I)V1£SI.J RETURNING MATERIALS: P1ace in book drop to ngaAmgs remove this checkout from .-3— your record. FINES will be charged if book is returned after the date stamped below. CHARACTERIZATIONS OF THE BLOCH SPACE AND RELATED SPACES By Karel Mattheus Rudolf Stroethoff A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1987 ABSTRACT CHARACTERIZATIONS OF THE BLOCH SPACE AND RELATED SPACES By Karel Mattheus Rudolf Stroethoff In the first chapter we give local and global Dirichlet-type characterizations for both the Bloch space and the little Bloch space, generalizing some of the characterizations for these spaces given in [2]. In the second chapter we characterize the Bloch space and the little Bloch space in terms of the pseudo—hyperbolic metric on the unit disk; it is shown that the Bloch space coincides with the class of analytic functions on the disk that are uniformly continuous with respect to the pseudo-hyperbolic metric. In chapter three we further develop some of the results obtained by Baemstein in [3], where he proved that an analytic function on the disk belongs to the space BMOA if and only if the Mobius transforms of the function form a bounded family in the Nevanlinna class. We give a description of the space VMOA in terms of the Nevanlinna characteristic. A description of VMOA cannot be obtained by simply replacing Baemstein's boundedness condition by the corresponding vanishing condition (as is usually the case). We then formulate and prove analogous characterizations for the Bloch space and the little Bloch space in terms of an area version of the Nevanlinna characteristic. In the fourth chapter we give a different proof of Baemstein's value distribution characterization for BM 0A [3], Theorem 3, and we formulate and prove the corresponding description of the space VMOA. Defining an area version of the counting function used in the value characterizations for BMOA and VM 0A, we obtain analogous results for the Bloch space and the little Bloch space. Karel Mattheus Rudolf Stroethoff In chapter five we give estimates for the growth of analytic functions in weighted Dirichlet space, which then are used to give necessary and sufficient conditions on the growth of an analytic function on the disk for inclusion in the Bloch space or the little Bloch space. Chapter six briefly discusses cyclic vectors in the little Bloch space. We generalize a theorem of Anderson, Clunie and Pommerenke [1], Theorem 3.8. In the seventh chapter we consider Hankel operators with integrable symbol. The Hankel operators that we study are defined by projecting onto the orthogonal complement of the Bergman space. We first prove that these Hankel operators transform in a unitarily equivalent way if the symbol is replaced by one of its Mobius transforms. We then restrict our attention to Hankel operators with conjugate analytic symbol, and show Sheldon Axler's results [2], Theorems 6 and 7, hold if the operator norm of the Hankel operator is obtained by putting a weighted lP-norm on both its domain and its range. REFERENCES : [1] J. M. Anderson, J.G. Clunie and Ch. Pommerenke, "On Bloch functions and normal functions," Journal fiir die Reine und Angewandte Mathematik 270 (1974), 12—37. [2] S. Axler, "The Bergman Space, The Bloch Space, and Commutators of Multiplication Operators," Duke Mathematical Journal 53 (1986), 315-332. [3] A. Baemstein, "Analytic Functions of Bounded Mean Oscillation," pages 3-36 in Aspects of Contemporary Analysis, edited by DA. Brannan and J.G. Clunie, Academic Press, London, 1980. ACKNOWLEDGMENT I would like to thank my advisor, Sheldon Axler. He has been an excellent teacher and he has provided me with very helpful guidance and advice. I thank him for the many valuable conversations; I particularly appreciate his interest, enthusiasm and encouragement. ii TABLE OF CONTENTS page Chapter 0 ..................................... 1 Chapter 1 .................................... 24 Chapter 2 .................................... 43 Chapter 3 .................................... 51 Chapter 4 .................................... 63 Chapter 5 .................................... 79 Chapter 6 ................................... 103 Chapter 7 ................................... 112 Bibliography ................................. 139 iii Chapter 0 In this chapter we give some background information, establish most of the notation for the chapters that follow, and list the major results of this thesis. Since we will be dealing with Bloch functions on the unit disk, we start with a theorem of the man whose name is attached to these functions. Let D = {z e G: : lzl < 1} denote the open unit disk in the complex plane. The basic idea of a Bloch function on D goes back to the following theorem of Andre Bloch [9]. Bloch 's Theorem : There exists a finite positive constant b such that if g is an analytic function on ID, normalized so that g (0) = 0 and g' (0) = 1, then there is a disk A contained in ID on which g is one-to-one and such that g (A) contains a disk of radius b. Forwe (I: and0 0 for which there exists an open connected neighborhood .(2 of z in [D such that f is one-to-one on {2 and f (.0) = A( f (z), r), unless f ' (z) = O (and thus there are no such r > 0), in which case we let df(z) = 0. If df(z) > 0, then necessarily df (z) < co, and it is easy to show that the supremum in the definition of quantity df(z) is actually attained, i.e., there exists an Open connected neighborhood .(2 of z in [D such that f is one-to—one on .Q and f (.(2) = A( f (z), df(z)). A disk A( f (z), r) that is the image under f of an open connected neighborhood of 2 on which f is one-to-one, is called a schlicht disk of f around f (2). Thus the number df(z) is the radius of the largest schlicht disk of f around f (z). The first systematic study of this quantity was done by W. Seidel and IL. Walsh in [34]. As an easy consequence of Schwarz's Lemma we have: df(0) s lf’(0)|. (0.1) Another easy property is that for y at 0: dyf (z) I y I df (z) . (0.2) For A e D let the Mobius function {P}. : ID —-) D be defined by (01(2) = A ._Z , 26113. (0.3) l-lz Then it is easily verified that for an analytic function f on D and for every )1 e D : df. be) = dfaple». (0.4) Using this quantity, Bloch's Theorem can now be restated as follows: There exists a finite positive constant b such that if g is an analytic function on D, normalized so that g (0) = 0 and g' (0) = 1, then there exists a point w e D for which dg(w) 2 b. If f is an analytic function on D and f ’ (0) at 0, then we can apply this version of Bloch‘s Theorem to the function g = ( f - f (0))/ f ' (0). Using the properties (0.1) and (0.2) it follows that there exists a point w e D (depending on f) such that 1 df(0) S lf'(0)| S 3 df(w). Observe that the above is trivially satisfied if f ' (0) = 0 (with any w e D), so that the initial restriction that f ' (0) :t 0 can be removed. Now take A e D. It is elementary to verify that (p110) = I A I2 - 1, so that the above inequality and (0.4) give that for every 1 in D there exists a point w}. e D for which 2 , l df(l) S (1 -IM )lf (2.)l _<_ 7)- df(w/1). (0.5) Now, for an analytic function f on D we set IIfIIB = sup (1-1212) If'(z)l. ZED The Bloch space 58 is the set of all analytic functions f on D for which ll f “B < oo. Even though || . "EB is not a norm, we will refer to II f "13 as the Bloch norm of function f. The quantity If (0) I + II f “9 defines a norm on the linear space $3, and we will see later that $8 equipped with this norm is a Banach space. Two quantities Af and Bf, both depending on an analytic function f on D, are said to be equivalent, written as Afz Bf , if there exists a finite positive constant C not depending on f such that for every analytic function f on D we have: If the quantities Af and Bf are equivalent, then in particular we have Af < co if and only if Bf < co. It follows from (0.5) that for an analytic function f on D we have the equivalence ||f|| = sup d (z). (0.6) 13 26D f For a region {2 C (I: let H°°(.Q) denote the algebra of all bounded analytic functions on .Q . We will simply write H°° for H°°(D). It is clear that the image of a bounded analytic function cannot contain arbitrarily large schlicht disks, so that the equivalence (0.6) immediately gives us the inclusion H°° C 13. In the argument leading from Bloch's Theorem to the equivalence (0.6) the Mobius functions on the disk played an important role. For an analytic function f on D and a point 2. e D, we will call the function f o (P). - f (A) a Mobius transform of function f. It follows from (0.4) and equivalence (0.6) that SE is invariant under Mobius transforms, i.e., iffe i3 and l e D, then fo (PA —f(/'l) e 13. This is also easy to see from the definition of the Bloch norm. Let f be an analytic function on D, and let )1 e D. We have already observed that 4,110) = l/‘l I2 - 1, so that by the chain rule we have (to 49,1 >' (0) =f'(l) (pg (0) = (I MZ- 1)f'(/1). It follows that Ilfll = sup |(fo€p )'(0)|, B 16D '2' hence for every 2. e D: "fHEB ="f°(p}_"33° (0.7) In [31] Rubel and Timoney showed that the Bloch space ‘13 is maximal among all Mebius-invariant Banach spaces of analytic functions on D which have a decent linear functional. Contained in the Bloch space is the little Bloch space 380 , which is by definition the set of all analytic functions f on D for which 2 , - (1-lz|)f (z) —> 0 as Izl—>1. It follows immediately from (0.5) that if f is in $80 then df (z) —) 0 as I2 I —-) 1’. That the converse is also true follows from the following result of Pommerenke ([27], Theorem 1): If f is analytic on D and df(z) S l for all z e D, then for all z e D: 2 (1-1212) If'(z)| s— [df(z) (3-df(z)). (0.8) J3 We can actually obtain a simpler proof as a result of the following theorem: Theorem 0.1 : Let f be an analytic function on D. Then for every 2 e D we have: (1 -1212) If'(z)l 5 4de(2) IIfIISB . (0.9) This theorem has the following corollary: Corollary 0.2 : Let f be an analytic function on D. Then there exists a point w e D for which 1 df(w) 2 50 IIfII13 . (0.10) If for an analytic function g on the unit disk g' (0) = 1, then H g ":8 2 1, and we see that Bloch's Theorem is a consequence of Corollary 0.2 (and conversely, it is easy to show that Corollary 0.2 is a consequence of Bloch's Theorem). A proof of Theorem 0.1 can be based on the following lemma which Edmund Landau used to give a proof of Bloch's Theorem (see [20], Satz 2). Lemma 0.3 : Let 0 < R < 00. Let g be analytic on the disk A(0, R), such that g (0) = 0 and a = lg' (0) l > 0. Suppose that lg (z) I s M for all | z I < R. Then: R202 6M dg (0) 2 (0-11) The following proof is derived from Landau's proof. Proof: Without loss of generality we can assume that R = l and M = 1 (otherwise consider the function h on A(0,l) = D defined by h(z) = g(Rz)/M for z e D). Suppose that g is analytic on D, such that g(0) = 0, a = lg' (0) I > 0, and lg (z) | s 1 for all z 6 ID. We must show that dg(0) 2 a2/6. Let W g(z) = 2 anz" ,z e D, n=l be the Taylor series expansion of function g. Then it is easy to show that Ian I S 1 for all n in IN. In particular a = lal | S 1. So if we put p = a/4, then 0 < p S '/4. Take a point w in A(0, a2/6), and consider the function gw defined on D by gw(z) = alz - w for z e D. For |z|=p wehave oo oo 2 2 Ig(z)-w-gw(z)lS E lanllzlns E owl—3%. n=2 n=2 l-p Since lw I < a2/6 we also have that for lz I = p , lgw(z) I 2 lalz I - lw l > a2/4 - a2/6 = (22/12. Thus for all Izl = p we have |g(z) - w - gw(z) l < lgw(z)l . By Rouché's Theorem the number of zeros of g - w in A(0,p) is equal to the number of zeros of gw in A(0,p), which is easily seen to be one. This shows that A(0, a2/6) is a schlicht disk around g(0) = 0, so that (18(0) 2 a2/6, as was to be shownD Proof of Theorem 0.1: Let f be an analytic function on D. We must show that (0.9) holds. In view of (0.4) and (0.7) it suffices to show lf'(0)l s 4de(0) Ilfu33 FmflzlSké , 1 4 If (z)l s __ llfllfa s 3 llflLB, (1-121 sothatforlzlS'/2: |f(z) -f(0)l s Izl lf’(tz)| dt IA 4 2 3 ufn,B = .5 IlfIIB. Apply Lemma 0.3 with g =f-f(0), R = %t,M = (2/3) llfllg , and a = lf'(0)l. It follows from (0.11) that 6M R2 . 2 _ If (0)1 .<_ df(0) _ 16df(0) IIfIIE, and the proof is completeD Proof of Corollary 0.2: Let f be an analytic function on D. If N f IIB = 0 then there is nothing to show, so assume that IlfIISB > 0. Let 0 < y < 1. Choose a w e D for which 2 . (1..le )lf (w)! 2 yIIfIIEB. Then it follows from (0.9) that 2 df(w) 2 (.3!) llfllg, from which (1.10) follows by taking 7 sufficiently largefl We now turn from the geometric aspects of Bloch functions to the functional analytic aspects of the linear space 33. In [13] the Bloch space is identified as the dual space of a Banach space whose norm is defined by an area integral. This implies that the Bloch space is a Banach space (which can also be proved directly from the definition). We will now introduce the Bergman spaces on the unit disk. Let A denote the usual Lebesgue area measure on the complex plane (II. For an analytic function f on D and 0 < p < no we define l/p llfll = JlflpdA/tt LP a [D The Bergman space L up is defined to be the set of all analytic functions f on D for which II f II L 0p < oo . The subscript a stands for "analytic." Clearly each Bergman space Lap is a linear space. For 1 S p < oo , || . "Lap is a norm on La? , and equipped with this norm Lap becomes a Banach space. For 0 < p < 1, ll . "Lap is no longer a norm, but N f - g IIPLap defines a translation invariant complete metric on Lap , so that Lap is a Fréchet space. If 1 < p < oo , let p’ = p /( p - 1) denote the conjugate index. The dual space of Lap can be identitied as Lap' : defining the pairing = J f(Z) EYES dA(Z)/7t . (0.12) [D for f 6 Lap, g 6 Lap', every bounded linear functional on Lap is of the form f 1+ (f, g>.