Relative elliptic operators and the Sobolev problem. I (Q2459776)

Relative elliptic theory is a sort of elliptic theory associated with a pair \((M,X)\), where \(M\) is a smooth manifold without boundary and \(X\) is a smooth or stratified submanifold of \(M\). An operator algebra associated with a smooth embedding \(i\colon X\to M\) has been constructed in [\textit{B.\,Y.\thinspace Sternin} and \textit{V.\,E.\thinspace Shatalov}, Sb.\ Math.\ 187, No.\,11, 1691--1720; translation from Mat.\ Sb.\ 187, No.\,11, 115--144 (1996; Zbl 0882.58053)]. That algebra contains all operators which can be represented as admissible compositions of pseudodifferential operators and the boundary and coboundary operators induced by \(i\). In the paper under review, the author generalizes this theory to the case when the submanifold \(X\) is a stratified submanifold represented as a union of transversally intersecting manifolds. This first part of the paper describes in detail all integral operators obtained as arbitrary admissible compositions of boundary and coboundary operators and pseudodifferential operators acting on various submanifolds. It is shown that the set of these operators is closed and contains only operators of seven types, five of which have been comprehensively studied earlier, while the remaining two types are analyzed here. [For Part II, see the following review Zbl 1133.58022.]