Given f 2 L1(T) we denote by wmo(f) the modulus of mean oscillation given

by

wmo(f)(t) = sup

0<|I| t

1

|I|

Z

I

|f(ei ) − mI (f)|

d 

2 

where I is an arc of T, |I| stands for the normalized length of I, and mI (f) =

1

|I|

R

I f(ei ) d 

2  . Similarly we denote by who(f) the modulus of harmonic oscillation

given by

who(f)(t) = sup

1−t |z|<1

Z

T

|f(ei ) − P(f)(z)|Pz(ei )

d 

2 

where Pz(ei ) and P(f) stand for the Poisson kernel and the Poisson integral

of f respectively.

It is shown that, for each 0  0 such that

Z 1

0

[wmo(f)(t)]p dt

t

 

Z 1

0

[who(f)(t)]p dt

t

  Cp

Z 1

0

[wmo(f)(t)]p dt

t.-Oscar.Blasco@uv.e

Blasco de la Cruz, Óscar

Pérez, María Amparo

Given f 2 L1(T) we denote by wmo(f) the modulus of mean oscillation given
by
wmo(f)(t) = sup
0<|I| t
1
|I|
Z
I
|f(ei ) − mI (f)|
d 
2 
where I is an arc of T, |I| stands for the normalized length of I, and mI (f) =
1
|I|
R
I f(ei ) d 
2  . Similarly we denote by who(f) the modulus of harmonic oscillation
given by
who(f)(t) = sup
1−t |z|<1
Z
T
|f(ei ) − P(f)(z)|Pz(ei )
d 
2 
where Pz(ei ) and P(f) stand for the Poisson kernel and the Poisson integral
of f respectively.
It is shown that, for each 0  0 such that
Z 1
0
[wmo(f)(t)]p dt
t
 
