Sobolev problems with spherical mean conditions and traces of quantized canonical transformations (Q2304179)

Let \(M\) be a closed Riemannian manifold of dimension \(n\), \(X\) its closed submanifold of codimension \(\nu\) and \(i: X \rightarrow M\) the corresponding embedding. Let \(D\), \(B\) and \(A\) be pseudodifferential operators on \(M\), and \(i^*\) the boundary operator associated with the embedding \(i\) The authors consider the following problem \begin{align*} D u = f \quad & \text{ on} \quad M \setminus X, \\ i^*\big(B+A\mathcal{M}^r(\mu)\big)u = g\quad &\text{ on} \quad X, \end{align*} where \(\mathcal{M}^r(\mu)\) is the spherical mean operator on \(M\) over geodesic spheres of radius \(r\). The main aim is to find the sufficient conditions to guarantee that the problem is Fredholm. In particular it is assumed that \(D\) is elliptic and \(i\) is totally geodesic, but some other technical assumptions are also needed. In some special cases index formulas are obtained.