Median-Type John–Nirenberg Space in Metric Measure Spaces

We study the so-called John–Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John–Nirenberg inequalities, which give weak-type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón–Zygmund decomposition and a good-λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} inequality for medians. A John–Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John–Nirenberg spaces coincide under a Boman-type chaining assumption.


Introduction
The space of functions of bounded mean oscillation (BMO) was introduced by John and Nirenberg in [27].In that article, they also discussed a larger BMO-type space called the John-Nirenberg space, denoted by JN p with 1 < p < ∞.The John-Nirenberg space contains BMO, and BMO is obtained as the limit of JN p as p → ∞.The John-Nirenberg lemma in [27] states that a logarithmic blowup is the worst possible for a BMO function.Moreover, another version of the John-Nirenberg lemma in [27] implies that JN p is a subset of weak L p .This inclusion is strict, as shown by a one-dimensional example in [1].It is also known that L p is a proper subset of JN p ; see [8].
In [26], John discussed BMO using medians instead of integral averages.He also proved the analogous John-Nirenberg inequality for the median-type BMO in a Euclidean space.Later, Strömberg proved the inequality for a larger class of medians and also gave a proof in a metric measure space with a doubling measure [35,36].In particular, this implies that the ordinary BMO and the median-type BMO are equivalent.While BMO has been studied extensively, the John-Nirenberg space is not equally well understood.However, some results related to John-Nirenberg inequalities and interpolation of operators can be found in [6,[12][13][14]34].
Our main results are John-Nirenberg inequalities for the median-type John-Nirenberg space.We carry out the analysis in a metric measure space with a doubling measure, but the main results are new even in the Euclidean setting.The main novelty is that we consider medians instead of integral averages throughout.
One of the advantages of using medians is that we do not need to assume that the function is locally integrable.Thus, the definition via medians applies to any measurable function.There exist different definitions of John-Nirenberg spaces in metric measure spaces; see [1,2,32].The difference is whether the balls in the definition are required to be pairwise disjoint or allowed to overlap.Definitions with disjoint balls can be found in [2,32] and with overlap in [1].We adopt the definition in [2,32] which leads to a more general theory.
The proof of the local John-Nirenberg inequality with medians (Theorem 4.4) in Section 4 is based on a Calderón-Zygmund decomposition (Lemma 4.2) and a good-λ inequality (Lemma 4.3) for medians.This is inspired by [1,2], where the authors examine integral-type John-Nirenberg spaces in metric measure spaces with a doubling measure.A challenge in proving results for the median-type John-Nirenberg space is that medians lack subadditive properties and monotonicity on sets compared to integral averages.Our Theorem 4.4 implies a related John-Nirenberg inequality in [30,Theorem 1.3] where a discrete summability condition is considered; see also [2,10,31].A simple one-dimensional example shows that our result is more general than [30,Theorem 1.3].In particular, our approach does not depend on the discrete summability condition and rearrangements.
In Section 5, we prove a John-Nirenberg inequality with medians up to the boundary (Theorem 5.3) in Boman sets by applying the local John-Nirenberg inequality together with chaining arguments.Boman sets are closely related to the Boman chain condition, introduced in the unpublished paper [4].The connection is discussed in [32].In particular, the two are equivalent in a geodesic space with a doubling measure.The Boman chain condition characterizes John domains in many metric measure spaces, including Euclidean spaces; see [5].The corresponding results for the integral-type John-Nirenberg space can be found in [23,32].We apply the arguments in [7,23,32] for medians.As a corollary of the global John-Nirenberg inequality, we show that the integral-and median-type John-Nirenberg spaces coincide in every open set under the assumption that balls are Boman sets with uniform parameters (Corollary 5.4).This means that the median-type John-Nirenberg condition is possibly the weakest for a function to be in JN p .The uniform Boman condition on balls holds, for example, in geodesic spaces [16].

