Corner Operators with Symbol Hierarchies

This paper outlines an approach for studying operators on stratified spaces $$M \in \mathfrak {M}_k$$ with regular singularities of higher order k. Smoothness corresponds to $$k=0.$$ Manifolds with smooth boundaries belong to the category $$\mathfrak {M}_1.$$ 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 $$\{\sigma _j(A)\}_{ j=0,1},$$ consisting of interior and boundary symbols. Singularities of $$M \in \mathfrak {M}_k$$ for higher order k give rise to a sequence of strata $$s(M)=\{s_j(M) \}_{ j=0,\ldots ,k},$$ where $$s_j(M)\in \mathfrak {M}_0.$$ Operators A in corresponding algebras of operators (corner-degenerate in stretched variables) are determined by a hierarchy of symbols $$\sigma (A)=\{\sigma _j(A)\}_{ j=0,\ldots ,k},$$ modulo lower order terms. Those express ellipticity and parametrices $$A^{(-1)}$$ in weighted corner Sobolev spaces, containing sequences of real weights $$\gamma _j.$$ Components $$\sigma _j(A)$$ for $$j>0,$$ depending on variables and covariables in $$T^*(s_j(M))\setminus 0,$$ act as operator families on infinite straight cones with compact singular links in $$\mathfrak {M}_{j-1},$$ and $$\sigma _0(A)$$ is the standard principal symbol on $$T^*(s_0(M))\setminus 0.$$