Fully Measurable and Approximation Problems with Modular Inequalities in Variable Lebesgue Spaces

Abstract

We show the new properties and weighted fully measure and fully measurable of small, grand and iterated grand Lebesgue spaces with their applications and the maximal theorem .Direct and inverse theorems of approximation theory in variable Lebesgue and Smirnov spaces are discussed . The trigonometric and polynomial approximation of functions and problems in generalized Lebesgue spaces with variable exponent and Smirnov spaces with nonstandard growth are studied . The maximal function and atomic decomposition of Hardy spaces with variable exponents and its application to bounded linear operators are considered .We characterize the modular inequalitis for the Calderon and maximal operators in variable Lebesgue spaces.