奇异测度的特征值与Connes非交换积分
Eigenvalues of singular measures and Connes noncommutative integration
摘要：
对于奇异测度$μ$，在$\mathbb{R}^{\mathbf{N}}$中具有紧支集的$α>0,$阶Ahlfors正则算子和$-l=-\mathbf{N}/2$阶伪微分算子$\mathbf{A}$，我们考虑紧算子$\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$这里$P$是符号测度，密度为$V$的$P=Vμ$属于Orlicz类$L^{Ψ,μ}$，$Ψ(t)=(t+1)\log(t+1)-t.$使用在\texttt{arXiv:2011.14877}中得到的这些算子的特征值估计，我们建立了$\mathbf{T}(P,\mathbf{A})$对一类测度的特征值渐近性，包括一致可求长集上的测度。这些结果导出了算子$\mathbf{T}(P,\mathbf{A})$在A.Connes意义下的可测性和这些算子的奇异迹的公式，从而产生了关于奇异测度的积分的非交换形式.
英文摘要：
For a singular measure $μ$, Ahlfors regular of order $α>0,$ 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, $P=Vμ$ with density $V$ belonging to the Orlicz class $L^{Ψ,μ}$ with $Ψ(t)=(t+1)\log(t+1)-t.$ 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.
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