aa r X i v : . [ m a t h . C V ] S e p Bloch functions and Bekoll´e-Bonamiweights . Adem Limani & Artur Nicolau ∗ Abstract
We study analogues of well-known relationships between Muckenhouptweights and
BM O in the setting of Bekoll´e-Bonami weights. ForBekoll´e-Bonami weights of bounded hyperbolic oscillation, we providedistance formulas of Garnett and Jones-type, in the context of
BM O on the unit disc and hyperbolic Lipschitz functions. This leads to acharacterization of all weights in this class, for which any power ofthe weight is a Bekoll´e-Bonami weight, which in particular reveals anintimate connection between Bekoll´e-Bonami weights and Bloch func-tions. On the open problem of characterizing the closure of boundedanalytic functions in the Bloch space, we provide a counter-exampleto a related recent conjecture. This shed light into the difficulty ofpreserving harmonicity in approximation problems in norms equiva-lent to the Bloch norm. Finally, we apply our results to study certainspectral properties of Cesar´o operators.
Let D denote the unit disc and ∂ D the unit circle of the complex plane. Let L ∞ ( ∂ D ) be the algebra of essentially bounded real-valued functions definedon the unit circle and let BM O ( ∂ D ) be the space of integrable functions f : ∂ D → R such that k f k BMO ( ∂ D ) = sup I | I | Z I | f − f I | dm < ∞ , ∗ The second author is supported in part by the Generalitat de Catalunya (grant 2017SGR 395) and the Spanish Ministerio de Ciencia e Innovaci´on (project MTM2017-85666-P). I ⊂ ∂ D , m denotes the Lebesguemeasure on ∂ D and f I denotes the mean of f over I . A positive integrablefunction w on the unit circle is called an A weight, ifsup I (cid:18) m ( I ) Z I wdm (cid:19) (cid:18) m ( I ) Z I w − dm (cid:19) < ∞ , where the supremum is taken over all arcs I ⊂ ∂ D . Functions satisfy-ing this condition are called Muckenhoupt A weights and appear naturallywhen studying the boundedness of Calder´on-Zygmund operators on weighted L spaces. Muckenhoupt weights are also intimately related to BM O ( ∂ D )functions. The well-known equivalence between the A condition and theHelson-Szeg¨o Theorem gives the following beautiful distance formula. For f ∈ BM O ( ∂ D ) consider the quantity α ( f ) = inf { t > e f/t ∈ A } . Thenthere exists a universal constant
C > f ∈ BM O ( ∂ D ), wehave C − α ( f ) ≤ inf {k f − h k BMO ( ∂ D ) : h ∈ L ∞ ( ∂ D ) } ≤ Cα ( f ) . (1)In particular, a function f ∈ BM O ( ∂ D ) belongs to the closure of L ∞ ( ∂ D ) in BM O ( ∂ D ) if and only if e λf ∈ A , for any λ >
0. See Chapter VI of [11]. Themain purpose of this paper is to treat analogous results for Bekoll´e-Bonamiweights on D . To this end, we equip the unit disc D with the hyperbolicmetric β ( z, ζ ) = 12 log (cid:12)(cid:12) φ z ( ζ ) (cid:12)(cid:12) − (cid:12)(cid:12) φ z ( ζ ) (cid:12)(cid:12) , z, ζ ∈ D . Here φ z denotes the M¨obius automorphism on D , given by φ z ( ζ ) = z − ζ − zζ , z, ζ ∈ D . Let dA denote the normalized Lebesgue area-measure on D . An integrablepositive function w defined on D is called a B -weight if[ w ] B = sup I A ( Q I ) Z Q I wdA ! A ( Q I ) Z Q I w − dA ! < ∞ , (2)where the supremum is taken over all arcs I ⊂ ∂ D and Q I denotes theCarleson square associated to I , defined by Q I = (cid:26) z ∈ D \ { } : z | z | ∈ I , − | z | < m ( I ) (cid:27) . P ( f )( z ) = Z D f ( ζ )(1 − ζz ) dA ( ζ ) , z ∈ D , on the space L ( D , wdA ), which consists of square-integrable functions on D with respect to the measure wdA . It was proven in 1978, by Bekoll´eand Bonami that the Bergman projection P : L ( D , wdA ) → L ( D , wdA ) isbounded if and only if w is a B -weight. Actually, they introduced the notionof so-called B p -weights, when characterizing weights for which the Bergmanprojection is bounded on L p ( D , wdA ), 1 < p < ∞ . See [4]. In this sequel,we will mainly restrict our attention to a specific class of weights, that is, aspecific class of integrable and positive functions. A weight w : D → (0 , ∞ )is said to be of bounded hyperbolic oscillation (on hyperbolic discs withfixed radii), if there exists a constant C ( w ) >
0, such that (cid:12)(cid:12) log w ( z ) − log w ( ζ ) (cid:12)(cid:12) ≤ C ( w ) (cid:0) β ( z, ζ ) (cid:1) , z, ζ ∈ D . (3)Notice that condition (3) is conformally invariant, that is, invariant underM¨obius automorphisms. This particular class of weights has been recentlystudied in [2], where the authors showed that within this class of weights, onecan develop a fruitful theory of B ∞ -weights, satisfying the H¨older inequal-ity, similar to the well-studied Muckenhoupt A ∞ -weights on ∂ D . In general,Bekoll´e-Bonami weights are far more ill-behaved and lack self-improvementproperties, see [6]. Functions satisfying (3) have also naturally appeared in[5], when studying symbols of Hankel operators on the Bergman spaces.A function v : D → C will be called a hyperbolic Lipschitz function if k v k HLip = sup z,ζ ∈ D z = ζ (cid:12)(cid:12) v ( z ) − v ( ζ ) (cid:12)(cid:12) β ( z, ζ ) < ∞ , (4)that is, if it is Lipschitz continuous when D and C are endowed with thehyperbolic metric and the euclidean metric, respectively. We denote the spaceof hyperbolic Lipschitz functions by HLip( D ). The subspace of holomorphicfunctions on D which belong to HLip( D ) is precisely the classical Bloch space B , which consists of analytic functions g in D , such that k g k B = sup z ∈ D (1 − | z | ) | g ′ ( z ) | < ∞ . Indeed, is not difficult to show that k g k B = k g k HLip . In fact, the Bloch space B is a M¨obius invariant Banach space (modulo constants) and plays a crucial3ole in the theory of conformal mappings and in the theory of Bergmanspaces. See [18], [8] and [14] for further details on this matter. By takingthe closure of analytic polynomials in B , we obtain the so-called little Blochspace B , which consists of analytic functions g ∈ B , satisfyinglim | z |→ − (1 − | z | ) | g ′ ( z ) | = 0 . Let L ∞ ( D ) be the algebra of essentially bounded real-valued functions definedon the unit disc and let BM O ( D ) denote the space of functions of boundedmean oscillation with respect to the Lebesgue measure restricted to the unitdisc. In other words, an integrable real-valued function f on D belongs to BM O ( D ) if k f k BMO ( D ) = sup 1 A ( D ∩ D ) Z D ∩ D | f − f D ∩ D | dA < ∞ , where the supremum is taken over all discs D centered at points in D and f D ∩ D denotes the mean of f over D ∩ D . Taking the closure of polyno-mials in BM O ( D ), we obtain the subspace V M O ( D ). These spaces havepreviously been studied in [7] by Coifman, Rochberg and Weiss, where it isfor instance showed that the space of analytic functions in D whose real orimaginary parts belong to BM O ( D ) (respectively V M O ( D )) coincide with B (respectively B ), with equivalent norms (modulo constants). It is notdifficult to see that if the weight w has bounded hyperbolic oscillation, thenlog w ∈ BM O ( D ).In Lemma 2.4 we shall show that Bekoll´e-Bonami weights and BM O ( D )are intimately related. In fact, if w ∈ B then A ( { z ∈ Q : | log w ( z ) − (log w ) Q | > λ } ) ≤ e [ w ] B e − λ A ( Q )for any Carleson square Q and any λ > w ] B . This is a uniform John-Nirenberg type estimate which is distinctive of BMO-functions of boundednorm. Conversely, there exists a universal constant c > e cf ∈ B ,for any f ∈ BM O ( D ) with k f k BMO ( D ) = 1. Moreover, we shall in Proposition2.5 illustrate a close connection between B -weights which are exponentials ofharmonic functions and the Bloch space. Our first main result is an analogueof formula (1). Given f ∈ BM O ( D ), we set γ ( f ) = inf { t > e f/t ∈ B } . (5) Theorem 1.1.
