Superposition operators between the Bloch space and Bergman spaces

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Ark. Mat., 42 (2004), 205-216

Superposition operators between the Bloch space and Bergman spaces

V e n a n c i o A l v a r e z , M. A u x i l i a d o r a M ~ r q u e z a n d D r a g a n Vukotid

A b s t r a c t . Using a geometric method, we characterize all entire functions that transform the Bloch space into a Bergman space by superposition in terms of their order and type. We also prove that all superposition operators induced by such entire functions act boundedly. Similar results hold for superpositions from BMOA into Bergman spaces and from the Bloch space into certain weighted Hardy spaces.

I n t r o d u c t i o n

If X a n d Y a r e m e t r i c s p a c e s of a n a l y t i c f u n c t i o n s in t h e d i s k a n d ~ is a c o m p l e x - v a l u e d f u n c t i o n in t h e p l a n e such t h a t ~ o f E Y w h e n e v e r f E X , we s a y t h a t acts by superposition f r o m X i n t o Y. If X a n d Y c o n t a i n t h e l i n e a r f u n c t i o n s , t h e n m u s t b e a n e n t i r e f u n c t i o n . T h e superposition operator S~: X - - ~ Y w i t h sy'mbol is t h e n d e f i n e d b y S ~ ( f ) = p o f . N o t e t h a t if X a n d Y are also l i n e a r spaces, t h e o p e r a t o r S~ is l i n e a r if a n d o n l y if ~ is a l i n e a r f u n c t i o n t h a t fixes t h e origin. T h e c e n t r a l q u e s t i o n s are:

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

E v e n t h o u g h a n a l o g o u s c o n c e p t s also m a k e sense in t h e c o n t e x t of r e a l - v a l u e d f u n c t i o n s a n d t h e i r t h e o r y h a s a long h i s t o r y (see [AZ]), t h e s t u d y of such n a t u r a l q u e s t i o n s oi1 s p a c e s o f a n a l y t i c f u n c t i o n s h a s o n l y b e g u n f a i r l y recently. T h e o p e r - a t o r s S~ t h a t m a p one B e r g m a n s p a c e i n t o a n o t h e r , or i n t o t h e a r e a N e v a n l i n n a class, were c h a r a c t e r i z e d in t e r m s of t h e i r s y m b o l s in [CG]; cf. [C] for H a r d y spaces.

R e s u l t s of s i m i l a r n a t u r e were o b t a i n e d in [BFV] for s u p e r p o s i t i o n s b e t w e e n v a r i o u s s p a c e s of D i r i c h l e t t y p e .

I n t h i s p a p e r we give a c o m p l e t e d e s c r i p t i o n of t h e s u p e r p o s i t i o n s b e t w e e n a B e r g m a n s p a c e A p a n d t h e B l o c h s p a c e B (in b o t h d i r e c t i o n s ) in t e r m s of t h e o r d e r a n d t y p e of ~. T h a t is, we m e a s u r e in a p r e c i s e w a y "how m u c h slower" t h e B l o c h

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206 Venancio Alvarez, M. Auxiliadora Mgrquez and Dragan Vukotid

functions grow than those in A p. We recall the basic definitions and facts which motivate our results.

Let d A denote the Lebesgue area measure in the unit disk D, normalized so that A ( D ) = I . If 0 < p < o c , the Bergrnan space A p is the set of all analytic functions f in the unit disk D with finite LP(D, dA) norm:

,,fH~ = / D If(z), p d A ( z ) = _ 1 / i [ 2 ~ I f ( r c i ~

7r J o ,]o

Note that ]]fl]p is a true norm if and only if l_<p<oe. When 0 < p < l , A p is a complete space with respect to the translation-invariant metric defined by dp(f, g ) = II/-glIG. It is clear that A q C A p if 0 < p < q < o c . The monograph [HKZ] contains plenty of information about Bergman spaces and so does [DS]; see also [Z].

The maximum growth of functions in Bergman spaces will be essential. If f c A p, by the subharmonicity of Ill p and the area version of the sub-mean value property applied to the disk of radius 1 ]z] centered at z, we readily obtain the standard estimate

(1)

)f(z)) <

[Ifl[~) for all z in D.

