Eigenvalues of singular measures and Connes noncommutative integration

TL;DR

This paper investigates the eigenvalue asymptotics of certain compact operators associated with singular measures and pseudodifferential operators, establishing noncommutative integration formulas for measures supported on complex geometric sets.

Contribution

It introduces a novel approach to eigenvalue asymptotics for operators linked to singular measures, extending Connes' noncommutative integration to new classes of measures.

Findings

Eigenvalue asymptotics for operators with singular measures

Measurability and singular trace formulas in noncommutative geometry

Extension of noncommutative integral to measures on rectifiable sets

Abstract

For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=V\mu$ with density $V$ belonging to the Orlicz class $L^{\Psi,\mu}$ with $\Psi(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$ and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.