Variable Exponent Function Spaces related to a Sublinear Expectation

and Applied Analysis 2 Denition 1. A set is a polar if ( ) = 0 and a property holds “quasi-surely” (q.s.) if it holds outside a polar set. e upper expectation of P is dened as follows: for each ∈ 0(Ω) such that [ ] exists for each ∈P, About E, we know the following properties. Theorem 2 (see  eorem 9 in [12]). e upper expectation E[⋅] of family P is a sublinear expectation on (Ω) as well as on (Ω), i.e., (1) For all , in (Ω), ≥ , E[ ] ≥ E[ ]. (2) For all , in (Ω), E[ + ] ≤ E[ ] + E[ ]. (3) For all ≥ 0, ∈ (Ω), E[ ] = E[ ]. (4) For all ∈ R, ∈ (Ω), E[ + ] = E[ ] + . Let a B(Ω)-measurable real function : Ω→ [1,∞), be a variable exponent. In the space 0(Ω), the moduli are dened by Denition 3.  e space L (⋅) is the set of ∈ 0(Ω) satisfying (⋅)( ) < ∞, and it is endowed with the Luxemburg norm: We set N (⋅) := { ∈ 0(Ω) : ( ) = 0}, and denote L (⋅) := L (⋅)/N (⋅). As usual, we do not take care about the distinction between classes and their representatives. Proposition 4. If the variable exponent satises 1 < − ≤ ( ) ≤ + < ∞, then the inequality holds for every ∈ L (⋅) and ∈ L (⋅), where 1/ ( ) + 1/ ( ) = 1. Proof. By Young inequality, we have By the monotonicity, sub-additivity and positive homogeneity of E, and the property of the norm, we have (3) E[ ] := sup ∈P [ ]. (4) (⋅)( ) := [| |] = sup ∈ [| |] = sup ∈ ∫ Ω | |( ) ( ) ( ). (5) ‖ ‖ (⋅) = inf{ > 0 : (⋅)( ) ≤ 1}. (6) E[| |] ≤ (1 + 1 − − 1 +)‖ ‖ (⋅)‖ ‖ (⋅) (7) | | ‖ ‖ (⋅) | | ‖ ‖ (⋅) ≤ 1 ( )( | | ‖ ‖ (⋅) ) + 1 ( ) ( | | ‖ ‖ (⋅) ) . us, the inequality follows. ☐ Proposition 5. Suppose that the variable exponent satises 1 < − ≤ ( ) ≤ + < ∞. If , ∈ L (⋅), then (1) If ‖ ‖ (⋅) ≥ 1, then ‖ ‖ − (⋅) ≤ (⋅)( ) ≤ ‖ ‖ + (⋅). (2) If ‖ ‖ (⋅) ≤ 1, then ‖ ‖ + (⋅) ≤ (⋅)( ) ≤ ‖ ‖ − (⋅). (3) lim →∞ (⋅) = 0, if and only if lim →∞ (⋅)( ) = 0 (4) lim →∞ (⋅) = ∞, if and only if lim →∞ (⋅) ( ) = ∞. In particular, the linear expectation also follows this proposition. Proof. (1) By ‖ ‖P (⋅) ≥ 1 and the denition of the norm, us, (⋅)( ) ≤ ‖ ‖ + (⋅). As


Introduction
Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering. Basic properties of the spaces are rst given by Kováčik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2,3]. Harjulehto et al. present an overview of applications to di erential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of variable exponent function spaces and applications to partial di erential equations. In [6], Aoyama proves some important probability inequalities in variable exponent Lebesgue spaces.
Nonlinear expectations play an important role in the research of nancial markets. 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. In this paper, we are interested in behavior of sublinear expectation spaces with variable exponents. By the following representation theorem which can be found in Peng ([8], p. 4), we know that a sublinear expectation can be expressed as a supremum of linear expectations, i.e., there exists a family of linear expectations { } ∈Θ such that us, we consider the upper expectation only in this paper. Some other important theories about nonlinear expectations can be found in Peng's [9,10]. e remainder of the paper is divided as follows: in Section 2, motivated by Fu [11] and Denis et al. [12], we rst introduce L (⋅) , L (⋅) and L (⋅) and give some properties of these spaces.
And Each element of ‖⋅‖ (⋅) -completion of (Ω) has a quasi-continuous version is proved. In Section 3, applying the results of Section 2, we discuss a version of Kolmogorov's criterion for continuous modi cation of stochastic processes, which are in the variable exponent function space related to a sublinear expectation, a er proving the situation under a linear expectation.

Variable Exponent Function Spaces
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)).  e upper expectation of P is de ned as follows: for each ∈ 0 (Ω) such that [ ] exists for each ∈ P, About E, we know the following properties.

Proof. By Young inequality, we have
By the monotonicity, sub-additivity and positive homogeneity of E, and the property of the norm, we have us, the inequality follows. ☐

Proposition 5. Suppose that the variable exponent satis es
In particular, the linear expectation also follows this proposition.

Proposition 6. Suppose that the variable exponent satis es
Proof. For each ∈ P, by Markov inequality, we have Take the supremum in P both sides, us, ☐ Lemma 7 (Proposition 17 in [12]). Let ∈ (0, ∞] and { } be a sequence in L which converges to in L . en there exists a subsequence { } which converges to quasi-surely in the sense that it converges to X outside a polar set. where is a constant. at is to say { } is a Cauchy sequence in L 1 . By Proposition 14 in [12], L 1 is a Banach Space. us, { } converges in L 1 . Suppose that → , ∈ L 1 and further by Lemma 7, we suppose → quasi-surely (subtracting a subsequence if necessary). For each 0 < < 1, there exists 0 such that , that is to say, − ∈ L (⋅) , and ∈ L (⋅) . e proof is completed. ☐ We denote by L (⋅) the completion of (Ω) and by L (⋅) the completion of (Ω). By Proposition 8, we have Proposition 9. Suppose that the variable exponent satis es . On the other hand, for any ∈ L (20) (21)
Proof. For each > 0, there exists > 0 which is independent of such that for all ≥ ,

Kolmogorov's Criterion on Variable Exponent Function Spaces
De nition 16. Let be a set of indices. { } ∈ and { } ∈ be two processes indexed by . We say that is a quasimodi cation of if for all ∈ = q.s.
To prove a Kolmogorov's criterion for a process indexed by R with ∈ N on variable exponent function spaces, we give the following lemma rst. (34) ( ) ≥ 1 .
(28)   where ∈ 0, / + . By Lemma 17, we know that for any ∈ P, is nite and uniformly bounded with respect to so that us, is uniformly continuous on q.s. and set In the similar way in Lemma 17, ̃ = q.s. and ̃ is a modi cation. ☐

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no con icts of interest. (42) ≤ sup