(1-I~l):/p

An analytic function f in D is said to belong to the Bloch space 13 if IlfllB = If(0) l + s u p ( 1 - I z l 2 ) l f ' ( z ) l < oe.

z G D

Any such function satisfies the growth condition

(2)

I f ( z ) l ~

(

l ~

1 1

Ilfll~ for all z in D.

This follows by integrating the inequality l f ' ( C ) l < ( l l f l l u - l f ( O ) l ) ( 1 - 1 ~ l 2) 1 along the line segment from 0 to z. For further basic facts about the Bloch space the reader may consult [ACP] or Chapter 5 of [Z].

The inclusion B c A p ( 0 < p < o c ) follows easily from (2). Moreover, since the exponents in both (1) and (2) are sharp, it becomes intuitively clear that the Bloch space should be contained in any Bergman space "exponentially", thinking in terms of superpositions. However, our main result, Theorem 3, will show that functions such as p(z) = e x p ez, c r fail to map B into A p. In fact, the superposition operator S~ maps B into A p if and only if the entire function p is either of order less than one, or of order one and type zero. Thus, the characterization of the superpositions does not depend on p. Our proof resembles the ideas employed in the proofs of

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S u p e r p o s i t i o n o p e r a t o r s between the Bloch space and B e r g m a n spaces 207

some of the main results of [BFV] ( T h e o r e m 12 and T h e o r e m 19). Although here we no longer need the inequalities of Trudinger type, our argmnent still requires several delicate technical details. T h e key point is a geometric construction of a special univalent function in the Bloch space ( L e m m a 2) which m a y have some independent interest.

1. S u p e r p o s i t i o n f r o m a B e r g m a n s p a c e i n t o t h e B l o c h s p a c e Given t h a t any B e r g m a n space Ap is larger t h a n the Bloch space B, it should be intuitively obvious t h a t a superposition from the former to the latter is possible via constant functions only. However, some work is still required. This is due to the fact t h a t no uniform bounds are available (because we do not have any a priori information on the boundedness of the superposition operator).

P r o p o s i t i o n 1. Let 0 < p < c o and let ~ be entire. Then the superposition operator S~ maps A p into B if and only if ~ is a constant function.

Proof. Suppose t h a t S~ m a p s A p into B. To show t h a t ~ is constant, it suffices to prove t h a t y)~ is constant. Indeed, if ~ - a then ~ is a linear function: ~(z)=az+b.

Taking f c A P \ B , we have t h a t S ~ ( f ) = a f + b c B , which implies a = 0 and therefore is constant.

N o w there are t w o possibilities for the function ~ : either

(3) 0, as co,

~W o r

for some fixed 5 > 0 and some sequence {w,~}n~_l such t h a t Iw~l--+co.

Note t h a t (3), in particular, yields a s t a n d a r d Cauchy estimate

I~'(w)l <_Alwl

for some A > 0 and all sufficiently large w, which in t u r n implies t h a t p1(w) is a linear function. By letting w--+co and applying (3), it follows t h a t ~ ' is constant and we are done.

It is, thus, only left to show t h a t (4) is impossible. To this end, let c~E(0, 2/p) and consider the function

1

f ( z ) = ( l _ z ) ~ , z c D .

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208 V e a a n c i o Alvarez, M. A u x i l i a d o r a M g r q u e z a n d D r a g a n Vukotid

By integrating in polar coordinates centered at z = 1, it is easily checked t h a t f E A p.

Hence ~ o f E B , and therefore

M f : sup ( 1 - I z l 2 ) l f ' ( z ) l I~'(f(z))l < oo.

z E D

From here we deduce that

(5) I~'(f(z))l <_ M f l l - z [ ~ + l for all z in D.

Partition the complex plane into a finite number of equal sectors with common vertex at the origin and of aperture at most 89 each. At least one of them will contain infinitely many points w~> From now on we consider only this infinite

W oo

subsequence and use for it the same labelling { ~}~=1 in order not to burden the notation. We may rotate the sequence if necessary since the function ~t given by 99t(z)=99(eitz) has the same properties as ~. Thus, after eliminating the terms of modulus at most one, we may assume that

(argw, d < -~r~ and [w~ 1 > 1 for all n.

Consider the preimages of w~ under the function f :

(6) z~ = 1 - w X 1/~.

By our assumptions on {w~}n~_l, each z,~ belongs to the Stolz domain S = { z E D : l l - z l < 1 and l a r g ( 1 - z ) l < 17r}.

