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DATE DUE DATE DUE DATE DUE MSU in An Affirmative Adlai/Equal Opporanlty Intuition Walla-9.1 COMPACT COMPOSITION OPERATORS ON SOME MOBIUS INVARIANT BANACH SPACES By Maria Tjani A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSHOPHY Department of Mathematics 1996 ABSTRACT COMPACT COMPOSITION OPERATORS ON SOME MOBIUS INVARIANT BANACH SPACES By Maria Tj ani Let 8,, (1 < p < 00) be a Besov space and B the Bloch space. We give Carleson type measure characterizations for compact composition operators C4, : Bp —> Bq (1 < p S q < 00), C.» : Bp —+ BMOA, and Cd, : B —> VMOA. We show that if Cd, is bounded on some Besov space then Cd, is compact on larger Besov spaces if and only if it is compact on the Bloch space. Also, if d) is a boundedly valent holomorphic self-map of the unit disc U such that q5(U) lies inside a polygon inscribed in the unit circle, then C.» is compact on BMOA, and on VMOA if and only if it is compact on the Bloch space. ACKNOWLEDGMENTS I am deeply grateful to my advisor, Professor Joel Shapiro, for his guidance, teaching and encouragement. His advice and assistance in the preparation of this thesis were invaluable. I would like to thank Professor Wade Ramey for the courses he taught. I also would like to thank Professors Sheldon Axler, Michael Frazier, Wade Ramey, and William Sledd for all the seminar talks they gave through the years, and for serving in my committee. iii TABLE OF CONTENTS 1 Besov spaces, BMOA, and VMOA 8 2 Carleson measures and compact composition operators on Besov spaces and BMOA 20 3 Besov space, BMOA, and VMOA compactness of C¢ versus Bloch compactness of C.» 40 4 Final remarks and questions 64 BIBLIOGRAPHY 66 Introduction Let 45 be a holomorphic self-map of the open unit disc U, H 2 the Hilbert space of functions holomorphic on U with square summable power series coefficients. Associate to <15 the composition operator C4, defined by C¢f=f0¢, for f holomorphic on U. This is the first setting in which composition operators were studied. By Littlewood’s Subordination Principle every composition operator takes H 2 into itself. A natural question to ask is which composition operators on H 2 are compact. Shapiro in [31], using the Nevanlinna counting function, characterized the compact composition operators on H 2 as follows: 0,, is a compact operator on H 2 if and only if N¢>(w) [ml-+1 —log |w| — A natural follow up question is about the boundedness and compactness of compo— sition operators on other function spaces. we know the answer to this question in a variety of spaces. MacCluer in [20], Madigan in [21], Roan in [25], and Shapiro in [30] have charac— terized the boundedness and compactness of C¢ in “small” spaces. In “large” spaces, MacCluer and Shapiro show in [19] that 0,, is compact on Bergman spaces if and only if d) does not have an angular derivative at any point of 2 EU. The angular derivative criterion is not sufficient, in general, in smaller spaces unless we put extra conditions on the symbol. For example they showed that it is sufficient on Hardy spaces, if the symbol is boundedly valent. The Bloch space 3 is the space of holomorphic functions f on U such that [I f [I 3 = supzeU |f’(z)|(1—|z|2) < 00. It becomes a Banach space with norm |f(0)| + Ilfllg. A linear subspace X of B with a seminorm ||.||x is Mo'bius invariant if for all Mobius transformations ab and all f 6 X, f o <13 6 X and ||f o ¢||X = ||f||x, and there exists a positive constant c such that HfHB _<_ llfllX- It is easy to see that B is a Mobius invariant space. A Mo'bius invariant Banach space X is a Mobius invariant subspace of the Bloch space with a seminorm ||.||X, whose norm is f —> ||f||x or f -—> |f(0)|+ llfllX- Rubel and Timoney showed in [26] that B is the largest Mobius invariant Banach space that possesses a decent linear functional. Other Mobius invariant Banach spaces include the Besov spaces, the space of holomorphic functions with bounded mean oscillation BM 0A, and the space of holomorphic functions with vanishing mean oscillation VMOA. We will define and discuss properties of these spaces in chapter 1. Madigan and Matheson show in [22] that Cd, is compact on the Bloch space if and only if |¢’(Z)l(1-|Z|2) 1m 2 0. I¢>1 1 - |<15(Z)|2 They also show that if Cd, is compact on B then it can not have an angular derivative at any point of 8U. In this thesis we study the compact composition operators on B1,, (1 < p < 00), on BM 0A, and on VMOA. For the rest of this introduction let X denote one of these spaces, unless otherwise stated. One way to approach this problem is to relate it to properties of 45. That is to see how fast or how often ¢(U) touches 8U. In every function space that compact composition operators have been studied, the first 3 class of examples were provided by symbols 4) such that ¢(U) is a relatively compact subset of U. For the spaces that we study this is not an exception. Moreover, if C, is compact on X then 0,, can not have an angular derivative at any point 8U since, if Ca is compact on X then C, is compact on the Bloch space (see Proposition 3.2). In Chapter 2, using counting functions, we give a Carleson measure character- ization of compact operators 0,, : B? —-> 8,, (1 < p S q < 00) and C.» : Bp —> BM 0A (1 < p S 2). MacCluer and Shapiro give in [19] Carleson measure char- acterization of compact composition operators on the Dirichlet space D, which is a Besov space (p = 2). Let a) (A E U) be the basic conformal automorphism defined by a,\(z) = $33. We prove the following theorems. Theorem 2.7 Let 1 < p S q < 00. Then, the following are equivalent: 1. Cd, : 8,, —-> 3., is a compact operator. 2. Nq(w,¢)dA(w) is a vanishing q-Carleson measure. 3. ||C¢aA||Bq —-> 0, as IA] —> 1. Theorem 2.8 The following are equivalent: 1. Cd, : D —> BMOA is a compact operator. 2. ”Czar“... —> 0, as [A] —> 1. The main steps in the proof of the two theorems above are the following. First we characterize the vanishing p-Carleson measures (see Proposition 2.5). Then we give a general characterization of compact composition operators on certain Banach spaces of analytic functions in terms of bounded sequences that converge to 0 uniformly on compact subsets of U (see Lemma 2.10 and Lemma 2.11). Lastly a technique given by Arazy, Fisher, and Peetre in [2] and by Luecking in [17] and [18]. 4 In Chapter 3 we first give another characterization of compact composition ope- rators on the Bloch space. We prove the following theorem. Theorem 3.1 Let a5 be a holomorphic self-map of U. Let X = Bp (1 < p < oo), BMOA, or 8. Then C¢ : X —> B is a compact operator if and only if lim ||C¢aA||3 = 0. [Al-+1 Next we show that if Q, : X —-> X is compact then so is Cg, : B —-> [3. Moreover we give conditions on the symbol under which the converse is valid as well. If X is a Besov space then the converse holds if we suppose that 0,, is bounded on a smaller Besov space. We prove the following theorem. Theorem 3.7 Let 1 < r < q, 1 < p S q, suppose that Q, : B, —-) B, is a bounded operator. Then the following are equivalent: 1. Ca, : B —> B is a compact operator. §e Q S : 3,, -—> B, is a compact operator. 3. Ca, : D —> BMOA is a compact operator. 