Z 1
0
[who(f)(t)]p dt
t
  Cp
Z 1
0
[wmo(f)(t)]p dt
t

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

On Functions of Integrable Mean OscillationOscar BLASCO * and M. Amparo PE´REZDepartamento de Ana´lisis Matema´ticoUniversidad de Valencia46100 BurjassotValencia — Spainoblasco@uv.esRecibido: 1 de diciembre de 2004Aceptado: 21 de abril de 2005ABSTRACTGiven f ∈ L1(T) we denote by wmo(f) the modulus of mean oscillation givenbywmo(f)(t) = sup0<|I|≤t1|I|∫I|f(eiθ)−mI(f)| dθ2piwhere I is an arc of T, |I| stands for the normalized length of I, and mI(f) =1|I|∫If(eiθ) dθ2pi. Similarly we denote by who(f) the modulus of harmonic oscilla-tion given bywho(f)(t) = sup1−t≤|z|<1∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2piwhere Pz(eiθ) and P (f) stand for the Poisson kernel and the Poisson integralof f respectively.It is shown that, for each 0 < p <∞, there exists Cp > 0 such that∫ 10[wmo(f)(t)]p dtt≤∫ 10[who(f)(t)]p dtt≤ Cp∫ 10[wmo(f)(t)]p dtt.Key words: mean oscillation, BMO, modulus of continuity.2000 Mathematics Subject Classification: 46B25.*Partially supported by Proyecto BMF2002-0401 and MTN2004-21420-ERev. Mat. Complut.2005, 18; Nu´m. 2, 465–477465ISSN: 1139-1138O. Blasco/M. A. Pe´rez On functions of integrable mean oscillation1. Introduction.As usual we denote by BMO the space of functions f ∈ L1(T) such that‖f‖∗ = supI⊆T1|I|∫I|f(eiθ)−mI(f)| dθ2pi <∞,where I is an arc of the circle T, |I| stands for the normalized length of I andmI(f) = 1|I|∫If(eiθ) dθ2pi . We write ‖f‖BMO = |fˆ(0)|+ ‖f‖∗ .If f ∈ L1(T) and 0 < t ≤ 1, we define the modulus of mean oscillation of f at thepoint t aswmo(f)(t) = sup0<|I|≤t1|I|∫I|f(eiθ)−mI(f)| dθ2pi .Clearly, for 0 < t ≤ s < 1, one hassupt<|I|≤s1|I|∫I|f(eiθ)−mI(f)| dθ2pi ≤2t‖f‖1.Hence, for 0 < t ≤ s < 1, one haswmo(f)(t) ≤ wmo(f)(s) ≤ max{wmo(f)(t),2‖f‖1t}. (1)In particular, f ∈ BMO if and only if wmo(f)(t) < ∞ for some (or for all)0 < t ≤ 1.It is known that one can consider other equivalent moduli to define BMO. Forinstance, for 0 < q <∞,wmo,q(f)(t) = sup0<|I|≤t(1|I|∫I|f(eiθ)−mI(f)|q dθ2pi)1/q.It is also well known, by the John-Nirenberg lemma (see [7, 8]), that there existC1, C2 > 0 such that for all λ > 0 and any arc I with |I| ≤ t,|{ θ ∈ T : |f(eiθ)−mI | > λ }||I| ≤ C1e− C2λwmo(f)(t) .From here one gets that, for all t > 0,wmo(f)(t) ≈ wmo,q(f)(t). (2)One can also considerw′mo(f)(t) = sup|I|≤t(1|I|2∫I∫I|f(eiθ)− f(eiϕ)| dθ2pidϕ2pi)Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477466O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationorw˜mo(f)(t) = sup|I|≤t(infc(1|I|∫I|f(eiθ)− c| dθ2pi))Clearly one getswmo(f)(t) ≤ w′mo(f)(t) ≤ 2wmo(f)(t) (3)andw˜mo(f)(t) ≤ wmo(f)(t) ≤ 2w˜mo(f)(t).A function f is said to have vanishing mean oscillation, in short f ∈ VMO, iflim|I|→01|I|∫I|f(eiθ)−mI(f)| dθ2pi = 0.This is a closed subspace of BMO, which can be characterized in many ways(see [7, 8, 15]).Theorem 1.1. Let f ∈ BMO. The following statements are equivalent:(i) f ∈ VMO.(ii) limt→0+‖Ttf − f‖BMO = 0, where Ttf(eiθ) = f(ei(θ−t)).