Corner Operators with Symbol Hierarchies
Authors : D. -C. Chang, S. Khalil, and B.-W. Schulze
Abstract :This paper outlines an approach for studying operators on stratified spaces M ∈ Mk with regular singularities of higher order k. Smoothness corresponds to k = 0. Manifolds with smooth boundaries belong to the category M1. The case k = 1 generally indicates conical or edge singularities. Boutet de Monvel’s algebra of boundary value problems (BVPs) with the transmission property at the boundary may be interpreted as a special singular operator calculus for k = 1. Also, BVPs A with violated transmission properties belong to edge calculus and are controlled by pairs {σj(A)}j=0,1, consisting of interior and boundary symbols. Singularities of M ∈ Mk for higher order k give rise to a sequence of strata s(M) = {sj(M)}j=0,..,k, where sj(M) ∈ M0. Operators A in corresponding algebras of operators (corner-degenerate in stretched variables) are determined by a hierarchy of symbols σ(A) = {σj(A)}j=0,..,k, modulo lower order terms. Those express ellipticity and parametrices A(−1) in weighted corner Sobolev spaces, containing sequences of real weights γj. Components σj(A) for j > 0, depending on variables and covariables in T∗(sj(M))\0, act as operator families on infinite straight cones with compact singular links in Mj−1, and σ0(A) is the standard principal symbol on T∗(s0(M))\0.
Keywords :Boutet de Monvel’s calculus, pseudo-differential operators, stratified spaces, singular cones, Mellin symbols with values in the edge calculus, Kegel space
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