Proyecciones Vol. 27, No 2, pp. 163—177, August 2008. Universidad Cat´olica del Norte Antofagasta - Chile

FUNCTIONS OF BOUNDED (ϕ, p) MEAN OSCILLATION

RENE´ ERLIN´ CASTILLO OHIO UNIVERSITY, U. S. A.

JULIO CESAR´ RAMOS FERNANDEZ´ and EDUARD TROUSSELOT UNIVERSIDAD DE ORIENTE, VENEZUELA

Received : November 2006. Accepted : July 2008

Abstract 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 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

1. 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,iswellknownforitssev- eral applications in real analysis, harmonic analysis and partial differential equations. 1 The definition of BMO is that f BMO if supQ Q Q f(x) fQ dx = ∈ | | | − | 1 f BMO < where fQ = Q Q f(y)dy, Q is the LebesgueR measure of Q k k ∞ | | | | and Q is a cube in Rn, with sidesR parallel to the coordinate axes. In [1] Garnet and Jones gave comparable upper and lower bounds for the distance

(1.1) dist (f,L )= inf f g BMO . ∞ g L k − k ∈ ∞ 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 >0andλ0 = λ0() such that

1 λ/ (1.2) sup x Q : f(x) fQ >λ e− , Q Q |{ ∈ | − | }| ≤ | | whenever λ>λ0 = λ0(f,). Indeed, when f BMO, (1.2) holds with ∈ = C f BMO, where the constant c depends only on the dimension. k k Specifically, setting

(f)=inf >0:f satisfies (1.2) , { } Functions of Bounded (ϕ, p) Mean Oscillation 165

Garnett and Jones proved that

A1(f) dist (f,L ) A2(f), ≤ ∞ ≤ where A1 and A2 are constants depending only on the dimension. Also, they observed that dist (f,L ) can be related to the growth of ∞ 1 1 p p sup f(x) fQ dx Q Q Q | − | µ| | Z ¶ as p .Thisisbecause →∞ 1 p (f) 1 1 p (1.3) =lim sup f(x) fQ dx e p p Q Q Q | − | →∞ Ã | | Z ! p Our latter end is to extend (1.3) to BMOϕ (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].

2. Spaces of homogeneous type

Let us begin by recalling the notion of space of homogeneous type.

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.

2. d(x, y)=d(y, x) for all x, y X. ∈ 3. There exists a constant K such that

d(x, y) K [d(x, z)+d(z,y)] , ≤ for all x, y, z X. ∈ 166 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

A quasimetric defines a topology in which the balls B(x, r)= y X : d(x, y)

quasimetric d0 such that the d0-quasimetric balls are open (the existence of d0 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)isasetX 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.

Example 1. Let X Rn, X = 0 x : x =1 , put in X the euclidean ⊂ { }∪{ | | } distance and the following measure μ: μ is the usual surface measure on x : x =1 and μ ( 0 )=1.Thenμ is doubling so that (X, d, μ)isa { | | } { } homogeneous space.

n k Example 2. In R ,letCk (k =1, 2, )bethepoint(k +1/2, 0, , 0), ··· ··· for k 2, let Bk be the ball B(Ck, 1/2) and B1 = B(0, 1/2). Let ≥ X = Bk with the euclidean distance and the measure μ such that ∪k∞=1 k μ (Bk)=2 and on each ball Bk, μ is uniformly distributed.

Claim 1. μ satisfies the doubling condition. Let Br = B(P, r)with Functions of Bounded (ϕ, p) Mean Oscillation 167

P =(P1,...,Pn) and r > 0.

Case 1. Assume for some k, Bk Br and let k0 =max k : Bk Br . ⊂ { ⊂ } k0+1 k0 Then certainly P1 + r bk +1 =(k0 +1) +1 andμ(Br) 2 .But, ≤ 0 ≥ then

k0+1 P1 +2r 2 (k0 +1) +1 ≤ ³ k0+2 ´ (k0 +2) = ak +2. ≤ 0

Therefore B2r Ba (0) B0.But ⊂ k0+2 ≡

k0+1 k k0+2 μ(B0)= 2 2 4μ(Br). ≤ ≤ kX=0 Hence the doubling condition holds with A =4.