(f e La”) (0.13) for some unique g 6 Lap'. Moreover, the norm of the linear functional in (0.13) is 10 equivalent to the norm ll g “Lap' (see [7] for a proof). Very much in the same way the dual of the Bergman space La1 can be identified as the Bloch space 38. In [2] and in [13] the Bloch space is shown to be the dual of the space fl which is defined to be $1 = { f: f is analytic on D and f ' 6 L01}. The pairing used in both papers involves the derivative of the function in f]. This is not parallel to the pairing in (0.12); it seems more natural to pair a Bloch function with a function in Lal. This was done by Sheldon Axler in [7]. We will outline his results. There is however a problem with the pairing as defined in (0.12): there exist f 6 La1 and g e 13 such that the product f g' is not integrable over the disk D. To overcome this problem define the pairing by = lim Jf(z) 3(7) dA(z)/7r. (0.14) l—-)l- (ID If g e 13, then (0.14) is defined for every f 6 La1 and the map f H ,(f 6 La‘) (0.15) is a bounded linear functional on La1 with norm equivalent to II g "$8 + l g(0) | , and every bounded linear functional on La1 is of the form (0.15) for some unique g 6 $13. Finally, the dual space of the little Bloch space 80 can be identified with La1 : every bounded linear functional on 130 is of the form f l—> (f, g), (fe :80) (0.16) for some unique g 6 L01, and the norm of the linear functional in (0.16) is equivalent to the norm ll g “ml. 11 For A e D recall the definition of the Mebius function (PA defined in (0.3): (p(z) = ’1'] ,zeD. A 1-12 The function 4’1 is easily seen to be it own inverse under composition: (90,10 (0))(z)=z forallze D. The following identity can be obtained by straight forward computation: 1-3143(2) _ l-q)A(u)z __ ——_—, (u,).,z eD). (0.17) I'UA 1-12 The special case that u = 1 yields (1-itpl(z))(1-iz) = 1.1112, (11,26 D). (0.18) If we substitute u = 901(2) in (0.17) and make use (0.18) we obtain the identity: 2 =(1-I2I2)(1-lz|2) 2 (1.26 D). (0.19) ll-lzl 1- l (pl(z)| 12 A slightly different form in which we will frequently apply identity (0.19) is: 1 - ltpl(z) 12 2 =|(p'(z)| ,(/'l.,ze D). (0.20) A l—lzl For points A, z in the disk D the pseudo-hyperbolic distance d(/'L, 2) between A and z is defined by dd, 2) = I (pA(z) I . Then it can be shown that d is a metric on D (see, for example, [14], page 4). For each point 2. e D and 0 < r < l , the pseudo-hyperbolic disk D(/l,r) with pseudo-hyperbolic center A and pseudo-hyperbolic radius r is defined by D(/'l,r)={ze D: d().,z)/tt , (0.2220 Dal) 0m”) ll- 1 W l and 2 2 l—l I 1 (ho (pAXZ) dA(z)/7r = I h(w) .(——_—_L)—z dA(w)/7t . (0.22b) D(0,r) Dar) Il-Awl Many of the properties of the Bloch space and the little Bloch space are analogous to their counterparts in the classical Hardy space setting. Recall the definition of the Hardy spaces: for an analytic function f on D and 0 < p < oo define 1m 2n llfll = sup i I lf(re'6)lp d0 Hp 22: 0 0Sr = I fit) g’to (1110(4) . (0.23) 8D for fe HP and g GM), ([12], Theorem 7.3). Before we give Charles Fefferman's identification of the dual space of H1 we need to introduce more notation. A connected subset I C 8D for which you ) > 0 will be called an arc in 8D. For a function g e L1(8D,u0) and an are I in 3D let g, denote the average of g over I: 1 g, = —Jgd#0- #00), For a function g e L1(BD,/,10) let 1 . llgllBMO = sup { —Ilg-glldu0: I anarcrnBD] . 1 110(1) 15 A function g e L1(BD,u0) for which ll g "3M0 < oo is said to be of bounded mean oscillation. The set of all functions in L1(BD,/,LO) that are of bounded mean oscillation is denoted by BMO. The class BMO was first introduced by John and Nirenberg in [18] (in the context of functions defined on cubes in IR" ). Define BMOA = { fe H1: f* e BMO }, and for fe BMOA set * llfllBMOA =llf "BMO‘ Equipped with the norm N f ll BMO A + l f (0) l , BMOA is a Banach space. For 0 < p < oo it can be shown that for every analytic function f on D: "fIIBMOA = sup llfo (0/1- f(/1)IIH (0.24) t e D p Charles Fefferman proved that the dual space of H1 can be identified with the space BMOA . There is however a problem with the pairing as defined in (0.23): there exist functions f e H1 and g e BMOA such that f*g*' is not integrable over the circle 8D. Fefferman showed that if g e BMOA, then the map W(f) = lf*(€) 8*(0 du0(o,f e H°° (0.25) an) extends to a bounded linear functional on 1-11 with norm equivalent to H g "BMOA" lg (0) l, and every bounded linear functional on H1 is of the form (0.25) for a unique g e BMOA (for a proof see [8]). By using Taylor series it is easy to see that If ' (0) l S ll f "”2 for every analytic function f on D. It follows that for an analytic function f on D and a point A e D: 16 (1 -1212) If’(/l)l s llfo (p - f(/l)ll 2. (0.26) A H Thus we have the inclusion BMOA C 13 . Paley's integral inequalities (see Chapter 5) and a change-of-variable give us that for every analytic function f on D: 2 I lf’(z) 12(1 Jot/1(2)?) dA(z)/7t s llfo (pl - for)" 2 s [D H l/\ 2 I lf’(z)l2 (Mo/1(2)?) dA(z)/7t. (0.27) [D It follows from (0.24) and (0.27) that for every analytic function f on D: z sup lf’(z)l2(1 Ago/1(2) 12) dA(z)/7t . "fHBMOA tent) D In [32] Donald Sarason introduced the space VMO of functions of vanishing mean oscillation defined by 1 Ilg-glldpo -—> 0 as pO(I)—>0}. #00) , VMO ={g e L‘(alD,,u0): Define VMOA = { f 6 H1: f * e VMO }. Since clearly VMO is contained in BMO , we have that VMOA is contained in BMOA . It can be shown that analogous to equivalence (0.24), if 0 < p < co, then for every analytic function f on D: 17 feVMOA 4:)[llfotpl-f(l)ll p—rOaslAl—H']. (0.28) H From (0.26) and (0.28) we get the inclusion VMOA C 330 . From (0.27) and (0.28) we see that for every analytic function f on D: f e VMOA 4:» I lf’(z)l2(l-|(pl(z)|2) dA(z)/7r —+ 0 as 121—91‘ [D For an analytic function f on D and t e (0, l) the dilate f, is defined by f, (z) = f (tz) for z e D. It can be shown that an analytic function f on D belongs to VMOA if and only if ll f - ft "BMOA —) 0 as t —> 1'. Since each dilate of an analytic function is continuous on a neighborhood of D it is easy to see that the space VMOA is the closure in BMOA of the set of all polynomials. The dual space of VMOA can be identified with H1: if g 6 H1, then the map W) = like g'm duos) . f apolynomial. BID extends to a bounded linear functional on VMOA with norm equivalent to II g ll 111 , and every bounded linear functional on VMOA can be obtained in this way. 18 We new list the major results in this thesis. In the following two theorems we give local and global Dirichlet-type characterizations for the Bloch space and the little Bloch space, generalizing some of the characterizations for these spaces given in [6]: Theorem 1.7 : Let 0

