Variable Exponent Function Spaces related to a Sublinear Expectation

Bochi Xu

doi:10.1155/2020/1734174

Abstract

In this paper, variable exponent function spacesLp·,Lbp·, andLcp·are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.

Highlights

Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering
Let Ω be a complete metric space equipped with the distance, B(Ω) the Borel − algebra of Ω and M the collection of all probability measures on (Ω, B(Ω))
Let ∈

Summary

Introduction

Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering. Diening et al [5] summarize most of the existing literature of theory of variable exponent function spaces and applications to partial di erential equations. One of the most important application is that a coherent risk measure(the basic theory about coherent risk measure can be found in [7]) is a sublinear expectation E : H → R de ned on H , which is a linear space of nacial losses. We are interested in behavior of sublinear expectation spaces with variable exponents.

Variable Exponent Function Spaces

Kolmogorov’s Criterion on Variable Exponent Function Spaces