Let f be a real-valued function on D with the property thatthe weight e f has bounded hyperbolic oscillation. i) Then we have γ ( f ) ≤ inf {k f − h k HLip : h ∈ L ∞ ( D ) } ≤ γ ( f ) . (ii) There exists a universal constant C > , such that C − γ ( f ) ≤ inf {k f − h k B MO ( D ) : h ∈ L ∞ ( D ) } ≤ Cγ ( f ) . Note that in the statement of (i) the function f may not belong to HLip( D ).In fact, (i) of Theorem 1.1 says that there exists h ∈ L ∞ ( D ), such that f − h ∈ HLip( D ) and verifies the estimates in (i). From Theorem 1.1 we nowdeduce. Corollary 1.2.
Let w be a weight on D with bounded hyperbolic oscillation.Then the following statements are equivalent:(i) For any λ > , we have w λ ∈ B .(ii) For any ε > there exists h ε ∈ L ∞ ( D ) such that k log w − h ε k H Lip < ε. (iii) For any ε > there exists h ε ∈ L ∞ ( D ) such that k log w − h ε k BMO ( D ) < ε. (iv) For any ε > there exists C ( ε ) > such that (cid:12)(cid:12) log w ( z ) − log w ( ζ ) (cid:12)(cid:12) ≤ C ( ε ) + εβ ( z, ζ ) , z, ζ ∈ D . (6)A naturally occurring class of weights which have bounded hyperbolic oscil-lation are weights of the form e Re( g ) , for g ∈ B . These weights appear in thestudy of spectral properties of generalized Cesar´o operators on the Bergmanspace. See [1] and [15]. For our purposes, we highlight the following imme-diate, yet important Corollary of Theorem 1.1. Corollary 1.3.
Let g ∈ B . Then e Re( λg ) is a B -weight for all λ ∈ C , if andonly if for any ε > , there exists C ( ε ) > such that (cid:12)(cid:12) g ( z ) − g ( ζ ) (cid:12)(cid:12) ≤ C ( ε ) + εβ ( z, ζ ) , z, ζ ∈ D . (7)5et H ∞ be the algebra of bounded analytic functions in D . Schwarz’s Lemmagives that H ∞ ⊂ B . Observe that if g belongs to the closure of H ∞ in B ,then it satisfies condition (7). Indeed, if g ε ∈ H ∞ is an approximate of g in B , then | g ( z ) − g ( ζ ) | ≤ k g ε k ∞ + k g − g ε k B β ( z, ζ ) , z, ζ ∈ D . A natural question is whether a Bloch function g satisfying (7) must belongto the closure of H ∞ in B . It was recently conjectured in [15] that if e Re( λg ) is a B -weight, for all λ ∈ C , then g belongs to the closure of H ∞ in B .Unfortunately, this turns out to be false and is essentially the main contentof our next result. The problem of characterizing the closure of H ∞ in B wasinitially posed in 1974 by Anderson, Clunie and Pommerenke (see [3]) andremains open to this date. It is worth mentioning that several variants of thisproblem have been studied in [9], [10], [13], [17] and [19]. For 0 < p < ∞ , let H p denote the classical Hardy of analytic functions f in D , satisfyingsup There exists a function g ∈ B such that g satisfies condition (7) , but g does not belong to the closure of H p ∩ B in B , for any < p ≤ ∞ . It is worth mentioning that Theorem 1.1 and Theorem 1.4 indicate that har-monicity is not preserved in Corollary 1.2. That is, there exists a B -weight w with the property that log w is harmonic in D , but the L ∞ ( D )-approximatesin (i) or (ii) of Theorem 1.1 are not harmonic. In other words, there exists g ∈ B which belongs to the closure of L ∞ ( D ) in BM O ( D ) and in HLip( D ),but it is not in the closure of H ∞ in any of these (semi-)norms.The paper is organized as follows. Several auxiliary results which may beof independent interest, are collected in Section 2. Section 3 contains theproofs of Theorem 1.1, Corollary 1.2 and Theorem 1.4. Finally, in section 4,we briefly mention some applications to the spectra of Cesar´o operators onthe Bergman space. The letter C will denote an absolute positive constantwhose value may change from line to line and C ( w ) will denote a constantdepending on w . We start with an auxiliary Lemma whose proof is a tailor-made version of amore general result which can be found in [1] (See Lemma 2.1).6 emma 2.1. Let w be a B -weight on D of bounded hyperbolic oscillation.Then there exists a constant C ( w ) > such that Z D ( w ◦ φ z ) ( ζ ) dA ( ζ ) ≤ C ( w ) w ( z ) , z ∈ D . Proof. Notice that the change of variable ζ φ z ( ζ ) gives Z D ( w ◦ φ z ) ( ζ ) dA ( ζ ) = Z D (1 − | z | ) | − zζ | w ( ζ ) dA ( ζ ) . (8)Let D ( z ) denote the disc D ( z ) = (cid:8) ζ ∈ D : | ζ − z | < (1 − | z | ) (cid:9) and observethat by the mean-value theorem for harmonic functions, we can write1(1 − zζ ) = 1 A ( D ( z )) Z D ( z ) − ηζ ) dA ( η ) = P ( χ ( z, · ))( ζ ) , z, ζ ∈ D , where χ ( z, η ) = A ( D ( z )) D ( z ) ( η ) and P denotes the Bergman projection. Re-call that w ∈ B if and only if P : L ( D , wdA ) → L ( D , wdA ) is bounded,hence there exists a constant C ( w ) > w , such that Z D | P ( χ ( z, · ) | wdA ≤ C ( w ) Z D | χ ( z, · ) | wdA, z ∈ D . Going back to (8) with these observations in mind, we obtain Z D ( w ◦ φ z ) ( ζ ) dA ( ζ ) ≤ C ( w ) 1 A ( D ( z )) Z D ( z ) w ( ζ ) dA ( ζ ) . (9)Using the fact that w is of bounded hyperbolic oscillation, the rightmostintegral in (9) is comparable to w ( z ), which completes