ON A PROGRESS IN THE THEORY OF INTEGRAL OPERATORS IN WEIGHTED BANACH FUNCTION SPACES Vakhtang Kokilashvili Dedicated to Alois Kufner on the occasion of his 70th birthday Abstract. The paper presents the recent results concerning boundedness criteria in weighted Banach function spaces with non-standard growth both for classical integral operators and integral transforms defined, generally speaking, on metric spaces with measure. 1. Introduction It is a great pleasure for me to take this opportunity and pay a tribute to Professor Alois Kufner for his outstanding mathematical abilities and exceptional personal quality. In a series of books and papers Alois Kufner has given comprehensive treatment of the weight theory and its application to harmonic analysis, partial differential equations. He was an inspirer of my long-standing and very fruitful collaboration with the Function Spaces group in Prague. Our contacts gave a profound impact on my subsequent scientific activity. I would like to thank the organizers of the conference “Function Spaces, Differential Operators and Nonlinear Analysis” FSDONA 2004, and espe- cially Professors Pavel Dr´ abek and Jiˇ r´ ı R´ akosn´ ık for this opportunity, for their warm hospitality and creating friendly atmosphere in the meeting. The present paper is a survey of the very recent results in the theory of integral operators in weighted Banach functional spaces. For the consider- able achievement in this area we refer to [GR], [K1], [OK], [GGKK], [BK], [KP], [GM], [EKM]. The latter book focuses our attention on boundedness 2000 Mathematics Subject Classification. 42B20, 42B25, 42A50, 26A33, 46E30, 46E35, 47B38. Key words and phrases. Maximal function, singular integral operator, Riesz potential, variable exponent Lebesgue space, weight. 152