Preliminaries
Let (X, d, µ) be a metric measure space with a metric d and a doubling measure µ.A Borel regular measure is said to be doubling if for every ball B = B(x, r) = {y ∈ X : d(x, y) < r}, where c µ > 1 is the doubling constant.We use the notation λB = B(x, λr), λ > 0, for the λ-dilate of B. From the doubling property of the measure, it can be deduced that if where D = log 2 c µ is the doubling dimension of the space (X, d, µ).The proof can be found in [3, p. 6].We denote the integral average of a function f ∈ L 1 (A) in a set A ⊂ X by A basic tool in metric measure spaces is the 5-covering theorem.It is also sometimes referred to as the basic covering theorem or Vitali's covering theorem.Although, one must be careful since there is another covering theorem named after Vitali [22, pp. 3-4].One can check [22, pp. 2-3] for a proof.Lemma 2.1.Let F be a collection of balls of uniformly bounded radii in X.Then there exists a countable disjointed subcollection G such that The Lebesgue differentiation theorem states that the integral average of a function f over a ball B(x, r) approaches f (x) when the radius r tends to zero.Two different proofs can be found in [22, pp. 4-6, 12-13].
We follow the definition in [2,32] for the integral-type John-Nirenberg space JN p,q in metric measure spaces.Definition 2.3.Let Ω ⊂ X be an open set, 1 < p < ∞ and 0 < q < p.We say that a function f ∈ L q loc (Ω) belongs to the John-Nirenberg space JN p,q (Ω) if where the supremum is taken over countable collections of pairwise disjoint balls B i ⊂ Ω.
If q = 1, we write JN p instead of JN p,1 .We omit Ω from the norms if the considered set is clear from the context.
Definition 2.4.Let A ⊂ X be a set of finite and positive measure, 0 < s ≤ 1 and f an s-median of the function f over a set A. Note that a 1 2 -median is a standard median value of f over A. An s-median of a function is not always unique.For example, consider f = χ [1/2,1] on the interval [0, 1].Then any value between 0 and 1 is a 1  2 -median of f .Thus, we define the maximal s-median m s f (A) which is unique [33].
Definition 2.5.Let 0 < s ≤ 1 and A ⊂ X be such that 0 < µ(A) < ∞.The maximal s-median of a measurable function f : X → [−∞, ∞] over a set A is defined as It can be shown that the maximal s-median of a function is indeed an s-median [33].In the next lemma, we list the basic properties of the maximal s-median.Most of these properties are listed without proofs in [20,21].The proofs of properties (i), (ii), (v), (vii), (viii) and (ix) can be found in [33,Proposition 1.1] in the Euclidean setting.The proofs of these properties are practically same in metric measure spaces, and thus are omitted here.We give proofs for the remaining properties.
(iv) It holds that To prove the other inequality, assume that a < inf i m s f (A i ).We then get Remark.Assume that 0 < s ≤ 1/2.Then property (vii) assumes a slightly simpler form Suppose that f ∈ L 1 loc (X) and 0 < s ≤ 1/2.Using the Lebesgue differentiation theorem (Lemma 2.2) together with properties (v), (vii) and (ix) of the maximal s-medians, we obtain the following version of the Lebesgue differentiation theorem: as r → 0. However, there is a more general version of the Lebesgue differentiation theorem for medians [33] where we need to assume only that f is a measurable function.
The proof of Lemma 2.7 can be found in [33,Theorem 2.1] where the lemma is proven in the Euclidean setting.The proof is almost identical in metric measure spaces with a doubling measure, and thus is omitted here.Lemma 2.7.Let f : X → [−∞, ∞] be a measurable function which is finite almost everywhere in X and 0 < s ≤ 1.Then it holds that We give a definition for the median-type BMO which coincides with the classical BMO [26,35,36].Definition 2.8.We say that a measurable function f belongs to BMO 0,s (X) if 3. Definition and properties of JN p,0,s In this section, we give a definition of the median-type John-Nirenberg space in metric measure spaces.Moreover, we examine the basic properties of the space.Definition 3.1.Let Ω ⊂ X be an open set, 1 < p < ∞ and 0 < s ≤ 1/2.We say that a measurable function f belongs to the median-type John-Nirenberg space JN p,0,s (Ω) if where the supremum is taken over all countable collections of pairwise disjoint balls The zero in the notation JN p,0,s means that we do not need to assume any local integrability but only measurability.The range 0 < s ≤ 1/2 is necessary since f JNp,0,s = 0 for s > 1/2 and a two-valued function f .
The next lemma shows that the constants c i in the definition of JN p,0,s can be replaced by the maximal t-medians where s ≤ t ≤ 1 2 .Lemma 3.2.Let f be a measurable function.It holds that

