Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

We study a characterization of the precompactness of sets in variable exponent Morrey spaces on bounded metric measure spaces. Totally bounded sets are characterized from several points of view for the case of variable exponent Morrey spaces over metric measure spaces. This characterization is new in the case of constant exponents.


Introduction
In the present article, we investigate relatively compact sets in variable exponent Morrey spaces. In classical L p -spaces, the celebrated Riesz-Kolmogorov theorem nicely characterizes the relatively compact sets [22,30]. The aim of the paper is to characterize precompact sets in variable exponent Morrey spaces on an arbitrary doubling metric measure space. Our characterization is new even in the constant exponent case. Let us mention some generalizations of the Riesz-Kolmogorov theorem. Rafeiro characterized precompact sets in variable exponent Lebesgue spaces on Euclidean spaces [29]. For example, we refer to [3,10,11] for the characterization of precompact sets in variable exponent Lebesgue spaces on an arbitrary doubling metric measure space. Recently, the precompactness in quasi-Banach function spaces was studied in [6,12,13] (see [4,14] as well).
The study of variable exponent function spaces is now an extensively developed field due to the advent of two books [7,8]. Function spaces with variable exponents are a very active area of research and this theory finds many applications, e.g., in nonlinear elastic mechanics [36], electrorheological fluids [31], image restoration [24], and in differential equations with nonstandard growth. During the last decade, Lebesgue and Sobolev spaces with f L p(·) (Ω) + g L p(·) (Ω) for all f, g ∈ L p(·) (Ω). In particular We need the following lemma on the generalized Hölder inequality in variable exponent Lebesgue spaces. [19]. Let Ω be a measurable subset of X. Suppose that we have p(·), q(·), s(·) ∈ P(Ω, μ) be such that

2.
As is seen from the definition of L p(·),λ(·) (X), this function space depends on p(·) and λ(·)/p(·). The function p(·) itself can be considered modulo the sets of μ-measure zero. However, one can not do so for λ(·)/p(·). A prominent example reflecting this fact is the following example: Let (X, ρ, μ) be a the unit ball in R d with the Lebesgue measure, and let p(·) := 2 and λ(·) = 1 3 We carry over Lemma 2.1 to variable exponent Morrey spaces.
If we insert this estimate into the definition of f L p(·),λ(·) (X) , we obtain the desired result.
We transform Lemma 2.2 to the one in variable exponent Morrey spaces.
Under an additional conditions sup x∈X Lemma 2.8. Let (X, ρ, μ) be a metric measure space with finite measure, and let p(·) and λ(·) be non-negative variable exponents. Assume 0 < p − ≤ p + < ∞ and sup x∈X .
More generally, for any measurable set F Proof. Thanks to Lemma 2.1, we obtain .
Recall that sup x∈X λ(x) p(x) ≤ 1 p+ by assumption and hence for all x ∈ X. Thus .

Log-Hölder Continuity of Functions
Let (X, ρ) be a metric space, and let Ω be a measurable subset of X. We say that a function p(·) : Ω → R is locally log-Hölder-continuous on Ω, if there exists C 1 > 0, such that for all x, y ∈ Ω .
In addition, we say that the exponent p(·) satisfies the log-Hölder decay condition at infinity with a fixed point We also say that p(·) is globally log-Hölder-continuous on Ω if it is locally log-Hölder-continuous on and satisfies the log-Hölder decay condition at infinity. Then, the constant C log (p(·)) := max {C 1 ; C 2 } is called the log-Hölder constant for the exponent p(·). We also define the set of log-Hölder-continuous exponents by is globally log-Hölder-continuous .

Metric Measure Spaces
Let (X, ρ, μ) be a metric measure space equipped with a metric ρ and the Borel regular measure μ. We assume throughout the paper that the measure of every open nonempty set is positive and that the measure of every bounded set is finite. Additionally, we assume that the measure μ satisfies a doubling condition. It means that there exists a constant C μ > 0, such that for every ball B(x, r) It is well known (see [20] for example) that the doubling condition implies that there exists a positive constant D satisfying where s := log 2 C μ for all balls B (x 2 , r 2 ) and B (x 1 , r 1 ) , with r 2 ≥ r 1 > 0 and x 1 ∈ B (x 2 , r 2 ). It follows from the above inequality that if we fix a ball B (x 0 , R) , then there exists b > 0, such that the following lower Ahlfors condition holds for x ∈ B (x 0 , R) and for r ≤ R: In particular, if X is bounded, then there exists b > 0, such that the following inequality holds for r < diam(X) and x ∈ X

Space of Measurable Functions L 0 (X)
Let (X, μ) be a finite measure space. 1 Then, by L 0 (X), we denote the space of measurable functions on X. This space is a complete metric space with respect to the metric It is well known that the convergence in this metric is equivalent to the convergence in measure. Using the idea employed in [23], one can show the following theorem.
Theorem 2.10. Let (X, ρ, μ) be a totally bounded metric measure space. Any almost uniformly bounded and almost equicontinuous subset of L 0 (X, μ) is totally bounded. That is, a subset F of L 0 (X, μ) is totally bounded if for any ε > 0 and any function f ∈ F, there exists measurable subset E(f ) ⊂ X and constants δ > 0 and Λ > 0 independent of f with the following properties:
Theorem 3.1. Let (X, ρ, μ) be a metric measure space with finite measure. Let p(·) and λ(·) be non-negative variable exponents, such that 0 < p − ≤ p + < ∞ and that sup x∈X Assume that a subset F of L p(·),λ(·) (X) satisfies the following conditions: 1 Finite measure space means that the measure of the space is finite.
where φ is given by (2.6) and K is a finite set.
By the definition of δφ(α)-nets, for any and let Therefore, thanks to Lemma 2.6, we obtain Since thanks to Lemma 2.8 once again. Moreover, in view of the monotonicity of φ and the Markov inequality, 2 we deduce Hence, by condition (ii), Now, we prove the converse. We aim to find additional conditions on variable Morrey spaces that guarantee the validity of (i) and (ii). Now, we recall the definition of the absolute continuity.