(iii) limr→1‖Pr ∗ f − f‖BMO = 0, where Pr(eiθ) = <( 1+re−iθ1−re−iθ ).(iv) f belongs to the closure of C(T) in BMO.(v) limt→0+ wmo(f)(t) = 0.A generalization of BMO is the space BMO(ρ), consisting of functions f ∈ L1(T)such that wmo(f)(t) = O(ρ(t)) for a fixed function ρ with certain properties. Thespace BMO(ρ) has been considered by various authors (see [10,15,17]).Our aim will be to analyze spaces where the function ρ is not explicitly given, butwe do know its behavior at the origin in terms of certain integrability conditions.Given 0 < p < ∞, we shall denote by MOp(T) the space of integrable functionssuch that∫ 10[wmo(f)(t)]p dtt <∞.Due to (2), the spaces MOpq of functions such that∫ 10[wmo,q(f)(t)]p dtt <∞ are allthe same for 0 < q <∞.These spaces were considered in [12] (see page 74), under a different notation. Alsosome spaces MOαs,r, which are closely related to the ones considered in this paper,were introduced in [13].We use the notationsω∞(f)(t) = sup|θ−ϕ|≤t|f(eiθ)− f(eiϕ)|467 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationandωq(f)(t) = sup|u|≤t(∫T|f(ei(θ+u))− f(eiθ)|q dθ2pi)1/qfor 0 < q <∞.Now, for 0 < s < 1 and 0 < p, q ≤ ∞, the Besov space Bsq,p(T) consists offunctions in Lq(T) such that t−sωq(f) ∈ Lp((0, 2pi), dtt ). Of course the cases Bsq,∞where 0 < q ≤ ∞ correspond to Lipschitz or Ho¨lder classes, to be denoted Lips(T)instead of Bs∞,∞(T).We denote by Xp the space consisting of functions L∞(T) such that ω∞(f) ∈Lp((0, 2pi), dtt ).From (3) we easily obtain, for any t > 0,wmo(f)(t) ≤ Cω∞(f)(t).Hence Xp ⊂MOp(T) for any 0 < p <∞.On the other hand, if I, J are arcs on T such that I ⊂ J then|mJ(f)−mI(f)| ≤ |J ||I| wmo(f)(|J |). (4)Now, given I with |I| ≤ t, using the Lebesgue differentiation theorem, one getsf(eiθ) = limnmInf, for a.a. θ ∈ I,where In is a decreasing sequence of arcs containing θ such that |In| = 2|In+1|.Hence using (4), we have that for any f ∈ MO1(T)|f(eiθ)−mI | ≤ limn|mInf −mIf |≤∞∑k=1|mIkf −mIk−1f |≤ C∞∑k=1wmo(f)(2−kt)≤ C∫ t0wmo(f)(s)dss.Therefore we obtain, for any t > 0,ω∞(f)(t) ≤ 2C∫ t0wmo(f)(s)dss. (5)This implies that MO1(T) ⊂ Lipφ, where Lipφ stands for the space of continuousfunctions such that|f(eiθ)− f(eiϕ)| ≤ Cφ(|θ − ϕ|),Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477468O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationfor φ(t) = sup{∫ t0wmo(f)(s)dss :∫ 10wmo(f)(t)dtt ≤ 1}.BMO-type characterizations of these spaces have been extensively considered inthe literature. The reader is referred to [2, 3, 9] for the case p = ∞ and to [4, 5] forthe cases 0 < s and 1 < p <∞.We shall consider a description of MOp(T) where the averages over arcs are re-placed by averages with respect the Poisson kernel.We denote by BMOH the space of functions f ∈ L1(T) such that‖f‖∗∗ = supz∈∆∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2pi <∞,where ∆ denotes the open unit disc, P (f)(z) =∫T f(eiθ)Pz(eiθ) dθ2pi and Pz(eiθ) =<( 1+ze−iθ1−ze−iθ ). We write ‖f‖BMOH = |P (f)(0)|+ ‖f‖∗∗.It is not