Case 2. If for all k, Bk Br,thenr<1sothatBr and B2r intersect only 6⊆ one ball Bk. Then the doubling condition holds.

3. Preliminaries

In this section, we recall the definition of the space of functions of Bounded (p) (ϕ, p) Mean Oscillation, BMOϕ (X), where X is a space of homogeneous type (see [4]). Let ϕ be a nonnegative function on [0, ). A locally μ- ∞ (p) integrable function f : X R is said to belong to the class BMOϕ (X), → 1 p< ,if ≤ ∞ 1 1 p p sup p f(x) fB dμ(x) < , μ(B)[ϕ (μ(B))] B | − | ∞ µ Z ¶ where the sup is taken over all balls B X,and ⊂ 1 f = f(y)dμ. B μ(B) ZB 168 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

Remark 1. It is not hard to check that the expression

1 1 p p (3.1) f p =sup f(x) f dμ(x) < , BMOϕ p B k k μ(B)[ϕ (μ(B))] B | − | ∞ B µ Z ¶ (p) define a norm on BMOϕ (X).Forϕ 1 and p =1, p coincide ≡ k·kBMOϕ with BMO. k·k

4. John-Nirenberg inequality on homogeneous type space

The proof of this theorem follows along the same lines as the proof of [4].

Theorem 4.1. There exist two positive constants β and b such that for

any f BMOϕ(X) and any ball B X, one has ∈ ⊂

(4.1) μ ( x S : f fB >λ ) β exp bλ/ f BMO μ(B). { ∈ | − | } ≤ − k k ϕ © ª Proof. We follows the standard stopping time argument; that is, we

assume that λ is large enough and fixsomeλ1. Then we study the sets

x S : f(x) fS λ1 , x S : f(x) fS 2λ1 up to { ∈ | − | ≤ } { ∈ | − | ≤ }

x S : f(x) fS mλ1 λ { ∈ | − | ≤ ∼ } in showing (4.1), we assume f ϕ =1andfix S = B(a, R). We define a k k maximal operator associated to S (if we replace S by another ball, then the maximal operator changes)

1 MSf(x)= sup f(y) fS dμ(y) . B ball ,x B,B B(a,αR) ϕ (μ(B)) μ(B) B | − | ∈ ⊂ ½ Z ¾ Using a Vitali-type covering lemma, one can prove that

A μ ( x : MSf(x) >t ) μ(S), { } ≤ t Functions of Bounded (ϕ, p) Mean Oscillation 169

where A is a constant that depends only on K and k2 but not on S.Take

λ0 >Aand consider the open set U = x : MSf(x) >λ0 .Wehave { } A μ (U S) μ(S) <μ(S), ∩ ≤ λ0 and therefore S U c = .Define ∩ 6 ∅ 1 r(x)= dist (x, U c) . 5K

If x, y S,thend(x, y) 2KR.SinceS U c = ,ifx S,wehave ∈ ≤ ∩ 6 ∅ ∈ r(x) 2KR/(5K)=2R/5. ≤ Clearly,

U S B (x, r(x)) U. ∩ ⊂ x U S ⊂ ∈[∩ 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 (xj,rj) such that { } rj = rj(x)and

U S B (xj, 4Krj) U. ∩ ⊂ j ⊂ [ c On the other hand, B (xj, 6Krj) U = and B (xj, 6Krj) B (a, αR) ∩ 6 ∅ ⊂ because 6krj 12KR/5. Thus, we get ≤ 1 f fS dμ λ0, ϕ (μ (B (x , 6Kr ))) μ (B (x , 6Kr )) B(x ,6Kr ) | − | ≤ j j j j Z j j and consequently, if we write Sj = B (xj, 4Krj), we obtain

1 fS fSj f fS dμ − ≤ μ (Sj) S | − | ¯ ¯ Z j ¯ ¯ 2 ¯ ¯ ϕ (Sj) k2 λ0 := λ1 ≤ μ (B (xj,Krj)) because μ is a doubling measure. 170 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

By differentiation theorem, f(x) fS λ0 for μ-a.e. x S jSj. | − | ≤ ∈ \ ∪ Moreover,

