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Abstract. We establish criteria for both boundedness and compactness for some classes of integral operators with logarithmic singularities in weighted Lebesgue spaces for cases 1 < p q < ∞ and 1 < q < p < ∞ . As corollaries some corresponding new Hardy inequalities are pointed out.Mathematics subject classification (2010): 47G10, 47B38, 45P05.

Let 0 < α < 1. The operator of the form K α,ϕ f (x) = ϕ(x) a f (t)w(t)dt (W (x) −W (t)) (1−α) , x > 0, is considered, where the real weight functions v(x) and w(x) are locally integrable on I := (a,b) , 0 a < b ∞ and dW (x) dx ≡ w(x). In this paper we derive criteria for the operator K α,ϕ , 0 < α < 1 , 0 < p;q < ∞ , p > 1 α to be bounded and compact from the spaces L p,w to the spaces L q,v. Mathematics subject classification (2010): 46E30, 42A16.

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

Abstract. Inequalities of the formare considered, where K is an integral operator of Volterra type and H is the Hardy operator. Under some assumptions on the kernel K we give necessary and sufficient conditions for such an inequality to hold.Mathematics subject classification (2010): 26D10, 39B62.