Spectral asymptotic behavior of a class of integral operators

A. A. Laptev

Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 741-750

Résumé

Integral operators of the type $$ (Tf)(x)=\int_0^1\frac{x^\beta y^\gamma}{(x+y)^\alpha}f(y)\,dy, $$ the kernels of which have a singularity at a single point, are discussed. H. Widom's method and some of his results are used to show that, if $\alpha>0$, $\beta,\gamma>-\frac12$, $\rho\stackrel{def}=\beta+\gamma-\alpha+1>0$, then we have for the distribution function of the singular numbers of the operator, $$ \lim_{\varepsilon\to0}N(\varepsilon,T)ln^{-2}\frac1\varepsilon=\frac1{2\pi^2\rho}. $$