Proceedings of the Royal Society of Edinburgh, 127A, 157-170,1991 Boundedness of operators of Hardy type in AP* spaces and weighted mixed inequalities for singular integral operators* F. J. Martin-Reyes Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Malaga, 29071 Malaga, Spain P. Ortega Salvador Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Malaga, 29071 Malaga, Spain M. D. Sarrion Gavilan Departamento de Economia Aplicada (Estadistica y Econometria), Facultad de Ciencias Economicas, Universidad de Malaga, 29071 Malaga, Spain (MS received 8 March 1995. Revised MS received 4 January 1996) We consider certain n-dimensional operators of Hardy type and we study their boundedness in AJ*(w). These spaces were introduced by M. J. Carro and J. Soria and include weighted L pq spaces and classical Lorentz spaces. As an application, we obtain mixed weak-type inequalities for Calderon—Zygmund singular integrals, improving results due to K. Andersen and B. Muckenhoupt. 1. Introduction Weighted Lebesgue-norm inequalities for the Hardy operators Pf(x)= f f(t)dt and Qf(x)= [ f(t)dt Jo Jx have been characterised by many authors (see [2, 7,9,12]). As direct consequences of these results, the characterisation of the weighted strong-type inequalities for the modified Hardy operators f(t)dt and QJ(x) = x~° \ f(t)dt o Jx are obtained. However, the characterisations of the weighted weak-type inequalities for P x and Q x cannot be deduced from the corresponding ones for P and Q. Andersen and Muckenhoupt [1] succeeded in this problem. Sawyer [10] studied the boundedness of the operators P a and Q x in L pq spaces *This research has been supported by D.G.I.C.Y.T. grant (PB91-0413) and Junta de Andalucia.