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Spectral Estimates and Asymptotics for Integral Operators on Singular Sets

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For singular numbers of integral operators of the form

$$ u(x)\mapsto \int {F}_1(X)K\left(X,Y,X-Y\right){F}_2(Y)u(Y)u(dY) $$

with a measure μ singular with respect to the Lebesgue measure in ℝN we obtain ordersharp estimates for the counting function. The kernel K(X, Y, Z) is assumed to be smooth in X, Y, Z ≠ 0 and to admit an asymptotic expansion in homogeneous functions in the Z variable as Z → 0. The order in the estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure μ and involves integral norms of the weight functions F1 and F2. In the selfadjoint case, we obtain asymptotics of the eigenvalues of this integral operator provided that μ is the surface measure on a Lipschitz surface of some positive codimension 𝔡.

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Authors and Affiliations

  1. Chalmers University of Technology and University of Gothenburg, S-412 96, Gothenburg, Sweden

    G. Rozenblum

  2. The Euler International Mathematical Institute, 10, Pesochnaya nab, St. Petersburg, 197022, Russia

    G. Rozenblum

  3. The Bonch-Bruevich Saint-Petersburg State University of Telecommunications, 22, Bol’shevikov pr, Saint-Petersburg, 193232, Russia

    G. Tashchiyan

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  1. G. Rozenblum
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  2. G. Tashchiyan
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Correspondence to G. Rozenblum.

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To Volodya Maz’ya with admiration

Translated from Problemy Matematicheskogo Analiza 119, 2022, pp. 81-93.

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Rozenblum, G., Tashchiyan, G. Spectral Estimates and Asymptotics for Integral Operators on Singular Sets. J Math Sci 268, 493–508 (2022). https://doi.org/10.1007/s10958-022-06206-y

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