PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 6, June 1998, Pages 1739–1746 S 0002-9939(98)04247-6 ON WEIGHTED WEAK TYPE INEQUALITIES FOR MODIFIED HARDY OPERATORS F. J. MART ´ IN-REYES AND P. ORTEGA (Communicated by J. Marshall Ash) Abstract. We characterize the pairs of weights (w, v) for which the modified Hardy operator Tf (x)= g(x)  x 0 f applies L p (v) into weak-L q (w) where g is a monotone function and 1 ≤ q<p< ∞. 1. Introduction The purpose of this paper is to study weighted weak type inequalities for the modified Hardy operator T acting on measurable functions f : (0, ∞) → R and defined by Tf (x)= g(x)  x 0 f (y) dy, where g is a nonnegative measurable function on (0, ∞). Examples of these oper- ators are the Hardy operator 1 x  x 0 f (y) dy and the operators x α  x 0 f (y) dy which have been extensively studied in the last twenty years. Muckenhoupt [Mu] and Bradley [B] studied the weighted strong type inequality (p, q) with 1 ≤ p ≤ q in the case g(x) = 1. If w and v are nonnegative measur- able functions then the result in [B] implies directly that the weighted strong type inequality  ∞ 0 |Tf | q w  1/q ≤ C  ∞ 0 |f | p v  1/p for all f ∈ L p (v) (1) holds for 1 ≤ p ≤ q if and only if sup a>0  ∞ a g q w  1/q  a 0 v 1−p ′  1/p ′ < ∞. However, the characterizations of the weighted weak type inequalities λ   {|Tf |>λ} w  1/q ≤ C  ∞ 0 |f | p v  1/p (2) Received by the editors September 8, 1995 and, in revised form, December 1, 1996. 1991 Mathematics Subject Classification. Primary 26D15. Key words and phrases. Hardy operators, weights, inequalities. This research has been partially supported by D.G.I.C.Y.T. grant (PB94-1496) and Junta de Andaluc´ ıa. c 1998 American Mathematical Society 1739 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use