Trace expansions and the noncommutative residue for manifolds with boundary (Q2729304)

The authors extend the theory of the trace functional on the algebra of classical pseudodifferential operators on a closed manifold to the case of pseudodifferential boundary value problems in the Boutet de Monvel calculus. NEWLINENEWLINENEWLINETo put things in context, let us recall that, as discovered by Wodzicki, there is a trace on the algebra \(\Psi(x)\) of the classical pseudodifferential operators on a closed manifold, i.e. a linear functional \(\tau\) on \(\Psi(X)\) such that \(\tau(\varphi \psi)= \tau(\psi \varphi)\). The trace of \(A\in \Psi(x)\), called the non-commutative residue and denoted \(\text{res} (A)\), was defined by the following formula: NEWLINE\[NEWLINE\text{res} (A) =\text{ord} P\bullet {d\over du}|_{u=0} \text{Res}_{s=1} Tr\bigl((P+uA)^{-s} \bigr).NEWLINE\]NEWLINE In this formula, \(P\) is an auxiliary invertible classical pseudodifferential operator of sufficiently large order subject to some additional condition. Then, by Seeley, complex powers \((P+uA)^{-s}\) are well defined for small \(u\) and there exists a meromorphic extension to the complex plane \(C\) of \(Tr(P+uA)^{-s}\). Then \(\text{res}(A)\) as defined above is independent of \(P\) and given by the following formula NEWLINE\[NEWLINE\text{res} (A)=(2\pi)^{-n}\int_{S^*X} tr a_{-n}(x,\xi) d\sigmaNEWLINE\]NEWLINE where \(a_{-n}(x,\xi)\) is the \(-n\)-th term in the asymptotic expansion of the symbol of \(A\).NEWLINENEWLINENEWLINEThis formula was extended by Fedosov, Golse, Leichtnam and Schrohe to operators \(A\) in the Boutet de Monvel algebra of polyhomogeneous pseudodifferential boundary value problems on a compact manifold \(X\) with boundary \(X'\) thus proving that a trace functional exists also on this algebra. Said formula is similar to the formula 2 above, but necessarily more complicated as some ``boundary'' expressions are to be taken into account.NEWLINENEWLINENEWLINEThus an interpretation of the trace on the Boutet de Monvel algebra as the residue of a meromorphic function in the style of the formula 1 above was lacking to make its theory fully parallel to that of the trace on the algebra of \(\psi\)do's on closed manifolds. The main object of the present paper is to fill the gap. This goal is achieved by proving that for a suitable auxiliary invertible \(B\) (the Dirichlet realization of a second order strongly elliptic operator in the interior of \(X\) and an elliptic operator on the boundary \(X')\) the holomorphic function \(Tr(AB^{-s})\), \(\text{Re} s\gg 0\), has a meromorphic extension with at most double poles and the trace of \(A\) is given by the following formula: NEWLINE\[NEWLINE\text{res} (A)=\text{ord} B\bullet\text{Res}_{s=0} Tr(AB^{-s}).NEWLINE\]NEWLINE The proof of these facts is based on beautiful expansions of trace of certain operators related to the boundary value problem \(A\) and auxiliary operator \(B\) extending earlier results of Grubb and Seeley providing analogous expansions for pseudodifferential operators on closed manifolds. In fact these expansions are lovely results in themselves and proof of them and detecting the components of \(\text{res} (A)\) in said expansions form the bulk of the paper.