On Fredholm riggings of \(G\)-translators
From the summary: ``We consider riggings of \(G\)-translators on manifolds with multidimensional singularities. We derive a condition for the ellipticity of \(G\)-riggings, prove the finiteness theorem, and analyze the relationship between the notions of \(G\)-ellipticity and ordinary ellipticity of the riggings.'' As an example of application, the author studies the ellipticity of the following operator \(P\). Let \(T^3\) be a torus with coordinates \(t_1\), \(t_2\) and \(x\). Let \(Y^1= \{t_2=0\}\) and \(Y^2= \{t_1= 0\}\) be two two-dimensional tori in it. The intersection \(X= Y^1\cap Y^2\) is the circle \(\{t_1= t_2= 0\}\) with coordinate \(x\). The group \(\mathbb{Z}^2\) acts on \(T^3\). Then \[ P= \begin{pmatrix} 1 & \lambda\partial_{t_1} i^1\Delta^{-1} i_2\partial_{t_2}\Delta^{-1/2}_2\\ \lambda\partial_{t_2} i^2 \Delta^{-1} i_1\partial_{t_1}\Delta^{-1/2}_1 & 1\end{pmatrix}: \oplus_p H^s(Y^p)\to \oplus_p H^s(Y^p), \] where \(\Delta\) is the positive Laplace operator on \(T^3\), \(\Delta_p\) is the Laplace operator on \(Y^p\), \(p= 1,2\), \(\lambda\) is a positive parameter, \(i_p\) and \(i^p\) are the (co)boundary operators corresponding to the embeddings \(Y^p\subset T^3\) and \(H^s(Y^p)\) are the Sobolev spaces on \(Y^p\), \(-1< s< 0\).
scientific article; zbMATH DE number 6124957