Eigenvalues of singular measures and Connes’ noncommutative integration

Journal article, 2022

In a domain \Omega subset R^N we consider compact, Birman–Schwinger type operators of
the form T_{P;A}= A^*P A with P being a Borel measure in \Omega containing a singular part,
and 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