Elena P. Ushakova

Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions

Some Hardy-type integral inequalities in general measure spaces, where the corresponding Hardy operator is replaced by a more general Volterra type integral operator with kernel k(x,y), are considered. The equivalence of such inequalities on the cones of non-negative respective non-increasing functions are established and applied.

Hardy operator with variable limits on monotone functions

We characterize weighted Lp-Lq inequalities with the Hardy operator of the form Hf(x)=∫a(x)b(x)f(y)u(y) dy with a non-negative weight function u, restricted to the cone of monotone functions on the semiaxis. The proof is based on the Sawyer criterion and the boundedness of generalized Hardy operator with variable limits.

On the Geometric Mean Operator with Variable Limits of Integration

A new criterion for the weighted Lp−Lq boundedness of the Hardy operator with two variable limits of integration is obtained for 0 < q < q + 1 ≤ p < ∞. This criterion is applied to the characterization of the weighted Lp−Lq boundedness of the corresponding geometric mean operator for 0 < q < p < ∞.

On boundedness and compactness of a certain class of kernel operators

New conditions for Lp[0,∞)-Lq[0,∞) boundedness and compactness (1<p, q<∞) of the map f→w(x)∫a(x)b(x)k(x,y)f(y)v(y)dy with locally integrable weight functions v,w and a positive continuous kernel k(x,y) from the Oinarov’s class are obtained.

Full length article: Estimates for Schatten-von Neumann norms of Hardy-Steklov operators

In this paper we study the mapping properties of some Hardy-type operators with general bounds of integration. In particular, some estimates for Schatten-von Neumann norms of these Hardy-Steklov operators are derived and discussed.

On weighted Sobolev spaces on the real line

Precise descriptions of the spaces associated with weighted Sobolev spaces on the real line are given.

Localisation property of Battle–Lemarié wavelets' sums

Abstract Explicit formulae are given for a type of Battle–Lemarie scaling functions and related wavelets. Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii–Besov spaces on R .

Hardy–Steklov Operators and Duality Principle in Weighted Sobolev Spaces of the First Order

Boundedness criteria for the Hardy–Steklov operator in Lebesgue spaces on the real axis are presented. As applications, two-sided estimates for the norms of spaces associated with weighted Sobolev spaces of the first order with various weight functions and summation parameters are established.

Upper estimates for the approximation numbers of the generalized Laplace transform

A Laplace-transform-type operator acting in the Lebesgue spaces of real functions on the half-axis is considered. Sufficient conditions under which belongs to some Schattentype classes are found. Upper asymptotic estimates for the approximation numbers of are obtained.We deal with a real valued integral operator L of Laplace transformation type acting between Lebesgue spaces on the semi-axis. Sufficient conditions for belonging L to Schatten type classes are obtained. Some upper asymptotic estimates for the approximation numbers of L are also given.