Therefore there exists a constant c such that

(7) /1-z,~ / < c ( 1 - / z , ~ l ) for aI1 n in N.

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

alw~l _< [~'(/(z~))l <_ M z l l - z ~ l ~ + l < c M f { l _ z , d ~ _ cM], c~(1-1znl ~) - c~ c~lw,~ I"

It follows that

~lw~12 < c Ms ,

c~

which contradicts the assumption that Iw,,}~oo. This completes the proof. []

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Superposition operators between t h e Bloch space and B e r g m a n spaces 209

2. S u p e r p o s i t i o n f r o m t h e B l o c h s p a c e i n t o a B e r g m a n s p a c e Proposition 1 tells us that B is "very small" inside the Bergman space A p. The following questions are, thus, much more challenging t h a n the one answered in the previous section.

For which entire functions ~ does S~ map 13 into AP? Which ones among them induce a bounded operator S~ ?

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

2.1. T h e k e y l e m m a

A univalent function in D is an analytic function which is one-to-one in the disk. By the Riemann mapping theorem, for any given simply connected domain ft (other than the plane itself) there is such a function f (called a Riemann map) that takes D onto f~ and the origin to a prescribed point. Denoting by dist(w, Oft) the Euclidean distance of the point w to the boundary of the domain ft, the Riemann map f has the following property (see Corollary 1.4 of [P2], for example):

(8) al (1-[zl 2) If'(z) l _< dist(f(z), Oft) _< (1-Iz[ 2) If'(z) l for all z in D.

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

The auxiliary construction of a conformal map onto a specific Bloch domain with the maximal (logarithmic) growth along a certain polygonal line displayed below might be of some independent interest. Thus, we state it separately as a lemma. Loosely speaking, such a domain can be imagined as a "highway from the origin to infinity" of width 25. Somewhat similar constructions of simply connected domains as the images of functions in various function spaces can be found in the recent papers [BFV] and [DGV].

W oo

L e m m a 2. For each positive number 6 and for every sequence { ~}~ o of complex numbers such that w0=0, 1W11~55, a r g w l <57r,1 argwn"-s0, and

i n--1 I

(9) Iwn[ _> max 3/wn_ll, E Iwk--wk_l I f o r a l l n ~ _ 2 ,

k 1

there exists a domain ft with the following properties:

(i) f~ i~ simply connected;

(6)

L O0

(ii) ~ contains the infinite polygonal line = U n = l [wn-1, Wn], where [Wn-1, wn]

denotes the line segment from w,~-i to w~;

(iii) any Riemann map f of D onto ~ belongs to 13;

(iv) dist(w, 0 ~ ) = 5 .for each point w on L.

Proof. It is clear from (9) that lw~lffoc, as n-+oo. We construct the domain as follows. First connect the points w~ by a polygonal line L as indicated in the statement. Let D(z, 5)={w: I z - w I <5} and define

i.e. let 9 be a 6-thickening of L. In other words, f~ is the union of simply connected cigar-shaped domains

C,,~

= U { D ( z , a ) : z C [~n ~,*~n]}-

By our choice of w~, it is easy to check inductively that

Iw,~-wkl>5~

whenever' n>k. Since our construction implies that

wee see immediately that

(a) for all m and n, CmNC,r if and only if Im nl_<l;

(b) for all n, CnNCn+~ is either D(wn, a) or the interior of the convex hull of D(wn, (~)U{an} for some point an outside of D(w~, ~).

Thus, each f~x = U ~ = l c n is also simply connected. Since N

O O

f t = U f i n and f~NCf~N+I for a l l N ,

N 1

we conclude that ft is also simply connected (like in [DGV], Section 4.2, p. 56). By construction, dist(w,O~)<_5 for all w in f~, hence any Riemann map onto ft will belong to B. It is also clear t h a t (iv) holds. []

2.2. C h a r a c t e r i z a t i o n s o f s u p e r p o s i t i o n o p e r a t o r s

In what follows we will need a few basic properties of the hyperbolic metric.

Recall that the hyperbolic distance between two points z and w in the disk is defined

a s

z - w

inf [ I<1 1

1

1 ~'~c,

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Superposition operators between the Bloch space and B e r g m a n spaces 211

where the infimum is taken over all rectifiable curves ~/in D t h a t join z with w.