μ (Sj) k2 μ (B (xj, 2Krj)) ≤ j X X C μ (B (xj,rj)) ≤ j X Cμ(U) ≤ CA μ(S). ≤ λ0

Now, we do the same construction for each Sj. Again f(x) fS λ0 for | − | ≤ (2) μ-a.e. x Sj iS and therefore for these points ∈ \ ∪ i

f(x) fS f(x) fS + fS fS | − | ≤ − j j − ¯ ¯ ¯ 2 ¯ ¯ ϕ (¯Sj)¯k2 ¯ ¯λ0 + ¯ ¯ λ0 ¯ ≤ μ (B (xj,Krj)) 2 2ϕ (Sj) k2 λ0, ≤ μ (B (xj,Krj)) taking λ0 =2CA,itisclearthat CA μ S(2) μ (S ) k λ j à ! ≤ j 0 [k X CA 2 μ(S)=2 2μ(S). λ − ≤ µ 0 ¶ Continuing in this maner we get N =1, 2, a family of ball SN such ··· j that n o N N μ S 2− μ(S), j ≤ ³[ ´ finally

μ ( x S : f(x) fS >λ ) μ ( x S : f(x) fS >Nλ1 ) { ∈ | − | } ≤ { ∈ | − | } N N bλ μ S 2− μ(S)=e− μ(S). ≤ j ≤ ³[ ´ This complete the proof. 2 Functions of Bounded (ϕ, p) Mean Oscillation 171

5. Completeness

In this section we state some simple lemmas. The first one is showed by elementary calculations.

Lemma 5.1. Let B0 and B1 be two balls such that B0 B1 and f ⊂ ∈ BMOϕ. Then there exists a constant C depending on B0 and B1 such that

fB fB C f BMO . | 0 − 1 | ≤ k k ϕ

Proof. Indeed,

1 fB0 fB1 = (f(y) fB1 ) dμ(y) | − | μ (B ) B − ¯ 0 Z 0 ¯ ¯ ¯ ¯ 1 ¯ ¯ f(y) fB1 dμ(y) ¯ ≤ μ (B ) B | − | 0 Z 1 μ (B1) ϕ (μ (B)) = f(y) fB1 dμ(y) μ (B ) ϕ (μ (B)) μ (B ) B | − | 0 1 Z 1 μ (B1) ϕ (μ (B)) f BMOϕ . ≤ μ (B0) k k

This complete the proof of Lemma 5.1. 2

(p) Lemma 5.2 (John-Nirenberg type). Let f BMOϕ (X), 1 p< ∈ ≤ , then there exists a constant Cp such that ∞

f BMOϕ f (p) Cp f BMOϕ . k k ≤ k kBMOϕ ≤ k k

Proof. By H¨older’s inequality we have

1 1 1 p p f(y) fB dμ(y) supB [ϕ(μ(B))]pμ(B) B f(y) fB dμ(y) ϕ (μ(B)) μ(B) B | − | ≤ | − | Z for any ball, thus ³ R ´

f BMOϕ f (p) . k k ≤ k kBMOϕ 172 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

On the other hand

p ∞ p 1 f(y) fB dμ(y) pλ − μ ( x B : f(x) fB >λ ) dλ. B | − | ≤ 0 { ∈ | − | } Z Z By Theorem 4.1, we obtain

p ∞ p 1 f(y) fB dμ(y) pλ − exp bλ/ f BMOϕ μ(B)dλ. B | − | ≤ 0 − k k Z Z ¡ ¢ Therefore

1 p p f(y) fB dμ(y) pΓ(p)C f BMOϕ [ϕ (μ(B))] μ(B) B | − | ≤ k k Z and thus

f (p) Cp f BMOϕ . k kBMOϕ ≤ k k The Lemma is proved. 2

(p) Theorem 5.1. BMOϕ equipped with the norm (3.1) is a Banach space.