the proof.Our next result establishes the conformal invariance of Bekoll´e-Bonami weights. Lemma 2.2. The collection of B -weights is conformally invariant. In fact,if w is a B -weight, there exists a constant C ( w ) > , such that [ w ◦ φ z ] B ≤ C ( w ) for any z ∈ D .Proof. According to the well-known result of Bekoll´e-Bonami in [4], it sufficesto prove that there exists a constant C ( w ) > 0, independent of z ∈ D , suchthat Z D | P ( f )( ζ ) | ( w ◦ φ z )( ζ ) dA ( ζ ) ≤ C ( w ) Z D | f ( ζ ) | ( w ◦ φ z )( ζ ) dA ( ζ ) , (10)7or all f ∈ L (cid:0) D , ( w ◦ φ z ) dA (cid:1) , z ∈ D . Notice that the change of variable ζ φ z ( ζ ) gives Z D | P ( f ) | ( w ◦ φ z ) dA = Z D | (cid:0) P ( f ) ◦ φ z (cid:1) · φ ′ z | wdA. (11)We now claim that for any z, ζ ∈ D , the following identity holds: (cid:0) P ( f ) ◦ φ z (cid:1) ( ζ ) · φ ′ z ( ζ ) = P (cid:0) ( f ◦ φ z ) · φ ′ z (cid:1) ( ζ ) . (12)To prove this, we primarily observe that a straightforward calculation shows φ ′ z ( ζ )(1 − ηφ z ( ζ )) = φ ′ z ( η )(1 − φ z ( η ) ζ ) . Now using this, we can write (cid:0) P ( f ) ◦ φ z (cid:1) ( ζ ) · φ ′ z ( ζ ) = Z D f ( η ) · φ ′ z ( ζ )(1 − ηφ z ( ζ )) dA ( η ) = Z D f ( η ) · φ ′ z ( η )(1 − φ z ( η ) ζ ) dA ( η ) . Performing the change of variable η φ z ( η ) in the last integral, we cansimplify it as Z D ( f ◦ φ z ) ( η ) · φ ′ z ( φ z ( η )) · | φ ′ z ( η ) | (1 − ηζ ) dA ( η ) = P (cid:0) ( f ◦ φ z ) · φ ′ z (cid:1) ( ζ ) . This establishes the identity in (12). Now using this together with the resultof Bekoll´e-Bonami, we can find a constant C ( w ) > 0, such that Z D | P (cid:0) ( f ◦ φ z ) · φ ′ z (cid:1) ( ζ ) | w ( ζ ) dA ( ζ ) ≤ C ( w ) Z D | ( f ◦ φ z ) ( ζ ) · φ ′ z ( ζ ) | w ( ζ ) dA ( ζ )= C ( w ) Z D | f ( ζ ) | ( w ◦ φ z ) ( ζ ) dA ( ζ ) . In the last step, we again utilized the change of variable ζ φ z ( ζ ). Accordingto (11) and (12), we finally obtain Z D | P ( f ) | ( w ◦ φ z ) dA ≤ C ( w ) Z D | f | ( w ◦ φ z ) dA, (13)for all f ∈ L (cid:0) D , ( w ◦ φ z ) dA (cid:1) , which proves the desired claim in (10).Next, we state an elementary result for future reference. Given a Carlesonsquare Q = Q ( I ) ⊂ D , we denote by z Q the point z Q = (1 − m ( I )) ξ ( I ), where ξ ( I ) is the center of the arc I ⊂ ∂ D . 8 emma 2.3. Let v : D → R be a hyperbolic Lipschitz function with k v k HLip < . Then e v ∈ B .Proof. Note that there exists a constant M > Q and any z ∈ Q , we have (cid:12)(cid:12)(cid:12)(cid:12) β ( z, z Q ) − 12 log 1 − | z Q | − | z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ M. Hence for any Carleson square Q and any z ∈ Q , we can write e − M − | z | − | z Q | ! k v k HLip / ≤ e v ( z ) − v ( z Q ) ≤ e M (cid:18) − | z Q | − | z | (cid:19) k v k HLip / . (14)Since k v k HLip < 1, there exists a constant C = C ( k v k HLip ) > 0, such that Z Q (cid:18) − | z Q | − | z | (cid:19) k v k HLip ≤ CA ( Q ) . We deduce that 1 A ( Q ) Z Q e v dA ≤ Ce M e v ( z Q ) . Using the left inequality in (14) we also obtain1 A ( Q ) Z Q e − v dA ≤ Ce M e − v ( z Q ) . This finishes the proof.Functions in BM O ( D ) satisfy the John-Nirenberg inequality. That is, thereexist universal constants C > C > f ∈ BM O ( D )and any disc D centered at a point in D , we have A (cid:16)(cid:8) z ∈ D ∩ D : | f ( z ) − f D ∩ D | > λ (cid:9)(cid:17) A ( D ∩ D ) ≤ C e − C λ/ k f k BMO ( D ) , λ > . (15)Let us recall the well known relation between BM O ( ∂ D ) and Muckenhouptweights on ∂ D . If w is a Muckenhoupt A -weight, then log w ∈ BM O ( ∂ D ).Conversely there exists a universal constant c > f ∈ BM O ( ∂ D )with k f k BMO ( ∂ D ) = 1, then e cf is a A -weight. Our next result establishesan analogue for Bekoll´e-Bonami weights.9 emma 2.4. (i) There exist absolute constants C > , C > , such that [ e C f/ k f k BMO ( D ) ] B ≤ C , for any f ∈ BM O ( D ) .(ii) For any f ∈ V M O ( D ) , we have lim δ → sup A ( Q ) <δ A ( Q ) Z Q e f dA ! A ( Q ) Z Q e − f dA ! = 1 , where the supremum is taken over all Carleson squares Q ⊂ D .(iii) For any w ∈ B and any λ > w ] B , one has A ( { z ∈ Q : | log w ( z ) − (log w ) Q | > λ } ) ≤ e [ w ] B e − λ A ( Q ) , (16) for any Carleson square Q ⊂ D . Moreover, if w ∈ B is of boundedhyperbolic oscillation, then log w ∈ BM O ( D ) . It is worth mentioning that in contrast to the John-Nirenberg inequality (15),no constant depending on w appears in the exponent of (16). Roughly speak-ing, this indicates that the logarithm of a B weight behaves as a function offixed BM O -norm on Carleson squares. Proof. (i) We can assume k f k BMO ( D ) = 1. Observe that for any Carlesonsquare Q there exists a disc D centered at a point in D such that Q ⊂ D and A ( D ) ≤ A ( Q ). Note that | f D ∩ D − f Q | ≤ A ( Q ) Z Q | f − f D ∩ D | dA ≤ A ( D ) Z D ∩ D | f − f D ∩ D | dA ≤ . Then the John-Nirenberg inequality (15) provides two absolute constants C > C > A ( Q ) Z Q e C | f − f Q | dA ≤ C , for any Carleson square Q ⊂ D . Then1 A ( Q ) Z Q e C ( f − f Q ) dA ≤ C and 1 A ( Q ) Z Q e − C ( f − f Q ) dA ≤ C . Hence A ( Q ) Z Q e C f dA ! A ( Q ) Z Q e − C f dA ! ≤ C . f ∈ V M O ( D ) and notice that by the preceding argument in (i),it is enough to show thatlim δ → sup A ( Q ) <δ A ( Q ) Z Q e | f ( z ) − f Q | dA ( z ) = 1 . (17)To this end, let p be a polynomial and q = f − p . Observe that for anyCarleson square Q and z ∈ Q , we have | f ( z ) − f Q | ≤ | q ( z ) − q Q | + ω p ( Q ) . Here ω p ( Q ) := sup z,ζ ∈ Q | p ( z ) − p ( ζ ) | denotes the oscillation of p on Q . Thisgives 1 A ( Q ) Z Q e | f ( z ) − f Q | dA ( z ) ≤ e ω p ( Q ) A ( Q ) Z Q e | q − q Q | dA. We now apply the John-Nirenberg inequality in (15) to q , to find absoluteconstants c , c > 0, such that A (cid:16)(cid:8) z ∈ Q : | q ( z ) − q Q | > λ (cid:9)(cid:17) A ( Q ) ≤ c e − c λ/ k q k BMO ( D ) , λ > . for every Carleson square Q . With this at hand and by choosing p , such that k q k BMO ( D ) < c / 2, we get1 A ( Q ) Z Q e | q − q Q | dA ≤ c k q k BMO ( D ) c − k q k BMO ( D ) ≤ c c k q k BMO ( D ) . Now combining, we arrive at1 A ( Q ) Z Q e | f ( z ) − f Q | dA ( z ) ≤ e ω p ( Q ) (cid:18) c c k q k BMO ( D ) (cid:19) . By uniform continuity of polynomials, it follows lim δ → sup A ( Q ) <δ ω p ( Q ) = 0.Finally, choosing p to be an approximate of f in BM O ( D ) establishes (17),thus (ii) follows.(iii) We first prove the estimate in (16). Let Q be a Carleson square. ByChebyshev’s inequality, we have for any t > A (cid:0) { z ∈ Q : w ( z ) > tw Q } (cid:1) ≤ A ( Q ) t . Since w − Q ≤ [ w ] B /w Q , Chebyshev’s inequality also gives A { z ∈ Q : w ( z ) < w Q t [ w ] B } ! ≤ A ( Q ) t , t > . w ] B ≥ 1, we obtain for any t > A (cid:16) { z ∈ Q : | log w ( z ) − log( w Q ) | > log( t [ w ] B ) } (cid:17) ≤ A ( Q ) t . Set λ = log( t [ w ] B ) to deduce that for any λ > log [ w ] B , we have A (cid:0) { z ∈ Q : | log w ( z ) − log( w Q ) | > λ } (cid:1) ≤ w ] B e − λ A ( Q ) . (18)Observe that (18) gives | (log w ) Q − log( w Q ) | ≤ A ( Q ) Z Q | log w − log( w Q ) | dA = 1 A ( Q ) Z ∞ A (cid:0) { z ∈ Q : | log w ( z ) − log( w Q ) | > λ } (cid:1) dλ = Z log[ w ] B dλ + 2 [ w ] B Z ∞ log[ w ] B e − λ dλ = log [ w ] B + 2 := C ( w ) . With this at hand, we see that whenever λ > C ( w ), the following set inclusionholds { z ∈ Q : | log w ( z ) − (log w ) Q | > λ } ⊂ { z ∈ Q : | log w ( z ) − log( w Q ) | > λ − C ( w ) } . Applying (18), we obtain A ( { z ∈ Q : | log w ( z ) − (log w ) Q | > λ } ) ≤ e [ w ] B e − λ A ( Q ) , for all λ > C ( w ), hence proving (16). Now under the additional assump-tion that w is of bounded hyperbolic oscillation we will prove that log w ∈ BM O ( D ). Let D be a disc of radius r > z ∈ D . If r ≤ (1 − | z | ) / 2, we use the fact that w has bounded hyperbolic oscillation, to finda constant C ( w ) > 0, independent of D such that | log w − log w ( z ) | ≤ C ( w )on D . Since | log w ( z ) − (log w ) D | ≤ A ( D ) Z D | log w − log w ( z ) | dA ≤ C ( w ) , we deduce 1 A ( D ) Z D | log w − (log w ) D | dA ≤ C ( w ) . If r > (1 − | z | ) / 2, consider a Carleson square Q with D ∩ D ⊂ Q and A ( Q ) < A ( D ). Then1 A ( D ) Z D ∩ D | log w − (log w ) Q | dA ≤ A ( Q ) Z Q | log w − (log w ) Q | dA, which, by (16), is uniformly bounded. Hence log w ∈ BM O ( D ).12ecall that a locally integrable function f defined on R d belongs to BM O ( R d ),if k f k BMO ( R d ) = sup 1 m d ( R ) Z R | f ( x ) − f R | dm d ( x ) < ∞ , where the supremum is taken over all cubes R ⊂ R d and m d is Lebesguemeasure on R d . A crucial observation is that if f ∈ BM O ( D ), then theextension f ∗ defined by f ∗ ( z ) = ( f ( z ) , z ∈ D f (1 /z ) , | z | > , (19)belongs to BM O ( R ). Given f ∈ BM O ( R d ), denote by ε ( f ) the infimum of ε > 0, for which there exists a constant λ ( ε ) > 0, such that m d (cid:16)(cid:8) x ∈ Q : | f ( x ) − f Q | > λ (cid:9)(cid:17) ≤ e − λ/ε m d ( Q ) , for any λ > λ ( ε ) and any cube Q ⊂ R d . Garnett and Jones proved in [12]that there exists a constant C > 0, only depending on the dimension suchthat for any f ∈ BM O ( R d ), one has C − ε ( f ) ≤ inf {k f − h k BMO ( R d ) : h ∈ L ∞ ( R d ) } ≤ Cε ( f ) . (20)This result will be used in the proof of part (ii) of Theorem 1.1.In contrast to the theory of Muckenhoupt weights, where w ∈ A impliesthat log w ∈ BM O ( ∂ D ), one cannot expect the same to hold for general B -weights, since they can posses very wild behavior inside D . However,under some rigid assumptions one can in fact retrieve similar results. Infact, our next result provides a relationship between Bekoll´e-Bonami weightswhich are exponentials of harmonic functions and Bloch functions. Our nextProposition is reminiscent of the deep interplay between A -weights on ∂ D and BM O ( ∂ D ) as previously mentioned. Meanwhile, part (ii) gives a newcharacterization of B , in a similar spirit to the characterization of vanishingmean oscillation by Sarason in [20]. Proposition 2.5. Let f : D → R be analytic.