T h e hyperbolic metric L)a on an a r b i t r a r y simply connected domain ft (not the entire plane) is defined via the corresponding pullback to the disk: if f is a R i e m a n n m a p of D onto ft t h e n

Qa(.f(z), f(w)) = t~(z, w) = inf [

Idr

r J l ^1r l - I C ? '

where the infimum is taken over all rectifiable curves F in ft from f(z) to f(w). T h e metric t)a does not depend on the choice of the R i e m a n n m a p f . For more details we refer the reader to Sections 1.2 and 4.6 of [P2].

From the definition of the hyperbolic metric we notice the following i m p o r t a n t feature of R i e m a n n maps: if f ( 0 ) = 0 then

1 1

(10) ea(o,.f(z)) = ~ ( 0 , z) > ~ log l _ l z l , z c D .

Another fundamental property, which is easily deduced from (8), is t h a t (11) &f~(Wl, w2) < i ~ f f r Idol

- dist (w, Oft)'

where the infimum is taken over all rectifiable curves F in ~2 from Wl to w2.

Denote by DP~ (c~ > - 1 ) the weighted Dirichlet space of all analytic functions f in D for which

D If'(z)lP(1-1zl2)a dA(z) <

P l < p < o c ; see [DGV] for some and by B p the s t a n d a r d analytic Besov space Dp_2,

of its properties and [BFV] for superposition operators between such spaces. Since /3 can be viewed as a limit space of all B p as p - ~ o e , the results a b o u t superpositions from • into A q might be inspired by those for the superpositions from B p into A q by letting p--+oc. Also, it is well known t h a t

D If(z)lq dA(z)• + fD If'(z)lq(1-1zl2)q dA(z),

hence A q - D q as sets. One of the main results of [BFV], T h e o r e m 24 there, tells us t h a t S~(BP)cD~ if and only if either p has order less t h a n p / ( p - 1 ) or it has order p / ( p - 1 ) and finite type. This seems to suggest t h a t the characterization of all superpositions from B into A q should be related to the entire functions of order one (letting p--+ec), but in which way exactly? 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. T h e appearance of the functions of given order and type zero in the result below seems to be a novelty in this context. T h e intuitive reason behind it lies in the fact t h a t B is not the smallest space t h a t contains all B p spaces.

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T h e o r e m 3. Let 0 < p < o c and let ~ be entire. Then the following statements are equivalent:

(a) the superposition operator S~ maps 13 into AP;

(b) S~, is a bounded operator from B into AP;

(c) ~ 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 p is an entire function of order one and type zero. Writing M ( r ) = m a x { l ~ ( z ) [ : [zl=r}, this means that

lim sup log log M ( r ) _ 1 r ~ log r (see [Bo], for example) and the quantity

(12) E ( r ) -- log M ( r )

r tends to zero as r--+oc.

(la)

Let f be an arbitrary flmction in B of norm at most K . By (2), we have 1

(14) E(Iwl) < n 2 p ~ ' 1 whenever Iwl > Ro.

Thus, whenever

If(~)l>2~0

we get

199(f(z)) I < M ( I f ( z ) l ) --

elf(~)lE(If(z)l)

el/2p

(]-]zl)l/2P

by the definition of M ( r ) and by (12), (13), and (14), respectively.

If, on the contrary, If(z)l_<•0 then I~(f(z))l<_M(Ro) by the maximum modulus principle. Combining the two possible cases, we obtain

fD dA(z) +M(Ro)P=C,

II~~ (1 Iz[)l/2

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

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Superposition operators between the Bloch space and Bergman spaces 213 where C depends only on ~, p, and K , but not upon f . This shows t h a t S~ is a b o u n d e d o p e r a t o r from B into A p.

T h e reasoning is similar, but simpler, when ~ has order 0 < 1: use the estimate M(r)_<exp r ~ for a small enough e and large enough r.

For the proof of (a) ~ (c), let us assume t h a t S~: B--+A p but (c) is false. Thus, either ~ has order greater t h a n one, or it has order one and t y p e ere0. T h e former case is easier to handle, so we consider only the latter.

Since p has order one and t y p e different from zero, there exists an e and a sequence {r~}~~ 1 such t h a t r ~ - + o c , as n - + o c , and

(15) logM(r,~) > e > O for all n.