difficult to prove (see [7,8]) that f ∈ BMO if and only if f ∈ BMOH withequivalent norms.In this situation we define the modulus of harmonic oscillation of f at the point taswho(f)(t) = sup1−t≤|z|<1∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2pi .Hence, f ∈ BMO (respect. BMOH) if and only if wmo(f)(1) <∞ (respect. who(f)(1)<∞).For 0 < p < ∞, we denote by HOp(T) the space of f ∈ L1(T) such that∫ 10[who(f)(t)]p dtt < ∞.Of course one can also use other moduli to define this space. For instance, for0 < q <∞,who,q(f)(t) = sup1−t≤|z|<1(∫T|f(eiθ)− P (f)(z)|qPz(eiθ) dθ2pi)1/q,orw˜ho(f)(t) = sup1−t≤|z|<1infc∫T|f(eiθ)− c|Pz(eiθ) dθ2pi .The main objective of the paper is to show that, for 0 < p <∞, we have MOp(T) =HOp(T) and with equivalent “norms”.The paper is divided into two sections. The first one is devoted to introducingMOp(T) and proving some of its properties and the second one to introducing HOp(T)and to showing that HOp(T) coincides with MOp(T).2. Integrable mean oscillation.We will see first that the modulus of mean oscillation is continuous. We shall use thefollowing lemma.469 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationLemma 2.1. Let f ∈ L1(T). If {In} is a sequence of arcs such that limn→∞ In = Ifor some arc I with |I| > 0 thenlimn→∞1|In|∫In|f(eiθ)−mIn(f)|dθ2pi=1|I|∫I|f(eiθ)−mI(f)| dθ2pi .Proof. Let us first estimate1|In|∫In|f(eiθ)−mIn(f)|dθ2pi− 1|I|∫I|f(eiθ)−mI(f)| dθ2pi≤ 1|In|∫In|f(eiθ)−mI(f)| dθ2pi + |mIn(f)−mI(f)|− 1|I|∫I|f(eiθ)−mI(f)| dθ2pi≤ 1|I|(∫In|f(eiθ)−mI(f)| dθ2pi −∫I|f(eiθ)−mI(f)| dθ2pi)+ |mIn(f)−mI(f)|+ 2‖f‖1( 1|In| −1|I|).Notice that ν(A) =∫Af(eiθ) dθ2pi and µI(A) =∫A|f(eiθ)−mI(f)| dθ2pi are a complex anda positive measure respectively with integrable densities. Therefore the result followspassing to the limit as n goes to ∞.Proposition 2.2. Let f ∈ BMO. Then wmo(f) is increasing and continuous in (0, 1].Proof. Monotonicity has already been proved in our formula (1).Let 0 < t0 ≤ 1 and let us prove that it is left continuous at t0. Given ε > 0 wefind It0 ⊂ T such that 0 < |It0 | ≤ t0 andwmo(f)(t0) ≤ 1|It0 |∫It0|f(eiθ)−mIt0 (f)|dθ2pi+ε2.Let (tn) be a sequence such that tn ≤ t0 for all n ∈ N and converges to t0.If |It0 | = t0, we can find In ⊂ It0 such that limn→∞ In = It0 . Hencewmo(f)(t0)− wmo(f)(tn) ≤≤ 1|It0 |∫It0|f(eiθ)−mIt0 (f)|dθ2pi− 1|In|∫In|f(eiθ)−mIn(f)|dθ2pi+ε2.Now use Lemma 2.1 to get limn→∞ wmo(f)(t0)− wmo(f)(tn) = 0.If |It0 | < t0 there exists n0 such that |It0 | ≤ tn for n ≥ n0. Hence wmo(f)(t0) −wmo(f)(tn) < ε2 for n ≥ n0.Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477470O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationTo see that it is right continuous at t0, we shall argue as follows: Let (tn) bea sequence such that tn ≥ t0 for all n ∈ N and converges to t0. We shall find asubsequence (tnk) such that limk→∞ wmo(f)(tnk) = wmo(f)(t0).Given ε > 0 we find In ⊂ T such that 0 < |In| ≤ tn andwmo(f)(tn) ≤ 1|In|∫In|f(eiθ)−mIn(f)|dθ2pi+ ε.Let F = {n ∈ N : |In| > t0}. If F is finite then |In| ≤ t0 