Proof. It is clear that the first inequality holds. The other inequality follows from
. By taking the infimum over the constants c i , we observe that Therefore, the second inequality is attained as well.
Next, we list some basic properties of JN p,0,s spaces.
Lemma 3.3.Let f and g be measurable functions.Then the following properties hold true: Proof.(i) Using property (viii) of Lemma 2.6 and Minkowski's inequality, we estimate for any c f i , c g i ∈ R. Thus, we can take the infimum over the constants c f i and c g i to get where t 1 + t 2 ≤ s.
The next proposition tells that the space L p is contained in JN p,q which in turn is a subset of JN p,0,s .The first inclusion is strict in the Euclidean setting, that is, there exists a function in JN p \ L p [8].The second one holds in the other direction in many situations; see Corollary 5.4.
Proof.The first inequality follows straightforwardly from property (ix) of Lemma 2.6.The second one is obtained by a simple use of Hölder's inequality:
More precisely, we have The previous inequality follows from the estimates The median-type John-Nirenberg space JN p,0,s is a generalization of BMO in the sense that a function is in BMO if and only if the JN p,0,s norm of the function is uniformly bounded as p tends to infinity.Proposition 3.5.If Ω ⊂ X has finite measure, then it holds that as p → ∞.Hence, we have We can interchange the order of taking the supremum and the limit since is an increasing function of p which can be seen by Hölder's inequality.Thus, we conclude that lim p→∞ f JNp,0,s = f BMO0,s .