4. Cd, : Bp ——> BMOA is a compact operator. Next we describe the proof of the theorem above. At this point we have all the tools we need (see Lemma 2.11, Theorem 2.7, Theorem 2.8, and Theorem 3.1) to prove that 2, 3, 4—) I. The hypothesis that C.» : B, —+ B, is a bounded operator is not needed for these implications. To prove the rest of the implications we first give a partial case. We show that if cf) is a univalent function and 0.), is a compact Operator on the Bloch space then 0,, : B,, —> 8., (q > 2, 1 < p S q) is compact as well (see Theorem 3.5). Then we provide a general proof of this result for any C), such that 5 Cd, : B, —-> B, (1 < r < q) is bounded (see Proposition 3.6). The proof of Theorem 3.7 will now follow easily. We next note that a theorem of Arazy, Fisher and Peetre (see Theorem F) can be used to characterize the boundedness of composition operators with domain the Bloch space and range inside a variety of spaces. For example in any Besov space, in BM 0A, and in H 2 (see Proposition 3.8). Moreover we note that the integral condition of Shapiro and Taylor characterizing the Hilbert-Schmidt composition operators on the Dirichlet space (see [29]) also characterizes the bounded operators C¢ : B —> D. We show that such operators are compact on BM CA. More general examples of compact Q, on BM 0A are provided by integral conditions of this type. We prove the following proposition. Proposition 3.9 Let (b be a holomorphic self-map of U. Then, I. If1 B, is a compact operator (hence Cd, : 8—) BMOA is a compact operator as well). 2. If [95,03 [2(21‘aq( )l2 ) Iqi—n/u (1)—|d> )l") (W ): 0 then 0,, : B —> BMOA is a compact operator. Next we give a characterization of compact operators C¢ : X —> VMOA, where X is a Mobius invariant subspace of the Bloch space. We prove the following theorem. Theorem 3.11 Let qb be a holomorphic self-map of U, and X a Mb'bius invariant 6 Banach space. Then Cd, : X —> VMOA is a compact operator if and only if lim sup /( If’(¢ (2 ))||d>’( (z)|(1—|aq(z)|")dA(z)=0- | VMOA. The proof is similar to the one given by Arazy, Fisher, and Peetre in [2, Theorem 3] for characterizing Bloch Carleson measures. The main tools are Kintchine’s inequality for gap series and Theorem 3.11. We prove the following theorem. Theorem 3.13 Let gb be a holomorphic self-map ofU. Then the following are equiva- lent: 1. C4, : B —> VMOA is a compact operator. |¢’(z) 2(1-( [09(75 U2) Iq1311/(('|1_¢WH dA(z)— 0. Next we show that if <15 is a boundedly valent holomorphic self-map of U such that (15(U) lies inside a polygon inscribed in the unit circle then the compactness of C, on Besov spaces, BM 0A, and VMOA is equivalent to the compactness of C, on the Bloch space. More precisely we prove the following theorem. Theorem 3.15 Let (b be a boundedly valent holomorphic self-map of U such that ¢(U) lies inside a polygon inscribed in the unit circle. Then the following are equivalent: 1. Cd, : B —) VMOA is a compact operator. 2. C, : B —> BMOA is a compact operator. 3. Cd, : BMOA —) BMOA is a compact operator. 4. Cd, : B —-> B is a compact operator. 7 .5. Ca, : 30 —> 80 is a compact operator. 6. C¢ : VMOA —-> VMOA is a compact operator. The main tools of the proof are the following. First there is Madigan and Math— eson’s characterization of Bloch and little Bloch compactness. Next that boundedly valent holomorphic functions on the little Bloch space must belong to VMOA. Fi— nally, we use Proposition 3.12 and Theorem 3.13. In chapter 4 we give some final remarks and questions. CHAPTER 1 Besov spaces, BMOA, and VMOA Let U be the open unit disc in the complex plane and BU the unit circle. The one-to- one holomorphic functions that map U onto itself, called the Mb'bius transformations, and denoted by C, have the form Aap where A E 0U and a, is the basic conformal automorphism defined by _ 19“" —1—pz 012(2) for p 6 U. It is easy to check that the inverse of a, under composition is a, 0,, o ap(z) = z for z E U. Also, I ~ _ ITIPI2 iap("')i _ |1_-p-z|2 and 2 _ (1 -|P|2)(1-|Z|2) I 1— Iap(z)I — I, _W = (1— |2I2)|a,(z)l (1.1) for p,z E U. 9 The Bloch space 3 of U is the space of holomorphic functions f on U such that HfHB = sup(1— lzl2)lf’(z)I < oo. zEU It is easy to see that I f (0)I+ I I f I I 3 defines a norm that makes the Bloch space a Banach space. Using (1.1) it is easy to see that B is invariant under Mobius transformations, that is, if f E B then f o (15 E B, for all (b E G. In fact, llfO ¢|ls = llflls. The polynomials are not dense in the Bloch space. The closure of the polynomials in the Bloch norm is called the little Bloch space, denoted by 30. In [34, page 84] is shown that f 6 [30 if and only if|1]m1(l — IzI2)If'(z)I = 0. z-i A linear space X of holomorphic functions on U with a seminorm IIII X is Mb'bius invariant if 1. X C B and there exists a positive constant c such that for all f E X, llfllzs S Cllfllx- 2. ForalquEG'andalleX,foquXand Hf0¢llx = llfllx. A Mo'bius invariant Banach space is a Mobius invariant linear space of holomorphic functions on U with aseminorm II.|IX, whose norm is f —> IIfIIX or f —-> If(0)I+IIfIIX. For 1 < p < 00, the Besov space E, is defined to be the space of holomorphic 10 functions f on U such that Ilfllf’g, / lf’(z)|”(1-|z|2)"‘2dA(z) / |f’(z)l"(1-|zl2)”dA(z) < oo U where dA(z) is the Mobius invariant measure on U, namely 1 0””) = (T—Tzn’ dA(z). It is easy to see that If(0)I + IIflpr is a norm on B, that makes it a Banach space. It is easy to see that log(1 — z) E 8. Moreover Holland and Walsh show in [13, Theorem 1] that if 1 < p < 00, and 7 < i (q is such that fi+ % = 1) then (log 2 )7 6 BP. Other examples of functions in 3,30, and B, (1 < p < 00) are l-z provided by gap series. Let f(2) = Zanzha n=0 where (An) is a sequence of integers satisfying An+l An 2A> 1, (1.2) where A is a constant and n E N. Anderson, Clunie, and Pommerenke show in [1, Lemma 2.1] that f E B if and only if an = 0(1), as n —) 00, and that f 6 80 if and only if an —+ 0, as n —-> 00. Moreover, a description of Besov spaces that Peller gives in [23, page 450] easily yields that f E B, if and only if 2:0 AkIakIp < 00. Let be a holomorphic function on U. The Hardy space H 2 is the collection of functions f 11 holomorphic on U for which def. 00 IIfIIIII2 : 2 [an]2 < 00- n=0 The Dirichlet space is the collection of functions f holomorphic on U for which 00 d f. llfllzp é Enlanl" < oo. 71:] NI-fi Both H2 and D become Hilbert spaces with norms IIfIIH2 and (If(0)I2 + IIfII%) respectively. It is easy to see, using polar coordinates, that f E D if and only if / |f’(z)|"’dA(z) < oo. U Thus, the Besov-2 space is the Dirichlet space and 82 = D C H2. Let const. denote a positive and finite constant which may change from one occurence to the next but will not depend on the functions involved. Unlike the Hardy and Bergman spaces the Besov space with a smaller index lies inside the Besov space with a larger index. Lemma 1.1 For1< p < q, B, C B, C B, andfor any f 6 BP, llfllzs S const-llfllsq S constllfllay Proof. First, let us show that each Besov space lies inside the Bloch space. Fix p > 1, let f 6 B,; then, co > [U lf’(z)l”(1—lzl)2)”‘2dA(z) 12 > [I/QTM ‘6)Ipd6}<1 ,2,._2,.d,. Te — '- " R 0 27r 1 2n 2 f I] |f’(Re‘9)|”é£}(1—r"’)’"2rdr R 0 2a 2a . d0 1 ___ I :0 p__ _ 2 p—2 (l /0|f(Re)|2,T/R(1r) rr 1 2n .9 pdg l—R2 p—2 = _ I z _ d 2/0 If(Re)I,,/o r r > l2” If'(Re‘9)IP-‘£9(1 — RV“ where c is some positive constant, and 0 < R < 1. Above we used the fact that the integral means of an analytic function f, Mp(R, f) = {2—11; 02" If(Re‘glpdd} (0 < p < 00), are a non- decreasing function of R (Hardy’s Convexity Theorem [11, page 9]). Thus, 1 1 M (R,f) S const.