On the Boundedness of Classical Operators on Weighted Lorentz Spaces

Conditions on weights u(·), v(·) are given so that a classical operator T sends the weighted Lorentz space Lrs(vdx) into Lpq(udx). Here T is either a fractional maximal operator Mα or a fractional integral operator Iα or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for Mα and Iα when the weights have some usual properties and max(r, s) ≤ min(p, q). § 0. Introduction Let u(·), v(·), w1(·), w2(·) be weight functions on Rn, n ∈ N∗, i.e., nonnegative locally integrable functions; and let T be a classical operator. The purpose of this paper is to determine when T is bounded from the weighted Lorentz space Lrs v (w1) into L pq u (w2), i.e., ∥ ∥ ∥w2(·)(Tf)(·) ∥ ∥ ∥ L u ≤ C ∥ ∥ ∥w1(·) f(·) ∥ ∥ ∥ Lrs v for all functions f(·). (0.0) Here C > 0 is a constant which depends only on n, p, q, r, s, and on the weight functions. Recall that ‖g(·)‖qLpq u = q ∞ ∫