FUNCTIONS OF BOUNDED ( φ , p ) MEAN OSCILLATION

In this paper we extend a result of Garnett and Jones to the case of spaces of homogeneous type. 2000 Mathematics Subject Classification : Primary 32A37. Secondary 43A85. 164 René Erĺın Castillo, Julio C. Ramos and Eduard Trousselot


Introduction
The space of functions of bounded mean oscillation, or BMO, naturally arises as the class of functions whose deviation from their means over cubes is bounded.L ∞ functions have this property, but there exist unbounded functions with bounded mean oscillation, for instance the function log |x| is in BMO but it is not bounded.The space BMO shares similar properties with the space L ∞ and it often serve as a substitute for it.The space of the functions with bounded mean oscillation BMO, is well known for its several applications in real analysis, harmonic analysis and partial differential equations.
The definition of BMO is that f dy, |Q| is the Lebesgue measure of Q and Q is a cube in R n , with sides parallel to the coordinate axes.
In [1] Garnet and Jones gave comparable upper and lower bounds for the distance The bounds were expressed in terms of one constant in Jhon-Nirenberg inequality.Jhon and Nirenberg proved in [2] that f ∈ BMO if and only if there is > 0 and λ 0 = λ 0 ( ) such that sup whenever λ > λ 0 = λ 0 (f, ).Indeed, when f ∈ BMO, (1.2) holds with = Ckf k BMO , where the constant c depends only on the dimension.Specifically, setting

Garnett and Jones proved that
where A 1 and A 2 are constants depending only on the dimension.Also, they observed that dist (f, L ∞ ) can be related to the growth of sup Our latter end is to extend (1.3) to BMO p ϕ (see Preliminaries and Theorem 6.1) on spaces of homogeneous type.Also, we like to point out that (1.3) was announced in [1] without proof.Under the light of Remark 1 (see Preliminaries) we should note that if |B| = μ(B), then our main result coincide with the result of Garnett and Jones [1].

Spaces of homogeneous type
Let us begin by recalling the notion of space of homogeneous type.2. d(x, y) = d(y, x) for all x, y ∈ X.
3. There exists a constant K such that for all x, y, z ∈ X.
A quasimetric defines a topology in which the balls B(x, r) = {y ∈ X : d(x, y) < r} form a base.These balls may be not open in general; anyway, given a quasimetric d, is easy to construct an equivalent quasimetric d 0 such that the d 0 -quasimetric balls are open (the existence of d 0 has been proved by using topological arguments in [3]).So we can assume that the quasimetric balls are open.A general method of constructing families {B(x, δ)} is in terms of a quasimetric.Definition 2.2.A space of homogeneous type (X, d, μ) is a set X with a quasimetric d and a Borel measure μ finite on bounded sets such that, for some absolute positive constant A the following doubling property holds μ (B(x, 2r)) ≤ Aμ (B(x, r)) for all x ∈ X and r > 0.
Next, we are ready to give some example of a space of homogeneous type.
k=1 B k with the euclidean distance and the measure μ such that μ (B k ) = 2 k and on each ball B k , μ is uniformly distributed.Claim 1. μ satisfies the doubling condition.Let B r = B(P, r) with P = (P 1 , . . ., P n ) and r > 0.
Case 1. Assume for some k, B k ⊂ B r and let k 0 = max {k : B k ⊂ B r }.
Hence the doubling condition holds with A = 4. Case 2. If for all k, B k 6 ⊆ B r , then r < 1 so that B r and B 2r intersect only one ball B k .Then the doubling condition holds.

Preliminaries
In this section, we recall the definition of the space of functions of Bounded (ϕ, p) Mean Oscillation, BMO (p) ϕ (X), where X is a space of homogeneous type (see [4]).Let ϕ be a nonnegative function on [0, ∞).A locally μ- where the sup is taken over all balls B ⊂ X, and

John-Nirenberg inequality on homogeneous type space
The proof of this theorem follows along the same lines as the proof of [4]. Proof.
We follows the standard stopping time argument; that is, we assume that λ is large enough and fix some λ 1 .Then we study the sets in showing (4.1), we assume kf k ϕ = 1 and fix S = B(a, R).We define a maximal operator associated to S (if we replace S by another ball, then the maximal operator changes) Using a Vitali-type covering lemma, one can prove that where A is a constant that depends only on K and k 2 but not on S. Take λ 0 > A and consider the open set U = {x : M S f (x) > λ 0 }.We have and therefore S ∩ U c 6 = ∅.Define Clearly, Again by a Vitali-type covering lemma (e. g, see [1, Theorem 3.1]), we can select a finite or countable sequence of disjoint balls {B (x j , r j )} such that r j = r j (x) and On the other hand, B (x j , 6Kr j ) ∩ U c 6 = ∅ and B (x j , 6Kr j ) ⊂ B (a, αR) because 6kr j ≤ 12KR/5.Thus, we get 1 ϕ (μ (B (x j , 6Kr j ))) μ (B (x j , 6Kr j )) and consequently, if we write S j = B (x j , 4Kr j ), we obtain Now, we do the same construction for each and therefore for these points Continuing in this maner we get This complete the proof. 2

Completeness
In this section we state some simple lemmas.The first one is showed by elementary calculations.
Lemma 5.1.Let B 0 and B 1 be two balls such that B 0 ⊂ B 1 and f ∈ BMO ϕ .Then there exists a constant C depending on B 0 and B 1 such that Proof.Indeed, This complete the proof of Lemma 5.1.
Proof.By Hölder's inequality we have On the other hand By Theorem 4.1, we obtain The Lemma is proved.
ϕ equipped with the norm (3.1) is a Banach space.

Proof.
We just need to prove that BMO ϕ is complete.To this end, let us take B 1 to be the unit ball centered at the origin.Let and assume that Let B be any ball in X and let B 2 be a ball that contains both B 1 and B, then By Minkoswki's inequality and by (5.1), we have By Lemma 5.1, we have By Lemma 5.2 is easy to see that Therefore and from (5.2), we obtain Finally, we want to show that: ϕ (X).This proves part (a).
On the other hand,

Definition 2 . 1 .
A quasimetric d on a set X is a function d : X × X → [0, ∞) with the following properties: 1. d(x, y) = 0 if and only if x = y.

Theorem 4 . 1 .
There exist two positive constants β and b such that for any f ∈ BMO ϕ (X) and any ball B ⊂ X, one has