——_—l = —-——,-. ” (1—R)“’T 0—H)"F Then by the Hardy-Littlewood theorem ([11, Theorem 5.9, page 84]), the infinity means of f’, MOO(R,f') = max If’(Rei9)I, OSO<2u can not grow faster than 1 1 (1—R)‘-‘s+% l—R that is - l su ' Re'g < c 96[0.I21r] If ( )I _ 1_ R for some positive constant c. Now, it is easy to see that this implies that f belongs 13 to the Bloch space, and llfll’é, ZCIIflls. Therefore, B, C B for any p > 1. Next, for the containment among Besov spaces, fix p and q such that 1 < p < q and let f 6 BP. Then, Ilfll‘lg, = flf’(z)|"(1—lz|2)"dk(z) = f|f’(z)|”(1—IzIz)”(lf’(z)l(1—|zl2))‘°"’dA(z) S CllfIIEJPIIfIIZ, < 00- Thus, B, C B]. This finishes the proof of the lemma. CI Lemma 1.2 For 1 < p < 00, B, is a Mo'bius invariant Banach space. Proof. Let f E 8,, q E U. Then, Ilfoaqll’i'a, : ./U|(f ° aql'(2)|"(1- |Z|2)p’2dA(z) [U If’(aq(z))lpla;(z)lp(1 — Izl2)”‘2dA(z) = [U lf’(w)”|a;(aq(w))|”(1 — laq(w)12)”‘2|a;(w)|2dA(w) = /U |f’(w)|”—-1——(1-lwl2)”‘2|a;(w)|”‘2la;(w)l2dA(w) |01£,(W)|p = llf’(w)|”(1—Iw|2)"‘2dA(w) = “flit.- 14 Above we made the change of variables aq(z) = w and used basic properties of the M6bius transformations. This shows that B, is invariant under Mobius transfor- mations. Thus, by Lemma 1.1, B, is a Mobius invariant Banach space. CI A holomorphic function f on U belongs to BM 0A, the holomorphic members of BMO, if llfHG=sug||f00q(Z)-f(61)llm <00- (1-3) 06 Under the norm I f(0)I + II f ”G BM 0A becomes a complete normed linear space. This is not the traditional definition of BM 0A, it is actually a corollary of the John- Nirenberg theorem [4, page 15]. By the Littlewood—Paley identities (see [34, page 167]) and the fact that log Til ~ 1 — IzI2, for z away from the origin we see that a seminorm equivalent to the one defined in (1.3) is Ilfllf = sup/|()’(2)foaq WWI—(2|) A(z) QEU = sup/lf(( ((naqz Wllaw (1—I~ |)dA(z) Thus after the change of variables aq(z) = w we obtain llfllf— - sup /( |f’(w (1— laq( w)! )dA(w). (1.4) Notation S(h,6) : {z E U: Iz — ewI < h}, where 0 E [0, 27r), h 6 (0,1)}. Let A and B be two quantities that depend on a holomorphic function f on U. We say that A is equivalent to B, we write A ~ B, if const. A S B S const. A. The notion of BMOA first arose in the context of mean oscillations of a function 15 over cubes with edges parallel to the coordinate axes or equivalently over sets of the form S(h,0) ([28, pages 36-39]). That is, 2 1 I 2 2 llfll. ~ sup , [M If(z)| (1 — lzl >dA(z). (1.5) he(0,1) 06[0,21r) The function log(1 — z) E BMOA. In fact if f is any holomorphic, univalent, and zero free function then log f 6 BM 0A. (this result first appeared in [3] and [6]). Other examples of B M GA functions include the following. If (an) is a bounded sequence then 220:0 fianz" E BMOA, and if 22:0 IanI2 < 00 then 220:0 an:‘\" 6 BMOA, where the sequence (An) satisfies (1.2). One of the many similarities between the Bloch space and BM 0A is that poly- nomials are not dense in either space. The closure of the polynomials in the B M GA norm forms VMOA, the space of holomorphic functions with vanishing mean oscil- lation. The space VMOA can be characterized as all those holomorphic functions f on U such that lim L lf’(w)|2(1-laq(w)l2)dA(w) = o (1.6) lei-*1 (the “little-oh” version of (1.4) ). Moreover the “little-oh” version of (1.5) is equivalent 0 (1.6) ([28, pages 36-37, page 50]). An easy way to see that BMOA is a subspace of the Bloch space is the following: |f’(0)| S llfllm for any f holomorphic on U; therefore, |(f 00p - f(P))'(0)| S ||fO 0p - f(p)llm S Ilfllc 16 hence If'(ap(0)lla;(0)l S const-IIfIL that is If'(P)I(1-IPI2) S COIrlst-IIfII-- thus, IIfIIB S const-Ilflla- Therefore, B M GA C B. Let H °° denote the space of bounded holomorphic functions on U. Lemma 1.3 The space VMOA fl H°o is closed under pointwise multiplication. Proof. Let f,g E VMOA fl H°°. Then, / I( fg)’( 1(— Iaq(z II )dA(z) =/|fW()VzI|gz(1(—laq(2r (z)+flg’()"’()zllfz|2( (1—|a( (2I(IIdAzI S const {/11 If’2( (1- Iaq(2 )|)dA(2)+/UI9’(Z)|2(1 - laq(Z)I2)dA(z)} - The righthand side of the above equation tends to zero as IqI ——> 1, since f, g 6 VMOA. Hence, fg E VMOA. El Lemma 1.4 For any p > 1, B, is a subspace of VMOA. Proof. Fix p > 2; first we will show that B, C H2. Let f E 8,. Then, flf’(2)l2(1-lz|2 /=|f()zl(“1—1—Iz|)( —I2~I )sz) 17 _2_ —2 s |lf|l23p( [U (1 — I2I2)P-2d/\(z))LP‘ by Holder’s inequality. Since, —:523;L2d/\z= —22i1—3dAz 00 [yo III (I [(1 III (I< U for any p > 2, /UIf’(Z)I2(1—Izlz)dA(z) < 00. Therefore, 8,, C H2, for any p > 2. Since D C H2,if1< p S 2 then, Bp E D C H2. Therefore Bp C H 2, for any p > 1. By the Mobius invariance of Besov spaces we obtain IUOGm-fMHfizSdUbOa-fMH%;=dUWE for some positive constant c and for any q E U. Therefore, IIfII3 S cIIfII'Zc;,,- (1-7) This shows that Bp C B M 0.4. Next we show that polynomials are dense in Bp. This together with (1.7) then shows that 8,, C VMOA. Let f E Bp, f(2) = Zanz" and 0,,(f) the n-th Fejer mean of f, that is: " I/\| A 2" ‘9 d9 = ' K, 0 — 1:0 n+1)a*z 0 (26 ) I )27r 18 where Kn(0) is Fejer’s kernel, K..(0) = Z (1 — Eli—IT)6_M‘ (1.9) We will show that 0,,(f) —> f in Bp; Fubini’s theorem yields, IIa.(fI — flI’E, = [U lan(f)’(z) — f’(z)l”(1 — lzlz)p‘2dA(z) 21r . _ d0 s /U / Ie’of’(ze"') — f’(z)lpKn(9)§7;(1 — Izlz)”‘2dA(z) 21r . d0 = / ||f(ze'9)-f(z)II’iapKn(0)§; 27r . d0 = / g(e'g)l1’n(0)2—7r where g(e‘9) = ||f(ze‘9) — f(z)||’gp. It is easy to see that g is a continuous function on 8U. Therefore, by Theorem 2.11 in [14, page 15] lim max |0n(g)(eit) — g(e“)| = 0. n—ioo OStSZW Hence, 0n(g)(1) —+ 9(1) 2 0, as n —> 00. Thus (1.10) yields, 3320 IIa.(fI —fII'2.,, = . Therefore we obtain that Bp C VMOA. CI 19 We have shown that for p < q 3,, c B, c: VMOA c BMOA c 8. Similarly to Lemma 1.2 we can show that BM 0A and VAIOA are also Mobius invariant Banach spaces. In fact, the reason for insisting that a Mobius invariant Banach space be a subspace of the Bloch space is that Rubel and Timoney proved in [26] that if a linear space of analytic functions on U with a seminorm IIHX is such that for all f E X, fo (b 6 X and ||fo qux = ||f||x, and it has a non-zero linear functional L that is decent (that is L extends to a continuous linear functional on the space of holomorphic functions on U) then, X has to be a subspace of the Bloch space and the inclusion map is continuous. CHAPTER 2 Carleson measures and compact composition operators on Besov spaces and BMOA If o is a holomorphic self-map of U, then the composition operator Cd, C¢f = f 0 <19 maps holomorphic functions f to holomorphic functions. Shapiro and Taylor show in [29], using the Riesz Factorization theorem and Vitali’s convergence theorem that Cd, is compact on H p , for some 0 < p < 00 if and only if C,» is compact on H 2. Moreover, Shapiro solves the compactness problem for composition operators on H1!) in [31] using the Nevanlinna counting function N¢(w)= Z —log[w|. d>(z)=w The following theorem is proved there; Theorem A Let (I) be a holomorphic function on U. Then Gas is a compact operator on H2 if and only if lim N¢(w) — 0. [wI-H —log [w] _ Madigan and Matheson characterize compact composition operators in the Bloch space in [22]. The following theorem is proved there; 20 21 Theorem B Let to be a holomorphic function on U. Then, Cd, is a compact operator on B if and only if lcb’(z)l(1—|z|2) _ win”; 1- we)? ‘ 0' In this chapter we will use some Nevanlinna type functions to characterize the compact composition operators on Besov spaces BM 0A, and VMOA. Definition 2.1 The counting function for the p-Besov space is Np(w,¢> = Z {l<25’(z)|(1—lzl‘*)}”_2 ¢(z)=w foerU,p>1. Definition 2.2 The counting functions for BMOA are N(z)=w for w,q E U. The above counting functions come up in the change of variables formula in the respective spaces as follows: First, for f 6 Bp and p > 1 ||C¢f||iap = (f 0 ¢)’(Z)|p(1-|z|2)p‘2dA(z) l U = Llf’(¢(z))l”l¢’(z)lp(1-Izl2)""2dA(z). (2.1) By making a non-univalent change of variables as done in [32, page 186] we see that Howl's, = fu lf’(w)l”Np(w,¢)dA(w). (2.2) 22 Similarly, for BMOA IICifllf = sup / |()’(fo¢ l—laq z)l )dA(:) qu = su — a z 2 dA z . .et’f'fW 11¢ )I(1 |q()l) () Thus, IIC¢f||3=supf lf’(w)|2N(w,q,¢)dA(w)- (2.3) qu U Arazy, Fisher, and