(i) Then, f ∈ B if and only if e δ Re( f ) is a B -weight, for some δ > .(ii) Then, f ∈ B if and only if lim δ → sup A ( Q ) <δ A ( Q ) Z Q e Re( f ) dA ! A ( Q ) Z Q e − Re( f ) dA ! = 1 . (21)13otice that analyticity seems to compensate for the a priori assumptionof the weights being of bounded hyperbolic oscillation. In order to proveProposition 2.5, we will need to establish two lemmas. Lemma 2.6. Let f : D → C be an analytic function and u = Re( f ) . Thenfor any / < | z | < , there exists a Carleson square Q z ⊂ D with A ( Q z ) ≤ − | z | ) , such that (1 − | z | ) | f ′ ( z ) | ≤ CA ( Q z ) Z Q z | u − u Q z | dA, for some universal constant C > .Proof. Notice that by Cauchy’s integral formula, we can write f ′ ( z ) = 12 π Z π f ( z + re it ) − αre it dt, z ∈ D , for 0 < r < − | z | and all α ∈ C . Applying Cauchy-Schwartz inequality, weget | f ′ ( z ) | ≤ πr Z π | f ( z + re it ) − α | dt. Since the conjugate operator is an isometry on the Hardy space H , we canfind an absolute constant C > 0, such that | f ′ ( z ) | ≤ Cr Z π | u ( z + re it ) − Re( α ) | dt. Now integrating this inequality with respect to r ∈ [(1 − | z | ) / , − | z | ], weget(1 − | z | ) | f ′ ( z ) | ≤ C (1 − | z | ) Z { (1 −| z | ) / ≤| z − ζ | < −| z |} | u ( ζ ) − Re( α ) | dA ( ζ ) . Since 1 − | z | < / 4, we can take Q z to be a Carleson square containing thedisc { ζ : | z − ζ | < − | z |} with A ( Q z ) ≤ − | z | ) . Moreover, choosing α = f Q z , we obtain(1 − | z | ) | f ′ ( z ) | ≤ CA ( Q z ) Z Q z | u ( ζ ) − u Q z | dA ( ζ ) . Our next lemma is a refined version of an abstract measure theoretic lemma,attributed to D. Sarason (See [20], Lemma 3).14 emma 2.7. Let (Ω , µ ) be a probability space and let w be a positive inte-grable function on Ω such that w − is also integrable. Assume (cid:18)Z Ω wdµ (cid:19) (cid:18)Z Ω w − dµ (cid:19) = 1 + ε, (22) for some < ε < Then Z Ω | log w − Z Ω log w dµ | dµ ≤ ε. (23) Proof. By means of multiplying w with a positive scalar, thus not affectingthe quantity in (22), we may assume that R Ω wdµ = 1. Consider the set S ε = (cid:26) ω ∈ Ω : 11 + ε ≤ w ( ω ) ≤ ε (cid:27) , and notice that for ω ∈ S ε , we have that | log w ( ω ) | ≤ log(1 + ε ) ≤ ε .Moreover, from the elementary inequality 2 + t ≤ e t + e − t , t ∈ R , we getthat for every positive function w , the estimate log w ≤ w + w − − Z Ω log wdµ ≤ ε + Z Ω \ S ε (cid:0) w + w − (cid:1) dµ − µ (Ω \ S ε ) . (24)According to (22) and the assumption R Ω wdµ = 1, we have Z Ω \ S ε (cid:0) w + w − (cid:1) dµ = 2 + ε − Z S ε ( w + w − ) dµ ≤ ε − 21 + ε µ ( S ε ) . Hence going back to (24), we obtain Z Ω log wdµ ≤ ε + ε + 2(1 − 11 + ε ) µ ( S ε ) ≤ ε. It now immediately follows that Z Ω | log w − Z Ω log wdµ | dµ = Z Ω log wdµ − (cid:18)Z Ω log wdµ (cid:19) ≤ ε. Proof of Proposition 2.5. Note that there exists a constant C > k f k BMO ( D ) ≤ C k f k B . Similarly f ∈ V M O ( D ) if f ∈ B . Hence the left toright implications of (i) and (ii) are immediate consequences of Lemma 2.415nd the intrinsic content of Proposition 2.5 is the ”if” part of the statements.(i) In order to prove the converse implication, we assume e u is a B -weightwith u : D → R harmonic. Notice that for any Carleson square Q ⊂ D , wehave 1 A ( Q ) Z Q e | u − u Q | dA ≤ A ( Q ) Z Q e u − u Q dA + 1 A ( Q ) Z Q e u Q − u dA. From Jensen’s inequality and the B -condition on e u , it follows that e u Q ≤ A ( Q ) Z Q e u dA ≤ [ e u ] B e u Q . We deduce that there exists C ( u ) > 0, such thatsup Q A ( Q ) Z Q e | u − u Q | dA ≤ C ( u ) . Let Re( f ) = u and observe that by Lemma 2.6 and the simple inequality t ≤ e t , for t > 0, there exists a constant C ( u ) > / < | z | < (1 − | z | ) | f ′ ( z ) | ≤ C ( u ) . This is enough to conclude that f ∈ B .(ii) Again, let u = Re( f ) and notice that by Lemma 2.6, it follows that forsufficiently small δ > 0, we havesup −| z |≤ δ (1 − | z | ) | f ′ ( z ) | ≤ sup A ( Q ) < δ CA ( Q ) Z Q | u ( ζ ) − u Q | dA ( ζ ) , Now assuming that condition (21) holds, we have according to Lemma 2.7,that lim δ → sup A ( Q ) <δ A ( Q ) Z Q | u ( z ) − u Q | dA ( z ) = 0 . This proves that f ∈ B . We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. (i) We first show thatinf {k f − h k HLip : h ∈ L ∞ ( D ) } ≤ γ ( f ) . (25)16iven ζ ∈ D , let D ζ be the disc centered at ζ of radius (1 − | ζ | ) / 4. Sincethe weight w = e f is of bounded hyperbolic oscillation, so is ( w ◦ φ z ) λ , forall z ∈ D , λ > 0, by conformal invariance of (3). Consequently, there exits aconstant C ( λ ) > 0, such that( w ◦ φ z ) λ ( ζ ) ≤ C ( λ ) 1 A ( D ζ ) Z D ζ ( w ◦ φ z ) λ dA ≤ C ( λ ) | ζ | − | ζ | ! Z D ( w ◦ φ z ) λ dA, for all z, ζ ∈ D . Now assuming that w λ is a B -weight, we may apply Lemma2.1 to w λ . This provides a constant C ( λ ) > 0, such that( w ◦ φ z ) λ ( ζ ) ≤ C ( λ ) | ζ | − | ζ | ! w ( z ) λ , z, ζ ∈ D . Now performing the change of variable ζ φ z ( ζ ) and taking logarithms, weconclude that | log w ( ζ ) − log w ( z ) | ≤ log C ( λ ) λ + 4 β ( ζ , z ) λ , ζ , z ∈ D . (26)Set ε = 1 /λ and C ε = log C ( λ ) /λ . Fix η > (cid:8) z j (cid:9) ∞ j =1 be asequence of points in D , with the properties that β ( z k , z j ) ≥ C ε /η , for all k = j , and such that inf λ ∈ Λ β ( z, λ ) ≤ C ε /η , for any z ∈ D . See [8] fordetails on such constructions. By (26), we have that (cid:12)(cid:12) log w ( z k ) − log w ( z j ) (cid:12)(cid:12) ≤ C ε + 4 εβ ( z k , z j ) ≤ (4 ε + η ) β ( z k , z j ) , j = k. This means that the map log w : Λ → R defined by z j log w ( z j ) isLipschitz continuous with respect to the hyperbolic metric β on Λ ⊂ D and the euclidean metric in R , with norm less than (4 ε + η ). We nowuse the McShane-Valentine extension Theorem (see [16]) to find a function g = g ε,η : D → R , with k g k HLip ≤ ε + η , such that g ( z j ) = log w ( z j ) for j = 1 , , . . . . We claim that h := log w − g belongs to L ∞ ( D ). To thisend, fix an arbitrary z ∈ D and observe that by construction of Λ, we canalways pick z j ∈ Λ, with β ( z, z j ) ≤ C ε /η . Then, using (26), we obtain | h ( z ) | = | h ( z ) − h ( z j ) | ≤ C ε + (8 ε + η ) β ( z, z j ) ≤ C ε + 10(8 ε + η ) C ε /η and wededuce h ∈ L ∞ ( D ). Since k log w − h k HLip ≤ ε + η and η > < t < h ∈ L ∞ ( D ) such that f − h ∈ HLip( D ),we have e t ( f − h ) / k f − h k HLip ∈ B . Hence γ ( f ) ≤ k f − h k HLip / t . We deduce that γ ( f ) ≤ 12 inf {k f − h k HLip : h ∈ L ∞ ( D ) } . We now turn our attention to proving (ii). Note that γ ( f ) = γ ( f − h ) for any h ∈ L ∞ ( D ). Part (i) of Lemma 2.4 gives that γ ( f ) = γ ( f − h ) ≤ C − k f − h k BMO ( D ) , for any h ∈ L ∞ ( D ). Hence γ ( f ) ≤ C − inf {k f − h k BMO ( D ) : h ∈ L ∞ ( D ) } . Note that in this step, we did not need the assumption that e f has boundedhyperbolic oscillation, thus this part of the proof holds for any f ∈ BM O ( D ).We now establish the converse estimate. Fix α > w α = e f/α ∈ B . Part (iii) of Lemma 2.4 gives that A { z ∈ Q : (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) α − (log w α ) Q (cid:12)(cid:12)(cid:12)(cid:12) > t } ! ≤ e [ w α ] B e − t A ( Q ) , for any t > w α ] B and any Carleson square Q ⊂ D . Writing t ∗ = αt ,we deduce A (cid:0) { z ∈ Q : | f ( z ) − f Q | > t ∗ } (cid:1) ≤ e [ w α ] B e − t ∗ /α A ( Q ) , (27)for any t ∗ > α (2 + log [ w α ] B ) =: t ∗ and any Carleson square Q ⊂ D . Since w = e f has bounded hyperbolic oscillation, there exists a constant C ( w ) > D of center z ∈ D and radius smaller than (1 − | z | ) / | f − f ( z ) | ≤ C ( w ) on D . This observation and the estimate in (27)imply the existence of a constant C = C ( α, w ) > 0, such that for any disc D centered at a point in D , we have A (cid:0) { z ∈ D ∩ D : | f ( z ) − f D ∩ D | > t ∗ } (cid:1) ≤ C e − t ∗ /α A ( D ) , (28)for any t ∗ > max { t ∗ , C ( w ) } =: t ∗ . Consider the extension f ∗ of f given by(19). The estimate (28) gives that there exists a constant C > 0, such thatfor any square R ⊂ R , we have A (cid:0) { z ∈ R : | f ∗ ( z ) − f ∗ R | > t ∗ } (cid:1) ≤ C e − t ∗ / α A ( R ) , t ∗ > t ∗ . Now the result of Garnett and Jones (20) implies theexistence of an absolute constant C > 0, such thatinf {k f ∗ − h k BMO ( R ) : h ∈ L ∞ ( R ) } ≤ Cα. We deduce inf {k f − h k BMO ( D ) : h ∈ L ∞ ( D ) } ≤ Cα, which finishes the proof.Regarding the proof of Theorem 1.1, a valuable remark is in order. First,notice that as a byproduct of the proof of part (i), an application of theMcShane-Valentine theorem [16] gives the following description:A weight w has bounded hyperbolic oscillation if and only if there exists u ∈ L ∞ ( D ) and v ∈ HLip( D ), such thatlog w = u + v. (29)Now consider the linear space S = { f : e f has bounded hyperbolic oscillation } equipped with the norm k f k S = inf {k u k L ∞ ( D ) + k v k HLip : f = u + v } . It is straightforward to check that S is a conformally invariant Banach space,contained in BM O ( D ). We now turn to the proof of Corollary 1.2. Proof of Corollary 1.2. Notice that the equivalence of (i), (ii) and (iii) is animmediate consequence of the distance formulas from Theorem 1.1. Nowsuppose (i) holds. Then a verbatim implementation of the proof of part(i) of Theorem 1.1 yields (26), i.e, for any λ > 0, there exists a constant C ( λ ) > 0, such that (cid:12)(cid:12) log w ( z ) − log w ( ζ ) (cid:12)(cid:12) ≤ C ( λ ) + 4 λ β ( z, ζ ) , z, ζ ∈ D . This precisely condition (6) in (iv). Conversely, assume condition (6) in (iv)holds. Fix an arbitrary arc I ⊆ ∂ D with center ξ I and set z I = (1 − m ( I )) ζ I .Now for any subarc J ⊂ I , a straightforward calculation shows that thereexists an absolute constant M > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β ( z J , z I ) − 12 log (cid:18) − | z I | − | z J | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M. I T I Figure 1: Tiling the Carleson box Q I into top-halves { T J } J ∈D ( I ) .With this at hand, we may rephrase condition (6) as follows: for any ε > c ε > 0, such that for any arc J ⊂ I , we have e − c ε (cid:18) m ( I ) m ( J ) (cid:19) − ε ≤ w ( z J ) w ( z I ) ≤ e c ε (cid:18) m ( I ) m ( J ) (cid:19) ε . (30)We will show that w λ ∈ B for any λ > 0. To this end, fix an arbitrary λ > D ( I ) denote the dyadic decomposition of I . Since w hasbounded hyperbolic oscillation, it follows that all values of w on top-halves T J = { z ∈ Q J : 1 − | z | ≥ m ( J ) / } of a Carleson square Q J are comparableto w ( z J ). Observing that the collection { T J } J ∈D ( I ) forms a tiling of Q I (SeeFigure 1), we can find a constant C ( λ ) > 0, such that1 A ( Q I ) Z Q I w λ dA = 1 A ( Q I ) X J ∈D ( I ) Z T J w λ dA ≤ C ( λ ) A ( Q I ) X J ∈D ( I ) A ( T J ) w λ ( z J ) . Now applying (30) to this, we actually get1 A ( Q I ) Z Q I w λ dA ≤ w λ ( z I ) C ( λ ) A ( Q I ) X J ∈D ( I ) A ( T J ) (cid:18) m ( I ) m ( J ) (cid:19) λε . For 0 < ε < /λ , a straightforward computation shows that1 A ( Q I ) X J ∈D ( I ) A ( T J ) (cid:18) m ( I ) m ( J ) (cid:19) λε = 12 ∞ X n =0 (2 λε − ) n < ∞ . It now follows that there exists C λ > 0, such that1 A ( Q I ) Z Q I w λ dA ≤ C λ w λ ( z I ) . (31)20n a similar way, using the leftmost inequality of (30), we can also prove anestimate for the dual-weight w − λ , identical to that of (31). This is enoughto conclude that w λ is a B -weight, for any λ > ξ ∈ ∂ D , let Γ( ξ ) := { z ∈ D : | z − ξ | < σ (1 − | z | ) } be the Stolz angleof vertex ξ and fixed aperture σ > 1. Given a function g ∈ B and ε > 0, weconsider the set K ( ε, g ) = { z ∈ D : (1 − | z | ) | g ′ ( z ) | > ε } . In the proof of Theorem 1.4, we will use the following description of theclosure of H p ∩ B in B . Fix 0 < p < ∞ and g ∈ B . Then g belongs to theclosure of H p ∩ B in B if and only if, for any ε > 0, the area-function A ε ( g )( ξ ) = Z Γ( ξ ) ∩ K ( ε,g ) dA ( z )(1 − | z | ) ! / , ξ ∈ ∂ D , (32)belongs to L p ( ∂ D , dm ). See [10] and [9]. We will also use the followingdescription of lacunary series in the Bloch space. A lacunary series is apower series g ( z ) = ∞ X k =1 a k z n k , z ∈ D , (33)with the property that there exists a constant ρ > 1, such that the positiveintegers { n k } satisfy n k +1 ≥ ρ n k , k = 1 , , . . . . It was proved in [3] that alacunary series of the form (33) belongs to B if and only if sup k ≥ | a k | < ∞ . Proof of Theorem 1.4. Fix a sequence { a k } ∞ k =1 of complex numbers whichdoes not tend to 0 and such that sup k ≥ | a k | ≤ 1. Let { n k } ∞ k =1 be an increasingsequence of positive integers such thatlim k →∞ n k +1 n k = ∞ . (34)Consider the lacunary series g defined in (33). For M > k = 1 , , . . . ,consider the annulus A k ( M ) := { z ∈ D : 1 M n k ≤ − | z | ≤ Mn k } . Fix a point z ∈ A j (2) and notice that | z || g ′ ( z ) | ≥ n j | a j || z | n j − j − X k =1 n k − ∞ X k = j +1 n k (1 − n j ) n k . / n j ≤ − | z | ≤ /n j , we deduce(1 −| z | ) | z || g ′ ( z ) | ≥ | a j | − n j ) n j − n j j − X k =1 n k − n j ∞ X k = j +1 n k (1 − n j ) n k . (35)We claim that the last two terms on the righthand side of (35) tend to zero,as j → ∞ . Indeed, since { n j } is lacunar, there exists a constant ρ > 1, suchthat for all j = 1 , , . . . , we have n j +1 ≤ ρ ( n j +1 − n j ). The assumption (34)shows that the first sum tends to zero, via1 n j j − X k =1 n k ≤ n n j + ρ ( n j − − n ) n j −→ , j → ∞ . (36)A similar argument also gives1 n j ∞ X k = j +1 n k (1 − n j ) n k ≤ n j +1 n j (1 − n j ) n j +1 + ρn j ∞ X k = j +2 ( n k − n k − )(1 − n j ) n k ≤ n j +1 n j exp( − n j +1 / n j ) + ρn j ∞ X k = n j +1 (1 − n j ) k −→ , j → ∞ . (37)We have thus proven that the last two terms in (35) tend to 0 as j → ∞ .Consequently, given ε > 0, there exists j > j > j with | a j | > ε , we have (1 − | z | ) | z || g ′ ( z ) | > Cε, z ∈ A j (2) . (38)Here C > { a j } ∞ j =1 does not tend to zero, thereexists ε > 0, such that the set K ( ε , g ) = { z ∈ D : (1 − | z | ) | g ′ ( z ) | > ε } contains infinitely many annulus A j (2) .We deduce that the function A ε ( g )defined in (32) satisfies A ε ( g )( ξ ) = ∞ , for every ξ ∈ ∂ D . Hence the function g does not belong to the closure of H p ∩ B in B , for any 0 < p ≤ ∞ . Wenow prove that g satisfies condition (7). Let M > A ( M ) = ∞ [ k =1 A k ( M ) . Fix z ∈ D \ A ( M ) and let j = j ( z ) be the unique positive integer, satisfying Mn j +1 < − | z | < M n j . − | z | ) | z || g ′ ( z ) | ≤ M n j j X k =1 n k + (1 − | z | ) ∞ X k = j +1 n k | z | n k . According to (36), there exists a constant C > j , such that1 n j j X k =1 n k ≤ C . An argument similar to (37) shows that there exists a constant C > 0, suchthat (1 − | z | ) ∞ X k = j +1 n k | z | n k ≤ C (1 − | z | ) n j +1 | z | n j +1 . Now it is straightforward to check thatsup Mnj +1 < −| z | < Mnj (1 − | z | ) n j +1 | z | n j +1 ≤ M − Mn j +1 ! n j +1 ≤ M e − M . Combining, we conclude that there exists an absolute constant C > 1, suchthat (1 − | z | ) | z || g ′ ( z ) | ≤ C (cid:18) M + M e − M (cid:19) , z ∈ D \ A ( M ) . Let 0 < ε < k g k B be arbitrary and pick M = M ( ε ) > 1, such that(1 − | z | ) | g ′ ( z ) | ≤ ε, z ∈ D \ A ( M ( ε )) . (39)Denote by l ( γ ) the hyperbolic length of the arc γ ⊂ D , given by l ( γ ) = Z γ | dz | − | z | . Note that for any hyperbolic geodesic Γ and any k ≥ l (Γ ∩ A k ( M ( ε ))) ≤ M ( ε ). Indeed, the shortest hyperbolic segment joiningtwo concentric circles in D centered at the origin, is a segment contained ina radius. Thus l (Γ ∩ A k ( M ( ε ))) ≤ Z − /M ( ε ) n k − M ( ε ) /n k dt − t = 2 log M ( ε ) . γ of hyperbolic length larger than K ( ε ) :=2 log M ( ε ) /ε , we have l ( γ ∩ A ( M ( ε )) l ( γ ) ≤ ε. (40)Let A ε = A ( M ( ε )) be an abbreviation of that set. Now let z, w ∈ D with β ( z, w ) > K ( ε ) and let γ be the hyperbolic segment joining z and w . Then | g ( z ) − g ( w ) | ≤ Z γ | g ′ ( ξ ) || dξ | . Using (40), we have Z γ ∩ A ε | g ′ ( ξ ) || dξ | ≤ k g k B l ( γ ∩ A ε ) ≤ ε k g k B β ( z, w ) . (41)Applying (39), we also have Z γ ∩ ( D \ A ε ) | g ′ ( ξ ) || dξ | ≤ εl ( γ ) = εβ ( z, w ) . (42)Now combining (41) and (42), it follows that | g ( z ) − g ( w ) | ≤ ε ( k g k B + 1) β ( z, w ) , if β ( z, w ) ≥ K ( ε ) . Since g ∈ B , we automatically have | g ( z ) − g ( w ) | ≤ K ( ε ) k g k B , for β ( z, w ) ≤ K ( ε ). This finishes the proof. In this final section, we shall briefly mention an application which initiallysparked the interest in characterizing weights with the property that everypower of the weight is in the class B . The background concerns the spectrumof generalized Cesar´o operators T g f ( z ) = Z z f ( ζ ) g ′ ( ζ ) dζ , z ∈ D . (43)on the Bergman spaces L pa ( D , dA ) of analytic functions f in D , which belongto L p ( D , dA ), p > 0. It is well known that the linear operator T g is boundedon L pa ( D , dA ) if and only if the symbol g belongs to B . For further detailson Cesar´o operators, we refer the reader to [1] and references therein. Now acomplete characterization of the spectrum of T g on weighted Bergman spaceshas been given in terms of a B ∞ -type condition (see [1], and [2] for alternative24eformulations). For L pa ( D , dA ) with p > 0, it goes as follows: A complexnumber λ = 0 does not belong to the spectrum of T g on L pa ( D , dA ) if and onlyif, there exists a constant C g/λ > 0, such that the weight e p Re( g/λ ) satisfies1 A ( Q ) Z Q e p Re( g/λ ) dA ≤ C g/λ exp A ( Q ) Z Q p Re( g/λ ) dA ! , for every Carleson square Q ⊂ D . From this, it follows that e p Re( g/λ ) isa B -weight, if and only if λ and − λ do not belong to the spectrum of T g on L pa ( D , dA ). Actually, if k σ p ( g ) k denotes the spectral radius of T g on L pa ( D , dA ), then we have k σ p ( g ) k = inf { λ > p Re( g/λξ )) ∈ B , ∀ ξ ∈ ∂ D } . From this observation and Lemma 2.2, it follows that the spectral radius isconformally invariant. More precisely, if g z = ( g ◦ φ z ) − g ( z ) denotes thehyperbolic translate of g , then for any z ∈ D , we have k σ p ( g ) k = k σ p ( g z ) k . We naturally lend the notations of L ∞ ( D ) and BM O ( D ) to include complex-valued functions. As a consequence of Theorem 1.1, we can now successfullygive a Bergman space analogue of Theorem 2.4 in [15]. Corollary 4.1. There exists an absolute constant C > , such that for any p > and any g ∈ B , we have Cp k σ p ( g ) k ≤ inf h ∈ L ∞ ( D ) k g − h k BMO ( D ) ≤ Cp k σ p ( g ) k . In particular, if the spectrum of T g on L pa ( D , dA ) does not contain any non-zero points of the real and imaginary axes, then g belongs to the closure of L ∞ ( D ) in BM O ( D ) , and thus the spectrum is { } . A worthy final remark is that from the proof of Corollary 1.2, it follows thatthe spectral radius is comparable to the infimum of ε > 0, for which thereexists C ( ε ) > 0, such that | g ( z ) − g ( ζ ) | ≤ C ( ε ) + εβ ( z, ζ ) , z, ζ ∈ D . In order to estimate the spectral radius, this serves as a more practical con-dition. For example, it is evident from the discussions surrounding Corollary1.3, that functions which belong to the closure of H ∞ in B , induce Cesar´o25perators with zero spectral radius. Meanwhile, Theorem 1.4 provides a non-trivial example of a Bloch function g , such that T g has zero spectrum, butsuch that g does not belong to the closure of H p ∩ B in B , for any 0 < p ≤ ∞ . Acknowledgement. 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Sarason , Functions of vanishing mean oscillation , Transactions ofthe American Mathematical Society, 207 (1975), pp. 391–405. A. Limani, Centre for Mathematical Sciences, Lund University, P.O Box118, SE-22100, Lund, Sweden E-mail address , A. Limani: [email protected] A. Nicolau , Departament de Matem`atiques, Universitat Aut`onoma de Barcelona,08193 Barcelona E-mail address , A. Nicolau: [email protected]@mat.uab.cat