0 as Ill —9 1'; -np/2 ID(2,r)I 0W) (0) I lf(n)(z)lp(1-|zl2)"p'2dA(z)/7t —-> 0 as 111—) 1'; D(l,r) (a) I lf(")(z)lp(1-|zl2)np'2(1-I(ol(z)|2)2dA(z)/7t —> 0 as 121—911 [D 19 In [8] Baernstein proved that an analytic function on D belongs to the space BMOA if and only if the Mebius transforms of the function form a bounded family in the Nevanlinna class. The following theorem gives a description of the space VMOA in terms of the Nevanlinna chacteristic T : Theorem 3.3 : For an analytic function f on D the following Statements are equivalent: (a) f e VMOA ,° f o (p, - f (A) p (b) foreveryp>0wehave that T( )—> 0 as Ill—>11 The following two theorems give analogous characterizations for the Bloch space and the little Bloch space, in terms of Ta , an area version of the Nevanlinna characteristic: Theorem 3.6 : For an analytic function f on D the following statements are equivalent: M) fefi; (B) sup TGUO (p, - f(7l)) < 0°. 26D Theorem 3.7 : For an analytic function f on D the following statements are equivalent: (a) f 6130; f° to, -f(/1) p (b) Foreveryp>0wehavethatTa( )—>0as Ill—+1-. 20 In [8] Baernstein gave a value distribution characterization for the space BMOA. The following theorem describes the space VMOA in terms of the counting function N (for which the definition is given in chapter 4): Theorem 4.3 : For a nonconstant analytic function f on D the following statements are equivalent: (a) fe VMOA ; (b) for every 6 > 0 we have: sup {N(w,}l,f):we (Land |f(/l)-wl26} -> 0 as Ill—9 1'. Defining N a , an area version of the counting function N (see chapter 4), we have analogous results for the Bloch space and the little Bloch space: Theorem 4.4 : For a nonconstant analytic function f on D the following statements are equivalent: (A) f 6 33; (B) sup {Na(w,l,f) : we (13,16 D and lf(/'l)-w|21} < oo. Theorem 4.5 : For a nonconstant analytic function f on D the following statements are equivalent: (a) f e 1’80; (b) for every 5 > 0 we have: sup {Na(w,/'L,f):we (I: and lf(/’I)-w|25} —-9 0 as Ill—9 1'. 21 co ForananalyticfunctionfonD and0 0 as Ill —->1'. The following result on cyclic vectors generalizes a theorem of Anderson, Clunie and Pommerenke ([2], Theorem 3.8). It is similar to a result of Brown and Shields for the Dirichlet space ([10], Theorem 1). Corollary 6.4 : Let f, g e 90 , such that l f (z) I 2 lg (z)| (z e D), and suppose that g is bounded and g 2 is cyclic for $30. Then f is cyclic for ‘18 0‘ Similar to Proposition 11 of [10] we have the following result: Corollary 6.6 : If f, g e 130 n H°°, and if fg is cyclic for ‘80, then both f and g are cyclic for $30. 22 The Hankel Operators Hf are defined by projecting onto the orthogonal complement of the Bergman space (see chapter 7). These Hankel operators transform in a unitarily equivalent way if the symbol is repaced by one of its Mobius transforms: Theorem 7.1 : Let f e L1(D,dA/7t). For each l e D the Hankel operators Hf and H are u nitarily equivalent. full/1 More precisely, there exist unitary operators U 1 : L 02 —-> La2 and U2 : (L 02% —> (L 02H such that U1(H°°) C H°° and UZOHfotp =Hfo U1. leD Sheldon Axler's results [6], Theorems 6 and 7 hold if the operator norm of the Hankel on both the domain and the operator is obtained by putting a weighted lP-norm ll . llp'a range of the Hankel operator: Theorem 7.3 : Let l

0 as 121—)1'; a 1 - (c) I lf(z) -f I” dA(z)’7t —) o as Ill—>1; D(l,r) ID(A.r)I our) (a) 1 I lf(z) - for)” dA(z)/7t —) 0 as 121—) 1'; |D(l,r)| D(l,r) ,H°°(D(2,r))) —+ 0 as 1,11 —> 1'; l,r) (e) distance (flm 0) area f(D(l,r)) -> 0 as Ill—9 1' ; (g) I lf’(z)|2 dA(z)/7t —> 0 as Ill—9 1-; 001;) (h) llf-ftllB—ro ast—el'. 26 Although the definition of the Bloch space only involves the first derivative of the function, the following lemma gives characterizations involving higher derivatives. Lemma 1.3 : Let n e IN. Then for an analytic function f on D the following quantities are equivalent: (A) llflleB ; n-l (B) sup (1-1212)"If(")(z)l + zlf(k)(0)l. ZEID k=l Proof : For n = 1 the equivalence of the two quantities is precisely the definition of the Bloch norm. By induction it suffices to show that for a fixed n e IN, for every analytic function f on D the quantities -1 (B) sup (14212)" If(")(z)l + S: lf(k)(0)| n 26D k=1 and ’1 (am) sup (142.12)"+1 If("“)(z)l + Z lf(k)(0)| ZED i=1 are equivalent. 27 Let g be an analytic function on D, and let w e D. Then: 1 lg(w) - g(0)l s I lwl |g'(tw)| dt 0 l I l = I w (1-Itwl)"+1lg'(tw)l dt 0 outset)“ l s 'W' 1dt .sup (l-lz|)n+1lg'(z)l 0 (1-thI)“ zeID .<_ 1 sup (l-lzl)n+llg'(z)l . n(1-le)" zeID Thus (l-le)n |g(w)| S —1- sup (l-Iz|)n+1lg'(z)| + Ig(0)l . n 26D Put g = f (n), multiply by (1 + lwl)" (which is less than 2"), and take the supremum over all w e D, to get sup (1 _ l2'2)r1 +1 If(rt +1) ZEID :tIN, sup (l-le2)n lf(n)(w)| s (2)1 + lf(")(0)l . w e D Hence quantity (3") is less than or equal 2"ln times quantity (3,, +1). For the converse, fix 2 e D and put r = (1 - lz l)/2 . Again let g be an analytic function on D. By the Cauchy Integral Formula 28 g.(z) = _1_ I g(w)2 dw ’ 27” lw-z|=r (W‘Z) sothat |g'(z)l S -:-sup[|g(w)l:lw-zl=r}. (1.1) Iflw-zl=r,thenlwlS|zl+r=(1+|zl)/2.Bytheanalyticityof g, sup {lg(w)l:|w-2l=r}Ssup{|g(w)|:|w|S(l+|zl)/2 }. Multiply both sides of inequality (1.1) by (1 - I z I)"+1 = 2M1 r "+1 to get 4. (l-lzl)"+llg'(z)l s 2" 1sup{rn|g(w)l:lwl=(1-I-lz|)/2}. Forlwl=(1+lzl)/2wehave l -lwl=(l-lzl)/2=r ,soitfollowsthat n+1 n+1 (1-lzl) lg'(z)| S 2 sup { (l -lwl)" lg(w)l:|wl=(1+lzl)2 } . (1.2) Put g = f I"), multiply by (1 + I z l)"+1 (which is less than 2"”), and take the supremum over all z e D, getting sup (1 .1212)"+1 If("“)(z)l s 22" 2sup (1 -IwI2)" lf(")(w)l . ZEID WEID + 29 It follows that quantity (3M1) is less than or equal 22M2 times quantity (8"). This completes the induction and the lemma is proved CI The equivalences of Lemma 1.3 carry over to the little Bloch space, as is shown in the following lemma. Lemma 1.4 : Let n e IN. Then for an analytic function f on D the following statements are equivalent: (a) f 6 130 ; (b) (1 -IzI2)" f(")(z) -> 0 as lzl—) 1'. Proof: For n = 1 the equivalence of the two statements is precisely the definition of the little Bloch space. By induction it suffices to show that for a fixed n 6 IN, for every analytic function f on D the statements (bn) (1 -lzl2)"f(”)(z) —> 0 as Izl —> 1- and (am) (1 -1212)"+1f<"+1>(z) —> 0 as Izl—> 1- are equivalent. Let n e IN be fixed. That statement (bu) implies statement (bu +1) follows easily from (1.2) (applied to g = f (’0). 