Eigenvalues of singular measures and Connes’ noncommutative integration

Abstract

In a domain $\Omega \subset {\mathbb{R}}^N$ we consider compact, Birman–Schwinger type operators of the form $T_{P,\mathfrak{A}}={\mathfrak{A}}^{\ast }P\mathfrak{A}$ with $P$ being a Borel measure in $\Omega ,$ containing a singular part, and $\mathfrak{A}$ being an order $-N/2$ pseudodifferential operator. Operators are defined by means of quadratic forms. For a class of such operators, we obtain a proper version of H. Weyl's law for eigenvalues, with order not depending on dimensional characteristics of the measure. These results lead to establishing measurability, in the sense of Dixmier–Connes, of such operators and the noncommutative version of integration over Lipschitz surfaces and rectifiable sets.