W oo

In other words, there exists a sequence { ~},~ 1 such t h a t ]wn[=r~ and

(16) Is(w )l >e lw' l for all

Let us fix a constant 5 so t h a t 5>12/r We can now choose an infinite subsequence, denoted again {w,~}~~176 so t h a t the sequence {argwn}n~176 in [0,27c] is convergent and all points wn lie in an angular sector of opening ~71-.1 We m a y further assume t h a t they are all located in the first quadrant and the arguments arg w,~

decrease to 0, by applying symmetries or rotations if necessary. There is no loss of generality in doing this because the entire functions ~b and ~t, defined by ~p(z)=~(2) and ~t(z)=~(eitz), respectively, have the same order and t y p e as ~.

Select inductively a further subsequence, labelled again {w,~}n~176 in such a way t h a t w0=0, [w~1_>55, and (9) holds. Next, use L e m m a 2 to construct a domain f~

with t h e properties (i)-(iv) indicated there. Let f be a R i e m a n n m a p of D onto f~

t h a t fixes the origin.

Now let zn be the points in D for which w~=f(z~). Since [w,~l-+oc , as n - + o c , it follows t h a t Iz~l--+l. By applying e s t i m a t e (10) for the hyperbolic metric, the triangle inequality, inequality (11) and p r o p e r t y (iv) from L e m m a 2, as well as the properties (9) of the points wn, respectively, we obtain the following chain of inequalities:

~ l o g l__lZn[ ~kgf2(0, W,z)~

cOa(Wk 1,Wk)<

k 1 k 1,~] d i s t ( w , 0 f t )

:5s

[dw[ __ 1

5 i~Uk__Wk_l] ~

glwn[- k=l k-l,wkJ (~ 5 k=l

This shows t h a t

5 1

(17) I ~ 1 - ) ~ log 1-Iz~-~"

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214 Venancio Alvarez, M. Auxiliadora MArquez and Dragan Vukotid

It follows from (16) and (17) that

( ~ 1 ) 1

(18)

I~(wn)/>exp log

1-1 nl -

On the other hand,

fCB,

hence by assumption (a):

9~ofcAP.

By (1), we have I1 o/11

(19) I~(w~)l =

I(~of)(z~)l <

for all n. However, (18) and (19) contradict each other since

~6>2/p

by out' initial choice of & The proof is now complete. []

2.3. S u p e r p o s i t i o n s b e t w e e n s o m e r e l a t e d s p a c e s

We would like to emphasize t h a t T h e o r e m 3 used very few facts typical only of Bergman spaces. One could consider instead the weighted Hardy spaces H ~ (sometimes also denoted A - ~ and called

Korenblum spaces).

A function f analytic in D is said to belong to H ~ , 0 < a < o c , if and only if

sup (1 - [ z l 2 ) a I.f(z) I < o o .

z~lD

The point here is that, by (1),

APcH~p,

while it can be easily seen that c ~ < l / p implies H~ ~ c A p. Thus, it follows that

(a) S ; maps H ~ into/3 if and only if ~o=const;

(b) S~ m a p s / 3 into H ~ if and only if the entire function ~ is either of order one and type zero, or of order less t h a n one; also, whenever this happens, S~ acts boundedly f r o m / 3 into H ~ .

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

It should especially be worth pointing out that there is a variant of Theorem 3 with another well-known space instead of B. Recall that BMOA is the space of all Hardy H 1 functions whose boundary values have bounded mean oscillation. For the precise definition and the basic properties of BMOA we refer the reader to the survey papers [Ba] and [G].

Since B M O A c B , it makes sense to ask whether B can be replaced by the smaller space BMOA in T h e o r e m 3. This is indeed the case. Namely, a weE-known result of Pommerenke [P1] states that every univalent function in the Bloch space also belongs to BMOA, so the function used in the proof above works for this space as well. All standard norms on BMOA are equivalent and the injection operator fl'om BMOA into /3 is bounded, so an inequality for BMOA totally analogous to

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Superposition operators between the Bloch space and Bergman spaces 215

(13)

also holds. Thus, we can either give a proof t h a t (c) implies (b) as above or, alternatively, factor the superposition o p e r a t o r from B M O A into A v t h r o u g h 13 and use the boundedness of S~: 13--+A p. T h e rest of the proof is obtained by following the same steps as in the proof of T h e o r e m 3.

