Singular functions and relative index for elliptic corner operators (Q2756132)

The paper studies asymptotics of solutions to elliptic differential equations on a closed manifold \(M\) with singular points. Near any such point, the authors view the equation as an ordinary differential equation on the half-axis \(r \in \overline{R}_{+}\), with coefficients which take their values in a pseudodifferential algebra on a closed compact manifold \(X\). The coefficients will have degeneracies at \(r=0\) which depend on the geometry at the singular point. In generalized polar coordinates near \((0, \varepsilon)\times X\) a typical differential operator will have the form NEWLINE\[NEWLINE A=( \delta '(r)) ^{m}\sum_{j=0 }^{ m } A _{j} (r)(( 1/ \delta '(r))D _{r}) ^{j}, NEWLINE\]NEWLINE where \( \delta (r)\) is a \( {\mathcal C}^{\infty} \)-function on \((0,\varepsilon)\) such that \( \delta (r) \searrow - \infty\) as \( r \rightarrow 0\) and the coefficients \(A _{j} \) are \( {\mathcal C}^{\infty} \)-functions on \((0,\varepsilon)\) with values in \(\Psi ^{m-j}(X,w-j)\), a pseudodifferential algebra on \(X\). (The coefficients \(A _{j} \) are assumed to be continuous up to \(r=0\).) As a corollary, the authors obtain results on the relative index of the operator when viewed in two different weighted Sobolev spaces.