(p) Proof. We just need to prove that BMOϕ is complete. To this end, (p) let us take B1 to be the unit ball centered at the origin. Let fk BMOϕ , ∈ for each k =1, 2, 3, , such that ··· ∞ fk (p) < , k kBMOϕ ∞ kX=1 and assume that

(5.1) fk(y)dμ(y)=0. ZB1 Let B be any ball in X and let B2 be a ball that contains both B1 and B, then

1 1 1 p ∞ 1 p μ(B2) p 1 p p fk(y) dμ(y) = μ(B) k∞=1 μ(B ) B fk(y) dμ(y) . μ(B) B | | 2 2 | | k=1 µ Z ¶ ³ ´ ³ ´ By Minkoswki’sX inequality and by (5.1), we haveP R Functions of Bounded (ϕ, p) Mean Oscillation 173

1 p 1 ∞ 1 p p [ϕ (μ (B2))] μ (B2) p fk(y) dμ(y) μ(B) B | | μ(B) kX=1 µ Z ¶ µ ¶ 1 ∞ 1 p p p fk(y) fB2 dμ(y) + ≤ [ϕ (μ (B2))] μ (B2) B2 | − | kX=1 µ Z ¶ 1 1 μ (B ) p ∞ 1 p p + 2 (f ) (f ) dμ(y) k B2 k B1 μ (B) μ (B2) B2 − µ ¶ kX=1 µ Z ¯ ¯ ¶ 1 ¯ ¯ 1 p p ¯ ¯ [ϕ (μ (B2))] μ (B2) ∞ μ (B2) p fk (p) + (fk)B (fk)B . ≤ μ(B) ⎡k kBMOϕ μ (B) 2 − 1 ⎤ µ ¶ k=1 µ ¶ ¯ ¯ X ¯ ¯ ⎣ ¯ ¯⎦ By Lemma 5.1, we have

1 ∞ 1 p p fk(y) dμ(y) μ(B) B | | kX=1 µ Z ¶ 1 μ(B2) p p k∞=1 fk +[ϕ (μ (B2))] fk (p) . μ(B) BMOϕ BMOϕ ≤ ∙k k k k ¸ ³ ´ P By Lemma 5.2 is easy to see that

1 ∞ 1 p p fk(y) dμ(y) μ(B) B | | kX=1 µ Z ¶ 1 μ(B2) p p μ(B) k∞=1 (1 + [ϕ (μ (B2))] ) fk (p) . ≤ k kBMOϕ ³ ´ P 1 1 p p Therefore ∞ fk(y) dμ(y) . This means k=1 μ(B) B | | ≤∞ ³ ´ P R 1 1 p ∞ (5.2) fk < , μ(B) k kLp ∞ µ ¶ kX=1 and from (5.2), we obtain

m f = lim fk, a. e. m →∞ kX=1

For f Lp(B), clearly fB = ∞ (fk) . ∈ k=1 B Finally, we want to showP that: 174 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

(p) (a) f BMOϕ (X), ∈

m (b) k=1 fk f (p) 0asm 0. k − kBMOϕ → → P To this end, observe that

1 1 p p p f(y) fB dμ(y) [ϕ (μ (B ))] μ (B) B | − | µ 2 Z ¶

1 p p 1 ∞ = p (fk(y) (fk)B) dμ(y) Ã[ϕ (μ (B2))] μ (B) B ¯ − ¯ ! Z ¯kX=1 ¯ ¯ ¯ 1 ¯ ¯ p ∞ 1 ¯ p ¯ p fk(y) (fk)B dμ(y) ≤ [ϕ (μ (B2))] μ (B) B | − | kX=1 µ Z ¶ ∞ fk (p) < , ≤ k kBMOϕ ∞ kX=1 (p) thus f (p) < ,thenf BMOϕ (X). This proves part (a). k kBMOϕ ∞ ∈ On the other hand,

p 1 m p 1 ∞ p fk f (y) fk f dμ(y) ⎛[ϕ (μ (B2))] μ (B) B ¯ − − − ¯ ⎞ Z ¯Ãk=1 ! Ãk=1 !B¯ ¯ X X ¯ ⎝ ¯ ¯ ⎠ ¯ ¯ 1 ¯ p ¯ p 1 ∞ = p (fk(y) (fk)B) dμ(y) ⎛[ϕ (μ (B ))] μ (B) B ¯ − ¯ ⎞ 2 Z ¯k=m+1 ¯ ¯ X ¯ ⎝ ¯ ¯ 1⎠ ∞ 1 ¯ p ¯ p p ¯ fk(y) (fk)B dμ¯ (y) ≤ [ϕ (μ (B2))] μ (B) B | − | k=Xm+1 µ Z ¶ ∞ fk (p) 0, as m . ≤ k kBMOϕ → →∞ k=Xm+1 m Hence k=1 fk f (p) 0asm 0. This proves part (b). This k − kBMOϕ → → completesP the proof of the Theorem 5.1. 2

