Eigenvalues of singular measures and Connes’ noncommutative integration

In the recent paper [32] the authors have considered the Birman-Schwinger (Cwikel) type operators in a domain Ω ⊆ R, having the form TP = A∗PA. Here A is a pseudodifferential operator in Ω of order −l = −N/2 and P = V μ is a finite signed measure containing a singular part. We found out there that for such operators, properly defined using quadratic forms, for a special class of measures, an estimate for eigenvalues λk = λ ± k (TP ) holds with order λ ± k = O(k −1) with coefficient involving an Orlicz norm of the weight function V . For a subclass of such measures, namely, for the ones whose singular part is a finite sum of measures absolutely continuous with respect to the surface measures on compact Lipschitz surfaces of arbitrary dimension, an asymptotic formula for eigenvalues was proved, with all surfaces, independently of their dimension, making the same order contributions. In the present paper we discuss some generalizations of these results and their consequences for introducing Connes’ integration with respect to singular measures. Our considerations are based upon the variational (via quadratic forms) approach to the spectral analysis of differential operators in a singular setting, in the form developed in 60-s and 70-s by M.Sh. Birman and M.Z. Solomyak. This approach enables one to obtain, for rather general spectral problems, eigenvalue estimates, sharp both in order and in the class of coefficients involved, this sharpness confirmed by exact asymptotic eigenvalue formulas. In the initial setting, this approach was applied to measures P absolutely continuous with respect to Lebesgue measure. Passing to singular measures, it was found that, for the equation −λ∆(X) = Pu(X), X ∈ Ω ⊆ R, if the singular part of P is concentrated on a smooth compact surface inside Ω (or on the boundary of Ω, provided the latter is smooth enough), it makes contribution of the order, different from the one produced by the absolutely continuous part, see, e.g., [1] or [18]. It happens always, with the only exception of the case N = 2, where the above orders are the same. For a class of singular self-similar measures P , K.Naimark and M.Solomyak established in [28] two-sided estimates for eigenvalues. And it turned out there that the order of two-sided eigenvalue estimates depends generally on the parameters used in the construction of the measure, in particular, on the Hausdorff dimension of its support. However, in the single case, again of the dimension