In fact, the above reasoning shows t h a t the Bloch space in T h e o r e m 3 can be replaced by the class of all univalent functions in 13 (that is, in BMOA).

Acknowledgments. T h e first two authors were partially supported by grants FQM-210 from J u n t a de Andalucfa and BFM2001-1736 from M C y T , Spain. This work was s t a r t e d in 2002 during their t h r e e - m o n t h stay at the Universidad Carlos I I I de Madrid, with periodical visits to the Universidad A u t d n o m a de Madrid.

T h e third author was partially s u p p o r t e d by M C y T grant BFM2003-07294-C02-01, Spain.

We would like to t h a n k Fernando P~rez-Gonzdlez and Daniel Girela for call- ing our attention to the question of superpositions between the space B M O A and B e r g m a n spaces and for some helpful comments.

R e f e r e n c e s

[ACP] ANDERSON, J. M., CLUNIE, J. and POMMERENKE, C., On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37.

[AZ] APPELL~ J. and ZABREJKO, P. P., Nonlinear Snperposition Operators, Cambridge Tracts in Mathematics 95, Cambridge Univ. Press, Cambridge, 1990.

[Ba] BAERNSTEIN, A. II, Analytic functions of bounded mean oscillation, in Aspects of Contemporary Cornplea Analysis (Durham, 1979), pp. 3 36, Academic Press, London New York, 1980.

[Bo] BOAS, R. P., .JR., Entire Functions, Academic Press, New York, 1954.

[BFV] BUCKLEY, S. M., FERN/tNDEZ~ J. L. and VUKOTIC, D., Superposition operators on Dirichlet type spaces, in Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of his 60th Birthday, pp. 41 61, Rep. Univ. Jyv/iskylg Dept. Math. Stat. 83, University of Jyvgskyl/i, JyvSskylg, 2001.

[C] CJ~MERA, G. A., Nonlinear superposition on spaces of analytic functions, in Har- monic Analysis and Operator Theory (Caracas, 1994), Contemp. Math. 189, pp. 103 116, Amer. Math. Soc., Providence, RI, 1995.

[CG] CJ~MERA, G. A. and GIMI~NEZ, J., The nonlinear superposition operators acting on Bergman spaces, Cornpositio Math. 93 (1994), 23 35.

[DGV] DONAIRE, J. J., GIRELA, D. and VUKOTIC, D., On univalent functions in some MSbius invariant spaces, J. Reine Angew. Math. 553 (2002), 43 72.

[DS] DUREN, D. L. and SCHUSTER, A. P., Bergman Spaces, Math. Surveys and Mono- graphs 100, Amer. Math. Soc., Providence, RI, 2004.

[G] GIRELA, D., Analytic functions of bounded mean oscillation, in Complex Function

@aces (Mekrijiirvi, 1999), Univ. Joensuu Dept. Math. Rep. Ser. 4, pp. 61 170, Univ. Joensuu, Joensuu, 2001.

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216 Venancio Alvarez, M. Auxiliadora Mgrquez and Dragan Vukotid:

Superposition operators between the Bloch space and Bergman spaces

[HKZ]

[el]

[e2]

[Z]

HEDENMALM, H., KORENBLUM, B. and ZHU, K. H., Theory of BeTyman Spaces, Grad. Texts in Math. 199, Springer-Verlag, New York, 2000.

POMMERENKE~ C., Schlichte Funktionen und analytisehe Funktionen von be- schrgnkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), 591-602.

POMMERENKE, C., Boundary Behaviour of Conformal Maps, Grundlehren Math.

Wiss. 299, Springer-Verlag, Berlin, 1992.

ZHU, K. H., Operator Theory in Function Spaces, Monographs Textbooks Pure Appl. Math. 139, Marcel Dekker, New Yor~Basel, 1990.

Received April 22, 2003

in revised form November 13, 2003

Venaneio Alvarez

Departamento de Anglisis Matemgtico Universidad de Mglaga

Campus de Teatinos ES-29071 Mglaga Spain

emaih [email protected] M. Auxiliadora MSrquez

Departamento de Anglisis Matemgtico Universidad de Mglaga

Campus de Teatinos ES-29071 Mglaga Spain

email: marqnez~anamat.cie.uma.es Dragan Vukotid

Departamento de Matemgticas Universidad Autdnoma de Madrid ES-28049 Madrid

Spain

email: [email protected]