6. Main Result Functions of Bounded (ϕ, p) Mean Oscillation 175

(p) Theorem 6.1. Let f BMOϕ , then there is a constant >0, such that ∈

λ/ (6.1) sup μ ( x B : f(x) fB >λ ) /μ(B) e− , { ∈ | − | } ≤ where λ>λ(, f). Indeed by Theorem 4.1, we have = C f (p) and k kBMOϕ λ(, f)=C f (p) .Nowlet k kBMOϕ

(f)=inf :(6.1) holds . { }

Then (f) 1 =lim f (p) . p BMOϕ eϕ (μ(B)) →∞ pk k

Proof. Since

p ∞ p 1 f(x) fB dμ(x)=p λ − μ (x B : f(x) fB >λ) dλ B(x,r) | − | 0 ∈ | − | Z Z ∞ p 1 λ/ pμ(B) λ − e− dλ ≤ 0 Z p ∞ p 1 u = μ(B) u − e du. Z0 Thus 1 p p f(x) fB dμ(x) pΓ(p). μ(B) B(x,r) | − | ≤ Z Next, we obtain

1 1 p 1 1 p p [pΓ(p)] sup p f(y) fB dμ(y) p [ϕ (μ (B))] μ (B) B | − | ≤ ϕ (μ(B)) p µ Z ¶ and then,

1 1 1 p p (f) (6.2)lim sup p f(y) fB dμ(y) . p p [ϕ (μ (B))] μ (B) B | − | ≤ eϕ (μ(B)) →∞ µ Z ¶

On the other hand, if <(f) then there exists B0 X, such that ⊂

λ/ e− μ ( x B0 : f(x) fB >λ ) /μ (B0) . ≤ { ∈ | − | } 176 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot

Thus

∞ p 1 λ/ ∞ p 1 pμ (B0) λ − e dλ < p λ − μ (x B : f(x) fB >λ) dλ 0 0 ∈ | − | Z Z and

1 1 p [Γ(p)] 1 1 p p < p f(y) fB dμ(y) . ϕ (μ(B)) p p [ϕ (μ (B))] μ (B) B | − | µ Z ¶ It is follows that

1 (f) 1 1 p p (6.3) < lim sup p f(y) fB dμ(y) . eϕ (μ(B)) p p [ϕ (μ (B))] μ (B) B | − | →∞ µ Z ¶ Combining (6.2) and (6.3), we obtain the desired result. 2

Remark 2. Theorem 6.1 together with Lemma 5.2 allow us to estimate (p) the distance from BMOϕ to L in the other words we can estimate ∞

inf f g (p) g L k − kBMOϕ ∈ ∞ (p) with f BMOϕ . ∈

Acknowledgments. The authors would like to thank the referee for the useful comments and suggestions which improved the presentation of this paper.

References

[1] J. Garnett and P. Jones. The distance in BMO to L , Ann. of Math., ∞ 108, pp. 373-393, (1978).

[2] F. Jhon and L. Nirenberg. On functions of bounded mean oscillation, Comm. on Pure and Appl. Math., XIV, pp. 415-426, (1961). Functions of Bounded (ϕ, p) Mean Oscillation 177

[3] R. Marcias and C. Segovia. Lipschitz functions on space of homogeneous type, Avd. Math., 33, pp. 257-270, (1979).

[4] J. Mateu, P. Mattila, A. Nicolau and J. Orobitg. BMO for non-doubling measure, Duke Math. Journal, 102, pp. 533-565, (2000).

Ren´eErl´ın Castillo Department of Mathematics Ohio University Athens, Ohio 45701 e-mail address: [email protected]

Julio C´esar Ramos Fern´andez Departamento de Matem´aticas Universidad de Oriente 6101 Cuman´a, Edo. Sucre, Venezuela e-mail : [email protected]

and

Eduard Trousselot Departamento de Matem´aticas Universidad de Oriente 6101 Cuman´a, Edo. Sucre, Venezuela e-mail : [email protected]