Peetre prove in [2, Theorem 12] that composition operators in BM 0A are bounded for any holomorphic self-map of U, and they are bounded on VMOA if and only if the symbol belongs to VMOA. Next, we provide a proof similar to their proof. Theorem C Let (b be a holomorphic self-map of U. Then, 1. Cd, is a bounded operator on BMOA. 2. C¢(VMOA) c VMOA if and only M e VMOA. Proof of (1.) Suppose that (b is a holomorphic self-map of U and f E BMOA. If (15(0) = q E U then (b = (1,, o it» for some holomorphic self-map if) of U such that 2M0) = 0. Then Littlewood’s Subordination Principle (see [32, page 13]) yields |lf0-f(q)||H2 = llfoaczOd’-f(€1llly2 S llfoaq - f(q)||n2 S ||f||-- (2-4) Thus replacing (b in (2.4) with (poaq yields [|f0gboaq —f(q)||H2 g ||f||. for all q E U. Thus, |f(¢(0))| + llf 0 ¢|l. S must-(If(0)I + ||f||-), 23 for all f E BMOA. This shows that Cd, is a bounded operator on BMOA. Proof of (2.) First suppose that C¢ : VMOA —+ VMOA is a bounded ope- rator. Then since the identity function f(2) = 2: belongs to VMOA, f 0 ab = (b E VMOA. Conversely, suppose that 45 E VMOA. Then, by Lemma 1.3, {¢" 6 VMOA : n E N} C VMOA. Therefore {p(¢) :p polynomial} C VMOA. Since polynomials are dense in VMOA part (1) above yields that f 0 ab 6 V M 0A, for any f E VMOA. This completes the proof of the theorem. C] Now consider the restriction of Cd, to Bp. Then 0,), is a bounded operator if and only if there is a positive constant c such that ||C¢f||i§p S cllfllia. for all f E Bp or equivalently by (2.2) [U lf’(w)|”Np(w,¢)dA(w) s cllfllfi. for all f 6 BP. This leads, as in [2], to the definition of Carleson type measures. Since we are interested in characterizing the compact composition operators we will also talk about vanishing Carleson measures. We would like to use the following operator theoretic wisdom; If a “big-oh” condition characterizes the boundedness of an operator then the cor- responding “little-oh” condition should characterize the compactness of the operator. Definition 2.3 Let p be a positive measure on U and let X = Bp (1 < p < oo), BMOA, or 8. Then p is an (X, p)-Carleson measure if there is a constant A > 0 so 24 that [U lf’(w)|”du(w) s Allflli’x», for alleX. In view of (2.2) and (2.3) above we see that C¢ is a bounded operator on 3,, if and only if the measure Np(w, ¢)dA(w) is a (Bp, p)-Carleson measure, and Cd, is a bounded ope- rator on BMOA if and only if N(w,q, ¢)dA(w) are uniformly (BMOA,2)- Carleson measures. Arazy, Fisher, and Peetre gave the following characterization of (8,, p) Carleson measures in [2, Theorem 13] (the equivalence of (I) and {2) was given by Cima and Wogen in [7]). Theorem D For 1 < p < 00, the following are equivalent: 1. p is a (Bp,p)-Carleson measure. 2. There exists a constant A > 0 such that u(5(h,9)) S Ah” for all 0 E [0,27r), all h E (0,1). 3. There exists a constant B > 0 such that / Ia;(z)|”du(z) s B U for all q E U. Hence Theorem D yields, Theorem E Let (f) be a holomorphic function on U. Then Cd; is a bounded operator 25 on Bp {l < p < 00) if and only if SUP HC¢aqlpr < 00 - QEU We prove a similar theorem for compact composition operators on Besov spaces. Definition 2.4 Forl < p < 00, u is called a vanishing p-Carleson measure if r #fiwflh 1m sup _— = 0 . h—+O gelo’zfl) hp Note It is easy to see that if,u is a vanishing p—Carleson measure then it is a (Bp,p)- C arleson measure. The proposition below characterizes vanishing p-Carleson measures. The proof is similar to the one for Carleson measures on H 2 (p = l), as given by Garnett in [12] and by Chee in [5]. Proposition 2.5 For 1 < p < 00, the following are equivalent: 1. p is a vanishing p-Carleson measure. 2. fU Ia;(w)lpdp(w) -—> 0, as |q| —+ 1. Proof. First, suppose that {2) holds. Then, given an e > 0 there is a (5 > 0 such thatfor1—5<|ql<1 / Ia;(w)l”d#(w) < e. U Fix 6 > 0 and let 5 > 0 be as above. Consider any 0 < h < 6, 0 E [0,27r), let q = (1 — h)ei9 and w E S(h,0). Then, I-MP u—amz l—U—hY |1-—(1-—Ine-wu42 I%WM 26 h(2 — h) |6‘6 - (1 — hlwl2 h(2 — [2.) (leg - w| + I’w -(1— hlwll2 h(2 - h) (h + hlwl)2 2 — h h(1-i-Iw|)2 l 4h° IV IV IV Hence, w E S(h,0) implies that Ia’ (w )I” Z 4—17,”. Then by our hypothesis, 1 > ' pd >/ pd _ Sh,0 . e fumwn u- wlagw w)| u> W ( ( >) This proves (1 ) Conversely, suppose that (1) holds. Then, given an e > 0 there is a 6 > 0 such that for any 0 < h < 6 and any 0 E [0,27r), ”(502m) < ch”. (2.5) Fix 6 > 0, let 6 be as above. Fix ho < 6 such that (2.5) holds. Also, fix q = lqleia E U with Iql > 1 — bf. We will show that for q large, / la;(w)lpdu(w) < e U Let E = {w E U: Ie'i‘9 — Iqlwl 2 54“}. Then for each q E U, / Ia;(w )(tvlpdu )=/ Ia;(w )(tvlpdu )+/ la;(w )(|de (w.) (2.6) 27 We will estimate each of the integrals above. First if w G E, 1-lql"’ P 1—lq|2 P ' p = . < 42 —— < ”9”” (le‘g-Iqle) - h: " for q large. Therefore (2.7) yields that for q large, /IQ;(w)|pdu(w) < (ME) S ,u(U)e < const. 6. E Let N = N(q) be the smallest positive integer such that 2N(1—lQI)< ho S 2N+l(1"l9l)- We will show that EC C 5(2N(1 — |q|),0) C S(ho,0). Let w E E“. Then, w - 6”I = lw - 6‘9 + Iglw - lqlwl S |w — IQle + It?” 461le % 4 < 1- lql + 2""‘(l - lql) <1—lql+ |/\ 2N(1-I 0, as |/\| —-> 1. Theorem 2.8 The following are equivalent: 1. C¢ : ’D —+ BMOA is a compact operator. 2. ||C¢cu||.. —> 0, as |/\| —) 1. In the proof of the two theorems above we will need the following lemmas. Lemma 2.9 Let X = 3,, (1 < p < oo), BMOA, or 3. Then, 1. Every bounded sequence (fn) in X is uniformly bounded on compact sets. 2. For any sequence (fn) on X such that llfnllX -> 0, fn - fn(0) —> O uniformly on compact sets. Proof. In [34, page 82] is shown that a Bloch function can grow at most as fast as log film, that is 1 |f..(z)-fn(0)| s consult”. 1°81_|.| 31 l 1—|z|. S const.||f,,||x log Hence the result follows. C] Lemma 2.10 Let X,Y be two Banach spaces of analytic functions on U. Suppose that 1. The point evaluation functionals on X are continuous. The closed unit ball ofX is a compact subset ofX in the topology of uniform f6 convergence on compact sets. 3. T : X —> Y is continuous when X and Y are given the topology of uniform convergence on compact sets. Then, T is a compact operator if and only if given a bounded sequence (fn) in X such that fn ——> 0 uniformly on compact sets, then the sequence (Tfn) converges to zero in the norm on. Proof. First, suppose that T is a compact operator and let (fn) be a bounded sequence in X such that fn —) 0 uniformly on compact sets, as n —> 00. For the rest of this proof let |.|y denote the norm of Y. If the conclusion is false then there exists an e > 0 and a subsequence in < n2 < 723 < such that len,lY _>_ e, for all j = 1,2,3,... (2.14) Since (fn) is a bounded sequence and T a compact operator we can find a further subsequence n,, < n], < and f E Y such that len,, — NY -> 0. (2.15) 32 as k —-> 00. By (1) point evaluation functionals are continuous, therefore for any 2 E U |(Tfan — f)(z)| S const.