Solution of Some Weight Problems for the Riemann–liouville and Weyl Operators

The necessary and sufficient conditions are found for the weight function v, which provide the boundedness and compactness of the Riemann–Liouville operator Rα from Lp to L q v . The criteria are also established for the weight function w, which guarantee the boundedness and compactness of the Weyl operator Wα from L p w to Lq . In this paper, the necessary and sufficient conditions are found for the weight function v (w), which provide the boundedness and compactness of the Riemann-Liouville transform Rαf(x) = ∫ x 0 f(t) (x−t)1−α dt (of the Weyl transform Wαf(x) = ∞ x f(t) (t−x)1−α dt) from L p to Lv (from L p w to L q) when 1 < p, q < ∞, 1 p < α < 1 or α > 1 ( q−1 q < α < 1 or α > 1). A complete description of the weight pairs (v, w) providing the boundedness of the operators Rα and Wα from Lp w to L q v when 1 < p < q < ∞ and 0 < α < 1 is given in [1]. For 1 < p ≤ q < ∞ and α > 1 a similar problem has been solved by many authors (see, e.g., [2, 3]). The necessary and sufficient conditions for pairs of weights, which provide the boundedness of the above-mentioned operators when 1 < q < p < ∞ and α > 1, are obtained in [4]. For 1 < q ≤ p < ∞ and 0 < α < 1, the two-weight problem for the operators Rα and Wα remains unsolved and in this context the results presented here are interesting. Let v and w be positive almost everywhere, locally integrable functions defined on R+. 1991 Mathematics Subject Classification. 42B20, 46E40, 47G60.