For singular numbers of integral operators of the form
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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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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DOI: https://doi.org/10.1007/s10958-022-06206-y