30 For the converse suppose that f is an analytic function on D satisfying condition (ban): Let 0 < r < 1 . Let g be an analytic function on D, and let w e D. Then as in the proof of Lemma 1.3 : l lg(w) - g(rw)l s I lwl lg'(tw)l dt T n+1 S ——1—— sup { (1 -|zl) lg'(z)l : rlwlSlzl 0, choose p e (0, 1) such that (1 -1212)"+11f<"+1)(z)l< 3 whenever p < Izl < 1. For p < r < 1 it follows from the above inequality that (1 - I w l)n l f(w) l s Eln + (l - lw I)" If (")(rw) I whenever we have p/r < M < 1 . Hence (1 — Itv12yl f(")(w) —> 0 as Izl —> 1-, i.e., f satisfies (on). This completes the induction, and the lemma is proved. III For the statement and proof of the following lemma we need more notation. For a point l e D and 0 (1 - r2)2 , thus I lfw(z)|p (1-lzl2)np-2(l-I¢A(z)l2)2 dA(z)/7t 2 ID 2 (1-r2)2 I 1f‘")(z)1” (1.1.212)"”'2 dA(z)/7t, (1.16) D(lr) and it follows that quantity (D) is greater than or equal a constant times quantity (C) . To complete the proof we will show that quantity (D) is less than or equal a constant times quantity (A) . Again we make use of Lemma 1.3. I 1f(")(z)1” (1.1.212)""'2 (1-I¢l¢)I2)2 dA(z)/7t s 11) 22 P 1.1 ()1 S(sup (1-1212)"1f(")(z)1) I J2— «(2)/a. ZEID ID l-IZI 40 Now, the integral at the right of the above inequality is l [by (0.20) the integrand is equal to l 0 as 121—) 1'; D(l,r) (d) I lf(n)(z)|p(l-|zl2)np'2(l-|tpl(z)l2)2dA(z)/7t —> 0 as Ill-+1-. ID Proof: Take 0

0 as 121—) 1'. (1.18) 5D Combining (1.17) and (1.18) yields that (d) holds. Cl Chapter 2 In this chapter we will give characterizations of the Bloch space and the little Bloch space in terms of the pseudo-hyperbolic metlic. It will be shown that the Bloch space consists of those analytic functions on the disk that are uniformly continuous with respect to the pseudo-hyperbolic metric. A similar description will be given for the little Bloch space. We will also consider the real harmonic Bloch space on the unit disk. First we will show that for an analytic function on the disk the Bloch norm and the supremum of the oscillations of the function over pseudo-hyperbolic disks of a fixed radius are equivalent quantities. Theorem 2.1 : Let 0 < r < 1 .For f analytic on D the following quantities are equivalent: (A) ll f "9 ; (B) sup sup |f(z) -f(l)|. leD zeD(l,r) Proof: Fix 0 < r < l , and let f be analytic on D . It follows from the identity f'(0) = 32- I Ere) dam/2r r D(0,r) If'(0)l S érI If(z)| dA(z)/7t r D (0,r) 2 < _ r sup If (z)| . zeD(0,r) 43 44 Replacing f by f o IPA. - f (l) , we get the inequality (1-1212) lf'(l)| 5 ”(IN sup |f(2) - f(l)|. (2.1) 26001:) and it follows that IIfIII3 S 3- sup sup If(z) -f(l)l. leD zeD(l,r) On the other hand, as in the proof of Lemma 1.3, for l wl < r we have lwl 2 dt . "flIEB l-t lwl l+r |f (W) - f (W S Replacing f by f 0 (Pl - f (l) yields 1 1 |f( 0 as r—-)0+. (2.3) leD zeD(l,r) Let UC denote the class of all functions f: D -> C which are uniformly continuous with respect to the pseudo-hyperbolic metric. Let H (D) denote the set of all analytic functions on D. Corollary 2.2 : EB = UC n H(D) Proof: If f 6 UC r) H(D), then f satisfies (2.3). In particular, for some r e (0, 1) we have sup sup |f(z) -f(l)| S 1. leD zeD(l,r) so that by Theorem 2.1 fe $3 . For the converse suppose that f e 33 . Taking the limit r —-) 0+ in (2.2) yields (2.3), hence f 6 UC n H(D), and the corollary is provedD Remark 2.3 : For an arbitrary function f: D -—) C to be uniformly continuous with respect to the pseudo-hyperbolic metric f must satisfy the little-o condition (2.3). However, for an analyticfimction f: D -) C to be uniformly continuous with respect to the pseudo-hyperbolic metric it is sufi‘icient (and of course necessary) that f satisfies the big-0 condition 46 sup sup |f(z) -f(l)| < w. 16D ZEMU) for some r e (O, l) . As usual, the equivalences of the previous theorem carry over to the little Bloch space. This is expressed in the following theorem. Theorem 2.4 : Let 0 < r < 1 . For an analytic function f on D the following statements are equivalent: (a) f 6 130 ; (b) sup lf(z) -f(h)| -—> O as Ill-9 1'. zeD(l,r) Proof: That (b) implies (a) follows immediately from (2.1). For the converse, suppose that f 6 I30 . From the proof of Theorem 2.1 we see that for te (0,1) and he D 1+r l-r sup |f(2) -f,(2) - (f(l) - 1;(A»Is§-Iog( llf-fll . (2.4) zeD(2.r) ) t B Using the triangle inequality it follows from (2.4) that for t e (O, 1) and A e D sup |f(z) -f(2.)l s ézogfi”) Ilf-ftllfB + sup lft(z) -ft(/'L)l. (2.5) zeow) " xenon 47 Let t e (O, 1) . The dilate ft is analytic in a neighborhood of the disk, so clearly sup If‘(z) -ft(/'l)l —) 0 as Ill—)l', 26004) and it follows from inequality (2.5) that lmsup sup |f(z) -f(h)l S-é—log(llf:) llf- ftll‘B. Ill—)1- WOW) Since f e EEO , we have ll f - ft "SB —) O as t -) 1‘, hence the above inequality yields imsup sup |f(z) —f(l)| = O, Ill—)1- 260(k) which implies that (b) holdsD Let h(D) denote the set of all real harmonic functions on D . Define the real harmonic Bloch space B to be the class of all real harmonic functions u on D for which llullB = sup (l-Izl2)l(Vu)(z)l < co, 26D where Vu denotes the gradient of u . If f is analytic on D, and u = Ref, then it follows from the Cauchy-Riemann equations that If ' l = I Vu I, and consequently ll f "SB = II u "3. It follows immediately that B = Re 13 . So if as B , thenfe SB, so that fe UC, and hence ue UC. Thus we have the inclusion B C UC n h(D). We claim that the converse is also true, i.e., in analogy to 48 Corollary 2.2 we have the following result: Theorem 2.5 : B = UC n h(D). Proof: We make use of the fact that the conjugate function operation is a bounded operator in the L1(D,dA/7t) norm (for a proof see [7]): there is a constant C such that for every real harmonic function u on D JlfildA/tr s c I luldA/tr. [D ID Let 0 < r < 1. Dilatin g the above inequality gives that for every real harmonic function u on D J IUI dA/Jr S C J lul dA/7r . (2.6) D(O,r) D(O,r) Suppose that u 6 UC n h(D). Let f be analytic on D such that u = Ref. Since u is uniformly continuous with respect to the pseudo-hyperbolic metric we can pick 0 < r < 1 such that sup sup |u(z) - u(h)l S 1. (2.7) 16D 260(k) Let A e D be fixed. Using the change-of-variable formula (0.22a) and formula (0.21) for the normalized area of a pseudo-hyperbolic disk, we have: 49 1 I D(lfl 22 J lf-f(/1)| dA/rt S WI lfo (DA-f(DI dA/tt D(AJ) r (141”) D(O,r) 1 S J lfo (o - f(l)| dA/n' . (2.8) 2 4 A r (1' r) D(O,r) Write f = u + i (7. It is easily seen that (u 0 (pl - u (1))”: [70 (P2. - (7(2), so by (2.6) we have I |Uo(p;L - U(A)l dA/n: S C J Iuoqu - u(l)| dA/tt. D(O,r) D(O,r) Since fo (PA - f(l) = u o ‘Pit - u (A) + i ((70 (P1. - (7(2)), the above inequality and the triangle inequality give us that I lfogpA-f(/l)|dA/7t .