Rate of decrease of eigenvalues of a Fredholm operator (Q757809)

Let K(x,y) be a continuous function on [a,b]\(\times [a,b]\). The corresponding integral operator in \(L^ 2(a,b)\) we denote also by K: \[ (Kf)(x)=\int^{b}_{a}K(x,y)f(y)dy. \] Let \(\omega\) (t,x) be a modulus of continuity of \(\partial^ mK(x,y)/\partial y^ m\) as a function of y. Let us suppose that \(\partial^ mK(x,y)/\partial y^ m\) is continuous on [a,b]\(\times [a,b]\) and we have \(\omega\) (t,x)\(\leq c(x)\omega (t)\), where c is a function from \(L^ p(a,b).\) If \(\omega (t)=t^ a\) \((0<a\leq 1)\), \(c\in L^ 2\) then the rate of decrease of eigenvalues of K is estimated by \textit{M. S. Birman} and \textit{M. Z. Solomyak} [Russ. Math. Surv. 32, 15-89 (1977; Zbl 0376.47023)]. The present author obtains estimates in the case when \(c\in L^ 1(a,b)\) and \(\omega\) (t) is a regularly varying function [in the sense of \textit{E. Seneta}, Regularly varying functions, Lect. Notes Math. 508, Springer Berlin etc. (1976; Zbl 0324.26002)].