On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators

This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003, Minsk, Belarus. We give a survey of investigations on ‘the variable exponent business’, concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p(·) and W m, p(·).