APPROXIMATION AND PROBABILITY BANACH CENTER PUBLICATIONS, VOLUME 72 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2006 TWO-WEIGHTED CRITERIA FOR INTEGRAL TRANSFORMS WITH MULTIPLE KERNELS VAKHTANG KOKILASHVILI International Black Sea University 2, Agmashenebeli Alley 13th km., Tbilisi 0131, Georgia E-mail: kokil@rmi.acnet.ge ALEXANDER MESKHI A. Razmadze Mathematical Institute, Georgian Academy of Sciences 1, M. Aleksidze St., 0193 Tbilisi, Georgia Current address: Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail: a.meskhi@sns.it Abstract. Necessary and sufficient conditions governing two-weight L p norm estimates for mul- tiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given. Introduction. The survey deals with the boundedness (compactness) criteria for a large class of classical integral operators in classical Lebesgue spaces. For the consid- erable achievement in this area we refer to [GR], [K2], [OK], [GGKK], [BK], [KP], [GM], [EKM]. The latter book focuses our attention on boundedness (compactness) criteria of integral operators arising naturally in boundary value problems for PDE, the spec- tral theory of differential operators, continuum and quantum mechanics, stochastic pro- cesses, etc. The paper is organized as follows: in Section 1 we present some well-known two-weight results for the classical Hardy and potential operators; weighted boundedness criteria for 2000 Mathematics Subject Classification : 47G10, 26A33, 31B15, 46E30. Key words and phrases : potential with multiple kernels, higher dimensional Hardy operators, two-weight inequalities, nonhomogeneous spaces. Research of the first author supported by KBN grant 00-000. The paper is in final form and no version of it will be published elsewhere. [119]