<_ C I luotpl- u(h)| dA/n'. (2.9) D(O,r) D(O,r) From (2.7) we see that the integral at the right of (2.9) is bounded by r2. Combining this with (2.8) yields 1 lD(ll,r)| I lfotpl 4(1): dA/tt s C” 4 D(A,r) (1 ' ') By Theorem 1.1 (D), we have fe 13, so that u =Ref e B , as was to be shownfl 50 Remark 2.6 : For an Mmfimction u : D —) R to be uniformly continuous with respect to the pseudo-hyperbolic metric u must satisfy the little-o condition (2.3). However, for an W function u : D —> R to be uniformly continuous with respect to the pseudo-hyperbolic metric it is suflicient (and of course necessary) that u satisfies the big-0 condition sup sup |u(z) - u(l)l < co, 16D zeD(h,r) for some re (0, 1). Just as BMO is closed under the conjugate function operation, so is B, the class of real harmonic functions on the disk that are uniformly continuous with respect to the pseudo-hyperbolic metric. Corollary 2.7 : If u 6 UC n h(D), then (7 5 UC n h(D). Proof: Suppose that u 6 UC n h(D). By Theorem 2.5, u e B . Thus u =Ref, withfe ‘B. Then -i f 6 I3, so that i] = Re (-i f ) e B , and by Corollary 2.2 we are done.El Chapter 3 In this chapter we describe some spaces of analytic functions on the unit disk in terms of Nevanlinna characteristics. Our starting point is Baemstein's characterization for the space BM 0A ; he proved that an analytic function on the unit disk belongs to the space BMOA if and only if the Mobius transforms of the function form a bounded family in the Nevanlinna class. We give a similar description of the space VMOA . This description cannot be obtained by simply repacing Baemstein's boundedness condition by the corresponding vanishing condition (as is usually the case). We then formulate and prove analogous characterizations for the Bloch space and the little Bloch space in terms of an area version of the Nevanlinna characteristic. For f analytic on D the Nevanlinna characteristic T( f ) is defined by T(f) = sup log+lf(rei9)l d9 . l OSr1. Note that(‘l'la(f))(ei9)2|f*(ei9)l iff hasanon-tangential limit f*(ei9) at cm. 53 In [8] Baernstein proved the following "John-Nirenberg type" of theorem: Theorem 3.1 : There exists an absolute constant K such that for each 0 < a < 7t/2 and f analytic on D the following statements are equivalent .' (A) { f 0 (Pl - f (l) : A e D} is bounded in the Nevanlinna class N ; (B) There exists a constant [3 = [3 (a, f) for which none”: nave ‘9; -f(t»r ))1”? (3.4) The answer is negative: the condition at the right of (3.4) is certainly necessary for f to be in VMOA (this follows from (3.2)). but not sufficient. That the condition is not 54 sufficient follows from the observation that it is trivially satisfied when ll f llm S 1/2 (because this implies that T (f 0 ‘Ph - f (2.)) = O for all 11 e D), but not every analytic function f on D for which N f II” S 1/2 is contained in VMOA . Let's return to BM 0A and rewrite the condition in Theorem 3.2. Let p>0.Iffe BMOA ,then also flpe BMOA ,sothat (fowl-f(l)) sup T 0:sup T l 0wehavethat T( )-—) O as Ill-+1-. Before the proof we need to relate the Nevanlinna characteristic and the H2 - norm of an analytic function. We'll do this not just for the H2 - norm, but for any HP - norm: 55 Lemma 3.4 : Let 0 < p < co. For an analytic function f on D: llfllpp =p2 Ipp'lr(£) dp. (3.6) H 0 p Proof: Let 0 < p < co. Integration by parts yields the formula: 1 - 1 It” ‘tog—d:=i. t 2 o p Thus for 0 S x < oo we have: 00 x 1 Jpp'llog+idp=Jpp'llogidp=xpJtp'llogldt=-l-xp. t 2 o p 0 p o P For an analytic function f on D and O < r < 1 an application of Fubini's Theorem gives: W 1 1 2” f(reig) 1 1 Jupp- —Ilog+| IdB dp=—2— o 2"o p p 2” |f(rei9)|p d6 . (3.7) O'——.§a Taking the limit as r —-) 1', and using the Monotone Convergence Theorem we get (3.6).[1 56 Now we are ready for the proof of Theorem 3.3. Proof of Theorem 3.3 : Let f be an analytic function on D. We have already seen that condition (b) in Theorem 3.3 is necessary. To prove the sufficiency, suppose that f satisfies condition (b). Our first step is in showing that f e BMOA. Choose an r e (O, 1) such that T (f o a); - f (2.)) < 1 whenever r < I ll < 1. Note that g e N 4:) g o ‘1’}. e N (This follows easily from the fact that each function in the Nevanlinna class N is the quotient of two H°°-functions). Pick w such that r<|w| < 1. Then T(fo (pw - f(w)) < 1, so that fo ow e N, and therefore fe N. Thus log+l f I has a harmonic majorant, call it h. Then for A e D, h o a); is a harmonic majorant of f o 491 , whence T(f° m) S (’1 ° mXO) = (1(1). Using the inequality log+(x + y) S log+x + log+y + log 2 , it follows that for I ll 5 r : T(fo (Pl -f(l))Sh(h)+log+lf(/'l)l+log2. Hence the family { f 0 ‘PA - f (A) : 2. e D} is bounded in N, and by Theorem 3.2 we have fe BMOA. Since f e BMOA we can apply Theorem 3.1. Let 3 be such that (3.3) holds. Then forle D andt>0: none”: If'wxeie» -f(l)| > r l) O: 27!: * i9 . -(t> I((e»-(A>I . T(f «)1 f )SLJ::+: «p, f (wow) 27: 0 p Now let 8 > 0 be given. Choose R > 0 such that K e ‘ B R < (e 2,6 2)/8. Then integrating the above inequality we get °° f0 -f(l) IpT( (p3 )dp 0 as Ill—)1”. Proof: For x 2 p > O , log+£ 5 log (1 +1/p) log (1 +x) , P 103(1 +10) so that T(f° (pa 4(1)) 5 log(1+1/p) T’Cfo (pl, - f(l)). P log(l+p) Soif T'(fo 4),-f(l)) -+ 0 as l/Il-)1',then fe VMOA. The inequality log (1 +x)$x implies that sothatT'(fo (PAC f(l)) ——) 0 as |ll—>1‘,when fe VMOAD For f analytic on D the area version of the Nevanlinna characteristic, Ta( f ), is defined by Ta(f) = Jlog+lfldA/n. D The area-Nevanlinna class is the set Na = { f e H(D) : Ta( f ) < co}. 