John-Nirenberg lemma for JN p,0,s
We need two lemmas to prove the John-Nirenberg inequality for JN p,0,s which implies that JN p,0,s is contained in L p,∞ , that is, weak L p .The first lemma is a Calderón-Zygmund decomposition and the second one is a good-λ inequality.
Throughout the argument let η > 0 and B 0 = B(x B0 , r B0 ) ⊂ X be fixed.We denote where c µ is the doubling constant and D = log 2 c µ is the doubling dimension.We define a maximal function with the understanding that M B f (x) = 0 if there is no ball B ∈ B such that x ∈ B.
In particular, M B f (x) = 0 for every x ∈ X \ B 0 .By the Lebesgue differentiation theorem for medians (Lemma 2.7), we have |f (x)| ≤ M B f (x) for µ-almost every x ∈ B 0 .Moreover, denote , then it holds that r B ≤ η 5 r B0 .Proof.By the assumption, we have Therefore, it holds that which implies r B ≤ η 5 r B0 by recalling (4.1).
The following lemma is a Calderón-Zygmund decomposition for medians in metric measure spaces with a doubling measure.Lemma 4.2.Let f ≥ 0 be a measurable function defined on B 0 .Assume that E λ = ∅ and m t/α f ( B 0 ) ≤ λ holds for some 0 < t ≤ 1, where α is given in (4.1).Then there exist countably many pairwise disjoint balls B i ∈ B such that By the assumption, the set over which the supremum is taken is non-empty.Moreover, Lemma 4.1 implies that r x (λ) ≤ η 5 r B0 .For every x ∈ E λ , we can find a ball B x,λ ∈ B with x ∈ B x,λ such that We then have m t f (σB x,λ ) ≤ λ whenever σ ≥ 2 and σB x,λ ∈ B. By applying the 5-covering theorem, we obtain a countable collection of pairwise disjoint balls Hence, so obtained balls B i are the Calderón-Zygmund balls at level λ and we denote them by B i,λ .
We have constructed the Calderón-Zygmund decomposition at level λ and now focus on λ ′ .Note that E λ ⊂ E λ ′ and r x (λ) ≤ r x (λ ′ ) for every x ∈ E λ .Thus, for every x ∈ E λ , we can choose a ball Whereas, for every x ∈ E λ ′ \ E λ , we choose the ball B x,λ ′ in the similar way expect we do not have B x,λ ⊂ B x,λ ′ .We then apply the 5-covering theorem to the balls B x,λ ′ to obtain the Calderón-Zygmund balls B j,λ ′ at level λ ′ .Moreover, the 5-covering theorem states that for every ball B x,λ ′ there is an enlarged ball 5B j,λ ′ such that We move on to a good-λ inequality which is crucial for the proof of the John-Nirenberg inequality.In the proof of the good-λ type inequality for the integral-type JN p in [1], all Calderón-Zygmund balls can be treated in the same way.However, in the case of medians, we need to divide Calderón-Zygmund balls into two collections which must be considered separately.This is due to the fact that medians lack the monotonicity on sets that integrals have.
where α is given in (4.1).Consider collections of Calderón-Zygmund balls {B i,λ } i and {B j,Kλ } j for the function |f | such that each B j,Kλ is contained in some 5B i,λ .Then it follows that for every i ∈ N, and Particularly, the set J i contains those indexes j / ∈ i−1 k=1 J k for which B j,Kλ is contained in 5B i,λ .Since every B j,Kλ is contained in some 5B i,λ , we get the partition Using properties (ii), (v), (vii) of Lemma 2.6 and (iii), (iv) of Lemma 4.2 in this order, we obtain Since B j,Kλ are pairwise disjoint, property (x) of Lemma 2.6 implies that Hence, by property (iii) of Lemma 2.6, we have for every i ∈ I, where Then for i ∈ I, we have For every x ∈ {x ∈ B 0 : Applying the 5-covering theorem, we get a countable collection of pairwise disjoint balls B k such that We then have We now state our main result which is the John-Nirenberg inequality for JN p,0,s .It implies that JN p,0,s ( B) is contained in L p,∞ (B) for all balls B ⊂ X.

It follows that
We claim that for every n = 0, 1, . . ., N , where We prove the claim by induction.First, note that the claim holds for n = 0 since We show that this implies the claim for k + 1.By using Lemma 4.3 for K k λ 0 , we observe that Hence, we conclude that

Global John-Nirenberg inequality for JN p,0,s in Boman sets
We give a proof for the global John-Nirenberg inequality for JN p,0,s in Boman sets.For more detailed discussion about Boman sets, see [32] and references therein.
Parameters C 1 , C 2 , C 3 , ρ and M in the results below are the same as in Definition 5.1.The proof of the following lemma can be found in [32] for integral averages.The proof is identical for medians, and thus is omitted here.
, where the constant C 0 depends on p, the doubling constant c µ , C 1 , C 2 , C 3 , ρ and M .
For the next theorem, the global John-Nirenberg lemma, we fix the parameter η from Section 4 such that 1 + η = C2 C1 .The proof follows that of [32].Theorem 5.3.Let ∆ ⊂ X be a Boman set and 0 < s ≤ s 0 , where s 0 is given in Theorem 4.4.If f ∈ JN p,0,s (∆), then there exists a ∈ R such that for every λ > 0 it holds that If all balls are Boman sets with uniform parameters, then the median-type John-Nirenberg space coincides with the integral-type John-Nirenberg space in every open set.For example, geodesic spaces satisfy the uniform Boman condition on balls [16].
Proof.Let {B i } i be a countable collection of pairwise disjoint balls contained in Ω.The first inequality is stated and proven in Proposition 3.4.For the second inequality, since by the assumption the balls B i are Boman sets, we may apply

(
iii) There is a central ball B * ∈ F such that for each B ∈ F there exists a finite collection of balls C(B) = {B i } kB i=1 ⊂ F with B 1 = B * and B kB = B. (iv) In C(B), for each pair of balls B i and B i−1 corresponding to consecutive indices there exists a ball D