Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Boundedness of positive operators on weighted amalgams | Aguilar Canestro and Ortega Salvador Journal of Inequalities and Applications 2011 2011 13 3 Journal of Inequalities and Applications http content 2011 1 13 a SpringeiOpen Journal RESEARCH Open Access Boundedness of positive operators on weighted amalgams María Isabel Aguilar Canestro and Pedro Ortega Salvador Correspondence portega@uma. es Análisis Matemático Facultad de Ciencias Universidad de Málaga 29071 Málaga Spain Abstract In this article we characterize the pairs u v of positive measurable functions such that T maps the weighted amalgam Lp v q in Lp u lq for all 1 p q p q x where T belongs to a class of positive operators which includes Hardy operators maximal operators and fractional integrals. 2000 Mathematics Subject Classification 26D10 26D15 42B35 Keywords Amalgams Maximal operators Weighted inequalities Weights 1. Introduction Let u be a positive function of one real variable and let p q 1. The amalgam Lp u lq is the space of one variable real functions which are locally in Lp u and globally in lq. More precisely Lp u q f Ilfllp u q x where llfllp u q - q n 1 p Ị If Ipu 1 q ugZ n These spaces were introduced by Wiener in 1 . The article 2 describes the role played by amalgams in Harmonic Analysis. Carton-Lebrun Heinig and Hoffmann studied in 3 the boundedness of the Hardy operator Pf x - f-m If I in weighted amalgam spaces. They characterized the pairs of weights u v such that the inequality IlPf p u q Cllf p v q holds for all f with a constant C independent of f whenever 1 q q X . The characterization of the pairs u v for to hold in the case 1 q q TO has been recently completed by Ortega and Ramirez 4 who have also characterized the weak type inequality pf w . C f p v q SpringerOpen0 2011 Canestro and Salvador licensee Springer. This is an Open Access article distributed underthe terms of the Creative Commons Attribution License http licenses by which permits unrestricted