On Sobolev problems associated with actions of Lie groups
By considering \(M\) an \(n\)-dimensional closed manifold equipped with an action of Lie group \(G\) and \(X\) a submanifold of codimension \(\nu=n/2\), the author considers the following problem \[ (*)\qquad Du\equiv f \pmod{X},\qquad i^*Bu=\varphi, \] such that \(u\in H^s(M)\) (the fractional Sobolev space), \(i^*\) is the \(H^{s-b-\nu/2}(X)\)-valued operator, and \(D\) is a pseudo-differential operator (for short PDO) of order \(m\). The author uses the notation \(\equiv\) for expressing that \(Du=f\) on \(X^c\), the complement of \(X\) in \(M\), \(Bu=B_0u+\int_GB_gT_gudg\) such that \(B_0\) and \(B_g\) are PDOs, on \(M\), with the same order \(b\), and \(T_g\) is the shift operator corresponding to the representation \(g\to T_g\) of the group \(G\), and \(dg\) stands for the Haar measure on \(G\). First, the author shows that the \(H^{s-m+\nu/2}(M)\)-valued operator \(\widehat{A}:=1+A\) is Fredholm whenever it is elliptic (Definition 3) such that \(A=\mathcal{D}^{-1}\int_Gi^*B_gT_gi_*dg\) such that \(\mathcal{D}=i^*B_0D^{-1}i_*\) and \(i_*\) is the coboundary operator (Theorem 1). Then through the admissibility condition (Definition 1), the author deduces that \((*)\) is Fredholm. Theorem 2 deals with the index of \(\widehat{A}\), the calculus relies on the symbol associated to \(A\) (Definition 2). The fifth section treats a concrete example.
scientific article; zbMATH DE number 6486221