories, beginning with conical and edge singularities, give rise to new Mellin operators and Mellin quantisations, see Eskin's book [6] or the monographs [5], [18].…”