Corner Operators with Symbol Hierarchies

This paper outlines an approach for studying operators on stratified spaces M ∈ M k with regular singularities of higher order k . Smoothness corresponds to k = 0 . Manifolds with smooth boundaries belong to the category M 1 . 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 { σ j ( A ) } j = 0 , 1 , consisting of interior and boundary symbols. Singularities of M ∈ M k for higher order k give rise to a sequence of strata s ( M ) = { s j ( M ) } j = 0 , … , k , where s j ( M ) ∈ M 0 . Operators A in corresponding algebras of operators (corner-degenerate in stretched variables) are determined by a hierarchy of symbols σ ( A ) = { σ j ( 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 γ j . Components σ j ( A ) for j > 0 , depending on variables and covariables in T ∗ ( s j ( M ) ) \ 0 , act as operator families on infinite straight cones with compact singular links in M j - 1 , and σ 0 ( A ) is the standard principal symbol on T ∗ ( s 0 ( M ) ) \ 0 .