Normalized Cauchy Projections and Boundary K-Homology for Higher-Order Elliptic Operators
Volume 4 (2025) Issue 1,
DOI:
https://doi.org/ 10.71448/jcm2025v4i18
Submission Received: 23 Mar 2025
Revised: 25 May 2025
Accepted: 06 Jun 2025
Published: 30 Jun 2025
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Abstract
The higher-order elliptic differential operator on a compact manifold with boundary defines a relative analytic K-homology class through the minimal realization, whereas the maximal realization has the boundary contribution. In this paper, we construct normal-trace compression of the boundary class of an operator
\[
D:C^\infty(\Omega;E)\longrightarrow C^\infty(\Omega;F)
\]
of order m. We start with the even operator
\[
\mathcal D_D=
\left(\begin{array}{cc}
0&(D^\dagger)_{\min}\\
D_{\min}&0
\end{array}\right),
\]
where the generalized Bergman space is the kernel \KerDmax, take the complete trace vector γu=(γ0u,…,γm−1u), and match the mixed Sobolev trace levels using the diagonal elliptic operator
\[
\lambda_{1/2}=\operatorname{diag}(\Lambda_{1/2},\Lambda_{3/2},\ldots,\Lambda_{m-1/2}).
\]
The normalized Cauchy range
\[
\widehat H_D=\lambda_{1/2}^{-1}\gamma(\Ker D_{\max})
\subset L^2(\partial\Omega;E\otimes\C^m)
\]
has the orthogonal projection ΠD, which defines an odd Fredholm module over C(∂Ω). We obtain the following identity:
\[
\partial[\mathcal D_D]=[\Pi_D]
=\varphi_*\big([E_+(D)]\cap[S^*\partial\Omega]\big).
\]
It shows that normal trace compression includes precisely the K-homology boundary operator’s image and the Calderón symbol. The analysis also establishes that admissible extensions do not alter the interior class, low order perturbations retain the boundary K-class provided that the highest boundary symbol remains intact, and formal self-adjointness implies vanishing of scalar Toeplitz pairing while preserving the Cauchy data that is not identically zero. The answer is thus specific: the higher-order boundary class is the Fredholm module associated with the normalized full Cauchy projection.
Keywords
higher-order elliptic operators,normal traces,Calderon projection,relative K-homology,boundary map,generalized Bergman space,Fredholm modules,elliptic boundary value theory