Definition 3.2.
We say that an element f in a quasi-Banach space B ⊂ L 0 (X) has absolutely continuous quasi-norm if lim j→∞ fχ Ej B = 0 for every se- We use the condition of the absolute continuity as follows: Motivated by [34], the set of all functions in L p(·),λ(·) (X) having absolutely continuous quasi-norm is denoted by Theorem 3.4. Let (X, ρ, μ) be a metric measure space with finite measure and let p(·) and let λ(·) be non-negative variable exponents, such that 0 < p − ≤ p + < ∞. Then, conditions (i) and (ii) of Theorem 3.1 hold for any subset F of L p(·),λ(·) (X) which is totally bounded in L p(·),λ(·) (X).

a subset F of L p(·) (X) is totally bounded in L p(·) (X) if and only if F is totally bounded in L 0 (X) and the family F is p(·)-equi-integrable.
Lemma 3.7. Let (X, ρ, μ) be a doubling metric measure space with finite measure, and let q ∈ (0, p − ). Assume that sup Proof. Let r n be a sequence such that r n → 0. We fix ε > 0. Then thanks to Lemma 2.8, there exists δ > 0, such that χ E L p(·),λ(·) (X) < ε for any measurable set E satisfying μ(E) < δ.
Since μ is doubling, we have for μ-almost all x ∈ X. By the Egoroff Theorem, there exist N 0 ∈ N and a measurable set F , such that as long as n ≥ N 0 . For this F , we have If we take the L p(·),λ(·) (X)-norm, then we obtain as long as n ≥ N 0 . Since ε > 0 is arbitrary, we obtain the desired result.
With this in mind, we prove (a) − (c).
(a) Let f ∈ F. Then, from Lemma 2.6, we deduce where we have applied the inequality We have In the next step, we use Now, we give the bounds for I 1 , I 2 and I 3 . Note that I 1 < ε as we saw above.
= 0 thanks to Lemma 3.7, for sufficiently small r > 0, we have Therefore, we obtain I 2 ≤ 4 as long as r is small enough. It remains to control I 3 . We have (c) For k = 1, . . . , N, we fix sufficiently large R k satisfying Then, for R = max {R k : k = 1, . . . , N} , we have We now show the reciprocal of the previous theorem. Theorem 3.9. Let (X, ρ, μ) be a bounded doubling metric measure space, and let p(·) ∈ P(X, μ) satisfy 0 < p − ≤ p + < ∞ and sup x∈X Then, F is totally bounded in L p(·),λ(·) (X).
Proof. We divide the proof into two lemmata.
Proof. Let q ∈ (0, p − ) as in Theorem 3.9. Let E be a measurable subset of X and fix r ∈ (0, 1) and for any y ∈ B(x, r) and f ∈ F. Averaging the above inequality with respect to y ∈ B(x, r), we obtain |f (y)| q dμ(y) .
Meanwhile, due to Lemma 2.1, we obtain provided that r 1. Thus, gathering the above estimates, we have where C := 2 Finally, by the scaling relation, we obtain (3.4) By condition (b) for given ε > 0, we can and do fix sufficiently small r > 0, such that the first term on the right-hand side of (3.4) is less than ε. Thanks to Lemma 2.8, we can choose δ ∈ (0, 1), such that r Lemma 3.11. The family F is totally bounded in L 0 (X).
Let us fix ε ∈ (0, 1). We take δ > 0, such that the following condition holds for all f ∈ F: By Lemma 2.7, one has where the constant A is defined in Lemma 2.7. Next, for any x, y ∈ X satisfying ρ(x, y) < δ, we have A geometric observation shows that B(x, δ) ⊂ B(y; 2δ). Since the measure μ is doubling, the last inequality implies that, for any x, y ∈ X satisfying ρ(x, y) < δ Thanks to condition (a) of Theorem 3.8, there exists M > 0 which is independent of f ∈ F, such that Then, μ(E(f )) < ε as long as Λ 1. Moreover Thus, by Theorem 2.10, we conclude that F is totally bounded in L 0 (X) . Now, we can finish the proof of the theorem. Indeed, we conclude from the previous lemmata that F is L p(·),λ(·) (X)-equi-integrable, since F is totally bounded in L 0 (X). Thus, by virtue of Theorem 3.1, we obtain that F is totally bounded in L p(·),λ(·) (X).
From Theorems 3.8 and 3.9, we get the following. 2 is new in the constant exponent case, but this question has also been considered in [5]. In [35], some sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Note that in variable exponent Lebesgue spaces (that is for λ(·) := 0), Theorem 4.2 was proved in [3].