|Tfan —- fly. (2.16) Hence (2.15) and (2.16) yield, Tfnu — f —> 0 (2.17) uniformly on compact sets. Moreover, since fn 1* —> 0 uniformly on compact sets , (3) yields, Tfnjk —> 0 uniformly on compact sets. Thus by (2.17) f = 0. Hence (2.15) yields ITfanly ——> 0 as k —> 00, which contradicts (2.14). Therefore we must have ITfnly —> O, as n —-> 00. Conversely, let (fn) be a bounded sequence in X. We will show that the sequence (T fn) has a norm convergent subsequence. Without loss of generality (fn) belongs to the unit ball of X. By (2) there is a subsequence n1 < n2 < such that f") —) f uniformly on compact sets, for some f 6 X. Hence, by our hypothesis, IT fn J —T f ly —> O, as j —-> 00. This finishes the proof of the lemma. CI Note (=>) Only uses (1) and (3). (4:) Only uses (2). Lemma 2.11 Let X,Y = 8,, (1 < p < 00), BNIOA, or 3. Then C43 : X —> Y is a compact operator if and only if for any bounded sequence (fn) in X with fn -—+ 0 uniformly on compact sets as n —> oo, ||C¢fn||y —-> 0, as n ——> 00. Proof. We will show that (I), (2), (3) of Lemma 2.10 hold for our spaces. By Lemma 2.9 it is easy to see that (1) and (3) hold. To show that (2) holds, let (fn) be a sequence in the closed unit ball of X . Then by Lemma 2.9, (fn) is uniformly bounded on compact sets. Therefore, by Montel’s Theorem ( [8, page 153]), there is a subsequence n1 < n2 < such that fnk -> g uniformly on compact sets, for some g E H(U). Thus we only need to show that g E X. 33 (a)IfX=B,,(1 Y is a compact operator if and only if for any bounded sequence (fn) in X with f,, —> 0 uniformly on compact sets as n —> oo, lfn(¢(0))l + I|C¢fn||y —> 0, as n ——> 00. Which is clearly equivalent to the statement of this lemma. This completes the proof of the lemma. C] 34 An immediate corollary of Lemma 2.11 is the following. Corollary 2.12 Ifqb is a holomorphic self-map ofU such that Ilgblloo < 1 then C¢ is compact on every Besov space, and on BMOA. Proof. First, let us show that C4, is compact on the Besov space Bp. Let (fn) be a bounded sequence in 8,, such that fn —> 0 uniformly on compact subsets of U. Suppose that e > 0 is given. Since EU)- is a compact subset of U, there exists a positive integer N such that if n 2 N then |f,’,(gb(z))|p < e, for all z 6 U. Then by (2-1), ||C¢fn||f9p < ellcbll’gp < const. 6. Thus, ||C¢fn||3p —> 0, as n —> 00, and Lemma 2.11 yields that C), is a compact operator on Bp. The proof of the BM 0A compactness of Cd, is similar to the proof above. [I] Now we are ready to prove Theorem 2.7 and 2.8. The technique is similar to the one given by Arazy, Fisher, and Peetre in [2, Theorem 13] and Luecking in [17], and [18]. Proof of Theorem 2.7. By (2.2), Mowing = [U la&(w)l"N.‘i(w)dA(w)- Thus Proposition 2.5 yields (2) 4:} (3). Next we show that (1) => (3). We assume that Cd, : 8,, —> B9 is a compact operator. Note that {01.x : /\ E U} is a bounded set in 8,, since, llaxllap = |le culls. = llzllsp, 35 and the norm of cm in Bp is lai(0)l + llaillsp <1+llzllsp < 00- Also Ox — /\ —> O, as [/\I —-> 1, uniformly on compact sets since, 1— [M2 [l -Xz|. [CUM — /\| = lzl Hence, by Lemma 2.11, [|C¢,(a,\ -— A)||Bq —> 0, as [M —> 1. Therefore ||C¢a,\[|3q -—> 0, as [A] —> 1. Finally, let us show that (2) => (1). Let (fn) be a bounded sequence in HP, that converges to 0, uniformly on compact sets. Then the mean value property for the holomorphic function f,’, yields, 4 ,',w-———— 'szz. 2.18 f() l. f() () < > — 7r(1—- lwl)2 w—z|<‘—‘§2L n Therefore by Jensen’s inequality ([27, Theorem 3.3, page 62] and (2.18)), lf.’.(w)l" < 4 _ ml _.|1—-+/|‘z|<1————(——_(1-lz)l) ./S(2(1-|z|,a) Q(w,¢) AW) dA(~)) = const.(1 + II) , (2.21) forany0<5<1. Fix 6 > 0 and let 6 > 0 be such that for any 0 E [0,27r] and any h < 6 / Nq(w,qb)dA(w) < eh". (2.22) S(h,0) By (2.21) and (2.22) I S 296/ M(1—|z|2)qd/i(2) zl>l—- (1—[Z[2)2 S const.e||f,,||‘[3q < const. 6. (2.23) By (2.21), 11 S const./“<1“é |f;(z)|q (L Nq(w,q‘>)dA(w)) dA(z) 37 = ./|| , 6 |f,’,(z)|q||¢[|§3qu(z) < const. 6 (2.24) 251—5 for n large enough, since f; —) 0 uniformly on compact sets. Combining (2.21), (2.23) and (2.24) we obtain that [[C¢fn[[3q < const. 6 for n large enough. Therefore ||C¢fn||Bq —-> 0, as n —> co and Lemma 2.11 yields, 0,, : B,, —> B, is a compact operator. This finishes the proof of Theorem 2.7. C] Proof of Theorem 2.8. (1) => {2). Since a; is a bounded set in D and a), — /\ —> 0 uniformly on compact sets, as [Al ——> 1, Lemma 2.11 yields |[C¢ayll. —) 0, as [Al —> 1. (2) => (1). The proof is similar to the proof of Theorem 2.7. We will use Lemma 2.11. Let (fn) be a bounded sequence in ’D such that fn —> 0 uniformly on compact sets. Our hypothesis is that |[C¢a,\||.. —+ 0, as [Al —-> 1. That is sup / |a&(w)I2N(w,q,¢)dA(w) a o qu U as [Al —-> 1. Hence, Corollary 2.6 yields 1 lim su —- N ,, dA =0. ’1‘“) web) ft? 501.9) (wqu) (w) 06[0,21r) Fix an e > 0 and let 5 > 0 be such that for any 0 E [0,27r) and any q E U, if h < 6 then / N(w, q, ¢)dA(w) < chz. (2.25) S(h,6) Fix q E U. Then by (2.19), [U If.’.(w>IPN(w,q,¢)dA(w) 4 I 2 S AW (,/|w_z|<1_2£l lfn(z)l (114(2)) N(waqa ¢)dA(w) 38 cons _If,’,(z)|2 ( w d w) .. ‘ S ti/f;(1—-|z|)2 [9(2(1_|2l)fl)N( ,q,d>) A( ) dA(/.). (2.26) The proof of (2.26) is the same as the proof of (2.21) in Theorem 2.7. Next split the integral in (2.26) into two pieces, one over the set {2 E U : |z| > 1 — g} and the other over the complementary set . Then , lf.’.(~"3)|2 ( w) 7 ‘/|;I>1—g(1"|2l)2 [5(2(1_|2D’6)N(w,q,¢)dA( ) dA(~) lfr';(2)l2 _ 22 7 < A»; (1 — |z|)"~’4(1 '2' l “‘A‘”) < const. e||f,.||';’, < const. 6 , (2.27) and |fh(z)l2 ( , w w ) 7 LISl-g-(l—IZI)2 /SI2I1-I2I).a)M ’q’¢)dA( ) dA(~) 3 mt (sup [U N(w,q,¢)dA(w)) [HM If.’.(z>IPdAIz) qu S const. 6, (2.28) for 72 large enough since (I) E BMOA and f,’, —> 0 uniformly on {2 E U: |z| S 1-— g}. Therefore (2.26), (2.27), and (2.28) yield that supL |f,’,(w)|2N(w,q,q§)dA(w) < const. 6 QEU for n large enough. Thus ||C¢fn||. —> O, as n —> 00. Hence by Lemma 2.11, (1) holds. 39 This finishes the proof of the theorem. C] Note It is easy to see that Theorem 2.8 yields that C4, : 8,, ——> BM 0A is a compact operator if and only if ||C¢a,\|l. —> 0 as [Al -—) 1, for 1 < p S 2. Moreover in chapter three we will show that if Cd, is bounded on some Besov space then this is valid for anyp> 1. The following is a corollary of the proof of the Theorem 2.8. Corollary 2.13 If supqeu fula’,(w)l3N(w,q,¢)dA(w) —> 0, as [M -—> 1, then Cc. : BMOA —> BMOA is a compact operator. Note Similarly to the proof of the above theorems we can easily see that, the above sufficient condition for BM 0A compactness is equivalent to at : H 2 —) BM 0A being a compact operator. CHAPTER 3 Besov space, BMOA, and VMOA compactness of Cd, versus Bloch compactness of C¢ In this chapter we give conditions that relate the compact composition operators on Besov spaces, BM 0A, and VMOA with those on the Bloch space, and the little Bloch space. Recall the characterization of compact composition operators on the Bloch space that Madigan and Matheson give in [22, Theorem 2]. Theorem B Let (I) be a holomorphic self-map of U. Then, Cd; is a compact operator on B if and only if . I¢’(z)I(1—I2IP)_ 23$: i—I¢(2)IP ‘0' Next we give another characterization of compact composition operators on the Bloch space. Theorem 3.1 Let (b be a holomorphic self-map of U. Let X = 8,, (l < p < oo), BMOA, or [3. Then C¢ : X —> B is a compact operator if and only if I. C = 0. Mllgll || Mulls Proof. First, suppose that C2,, : X —> B is a compact operator. Then {cu : A E U} is a bounded set in X, and a) — A —+ 0 uniformly on compact sets as [M —> 1. Thus 40 41 by Lemma 2.11 11m llCaSaAllB = 0- |A|->l Conversely, suppose that lIml|C¢OA|l5 = 0, as [/\l —> 1. Let (fn) be a bounded sequence in X such that fn —> 0 uniformly on compact sets, as n —> 00. We will show that Jig; llC¢fnllB = 0- Let c > 0 be given and fix 0 < 6 <1 such that if [M > 6 then |lC¢aAH3 < 6. Hence for any 20 E U such that [¢(zo)l > 6, [[C¢,a¢,(zo)[[3 < e. In particular, lain.)(¢(20))l |¢'(Zo)|(1 - IZoI2) < 6 that is, l¢’(zo)l 1 ‘ [4420”2 Then (3.1) yields that for any n E N and 20 E U such that [¢(zo)[ > 6, (1 — [20m < e. (3.1) [(fn0¢)'(zo)[(1—l20[2) = |fl.(¢(Zo))ll¢'(Zo)I(1-|Zo|2) < If.'.(¢(Zo))|(1 - |¢(Zo)l2)€ S llfnllBC S llfnllX e < const.c . (3.2) Since the set A = {w : [wl S 6} is a compact subset of U and f,’, —> 0 uniformly on compact sets, suplf,’,(w)[ -~>0, asn—>oo. wEA Therefore we may choose N large such that |f,’,(q$(z))[ < e, for any n 2 N and any T7 ’ \r—FWI— 42 z E U such that |q§(z)[ S 6. Then, for all such z, |(fn°)'(z)|(1—|z|2) = |f.’.(¢(z))||¢'(z)|(1-Isl?) < C|'(z)|(1—lzl2) < ll¢ll3€, (3.3) where n 2 N. Thus, (3.2) and (3.3) yield “fr. 0 (bllg < const. e, for n 2 N. (3,4) Thus (3.4) yields that [|C¢fn[[3 —-> O as n —> 00. Hence by Lemma 2.11 C¢ : X -—> B is a compact operator. E] Notes (a) It is easy to see that the proof of Theorem 3.1 yields that 1° C = 0 ”[1331” Willis if and only if m |¢’(z)|(1-|z|2) : 0. l¢>(z)I-+1 1 - |<15(Z)|2 Therefore we obtain another proof of Theorem B. (b) The above theorem is valid for any Banach subspace X of the Bloch space such that the point evaluation functionals on X are continuous and the closed unit ball of X is compact in the topology of uniform convergence on compact sets. An immediate consequence of Theorem 3.1 along with Lemma 2.11 and Theorems 2.7 and 2.8 is the following proposition. Proposition 3.2 Let 1 < p S q S 00. Then: 1. If C), : Bp —> 3,, is a compact operator then so is C,» : B —> B. 43 2. For1 BMOA is a compact operator then so is C), : B ——> B. The following proposition gives a sufficient condition for a composition operator to be compact on a Besov space. Proposition 3.3 Let 1 < p S q < 00. If lim N4(w9 ¢) [wl—)1 (1 — [wl”)‘l‘2 =0 then C¢ : 8,, —> B, is a compact operator. Proof. Let (fn) be a bounded sequence in 3,, such that f... -—) 0 uniformly on compact sets as n —-> 00. Let c > 0 be given and fix 6 > 0 such that if 1 — 6 < [wl < 1 then N,(w,¢) < C(1_|w|2)q—2. (3.5) By (2.2) “ennui. = / lf.’.(w)l"Nq(w,¢)dA(w) = + ’ w qu w, dA w [WW lwlSl_5If.( )l ( d») (> = [+11 (3.6) By (3.5), I < e / |f.’.(w)l"(1 — IwIP)P-PdA(w) l—6 B, is a compact operator. Cl Composition operators on Besov spaces are not bounded for all holomorphic self- maps of U. But if the Besov space contains the Dirichlet space and the symbol is boundedly valent then the induced composition operator is bounded. Lemma 3.4 Let qb be a boundedly valent holomorphic self-map of U, 2 S q < 00, and 1 < p S q. Then Cd, : B,, —> 8,, is a bounded operator. Proof. Let f E B,, (1 < p < 00). Applying the Schwarz Lemma ([27, page 254]) to the function (12 o 45 0 mm) yields |¢'(2)l(1-|2l2)51-|¢(2)|2, for any 2 E U. Hence by (2.2), IIcifIIPB, = fl] lf’(w)l" Z (I¢'(2)I(1—IzIP)°‘2dA(w) 6(2):!” 3 const./U If’(w)l" Z (1—|¢(z)|2)"’2dA(w). ¢(z)=w 45 Therefore, [[C¢f[|qu S const.] [f'(w)lq(1— |w[2)q'2dA(w) S COHSt-llflliap U for any holomorphic function f on U. Thus, Cd, : Bp —) B, is a bounded operator. C1 The following theorem and proposition give conditions under which compactness in the Bloch space is equivalent to compactness from a Besov space to some larger Besov space. Theorem 3.5 Let qb be a univalent holomorphic self-map of U. Then, for q > 2, Cd, : B, -—> B, is a compact operator if and only if C.) : B —-> B is a compact operator. Proof. First, suppose that Q, is a compact operator on the Bloch space. The sufficient condition of Besov space compactness in Proposition 3.3 for a univalent function is lim [wl—+1 {|d>’(d>‘1(w))l(1—le“(w)|2}q_2 = 0 1— [wl2 or equivalently, |¢’(Z)|(1-l2|2) 1m 2 wen-+1 1 - |¢(2)l =0. Which is a compactness condition for the composition operator on the Bloch space (Theorem B). Hence, by our assumption, C4, : B, —> 8,, is a compact operator. The converse follows from Proposition 3.2. This finishes the proof of the theorem. Note Theorem 3.5 is not valid when q = 2. There exists a univalent holomorphic self-map of U such that C), is compact on the Bloch space but not on the Dirichlet space. To describe such an example we will need some preliminaries. First, the Koebe Distortion Theorem (see [32, page 156]) which asserts that if d) is a univalent function 46 on U then for any 2 E U 5¢(v)(¢(2)) ~ |¢'(Z)|(1—lzl2), where 6¢(U)(q’>(z)) is the Euclidean distance from (6(2) to 8¢(U). Thus the Madigan and Matheson condition of Bloch compactness for a univalent d) is equivalent to 5¢(U)(¢(Z)) _ willln 1 _ |¢(z)|2 — 0' (3.9) Let D(0,a) denote the disc centered at 0 of radius a. A nontangential approach region (to, (0 < a < 1) in U, with vertex C 6 (9U is the convex hull of D(0,a) U {C} minus the point C. If 11) is a univalent holomorphic self-map of U such that I,b(U) = QC, (0 < a < 1) then infzeu W > 0. Thus by (3.9) Cd, is not compact on 8. But if we delete certain circular arcs from 00, then for the Riemann map (b from U onto the induced domain G, C), is compact on B. Let L, = {z E (to, : [z — 1| S 1,]; (n 2 1). Then L1 3 L2 3 L3 3 Remove from L, \ Ln.“ (n 2 1) arcs centered at 1, with one end point at (900,, in such a way so that the succesive radii are less than 5‘; apart, and the distance of each arc to 600, is less than 3%. Then the distance from each .2 6 Ln \ Ln.” to the boundary of the induced subdomain, Gn, of Ln \ Ln“ is less than 31—". Let G = UnzlG'n. Then, as [2] -—) 1, 60(2) = o(1—|z[). Therefore by (3.9) Cd, is compact on 8. Moreover Cd, is not compact on the Dirichlet space. This follows from Theorem 2.7. The theorem above is a special case of the following proposition. We show that if Cd, is bounded on some Besov space then the compactness of Cd, on larger Besov spaces is equivalent to the compactness of C.) on the Bloch space. This result is similar to the compactness of C.) on weighted Dirichlet spaces Do, (a > —1). These are spaces 47 of holomorphic functions f on U such that [f(0)[2 + fulf'(z)|2(1—[z[2)°dA(:) < oo. MacCluer and Shapiro show in [19, Main Theorem, page 893] that if C4, is bounded on some weighted Dirichlet space Do then the compactness of C¢ on larger weighted Dirichlet spaces is equivalent to ct having no angular derivative at each point of UU. Proposition 3.6 Let 1 < r < q, 1 < p S q. Suppose that C4, : B, —> B, is a bounded operator. Then, C¢ : B,D —> B, is a compact operator [if and only if C¢ : B —-> B is a compact operator. Proof . First, suppose that Cd, is a compact operator on the Bloch space. For any AEU, IlcialIIPB, = [Iag(¢)IPI¢' (3.10) = [U la’.\(¢(z))l’l¢’(z)l'(1-|zl2)"2(Ia&(¢(z))ll¢'(z)l(1-|z|2))°‘”dA(z) < llC¢OAIIb_rllC¢OA| ,. B. S const.][C¢,a,\[[‘£’;r (by Theorem E and since C.» : B, —+ B, is bounded) . Therefore (3.10) and Theorem 3.1 yield that llC¢aAlleq -> 0 as [Al —-> 1. Thus by Theorem 2.7, Cd, : B,D —-) B, is a compact operator. The converse follows from Proposition 3.2. This finishes the proof of the proposition. C] The following theorem summarizes the above. If a composition operator is bounded on some Besov space then the compactness of the operator on larger Besov spaces, and from any Besov space to BM 0A, is equivalent to the Bloch compactness of the operator. 48 Theorem 3. 7 Let 1 < r < q, 1 < p S q, suppose that Cd): B, —-> B, is a bounded operator. Then the following are equivalent: 1. C¢ : B —) B is a compact operator. {e O 6 : B, -> B, is a compact operator. 3. Cd, : D —> BMOA is a compact operator. 4. C, : B, -—> BMOA is a compact operator. Proof. The previous proposition yields (1) 41> (2). Proposition 3.2 yields (3 ) => (1). Theorems 2.7 and 2.8 yield (2) => (3). Theorem 2.8 yields (3) (it (4), if 1 < p < 2. pr > 2 then (4) => (3 ) lS trivial, since the inclusion map,i :Bp —> D, is bounded. Moreover (2 ) => (4) follows as well (when p > 2) since the inclusion map, i: B, —> BMOA, is bounded. We have shown (3) => (1) (it (2) => (3) 4:) (4). This completes the proof of the theorem. E] Arazy, Fisher, and Peetre prove the following theorem in [2, Theorem 16]. Theorem F Let p be a positive measure on U, 0 < p < 00. Then, / (#42) < 00 v (1 - [lelp if and only if there is a positive constant e such that / |f’(~ )(lpdit )< cIIfIIB. for all f E 3. Note The proof of Theorem F can be used to show that a similar result holds for a 49 collection of positive measures {ftq : q E U}. That is, if 0 < p < 00, then d/iq sup/ —— < oo q€U U (1 — [z[2)P if and only if sup / If(P )Ipduq s cIIfIIs QEU for all f E B. These results, along with a non-univalent change of variables, yield the following characterizations of bounded composition operators from the Bloch space to Bp (1 < p < oo), BMOA, and H2. Proposition 3.8 Let (,b be a holomorphic self-map of U. 1. C¢ : B —> D is a bounded operator if and only if M w z I¢’(z)|2 z 00 /u(1—IwI'P’)2dA() L(1-l¢(z)|2)2dA()< , where n(¢;w) denotes the number of times (b takes the value w. Ifw is not in ¢(U) then let 17(4); w) = 0. 2. C,» : B —> Bp (1 < p < 00) is a bounded operator if and only if N100”I’(ZI”IZ2I)”’2 - /(1—IwI2lpdAw):/U(1—I¢TZ) I) dA(‘)<°°‘ 3. C45 : B —> BMOA is a bounded operator if and only if Nw,,) ’( —aq 311p/U___( q Wieu/ulqsh zll:1¢(z) I ()])dA() <00. qu (1 “‘ |w|2)2d “22) 50 4. C¢ : B ——> H2 is a bounded operator if and only if M w = I B, is a compact operator (hence C¢ : B—> BMOA is a compact operator as well). I¢’(z aqz( )I) Iqi->I/U(()]1-(I¢()I) (buzz) 0 then Cd, : B ——> BMOA is a compact operator. 2. If 51 Proof of (1). Let (fn) be a bounded sequence in B such that f,, ——> 0 uniformly on compact sets, as n —> 00. Then, “0.).“ng = flf.’.(¢ z))|”l<1’>(z”)I(1-I2I2)”‘2dA(z) = f + / If.’.(¢(z))l”l¢’(z)l”(1—Iz|2)”‘2dA(z) {zEU:6(z)|<1} {z€U:|¢(z)[S6} = [+11 (3.11) for any 0 < 6 <1. Then , |¢’(z)l”(1-Izl2)P‘2dA ,, 3 ,, “Hf"“P/ceu.was...) (1—I¢(2)IP)P (”)’ I' ) for any 6 > 0. Hence, as 6 —> 0 , I —> 0 by the Lebesgue Dominated Convergence Theorem ([27, Theorem 1.34, page 26]) and our hypothesis. Let c > 0 be given. Choose 6 6 (0,1) such that if h < 6 then I < e. For such an h, 11 = / I¢'(2)IPIf:.(¢(2))IP(1—I2IP)P-PdA(2) < e||¢IIEp (3.13) {zeU l¢>(z)l 0 uniformly on {2 E U: [qb( 2)] S h}. Thus (3.11) and (3.13) imply that there exist a positive integer N such that if n 2 N then [[C¢fn|[3p < const. 6. Thus, [lC¢,fn|[3p -—> 0, as n -> 00, and Theorem 2.11 yields that C4, : B -) Bp is a compact operator. The converse follows from Proposition 3.8. Proof of (2). Let (fn) be a bounded sequence in B such that fn —> 0 uniformly on compact sets as n —) 00. Let c > 0 be given. Then by our hypothesis there is a 52 6 > 0 such that if [ql > 1 — 6 then |¢'(z) I (1 " Ioqu: /( (Ml—I4) )< e. (3.14) Fix q E U such that [ql > 1 — 6. Then [If.'.(2)2))PII2)(21-Ia.()I)dA(2) _ I z 2 _ z 22I¢'(2)I2(1-Iaq(le2l 7 — [U If.(¢( ))I (1 W )I) (,_l¢(z,l,). 2121(2) |¢’(z)l2 —Ia.(2 )IP ) _1_5 (1-|¢(Z)|"’)2 <' l ) Using (3.17) it is easy to see, similarly to the proof of part (I), that / |f.’.(¢ )I'(I2 (2)I(1—I2IP)2A(2) = f + / |f.’.(¢(z))l2|¢’(z)|2(1—I2|2)dA(z) I2)(z)|>1—5 |¢(z)|Sl—6 < const. 6 (3.18) for n large. Then (3.15), (3.16), and (3.18) yield sup/UM) |f,’,(d> 2))I |¢'( )2)2 (1 — |aq(z)|2)dA(z) < const. 6. Therefore by (2.3), HC¢ntI. < const. 6. Thus, ||C¢fn||. —> O as n —+ 00, and Lemma 2.11 implies that Cd, : B —-> BM 0A is a compact operator. This finishes the proof of the proposition. [3 In view of Propositions 3.8 and 3.9 we obtain the following corollary. Corollary 3.10 For 1 < p < 00, every bounded composition operator from B to Bp is compact. Corollary 3.10 also follows from some nontrivial Banach space theory. Here is an outline of the argument. First, Bp is isomorphic to 1", since the LP Bergman space of U is isomorphic to I” (see [15, Theorem 6.2, page 247]). Next 8 is isomorphic to l°°, since [3 is isomorphic to the dual of the L1 Bergman space of U (see [10, Theorem 10 page 49]), which in turn is isomorphic to l°°. Moreover, if 1 < p < 00 then every 54 bounded linear operator T : l” —> I1 is compact (see [16, Proposition 2.c.3, page 76]). Thus, T“ : l°° —+ l" is compact for any q 6 (1,00). Also a bounded operator is an adjoint if and only if it is weak-star continuous. It is not difficult to show that if Cd, : B —+ 3,, is bounded, then it is weak-star continuous, and hence by the above argument, also compact. Next we give a characterization of compact composition operators whose range is a subset of VMOA. Theorem 3.11 Let (b be a holomorphic self-map of U, and X a M5bius invariant Banach space. Then Cd, : X —> VMOA is a compact operator if and only if pgfigL/UPW Wlw 2nu—2mam2%o=a Proof. First suppose that C¢ : X —> VMOA is a compact operator. Then A = cl({f o q) E VMOA : ||f||x < 1}) , the VMOA closure of the image under 0,, of the unit ball of X, is a compact subset of VMOA. Let c > 0 be given. Then there is a finite subset of X, B = (fl, f2, f3, ..., fN}, such that each function in A lies at most 6. distant from B. That is, ifg E A then there exists j E J = (1,2,3, ..., N} such that Im-toat<§. (aw) Since (f, o (6 : j E J} C VMOA, there exists a 6 > 0 such that for all j E J and .LKEMWRWU—kMflVMMfl< (2m) 6 4. By (3.19) and (3.20) we obtain that for each [ql > 1 — 6 and f E X such that Hflllx < 1 there existsj E J such that ./|(fo¢MW1~fiad)lwAk) 55 s 2 U |(f o d> - f. o ¢)’|2(1—Iaq(z)|2)dA(-:)+ 2 U |(f. o ¢)’(z)|2(1-laq(z)|2)dA(:) e e — 2—=. <24+4 c This proves one direction. In order to prove the converse, let (fn) be a sequence in the unit ball of X. By Lemma 2.9 and Montel’s Theorem there exists a subsequence n; < n2 < and a function g holomorphic on U such that fn,‘ —+ g uniformly on compact sets, as k —) 00. By our hypothesis and Fatou’s Lemma it is easy to see that 04,9 6 VMOA. We will show that ||C¢(fnk — g)“. —) O, as k —> 00. In order to simplify the notation we will assume, without loss of generality, that we are given a sequence (fn) in the unit ball of X such that fn —> O uniformly on compact sets, as n —> 00. We will show that "15120 ||C¢fn||. = 0. (3.21) To prove (3.21) we will use the equivalent BM 0A norm as given by (1.5). Thus, our hypothesis is equivalent to . 1 llm sup 3] |(f o ¢)’(z)|2(1 — |z|2)dA(z) = 0. (3.22) h"0 Helm») 301,9) UGX:|UHX<1} Let c > 0 be given. By (3.22), there exists a 6 > 0 such that if n E N, 0 6 [0,271’), and h < 6 then 1 E/SW) |(f,, 04)) (2)} (1 — |z| )dA(z) < e. (3.23) Fix ho < 6, 0 E [0,27r), n E N, and h 2 6. It is easy to see that there exists {91,02,...,QN} C [0,27r) such that S(h,0) is the union of the sets {5(h0,0,-) : j = 56 1,2, ..., N} and K, a compact subset of U. Hence, 1 1 _ +—— ,,o 'zzl—zszz S E /S(ho.0,) |(f ¢)( )I( l l) ( ) = I + II. (3.24) Since f,’, —> O uniformly on K, as n —> 00, there exists an N E N such that for n 2 N 11 g i (1 — |z|2)dA(z) 3 const. 6. (3.25) Moreover (3.23) yields, N I S 2 c = const. 6. (3.26) Hence (3.23), (3.24), (3.25), and (3.26) yield (3.21). Thus Lemma 2.11 yields that 0,), : X —> VMOA is a compact operator. [:1 There are symbols ¢ such that C,» is compact on BM CA but not on VMOA. For example, consider the self-map (15(2) = %exp{%}. Since ll¢lloo < 1, Corollary 2.12 yields that C4, is a compact operator on BM 0A. Moreover since (1) g [30, Ct, is not even bounded on VMOA (Theorem C, page 22). If (I) E VMOA then compactness of C2,, on BM 0A implies the compactness of C,» on VMOA. If T is a compact operator on a Banach space X, and Y is an invariant subspace of X such that T : Y —-> Y is bounded, then T : Y —> Y is a compact operator as well. Thus we obtain the following proposition. Proposition 3.12 Let qS be a holomorphic self-map of U. Then, I. If(b E VMOA and C¢ : BMOA —+ BMOA is a compact operator then C,» : 57 VMOA —-) VMOA is a compact operator. 