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Relative elliptic theory. (English) Zbl 1090.58016
Gil, Juan (ed.) et al., Aspects of boundary problems in analysis and geometry. Basel: Birkhäuser (ISBN 3-7643-7069-6/hbk). Operator Theory: Advances and Applications 151, 495-560 (2004).
Let \( M\) be a smooth manifold and \(X\) a smooth embedded submanifold. Also consider an elliptic differential operator on \(M\). The “Sobolev problem” consists in finding and studying suitable boundary conditions on \(X\) so that the boundary problem \(Du=f\) on \(M \setminus X\) + boundary conditions is well-posed. The authors look for an algebra of operators in which the Sobolev problem and approximate solution operators for it can be conveniently stated and studied. Technically speaking, they study matrices of operators \[ \left( \begin{matrix} D _{MM} &D_{MX}\\ D_{XM} & D_{XX} \end{matrix} \right) : \begin{matrix} H ^{s _{1}}(M ) \\ \oplus\\ H ^{s _{2}}(X ) \end{matrix} \rightarrow \begin{matrix} H ^{\sigma _{1}}(M ) \\ \oplus\\ H ^{\sigma _{2}}(X ) \end{matrix}, \] where the diagonal entries are pseudodifferential operators and the off-diagonal entries are Fourier integral operators related, respectively, to the restriction operators \(H ^{s}(M) \rightarrow H ^{\sigma }(X)\) (when such restriction operators make sense) and “extension” operators \(H ^{s}(X) \rightarrow H ^{\sigma }(M)\); the \( H ^{\tau } \) denote Sobolev spaces. In order to obtain an algebra of operators, the Lagrangian manifolds to which these operators are associated must of course satisfy suitable conditions, and the respective Sobolev indices will also have to be subjected to some restrictions.
The paper is to a significant degree expository. It contains, on the other hand, very many discussions and examples.
For the entire collection see [Zbl 1050.58002].

58J05 Elliptic equations on manifolds, general theory
58J20 Index theory and related fixed-point theorems on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds