superposition operators between the bloch space and bergman spaces

SUPERPOSITION OPERATORS BETWEEN THE BLOCHSPACE AND BERGMAN SPACES

VENANCIO ALVAREZ, M. AUXILIADORA MARQUEZ, AND DRAGAN VUKOTIC

Abstract. Using a geometric method, we characterize all entire func-tions that transform the Bloch space into a Bergman space by superpo-sition in terms of their order and type. We also prove that all superpo-sition operators induced by such entire functions act boundedly. Similarresults hold for superpositions from BMOA into Bergman spaces andfrom the Bloch space into certain weighted Hardy spaces.

Introduction

If X and Y are metric spaces of analytic functions in the disk and ϕ is acomplex-valued function in the plane such that ϕ f ∈ Y whenever f ∈ X,we say that ϕ acts by superposition from X into Y . If X and Y containthe linear functions, then ϕ must be an entire function. The superpositionoperator Sϕ : X → Y with symbol ϕ is then defined by Sϕ(f) = ϕf . Notethat if X and Y are also linear spaces, the operator Sϕ is linear if and onlyif ϕ is a linear function that fixes the origin. The central questions are:

When does a superposition operator map one space into another? Whenis it bounded?

Even though analogous concepts also make sense in the context of realvalued functions and their theory has a long history (see [AZ]), the study ofsuch natural questions on spaces of analytic functions has only begun fairlyrecently. The operators Sϕ that map one Bergman space into another orinto the area Nevanlinna class were characterized in terms of their symbolsin [CG]; cf. [C] for Hardy spaces. Results of similar nature were obtainedin [BFV] for superpositions between various spaces of Dirichlet type.

In this paper we give a complete description of the superpositions betweena Bergman space Ap and the Bloch space B (in either of the two directions)in terms of the order and type of ϕ. That is, we measure in a precise way“how much slower” the Bloch functions grow than those in Ap. We recallthe basic definitions and facts which motivate our results.

Let dA denote the Lebesgue area measure in the unit disk D, normalizedso that A(D) = 1. If 0 < p < ∞, the Bergman space Ap is the set of allanalytic functions f in the unit disk D with finite Lp(D, dA) norm:

‖f‖pp =

∫

D|f(z)|p dA(z) =

1

π

∫ 1

0

∫ 2π

0

|f(reiθ)|pdθrdr < ∞ .

2000 Mathematics Subject Classification. 30D45, 32A36, 47B33.1

2 V. ALVAREZ, M. A. MARQUEZ, AND D. VUKOTIC

Note that ‖f‖p is a true norm if and only if 1 ≤ p < ∞. When 0 < p < 1, Ap

is a complete space with respect to the translation-invariant metric definedby dp(f, g) = ‖f − g‖p

p. It is clear that Aq ⊂ Ap if 0 < p < q < ∞. Themonograph [HKZ] contains plenty of information about Bergman spacesand so does the forthcoming text [DS]; see also [Z].

The maximum growth of functions in Bergman spaces will be essential. Iff ∈ Ap, by the subharmonicity of |f |p and the area version of the sub-meanvalue property applied to the disk of radius 1−|z| centered at z, we readilyobtain the standard estimate

(1) |f(z)| ≤ ‖f‖p

(1− |z|)2/p, for all z in D .

An analytic function f in D is said to belong to the Bloch space B if

‖f‖B = |f(0)|+ supz∈D

(1− |z|2)|f ′(z)| < ∞ .

Any such function satisfies the following growth condition:

(2) |f(z)| ≤(

log1

1− |z| + 1

)‖f‖B , for all z in D .

This follows by integrating the inequality |f ′(ζ)| ≤ (‖f‖B−|f(0)|)(1−|ζ|2)−1

along the line segment from 0 to z. For further basic facts about the Blochspace the reader may consult [ACP] or Chapter 5 of [Z].

The inclusion B ⊂ Ap (0 < p < ∞) follows easily from (2). Moreover,since the exponents in both (1) and (2) are sharp, it becomes intuitivelyclear that the Bloch space should be contained in any Bergman space “ex-ponentially”, thinking in terms of superpositions. However, our main result,Theorem 3, will show that functions such as ϕ(z) = exp(cz), c 6= 0, fail tomap B into Ap. In fact, the superposition operator Sϕ maps B into Ap if andonly if the entire function ϕ is either of order less than one, or of order oneand type zero. Thus, the characterization of the superpositions does notdepend on p. Our proof resembles the ideas employed in the proofs of someof the main results of [BFV] (Theorem 12 and Theorem 19). Althoughhere we no longer need the inequalities of Trudinger type, our argumentstill requires several delicate technical details. The key point is a geometricconstruction of a special univalent function in the Bloch space (Lemma 2)which may have some independent interest.

1. Superposition from a Bergman space into the Bloch space

Given that any Bergman space Ap is larger than the Bloch space B, itshould be intuitively obvious that a superposition from the former to thelatter is possible via constant functions only. However, some work is stillrequired. This is due to the fact that no uniform bounds are available(because we do not have any a priori information on the boundedness ofthe superposition operator).

Proposition 1. Let 0 < p < ∞ and let ϕ be entire. Then the superpositionoperator Sϕ maps Ap into B if and only if ϕ is a constant function.

SUPERPOSITION OPERATORS 3

Proof. Suppose that Sϕ maps Ap into B. To show that ϕ is constant, itsuffices to prove that ϕ′ is constant. Indeed, if ϕ′ ≡ a then ϕ is a linearfunction: ϕ(z) = az+b. Taking f ∈ Ap\B, we have that Sϕ(f) = af+b ∈ B,which implies a = 0 and therefore ϕ is constant.

Now there are two possibilities for the function ϕ′: either

(3) ϕ′(w)/w → 0 as |w| → ∞or

(4) |ϕ′(wn)| ≥ δ|wn|for some fixed δ > 0 and some sequence wn such that |wn| → ∞.

Note that (3), in particular, yields a standard Cauchy estimate |ϕ′(w)| ≤A|w| for some A > 0 and all sufficiently large w, which in turn implies thatϕ′(w) is a linear function. By letting w → ∞ and applying (3), it followsthat ϕ′ = const and we are done.

It is, thus, only left to show that (4) is impossible. To this end, letα ∈ (0, 2

p) and consider the function

f(z) =1

(1− z)α, z ∈ D.

By integrating in polar coordinates centered at z = 1, it is easily checkedthat f ∈ Ap. Hence ϕ f ∈ B, and therefore

Mf = supz∈D

(1− |z|2)|f ′(z)| · |ϕ′(f(z))| < ∞ .

From here we deduce that

(5) |ϕ′(f(z))| ≤ Mf |1− z|α+1

α(1− |z|2) , for all z in D .

Partition the complex plane into a finite number of equal sectors withcommon vertex at the origin and of aperture at most πα/2 each. At leastone of them will contain infinitely many points wn. From now on we consideronly this infinite subsequence and use for it the same labelling wn in ordernot to burden the notation. We may rotate the sequence if necessary sincethe function ϕt given by ϕt(z) = ϕ(eitz) has the same properties as ϕ. Thus,after eliminating the terms of modulus at most one, we may assume that

| arg wn| < πα

4and |wn| > 1 for all n .

Consider the preimages of wn under the function f :

(6) zn = 1− w− 1

αn .

By our assumptions on wn, each zn belongs to the Stolz domain

S =

z ∈ D : |1− z| < 1, | arg(1− z)| < π

4

.

Therefore there exists a constant c such that

(7) |1− zn| ≤ c (1− |zn|) , for all n in N .


From (4), (5), (7), and (6) respectively, we get

δ|wn| ≤ |ϕ′(f(zn))| ≤ Mf |1− zn|α+1

α(1− |zn|2) ≤ cMf

α|1− zn|α =

cMf

α|wn| .

It follows that

δ|wn|2 ≤ cMf

α,

which contradicts the assumption that |wn| → ∞. This completes theproof. ¤

2. Superposition from the Bloch space into a Bergman space

Proposition 1 tells us that B is “very small” inside the Bergman spaceAp. The following questions are, thus, much more challenging than the oneanswered in the previous section.

For which entire functions ϕ does Sϕ map B into Ap? Which ones amongthem induce a bounded operator Sϕ?

We remind the reader that a (possibly nonlinear) operator acting betweentwo metric spaces is said to be bounded if it maps bounded sets into boundedsets.

2.1. The key lemma. A univalent function in D is an analytic functionwhich is one-to-one in the disk. By the Riemann mapping theorem, for anygiven simply connected domain Ω (other than the plane itself) there is sucha function f (called a Riemann map) that takes D onto Ω and the originto a prescribed point. Denoting by dist(w, ∂Ω) the Euclidean distance ofthe point w to the boundary of the domain Ω, the Riemann map f has thefollowing property (see Corollary 1.4 of [P2], for example):

(8)1

4(1− |z|2)|f ′(z)| ≤ dist(f(z), ∂Ω) ≤ (1− |z|2)|f ′(z)| , for all z in D .

This estimate plays an important role in the geometric theory of functions.In particular, (8) tells us that a function f univalent in D belongs to B ifand only if the image domain f(D) does not contain arbitrarily large disks.

The auxiliary construction of a conformal map onto a specific Bloch do-main with the maximal (logarithmic) growth along a certain broken linedisplayed below might be of some independent interest. Thus, we state itseparately as a lemma. Loosely speaking, such a domain can be imaginedas a “highway from the origin to infinity” of width 2δ. Somewhat simi-lar constructions of simply connected domains as the images of functions invarious function spaces can be found in the recent papers [BFV] and [DGV].

Lemma 2. For each positive number δ and for every sequence wn∞n=0 ofcomplex numbers such that w0 = 0, |w1| ≥ 5 δ, arg w1 < π/2, arg wn 0,and

(9) |wn| ≥ max

3|wn−1|,

n−1∑

k=1

|wk − wk−1|

for all n ≥ 2 ,

there exists a domain Ω with the following properties:


(i) Ω is simply connected;(ii) Ω contains the infinite polygonal line L = ∪∞n=1[wn−1, wn], where

[wn−1, wn] denotes the line segment from wn−1 to wn;(iii) any Riemann map f of D onto Ω belongs to B;(iv) dist(w, ∂Ω) = δ for each point w on the broken line L.

Proof. It is clear from (9) that |wn| ∞ as n → ∞. We construct thedomain Ω as follows. First connect the points wn by a polygonal line L asindicated in the statement. Let D(z, δ) = w : |z − w| < δ and define

Ω =⋃D(z, δ) : z ∈ L ,

i.e. let Ω be a δ-thickening of the polygonal line L. In other words, Ω isthe union of simply connected cigar-shaped domains

Cn =⋃D(z, δ) : z ∈ [wn−1, wn] .

By our choice of wn, it is easy to check inductively that |wn − wk| ≥ 5 δwhenever n > k. Since our construction implies that

Cn ⊂ w : |wn−1| − δ < |w| < |wn|+ δ ,

wee see immediately that

(a) for all m, n, Cm ∩ Cn 6= ∅ if and only if |m− n| ≤ 1;(b) for all n, Cn ∩ Cn+1 is either D(wn, δ) or the interior of the convex

hull of D(wn, δ) ∪ an for some point an outside of D(wn, δ).

Thus, each ΩN = ∪Nn=1Cn is also simply connected. Since

Ω = ∪∞N=1ΩN and ΩN ⊂ ΩN+1 for all N,

we conclude that Ω is also simply connected (like in [DGV], Section 4.2,p. 56). By construction, dist(w, ∂Ω) ≤ δ for all w in Ω, hence any Riemannmap onto Ω will belong to B. It is also clear that (iv) holds. ¤

2.2. Characterizations of superposition operators. In what followswe will need a few basic properties of the hyperbolic metric. Recall thatthe hyperbolic distance between two points z and w in the disk is defined as

ρ(z, w) = infγ

∫

γ

|dζ|1− |ζ|2 =

1

2log

1 +∣∣ z−w1−zw

∣∣1−

∣∣ z−w1−zw

∣∣ ,

where the infimum is taken over all rectifiable curves γ in D that join z withw.

The hyperbolic metric ρΩ on an arbitrary simply connected domain Ω(not the entire plane) is defined via the corresponding pullback to the disk:if f is a Riemann map of D onto Ω then

ρΩ(f(z), f(w)) = ρ(z, w) = infΓ

∫

f−1(Γ)

|dζ|1− |ζ|2 ,

where the infimum is taken over all rectifiable curves Γ in Ω from f(z) tof(w). The metric ρΩ does not depend on the choice of the Riemann mapf . For more details we refer the reader to § 1.2 and § 4.6 of [P2].


From the definition of hyperbolic metric we notice the following importantfeature of Riemann maps: if f(0) = 0 then

(10) ρΩ(0, f(z)) = ρ(0, z) ≥ 1

2log

1

1− |z| , z ∈ D .

Another fundamental property, which is easily deduced from (8), is that

(11) ρΩ(w1, w2) ≤ infΓ

∫

Γ

|dw|dist (w, ∂Ω)

,

where the infimum is taken over all rectifiable curves Γ in Ω from w1 to w2.

Denote by Dpα (α > −1) the weighted Dirichlet space of all analytic

functions f in D for which∫

D|f ′(z)|p(1− |z|2)αdA(z) < ∞

and by Bp the standard analytic Besov space Dpp−2, 1 < p < ∞; see [DGV]

for some of its properties and [BFV] for superposition operators betweensuch spaces. Since B can be viewed as a limit space of all Bp as p → ∞,the results about superpositions from B into Aq might be inspired by thosefor the superpositions from Bp into Aq by letting p → ∞. Also, it is wellknown that∫

D|f(z)|qdA(z) ³ |f(0)|q +

∫

D|f ′(z)|q(1− |z|2)qdA(z) ,

hence Aq = Dqq as sets. One of the main results of [BFV], Theorem 24

there, tells us that Sϕ(Bp) ⊂ Dqq if and only if either ϕ has order less than

p/(p−1) or it has order p/(p−1) and finite type. This seems to suggest thatthe characterization of all superpositions from B into Aq should be relatedto the entire functions of order one (letting p → ∞), but in which wayexactly? In the earlier papers [CG] (in the results for the Nevanlinna class)and [BFV] the cut usually occurred at the level of functions of infinite type.The appearance of the functions of given order and type zero in the resultbelow seems to be a novelty in this context. The intuitive reason behind itlies in the fact that B is not the smallest space that contains all Bp spaces.

Theorem 3. Let 0 < p < ∞ and let ϕ be entire. Then the followingstatements are equivalent:

(a) The superposition operator Sϕ maps B into Ap;(b) Sϕ is a bounded operator from B into Ap;(c) ϕ is an entire function either of order less than one, or of order one

and type zero.

Proof. We need only show that (a)⇒(c) ⇒(b).In order to prove that (c) implies (b), let us suppose that ϕ is an entire

function of order one and type zero. Writing M(r) = max|ϕ(z)| : |z| = r,this means that

lim supr→∞

log log M(r)

log r= 1


(see [Bo], for example) and the quantity

(12) E(r) =log M(r)

r

tends to zero as r →∞.Let f be an arbitrary function in B of norm at most K. By (2), we have

(13) |f(z)| ≤(

log1

1− |z| + 1

)K .

By our assumption on E(r), for some sufficiently large R0 it follows that

(14) E(|w|) < 1/(2pK) , whenever |w| > R0 .

Thus, whenever |f(z)| > R0 we get

|ϕ(f(z))| ≤ M(|f(z)|)= e|f(z)|·E(|f(z)|)

≤ exp

(K ·

(log

1

1− |z| + 1

)· E(|f(z)|)

)

≤ e1/(2p)

(1− |z|)1/(2p)

by the definition of M(r) and by (12), (13), and (14) respectively.If, on the contrary, |f(z)| ≤ R0 then |ϕ(f(z))| ≤ M(R0) by the Maximum

Modulus Principle. Combining the two possible cases, we obtain

‖ϕ f‖pp ≤ e1/2

∫

D

dA(z)

(1− |z|)1/2+ M(R0)

p = C ,

where C depends only on ϕ, p, and K but not upon f . This shows that Sϕ

is a bounded operator from B into Ap.The reasoning is similar, but simpler, when ϕ has order ρ < 1: use the

estimate M(r) ≤ exp (rρ+ε) for a small enough ε and large enough r.

For the proof of (a)⇒(c), let us assume that Sϕ : B → Ap but (c) is false.Thus, either ϕ has order greater than one, or it has order one and typeσ 6= 0. The former case is easier to handle, so we consider only the latter.

Since ϕ has order one and type different from zero, there exists an ε anda sequence rn∞n=1 such that rn →∞ as n →∞ and

(15)log M(rn)

rn

≥ ε > 0 , for all n .

In other words, there exists a sequence wn∞n=1 such that |wn| = rn and

(16) |ϕ(wn)| = M(rn) ≥ eε|wn| , for all n .

Let us fix a constant δ so that δ > 12/(εp). We can now choose an infinitesubsequence, denoted again wn, so that the sequence arg wn in [0, 2π]is convergent and all points wn lie in an angular sector of opening π/2.We may further assume that they are all located in the first quadrant andthe arguments arg wn decrease to 0, by applying symmetries or rotationsif necessary. There is no loss of generality in doing this because the entire


functions ψ, ϕt, defined by ψ(z) = ϕ(z), ϕt(z) = ϕ(eitz) respectively, havethe same order and type as ϕ.

Select inductively a further subsequence, labelled again wn, in such away that w0 = 0, |w1| ≥ 5 δ, and (9) holds. Next, use Lemma 2 to constructa domain Ω with the properties (i)–(iv) indicated there. Let f be a Riemannmap of D onto Ω that fixes the origin.

Now let zn be the points in D for which wn = f(zn). Since |wn| →∞ as n → ∞, it follows that |zn| → 1. By applying estimate (10) forhyperbolic metric, the triangle inequality, inequality (11) and property (iv)from Lemma 2, as well as the properties (9) of the points wn respectively,we obtain the following chain of inequalities:

1

2log

1

1− |zn| ≤ ρΩ(0, wn)

≤n∑

k=1

ρΩ(wk−1, wk)

≤n∑

k=1

∫

[wk−1,wk]

|dw|dist (w, ∂Ω)

=n∑

k=1

∫

[wk−1,wk]

|dw|δ

=1

δ

n∑

k=1

|wk − wk−1|

≤ 3

δ|wn| .

This shows that

(17) |wn| ≥ δ

6log

1

1− |zn| .

It follows from (16) and (17) that

(18) |ϕ(wn)| ≥ exp

(εδ

6· log

1

1− |zn|)

=1

(1− |zn|)(εδ)/6.

On the other hand, f ∈ B, hence by assumption (a): ϕ f ∈ Ap. By (1),we have

(19) |ϕ(wn)| = |(ϕ f)(zn)| ≤ ‖ϕ f‖p

(1− |zn|)2/p,

for all n. However, (18) and (19) contradict each other since ε δ/6 > 2/pby our initial choice of δ. The proof is now complete. ¤

2.3. Superpositions between some related spaces. We would like toemphasize that Theorem 3 used very few facts typical only of Bergmanspaces. One could consider instead the weighted Hardy spaces H∞

α (some-times also denoted A−α and called Korenblum spaces). A function f analytic


in D is said to belong to H∞α , 0 < α < ∞, if and only if

supz∈D

(1− |z|2)α|f(z)| < ∞ .

The point here is that, by (1), Ap ⊂ H∞2/p, while it can be easily seen that

α < 1/p implies H∞α ⊂ Ap. Thus, it follows that:

(a) Sϕ maps H∞α into B if and only if ϕ ≡ const;

(b) Sϕ maps B into H∞α if and only if the entire function ϕ is either of

order one and type zero, or of order less than one; also, wheneverthis happens, Sϕ acts boundedly from B into H∞

α .

A similar statement can easily be formulated for weighted Bergman spaces.

It should especially be worth pointing out that there is a variant of Theo-rem 3 with another well known space instead of B. Recall that BMOA is thespace of all Hardy H1 functions whose boundary values have bounded meanoscillation. For the precise definition and the basic properties of BMOAwe refer the reader to the survey papers [Ba] and [G].

Since BMOA ⊂ B, it makes sense to ask whether B can be replaced bythe smaller space BMOA in Theorem 3. This is indeed the case. Namely, awell known result of Pommerenke [P1] states that every univalent functionin the Bloch space also belongs to BMOA, so the function used in theproof above works for this space as well. All standard norms on BMOAare equivalent and the injection operator from BMOA into B is bounded,so an inequality for BMOA totally analogous to (13) also holds. Thus, wecan either give a proof that (c) implies (b) as above or, alternatively, factorthe superposition operator from BMOA into Ap through B and use theboundedness of Sϕ : B → Ap. The rest of the proof is obtained by followingthe same steps as in the proof of Theorem 3.

In fact, the above reasoning shows that the Bloch space in Theorem 3 canbe replaced by the class of all univalent functions in B (that is, in BMOA).

Acknowledgments

The two first-named authors were partially supported by grants FQM-210 from Junta de Andalucıa and BFM2001-1736 from MCyT, Spain. Thiswork was started in 2002 during their three-month stay at the UniversidadCarlos III de Madrid, with periodical visits to the Universidad Autonomade Madrid. The third author was partially supported by MCyT grantBFM2003-07294-C02-01, Spain.

We would like to thank Fernando Perez-Gonzalez and Daniel Girela forcalling our attention to the question of superpositions between the spaceBMOA and Bergman spaces and for some helpful comments.

References

[ACP] J.M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normalfunctions, J. reine angew. Math. 270 (1974), 12–37.

[AZ] J. Appel and P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge Uni-versity Press, Cambridge 1990.


[Ba] A. Baernstein II, Analytic functions of bounded mean oscillation, in Aspects ofcontemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham,Durham, 1979), pp. 3–36, Academic Press, London-New York 1980.

[Bo] R.P. Boas, Jr., Entire functions, Academic Press Inc., New York 1954.[BFV] S.M. Buckley, J.L. Fernandez, and D. Vukotic, Superposition operators on Dirich-

let type spaces, in Papers on Analysis: a volume dedicated to Olli Martio on theocassion of his 60th Birthday , pp. 41–61, Rep. Univ. Jyvaskyla Dep. Math. Stat.83, Univ. Jyvaskyla, Jyvaskyla 2001. Available in .ps and .pdf formats from theweb site: http://www.math.jyu.fi/research/report83.html.

[C] G.A. Camera, Nonlinear superposition on spaces of analytic functions, in Har-monic analysis and operator theory (Caracas, 1994), pp. 103–116, Contemp.Math. 189, Amer. Math. Soc., Providence, Rhode Island 1995.

[CG] G.A. Camera and J. Gimenez, The nonlinear superposition operators acting onBergman spaces, Compositio Math. 93 (1994), 23–35.

[DGV] J.J. Donaire, D. Girela, and D. Vukotic, On univalent functions in some Mobiusinvariant spaces, J. reine angew. Math. 553 (2002), 43–72.

[DS] P.L. Duren and A.P. Schuster, Bergman Spaces, Graduate Studies in Mathemat-ics, American Mathematical Society, Providence, Rhode Island, to appear.

[G] D. Girela, Analytic functions of bounded mean oscillation, in Complex functionspaces (Mekrijarvi, 1999), pp. 61–170, Univ. Joensuu Dept. Math. Rep. Ser. 4,Univ. Joensuu, Joensuu 2001.

[HKZ] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, GraduateTexts in Mathematics 199, Springer, New York 2000.

[P1] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen vonbeschrankter mittlerer Oszillation, Comment Math. Helv. 52 (1977), 591–602.

[P2] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag,Grundlehren der mathematischen Wissenschaften 299, Berlin 1992.

[Z] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc., Pure andApplied Mathematics 139, New York and Basel 1990.

Departamento de Analisis Matematico, Universidad de Malaga, Campusde Teatinos, 29071 Malaga, Spain

E-mail address: [email protected]

Departamento de Analisis Matematico, Universidad de Malaga, Campusde Teatinos, 29071 Malaga, Spain


Departamento de Matematicas, Universidad Autonoma de Madrid, 28049Madrid, Spain


superposition operators between the bloch space and bergman spaces

Documents