for n ≥ n0 andwmo(f)(tn)− wmo(f)(t0) < ε for n ≥ n0.Without loss of generality we assume |In| > t0 for all n ∈ N.Call I0 = ∪∞n=1 ∩∞k=n Ik. It is easy to see that I0 is an arc and that |I0| = t0. Takea subsequence nk such that (Ink) converges to I0. We havewmo(f)(tnk)− wmo(f)(t0) ≤ wmo(f)(tnk)−1|I0|∫I0|f(eiθ)−mI0(f)|dθ2pi≤ 1|Ink |∫Ink|f(eiθ)−mInk (f)|dθ2pi− 1|I0|∫I0|f(eiθ)−mI0(f)|dθ2pi+ ε.The proof is complete invoking Lemma 2.1.Remark 2.3. Let f ∈ BMO and take a(f) = limt→0+ wmo(f)(t). Hence f ∈ VMO ifand only if a(f) = 0.For each 0 < p <∞, we define the quasi-norm (norm for p ≥ 1) on MOp(T) by‖f‖MOp = ‖f‖L1(T) +(∫ 10[wmo(f)(t)]pdtt)1/p.Although the next result is probably known, we include a proof for the sake ofcompleteness.Theorem 2.4. Let 0 < p <∞. Then (MOp(T), ‖.‖MOp) is a complete space.Proof. Let {fn} be a Cauchy sequence in MOp(T). In particular, there exists f ∈ BMOsuch that {fn} converges to f .Let |I| ≤ t, 0 < t ≤ 1. Using that fn → f in L1(T) we get that mI(fn)→ mI(f)and that there exists a subsequence (nk), such that fnk → f a.e.471 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationNow1|I|∫I|fn(eiθ)− f(eiθ)−mI(fn − f)| dθ2pi=1|I|∫I|fn(eiθ)− limkfnk(eiθ)− limkmI(fn − fnk)|dθ2pi=1|I|∫Ilimk|fn(eiθ)− fnk(eiθ)−mI(fn − fnk)|dθ2pi≤ lim infk1|I|∫I|fn(eiθ)− fnk(eiθ)−mI(fn − fnk)|dθ2pi≤ lim infkwmo(fn − fnk)(t).Thereforewmo(fn − f)(t) ≤ lim infkwmo(fn − fnk)(t).Hence ∫ 10[wmo(fn − f)(t)dtt]p≤∫ 10lim infk[wmo(fn − fnk)(t)]pdtt≤ lim infk∫ 10[wmo(fn − fnk)(t)]pdttFinally, using that fn is a Cauchy sequence we get limn→∞‖fn−f‖MOp = 0 and thatf ∈ MOp.Proposition 2.5. Let 0 < p ≤ q <∞ and s > 0.(i) MOp(T) ⊆ MOq(T).(ii) Lips(T) ⊂⋂p>0MOp(T) ⊂ MO1(T) ⊂ C(T).(iii)⋃p>0MOp(T) ⊂ VMO.Proof. (i) It is a consequence of the following fact:(∫ 10[wmo(f)(t)]pdtt)1/p≈( ∞∑k=0[wmo(f)(2−k)]p)1/p.(ii) Note that f ∈ Lips if and only if wmo(f)(t) ≤ Cts. This gives the firstinclusion.The fact that MO1(T) ⊂ C(T) follows by (5).(iii) Observe that, for any p > 0 and t > 0, one haswmo(f)p(t) log1t≤∫ 1twmo(f)p(u)duu≤ ‖f‖MOp .Hence limt→0+ wmo(f)(t) = 0 for f ∈ ∪p>0MOp(T).Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477472O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationLet us point out some properties of BMO that are shared by these spaces.Proposition 2.6. If f ∈ MOp(T) then |f | ∈ MOp(T).Proof. Let t ∈ (0, 1) and I ⊂ T with |I| ≤ t. Then1|I|∫I∣∣|f(eiθ)| −mI(|f |)∣∣ dθ2pi≤ 1|I|∫I∣∣|f(eiθ)| − |mI(f)|∣∣ dθ2pi+∣∣mI(|f |)− |mI(f)|∣∣≤ 2|I|∫I∣∣|f(eiθ)| − |mI(f)|∣∣ dθ2pi≤ 2|I|∫I|f(eiθ)−mI(f)| dθ2piThis shows that wmo(|f |)(t) ≤ 2 wmo(f)(t) and the proof is complete.Recall that Tt denotes the translation operator, that is Ttf(eiθ) = f(ei(θ−t)). Wehave the following result.Theorem 2.7. Let 0 < p <∞ and f ∈ MOp(T). Thenlims→0+‖Tsf − f‖MOp = 0.Proof. Due to (iii) in Proposition 2.5 f ∈ VMO. Now Theorem 1.1 gives thatlims→0+‖Tsf − f‖BMO = 0.Note that wmo(Tsf − f)(t) ≤ ‖Tsf − f‖BMO for all 0 < t ≤ 1.On the other handwmo(Tsf − f)(t) = sup|I|≤t1|I|∫I∣∣(Tsf − f)(eiθ)−mI(Tsf − f)∣∣ dθ2pi≤ sup|I|≤t1|I|∫I∣∣Tsf(eiθ)−mI(Tsf)∣∣ dθ2pi+ sup|I|≤t1|I|∫I∣∣f(eiθ)−mI(f)∣∣ dθ2pi= 2 wmo(f)(t)The Lebesgue dominated convergence theorem gives lims→0+‖Tsf − f‖MOp = 0.3. Integrable harmonic oscillation.Throughout this section, given z ∈ ∆ \ {0}, we denote by Iz the open arc in T withmidpoint z|z| and length |Iz| = 1 − |z|. Given an arc I ⊂ T and λ ≤ |I|−1 we shallwrite λI for the arc with the same midpoint and length λ|I|.Let us collect several known facts to be used later on.473 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationLemma 3.1. There exist constants 0 < C,C1, C2, C3 <∞ such that(i) 1− |z| ≤ |eiθ − z| ≤ C (1− |z|), eiθ ∈ Iz, and z ∈ ∆.(ii) C1 1|Iz| ≤ Pz(eiθ) ≤ C2 1|Iz| , eiθ ∈ Iz, and z ∈ ∆.(iii) 14k|Iz| ≤ Pz(eiθ) ≤ C3 14k|Iz| , eiθ ∈ 2kIz \ 2k−1Iz, k ∈ {1, 2, . . . , N + 1}, whereN = [log21|Iz| ] and z ∈ ∆.Proof. All the statements follow from the estimates1− |z| ≤ |eiθ − z| ≤∣∣∣eiθ − z|z| ∣∣∣+ (1− |z|)and ∣∣∣eiθ − z|z| ∣∣∣ ≤ |eiθ − z|+ (1− |z|).For 0 < p <∞ we define‖f‖HOp = ‖f‖L1(T) +(∫ 10[who(f)(t)]pdtt)1/pto get a quasi-norm in the space HOp(T).Proposition 3.2. If f ∈ L1(T) and 0 < t ≤ 1 then wmo(f)(t) ≤ cwho(f)(t).Proof. Let I ⊆ T be an arc such that |I| ≤ t. Consider z ∈ ∆ for which I = Iz. From|Iz| = 1− |z| ≤ t we have 1− t ≤ |z| < 1.Using (ii) in Lemma 3.1 we have1|I|∫I|f(eiθ)−mI(f)| dθ2pi ≤1|Iz|∫Iz|f(eiθ)− P (f)(z)| dθ2pi+ |mI(f)− P (f)(z)|≤ 2|Iz|∫Iz|f(eiθ)− P (f)(z)| dθ2pi≤ C(∫ pi−pi|f(eiθ)− P (f)(z)|Pz(θ) dθ2pi)≤ Cwho(f)(t)Now taking the supremum over all arcs we get wmo(f)(t) ≤ C who(f)(t).Theorem 3.3. Let 0 < p < ∞. Then HOp(T) = MOp(T) with equivalent quasi-norms.Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477474O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationProof. HOp(T) ⊆ MOp(T) follows from Proposition 3.2.Assume now that f ∈ MOp(T). Let us show that f ∈ HOp(T) and ‖f‖HOp ≤C‖f‖MOp .Let t ∈ (0, 1]. For any z ∈ ∆ with |z| = 1 − t we consider the arc I = Iz, whichgives |Iz| = 1 − |z| = t. Take N and Ik (k = 0, 1, . . . , N) so that Ik = 2kIz whereI0 = I, IN−1 ( T, and IN = T.Using (iii) in Lemma 3.1 we have∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2pi≤2∫T|f(eiθ)−mI(f)|Pz(eiθ) dθ2pi≤ C(∫I|f(eiθ)−mI(f)|Pz(eiθ) dθ2pi+N∑k=1∫Ik\Ik−1|f(eiθ)−mI(f)|Pz(eiθ) dθ2pi )≤ C(1|I|∫I|f(eiθ)−mI(f)| dθ2pi+N∑k=114k|I|∫Ik\Ik−1|f(eiθ)−mI(f)| dθ2pi)≤ C(wmo(f)(t) +N∑k=112k|Ik|∫Ik|f(eiθ)−mI(f)| dθ2pi).On the other hand, by (4),1|Ik|∫Ik|f(eiθ)−mI(f)| dθ2pi≤ 1|Ik|∫Ik∣∣∣∣f(eiθ)−mIk(f) + ( k∑j=1mIj (f)−mIj−1(f))∣∣∣∣ dθ2pi≤ 1|Ik|∫Ik|f(eiθ)−mIk(f)|dθ2pi+k∑j=1|mIj (f)−mIj−1(f)|≤ 1|Ik|∫Ik|f(eiθ)−mIk(f)|dθ2pi+k∑j=1|Ij ||Ij−1|wmo(f)(|Ij |)≤ wmo(f)(|Ik|) +k∑j=12wmo(f)(|Ij |)≤ (1 + 2k)wmo(f)(|Ik|).475 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationCombining both estimates one gets∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2pi ≤ C(wmo(f)(t) +N∑k=11 + 2k2kwmo(f)(|Ik|)).Taking the supremum over {z : |z| = 1− t} and using |Ik| = 2kt we obtainsup|z|=1−t∫T|f(eiθ)− P (f)(z)|Pz(eiθ) dθ2pi ≤ C(wmo(f)(t) +Nt∑k=11 + 2k2kwmo(f)(2kt))where N = Nt = [log21t ] + 1. This implies thatwho(f)(t) ≤ C(wmo(f)(t) +Nt∑k=11 + 2k2kwmo(f)(2kt)).For 0 < p < 1 we have[who(f)(t)]p ≤ Cp([wmo(t)(t)]p +Nt∑k=1(1 + 2k)p2pk[wmo(f)(2kt)]p).For p ≥ 1 we apply Ho¨lder’s inequality to obtain[who(f)(t)]p ≤ Cp([wmo(t)(t)]p +Nt∑k=1(1 + 2k)p2k[wmo(f)(2kt)]p).Now integrating, and taking into account that 1 ≤ k ≤ Nt = [log2 1t ] + 1 isequivalent to 0 < t ≤ 2−k, we get∫ 10[who(f)(t)]pdtt≤ Cp∫ 10[wmo(t)(t)]pdtt+ Cp∫ 10Nt+1∑k=1(1 + 2k)p2kmin{p,1}[wmo(f)(2kt)]pdtt≤ Cp‖f‖pMOp + Cp∞∑k=1(1 + 2k)p2kmin{p,1}∫ 2−k0[wmo(f)(2kt)]pdtt≤ Cp‖f‖pMOp + Cp∞∑k=1(1 + 2k)p2kmin{p,1}∫ 10[wmo(f)(t)]pdtt≤ C‖f‖pMOp .Putting together all the estimates we have the result.Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477476O. Blasco/M. A. Pe´rez On functions of integrable mean oscillationReferences[1] A. Baernstein II, Analytic functions of bounded mean oscillation, Aspects of contemporary com-plex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), 1980, pp. 3–36.[2] S. Campanato, Proprieta` di ho¨lderianita` di alcune classi di funzioni, Ann. Scuola Norm. Sup.Pisa (3) 17 (1963), 175–188.[3] , Proprieta` di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964),137–160.[4] J. R. Dorronsoro, Mean oscillation and Besov spaces, Canad. Math. Bull. 28 (1985), no. 4,474–480.[5] K. M. Dyakonov, Besov spaces and outer functions, Michigan Math. J. 45 (1998), no. 1, 143–157.[6] C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971),587–588.[7] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, AcademicPress Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981.[8] D. Girela, Analytic functions of bounded mean oscillation, Complex function spaces (Mekrija¨rvi,1999), 2001, pp. 61–170.[9] H. Greenwald, On the theory of homogeneous Lipschitz spaces and Campanato spaces, Pacific J.Math. 106 (1983), no. 1, 87–93.[10] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and boundedmean oscillation, Duke Math. J. 47 (1980), no. 4, 959–982.[11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math.14 (1961), 415–426.[12] J. Peetre, On the theory of Lp, λ spaces, J. Functional Analysis 4 (1969), 71–87.[13] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed norm spaces andtheir atomic structure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 1, 1–54.[14] D. Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State Uni-versity Department of Mathematics, Blacksburg, Va., 1978.[15] W. S. Smith, BMO(ρ) and Carleson measures, Trans. Amer. Math. Soc. 287 (1985), no. 1,107–126.[16] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. ScuolaNorm. Sup. Pisa (3) 19 (1965), 593–608.[17] B. E. Viviani, An atomic decomposition of the predual of BMO(ρ), Rev. Mat. Iberoamericana3 (1987), no. 3-4, 401–425.477 Revista Matema´tica Complutense2005, 18; Nu´m. 2, 465–477

On Functions of Integrable Mean Oscillation

Abstract

Similar works

Full text

Available Versions

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura