Weighted Variable Exponent Lebesgue Spaces on a Probability Space

Abstract

In this paper, we introduce the weighted variable exponent Lebesgue spaces defined on a probability space and give some information about the martingale theory of these spaces. We first prove several basic inequalities for expectation operators and obtain several norm convergence conditions for martingales in weighted variable exponent Lebesgue spaces. We discuss the Hölder inequality and embedding properties in these spaces. Finally, under some conditions we investigate Doob’s maximal function.