Properties of the Hardy operator in Lorentz spaces

For the Hardy operator from the weighted Lorentz space Lᵥr,s(R⁺) to Lωp,q(R⁺), where the parameters satisfy 1<s≤ min(p,q)<∞ and 1<r<∞, criteria for its boundedness and compactness are established. Two-sided estimates for s-numbers are obtained for compact operators. Conditions for the Hardy operator to belong simultaneously to the ideal of compact operators, the ideal of approximable operators, and the ideals generated by sequences of s-numbers are found. Estimates for the norms of the Hardy operator in ideals in terms of integral expressions involving the weight functions of the original operator are given. Bibliography: 27 titles.