2. Ifd> 6 Bo and C¢ : B —> B is a compact operator then C45 : 80 —> 30 is a compact operator. Next we show that the sufficient condition of compactness of Ca : B —> B M 0/1 in Proposition 3.9 is also necessary for the compactness of Cd, : B —+ VMOA. We will use Khintchine’s inequality for gap series (as done by Arazy, Fisher, and Peetre in [2, Theorem 16]), and Theorem 3.11. Theorem 3.13 Let (f) be a holomorphic self-map ofU. Then the following are equiva— lent: N . C¢ : B —-+ VMOA is a compact operator. N) |’()|(1—Iaq(z >12) , _ lclIlI-il,/U( ((1_'¢z )2I2) dA(4)—0. Proof. First, suppose that (1) holds. Then by Theorem 3.11 and since for all 9 E [0,27r) (see [1, Lemma 2.1)), 2 N(waq,)dA(w) = 0- f: 2n(ei0w)2"—l lim sup / |<1|-+l oe[o, 21r) ":0 Let c > 0 be given. Then there exists a 6 > 0 such that for any q E U with lql > 1 — 6 and any 0 E [0,27r), 22%” 11:0 N(w,q,q5)dA(w) < e . (3.27) 0dc_f. A/ 58 Upon integrating (3. 27) with respect to— :and using Fubini 8 Theorem, we obtain d6 2" °° . .. .. 2 d6 f2" A9— =/ / Z2716“? 'llw2 ’1 -— N(w,q,¢)dA(w) S e. (3.28) 0 27r U 0 "=0 27r Khintchine’s inequality (see [36, Theorem V.8.4]) for gap series yields that for any positive integer N /21r 0 Therefore (3.28) and (3.29) imply that N Z 2nei0(2"—1)w2"-1 n=0 2d9 N ——~ 22nlw|2"“-2. (3.29) n=0 f0? A92—N [U {222% Vim-2} N(w,q,¢)dA(w). (3.30) n-O It is shown in [2, Theorem 16] that 00 2,, 2n+l const. 2 2 lwl 2 my; , (3-31) 11:0 for any w E U such that |w| Z %. Hence (3.28), (3.30), and (3.31) yield N(w,q,¢) 2" d0 _— < . _ . . /U (1 _ lwI2)2clA(w) _ const [0 A92” < const 6 , (3 32) for any q E U with |q| > 1 — 6, and any 6 > 0. Thus (3.32) yields (2). Conversely, suppose that (2) holds. Fix f in the unit ball of the Bloch space. Then, [1 |f’(¢(z))l2|¢’(z)|2(1 — Iaq(z)l2)dA(z) we )|:(1|- laq(z)l"’) <|lf|l§/(( WW), dA 59 |¢’(z)|2(1 -— laq(z)|2) 5 U (1—1¢(z>|2)2 0“ (z). The righthand side of the above inequality tends to O, as |q| —> 1, by our hypothesis. Hence Theorem 3.11 yields that C4, : B —+ VMOA is a compact operator. This finishes the proof of the theorem. D Proposition 3.14 Let <15 be a holomorphic self- -map ofU. If C¢z BMOA —+ VMOA is a compact Operator then |¢'(z )l” (1 _ laq(z )lz) _ Iii-H/U 1_ l¢(z )l2 dA(z) _ 0 _ Proof. By Theorem 3.11 and since f9(2) = log 3% E BMOA for all 0 6 [0, 27r), lim /Up|f9(z )|2N( (w ,q, gbz)dA( )= 0. (3.33) I‘ll—*1 9€S[0, 21r) Let c > 0 be given. Then there exists a 6 > 0 such that for any q E U with |q| > 1 — 6 and any 0 E [0,27r), A. = [U |f5(z)I2N(w,q,¢)dA(z) = /-———— |-————1 _ el"9w (w ,,q ¢)dA(w )< (3.34) Integrating (3.34) with respect to;— :and Fubini’ 5 Theorem yield (10 27" 1 (19 A __ — _ _ /021r 92"- ./(ij{v/o l1 - 6"9wl2 2n} N(w,q, (Md/1h“) S 6- 60 Thus, / WdA(w) < C , U 1— |w|2 for all lql > 1 — 6, and all e > 0. Therefore . warm—lam?) Z _ IllTI/u 1-|¢(z)|2 “(l-0' Next we show that composition operators on BM 0A and VMOA, where the symbol is a boundedly valent holomorphic function whose image lies inside a polygon inscribed in the unit circle, are compact if and only if they are compact on the Bloch space. We will use Propositions 3.9, 3.12, and the following theorem of Pommerenke ([24, Satz 1]). Theorem H Let f be a holomorphic function on U such that sup v/Iw—woKl 17(f,w)dA(w) < oo , wO where the supremum is extended over all points wo in the complex plane. Then, fEBMOAéfEB, fEVMOAfifEBo. In page 46 we defined a nontangential approach region (to, (0 < a < 1) in U with vertex C 6 3U. The exact shape of the region is not relevant. The important fact that we will use in the theorem below is that there exists 0 < r < 1 and c > 0 such that if z E (to, and IC — 2] < r, then IC - 2| S c(1-|Z|2)- (3-35) 61 Theorem 3.15 Let ¢ be a boundedly valent holomorphic self-map of U such that ¢(U) lies inside a polygon inscribed in the unit circle. Then the following are equiva- lent: 1. C.» : B —> VMOA is a compact operator. f8 0 e : B —-> BMOA is a compact operator. 3. Cd; : BMOA —-> BMOA is a compact operator. 4. C43 : B ——> B is a compact operator. 5. 0.1, : 80 —> 80 is a compact operator. 6. C¢ : VMOA —> VMOA is a compact operator. Proof. (1)=> (2): (3) is clear. (3)=>(4). This is valid for all holomorphic self-maps of U (Proposition 3.2). (4): (5). Since gb is a boundedly valent holomorphic self-map of U, <15 6 D C VMOA C 80. Thus (15 6 30. The compactness of 0,, now follows from part (2) of Proposition 3.12. (5)=>( I ) By Madigan and Matheson’s Theorem 1 (see [22]) 0.), is a compact operator on the little Bloch space if and only if .m I¢’(z)l(1—lzI2) _ Ill»l 1-I<¢>(z)|2 ’0' It follows that log If,» 6 30 for each w 6 EU. By Theorem H each boundedly valent function in 130 must belong to VMOA, hence log 31:) E VMOA for each w E 8U. Thus <10 —1—->' gw—wo 2 . _ ~ , _ Iggy] U (1 |a,(.)| )dA(z) —0 62 hence |q$’(z Mia“ )le ~ _ (q141/|w(—1(z¢)|2 dA(.)_o, (3.36) for each w E 8U. Let {wj : 1 S j S n} be the vertices of the inscribed polygon containing qb(U). Break the unit disc up into a compact set K and finitely many regions E: = {2 € U1 le -¢(Z)| < 7‘} where r is chosen so that the regions are disjoint, and so that M - 45(2)] S Coust-(l-|<15(Z)|2) for each 2 6 EJ- and each j. Then for each q E U, l¢’(Z)l2(1—laq(2)l2) ons ]¢'(Z)]2(l _ laq(z)l2) ~ E: (1-l¢(zll2)2 dA(z) S C t. /E, ij — ¢(z)l2 CM“). Hence |¢’(z)|2(1-Iaq(z)|’) ,, la (1—|¢(2)l2)2 dA‘“) _Z/E+ +/h l¢'(z ((1l:1|¢,—|a|q]: 22)] )dA(z) cons ]¢'(Z)l-l01q(z)]2) ~ < t ;E,Iw.(1—¢ VMOA is a compact operator. Proposition 3.12 yields (3) => (6). If(6 ) holds, that IS C¢: VMOA —> VMOA 1s a compact operator, then C), is weakly compact on VMOA. Hence by Theorem V1 5.5 in [9, page 189], C¢(BMOA) C VMOA. Thus log 6 VMOA (w E 8U). 1 w—d>(z) Thus by the proof of (5) => (1) we obtain (6) => (1). This finishes the proof of the theorem. D Definition 3.16 A region G C U is said to have a nontangential cusp at C E 6U if , lIsz 11m = z-H ll — z] 260 Note Theorem 3.5, Theorem 3.15 and Madigan and Matheson’s Theorem 5 (see [22, page 2685]) yield that if d) is a univalent self-map of U such that ¢(U) has finitely many points of contact with 8U and such that at each of these points ¢(U) has a nontangential cusp, then 0,), is a compact operator on 8,, (p > 2), on B M GA, and on VMOA. CHAPTER 4 Final remarks and questions Madigan and Matheson showed that if Cd, : 30 —> [30 is weakly compact then it is compact. Is a similar statement valid for C¢ : VMOA —-> VMOA? That is, does C¢(BMOA) C VMOA imply that C), is a compact operator on VMOA? In Theorem 3.15 we showed that for certain boundedly valent holomorphic self- maps of U, compactness of Cd, on B M 0/1 is equivalent to the compactness of C2), on B. Is this true for all boundedly valent symbols? In Theorem 3.15 we used that 45 is boundedly valent to be able to conclude that if log 633(7) 6 80 then log w_.l¢,(z) E VMOA (w E 3U). We should mention here that Stroethoff, using an area version of the BM 0A counting functions, characterizes exactly when a function ab 6 Bo belongs to VMOA. He showed in [33, page 78] that a function d 6 Bo belongs to VMOA if and only if for every 6 > 0 1 lim sup / tn(q§o ap — w,t)dt = 0. o [pl—)1 w |¢(P)-wl25 In Theorem 3.13 we showed that the compactness of C4, : B —) VMOA is de- termined by the “behavior” of {C¢z(e‘9¢(z))2n) : 0 6 [0,21r)}. Does a similar statement hold for compact operators Cd) : BM CA —> VMOA? That is, is it true 64 65 that C¢ : BMOA —> VMOA is a compact operator if and only if 2 (1— Iaq(z)l2)dA(z) = o ? 1 I (1031—47,) (‘2) lim sup / lq|-+196[o,21r) U BIBLIOGRAPHY BIBLIOGRAPHY [1] J .M. Anderson, J. Clunie and Ch. Pommerenke, “On Bloch functions and normal functions,” Journal fiir die Reine und Angewandte Mathematik 270 (1974), 12- 37. [2] J. Arazy, S.D. Fisher and J. Peetre, “Mobius invariant function spaces,” Journal fiir die Reine und Angewandte Mathematik 363 (1985), 110-145. [3] A. Baernstein, “Univalence and bounded mean oscillation,” Michigan Mathe- matics Journal, 23 (1976), 217-223. [4] A. 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