60 Let 0 < p < co . Integrating both sides of inequality (3.1) gives, in analogy to (3.2): 1 P Ta(f)S-Ellflle,for00wehavethat Ta( )-—>Oas Ill—>1'. Proof: That (b) is implied by (a) follows easily from (3.10) and the Garcia-norm characterization for the little Bloch space [Theorem 1.2 (b) ]. For the converse, suppose that f is an analytic function on D for which (b) holds. Fix 0 < r < 1 . Let 2, A e D such that d(z, 2.) < r . Then, as in the proof of Theorem 3.6: 62 _ 1 f°¢ 'f(4) |f(z) f(l)| Sap 2T0( 1 ) , (3.11) p (l-r) Given £> O , choose 0 < p < 8/2 . Since (b) holds we can choose a 5e (0 , l) for which T (to so, 4(2) 0 ) <(1-r)210g2, (3.12) p whenever 0 <1 -l/'ll < 6. Combining (3.11) and (3.12) we get that for 0 <1 - | Al < 6 If (2) - f ()1) l 5 2p < e. We conclude that sup |f(z) -f(/'l)l -—> O as Ill—9 1', 26001!) so that by Theorem 2.4, f e 5130 , and we are doneD Chapter 4 In this chapter we will give a different proof of Baemstein's value distribution characterization for BM 0A [8], Theorem 3, and then formulate and prove the corresponding description for the space VMOA . Defining an area version of the counting function used in the value distribution characterizations for BM 0A and VMOA, we obtain analogous results for the Bloch space and the little Bloch space. The Green's function for the unit disk is given by g(z,/'l) = log , for z,/le D. | 431(2)! For a nonconstant analytic function f on D let {zn( f )} denote the zeros of f in D, listed in increasing moduli and repeated according to multiplicities. Following Baernstein we define N (w, A, f), the "counting function for value w started at h ", by N(w, M) = 2 gene-w)». n Note that g (2, O) = log (IA 2 I) , so that 1 N(W,O,f) = 2" log W s the usual counting function. It is clear from the definition of the counting function that 63 64 N(w,h,f)=ooiff(h)=w; (4.1a) N(w, 2, f) = 0 if f omits the value w . (4.1b) The following properties of the counting function, which are easily verified, are useful: For we C, ate C\{O], he D and f analytic on D we have: N(w,h,f) = N(w+a,h,f+a) (4.23) N(w, h,f)=N (aw, h, af) (4.2b) N(w,h,f)=N(0,0,foqol-w). (4.2C) The following theorem is due to Baernstein ([8], Theorem 3). We will give a simpler proof of his theorem. Theorem 4.1: For a nonconstant analytic fimction f on D the following statements are equivalent: (A) f e BMOA ; (B) sup {N(w,h,f):we C,he D and lf(h)-wl21} < oo. Just as in Baemstein's proof we will need to relate the Nevanlinna characteristic of an analytic function with its counting function. This is done in the following classical result. Cartan's Formula : For a nonconstant analytic fitnction f on D: T(f) = N(ei9,0,f) d6 + log+lf(0)|. (4.3) 1 2.7: 05.? 65 A proof of Cartan's Formula can be found in [17], pages 214-215, for the case that f is analytic on a neighborhood of D. The general case follows easily by looking at the dilates ft of f. Using the Monotone Convergence Theorem we see that T (f,) increases to T (f) and for each 9 in (0, 2n) we have that N (e ‘9, O,ft ) increases to N (e i9, 0, f) as we take the limit t -) 1'. For these dilates f, we know that (4.3) holds, so that another application of the Monotone Convergence Theorem gives that (4.3) holds for f. Proof of Theorem 4.1: Let f be a nonconstant analytic function on D. By Jensen's Formula we have: i6 _ r loglf(re )I d0 — 2 log W + Ioglf(0)|. 1 27‘ n:lzn(f)l 1', gives us the inequality T(f) 2N(0,0, f) + loglf(0)| . (4.4) Replacing f by f 0 ‘Ph - w , and making use of (4.20) the above inequality yields N(w,h,f) S T(fo (pA - w) - loglf(h) - wl. 66 Using the inequality log+(x+y) S log+x + log+y + log 2, we get N(w, h,f) _<. T(fo (pl - f(h)) + log+lf(h)-wl - loglf(h)-wl + log2. Soiflf(h)-w|21,thenwehave N(w,h,f) S T(fo (pA -f(h)) + log 2. The above inequality and Theorem 3.2 show that (A) implies (B). To prove the converse suppose that M =sup{N(w,h,f):we C,h e D and |f(h) - w|21}< 00. By Cartan’s Formula T(f°¢l'f(l)) = Gags) i N(e‘e. 0.f°¢ -f(l))d9 . 2” h Now, using (4.2a) and (4.2c), for every 0 S 6S 2n we have N (em, 0, f o (p;~ - f (h)) =N (ei9+f(h ), h,f ) SM, so it follows that T(fo (p;L -f(h)) S M, forall he D, and hence, by Theorem 3.2, f e BMOA . El 67 Before going to VMOA let's rewrite the condition in Theorem 4.1 for inclusion in BMOA . Suppose thatfe BMOA , and let 6 > 0. Since f/6e BMOA , it satisfies condition (B) of Theorem 4.1. By (4.2b), N (w, h, f) = N (w/6, h, 176 ). Therefore we must have that for an analytic function f on D: fe BMOA (=) [V6>0:sup{N(w,h,f):w e ¢,2 e [D and |f(h) - w|26} 0 : 21! I“) = —1— ] N(e‘9,o, I.) d6 p 2x 0 p 271: = L I N(pe‘9,o,f) d0. 27! 0 Multiply by pP ' 1 and integrate with respect to p over the interval (0, co). By the formula (3.6) of Lemma 3.4 we get 0° 711' Ilfllp =p2J,)1"'l —1—J‘N(pei9,0,f)d6 dp ”p o 2” o 2 2 = 5— ] IwIP' N(w,0.f) dA O we have: sup {N(w,h,f):we Cand |f(h)-wl26} —> O as |hl—) 1'. Proof: Let f be a nonconstant analytic function on D. Let 6> 0. Making use of Cartan's Formula and the equations (4.2) we see 69 - f° -f(h) Mm gas )dQ T(fo¢4-f(l))=-l—-N 5 2n 1 27: “P 2t I“ 0 2! I N(6e 9+f(h), h,f) d6 0 {N(w,,h f): weC and |f(h) -w|2 6}, so that, by Theorem 3.3, (b) implies (a) . To prove the other implication we make use of Lemma 4.2. In this lemma take p = 2, and replace f by f 0 (PA - f (h), we get the formula 2 2 llfo (oi - f(h)|lH2 = ; I N(w +f(h),h,f) dA(w). (4.6) a: We will also need Lehto's Theorem [21], which states that for a function g , analytic on a neighborhood of D, the function w H N (w, 0, g ) is subharmonic on C \{ g (0)]. Let g be an analytic function on D for which g (0) = 0. Let 0 < r < 1. Applying Lehto's Theorem to the dilate g, of g we get that for 6 > O and for I u I 2 6 N (u, 0, gr) S -1—2 I N (v, 0, gr) dA(v) . (4.7) 7‘6 lu- vl<6 Taking the limit where r —-) 1‘, we get N(u,0,g) S —1—2‘I N(v,0,g) dA(v). 71:5 lu- vl<6 70 Apply the inequality to g = f o a); - f (h). Using equations (4.2) we get N(u mutt) s #2 I N(V+f(h),h,f) dA(v) . n6lu-N<6 Replacing u + f (h) by w yields the formula N(w,h,f) S -1—J' N(z,h,f) dA(z) . (4.8) 2 ”slw-fl<6 Combining (4.6) and (4.8) gives us that for If (h) - WI 2 6 1 2 N , h, S — II o - h H , so that sup{N(w, h,f):we C and |f(h) — WI 2 6} S —1—Ilfoq) - f(h)||22 , 252 h H from which it follows that (a) implies (b) .12] Now we will turn to the Bloch space and the little Bloch space. Defining an area version of the counting function used in the value distribution characterizations for BM 0A and VMOA, we obtain analogous results for the Bloch space and the little Bloch space. Define an area version Na of the counting function N as follows: given an analytic function f on D we first define N a (0, O, f) by 71 l Na(0,0,f) = I 2rN(0,0,fr) dr, 0 and, mimicking (4.2c), for w e C and h e D define Na (w, h, f) by Na(w,h,f) = Na(0,0,fo (a). - w). Observe that N a (w, h, f ) = 0 if f omits the value w, but that (4.1a) is not necessarily true for counting function N a . It follows immediately from the definition that properties (4.2) do hold for counting function Na : for w e C, a e C\{O], h e D and f analytic on D we have: Na (w, h,f) = Na (w + a, h,f+ a) (4.9a) Na (w, 2, f) = Na(0tw, h, of) (4.9b) Na(w, h,f) = Na(0,0,fo (pl - w). (4.9c) Theorem 4.4 : For a nonconstant analytic function f on D the following statements are equivalent: (A) f 6 33; (B) sup {Na (w, h,f):we C,he D and |f(h)-w121} < 00. Proof: Let f be a nonconstant analytic function on D, and let 0 < r < 1. By inequality (4.4) we have: 72 N(0,0,fr) s T(fr) - log lf(O)|. Multiply the above inequality by Zr and integrate with respect to r over the interval (0, l) to get: Na (0,0,f) S Ta (f) - log |f(0)| . (4.10) Just as in the proof of Theorem 4.1 it follows that if If (h) - WI 2 1, then we have Nam. h,f) 5 ram 4), 4(4)) + Iogz. Theorem 3.6 and the above inequality show that (A) implies (B) . Note that integrating Cartan's Formula gives us the formula 1 Ta(f) = Z 21! I Na(e’9,o,f) d6 + log+|f(0)l. (4.11) 0 To prove the converse we use this formula and proceed as in the proof of Theorem 4.1.1:] The value distribution characterization for the Bloch space carries over to the little Bloch space in the same way as going from BMOA to VMOA. 73 Theorem 4.5 : For a nonconstant analytic function f on D the following statements are equivalent: (a) f 6 130; (b) for ever 6> 0 we have: sup {Na(w,h,f):we Cand |f(h)-wl26} -—> O as |h|—> 1'. Proof: Let f be a nonconstant analytic function on D. Let 6 > 0. Making use of (4.11) and the equations (4.9), as in the proof of Theorem 4.3, we have for every 6 > 0 (to «)0A - f(l) T a 5 ) S sup {Na(w,h, f):we C and |f(h) - WI .>. 6}, so that, by Theorem 3.7, (b) implies (a). To prove the other implication we need an area-version of Lemma 4.2. If 0 < p < co, the function f is analytic on D, and f (O) = 0, then applying Lemma 4.2 to the dilates f, of f and subsequently integrating with respect to r over the interval (0, 1) yields the formula 2 llfllp = P— LP 27: a I lwlp-2Na(w,O,f) dA(w) . a: In the above formula take p = 2 and for h e D replace f by f 0 ‘Ph - f (h) ;analogous to (4.6) we get: 74 2 "fotpl—f(l)"L 2: a alN I Na(w +f(h), h, f) dA(w) . (4.12) 0: Integrating (4.7) with respect to r over the interval (0, 1) gives that for an analytic function goanorwhich g(O)=Oandfor|u|26>Owehave l Na(u)0:g)s m Na(v:0rg) dA(V). lu—vl<6 As in the proof of Theorem 4.3 it follows that whenever If (h) - WI 2 6 we must have 1 n62 Na(w, h, f) s I Na(z, a, f) dA(z) . (4.13) lw-z|<6 Combining (4.12) and (4.13) we get sup {N (w,h,f):we C and |f(h) - WI 2 6] S i llfoqo -f(h)l|22, a from which it follows that (a) implies (b).El For a nonconstant analytic function f on D it is easy to compute Na(0, O, f). Let {2"} denote the zeros of f in D, as usual, listed in increasing moduli and repeated according to multiplicities. Then for every 0 < r < 1: , N(0,0.f,) = 2 xO: sup {Itn(fo (pl-w,t)dt :we C and |f(h)-wl26]—->OaslhI—)1'. 0 Chapter 5 In this chapter we give estimates for the growth of analytic functions in weighted Dirichlet spaces, which then are used to give necessary and sufficient conditions on the growth of an analytic function on the disk for inclusion in the Bloch space or the little Bloch space. For the Bloch space and the little Bloch space we establish certain weighted Dirichlet-type conditions, and we investigate the question of whether analogous results are true for the spaces BM 0A and VM 0A. We start with a lemma that gives estimates for the weighted Bergman norms of an analytic function and its derivative. Lemma 5.1 : Let -1 < a < oo . For an analytic function f on D we have 1 Ilf'(z)l2(l-|z|2)a+2dA(z)/7t s Ilf(z) -f(0)|2(1-|z|2)adA(z)/7t 5 (1+1 D D s “+3 If'(z)|2(1-|z|2)a+2dA(z)/7t . (5.1) a+1 ID Proof: Let -1 < a < oo . For an analytic function f on D with Taylor series expansion n f(2) = Ea zn , zeD, n=0 it is easily seen that 79 80 M I lf(z)|2(1-Iz|2)a dA(z)/7t = 2 Ian 12pm, a), (5.2) [D "=0 where 1 pm, a) = I 1212" (1.-1212)“ «(2)/z: = Ix"(1-x)a dx. [D 0 Then we have that n!F(a+ 1) 1"(n+a+2) 1302,01) = (5.3) For the derivative of f we have I lf'(z)|2(1-lz|2)a+2dA(z)/7t = 2 lan|2n2,6(n-l,a+2). n=l D Using (5.3) and the properties of the Gamma-function it is easy to verify that 1 n2/3(n-1,a+2) = 353;) )3(n, a). n+a+2 Thuswehave 1 (1+3 n2fi(n-1,a+2)s )3(n,a)s n2p(n-1,a+2), a+1 0+1 81 and (5.1) follows immediately. III In the above proof, for each n e IN, (a + 1) [3(n, a) = n p (n - 1, (1+ 1) increases to 1 as (1 decreases to -1. If we take f e H2 then we have 2 N IlfII 2: Elanlz, H n=0 so that for eachme IN, 2 lllfll 2 - (01+ 1) I If(z)|2(l-|zI2)a dA(z)/7t I H D = z Ianl2 (1 -(a+ 1)/3(n, (1)) n=1 m 2 m 2 SXIanI (l-(a+1)[3(n,a)) + 2 Z Ianl , n=l n=m+l which implies that (01+ 1) I |f(z)|2(1 -1212)“ dA(z)/7t —> Ilfll:2 as a —> -1*. ID Taking the limit in (5.1) where a -—) 1*, we thus obtain Paley‘s integral inequalities (see [14], Lemma 3.2), which we will use later in this chapter: 82 II , 2 2 2 , 2 I2 f (z)| (1-lzl)dA(z)/7t S llf -f(0)|| 2 S 2 If (z)| (l-lz )dA(z)/7t.(5.4) H D D For an analytic function f on D with Taylor series expansion f(z)=2a zn,ze D, n n=0 SCI M(r,f) = 2 lanlr" ,for OSr<1. n=0 The quantity r? (r, f ) is a very crude estimate on the growth of the modulus of the function f. In the following lemma we give an estimate on [‘7 (r, f ) in terms of a weighted Dirichlet norm of the function f. Lemma 5.2 : Let 0 < a < oo. For an analytic fitnction f on D for which f (O) = O, we have for all r e [0, 1) the inequality: in (1-r2)a/2M(r,f) s ””1 (Ilf'(z)|2(1-Iz|2)adA(z)/7t . (5.5) (1 ID Proof: Let 0 < a < oo , and r e [0, 1). For the derivative f' of f we have 83 Ilf’(z)|2(l-I2I2)ad4(z)/7t = zlanlzn2p(n-1,a). [D "=1 Using the Cauchy-Schwarz inequality we have 1/2 1/2 M(r,f)S 22 Elanlznzfim-La) n=l n 2-fi(n 1,0) n=l as 1/2 1/2 < 21,12 Ilf'(z)|2(1-Iz|2)adA(z)/tt . (5.6) n= Zfi(n- 1, a) [D We need to estimate the infinite sum in (5.6). It follows from (5.3) that I "2301-1,(1): "‘1 fl, n+a F(n+a) therefore °° r2" °° n+0: F(n+a) 2n 2 S ——r n=1n[3(n-l,a) n=1 "‘1 "!r(a) Soz+1 (1-2'ar) a which together with (5.6) gives the desired inequality. [I We will use Lemma 5.2 to obtain characterizations for the Bloch space and the little Bloch space in terms of quantity f1— . The lemma can also be used to prove a result due to V.S. Zakharyan [36]: 84 Let 0 < a < oo . If f is an analytic function on D for which Ilf'(z)l2 (1 -|2I2)adA(z)/7t < co, (5.7) [D Then (1-r2)a/2M(r,f) —+ 0 as r—41'. (5.8) Proof: Fix 0 < a < co . Let f be an analytic function on D with Taylor series expansion ll f(2) = Zanz , ze D. n = 0 For N e IN apply Lemma 5.2 to the function gN defined by n anz,zeD. gN(Z) =f(Z) ' M We get (1 -r2)a/2A—4(r,f) s N 1/2 5 (1 - r50"2 2 lanlr" + I? (I IgN'(z)I2 (1 -|z|2)adA(z)/7t . (5.9) n=0 D It follows from (5.9) that 85 1/2 limsup (1-r2)“’27)7(r,f) _<. Ia“ IlgN'(z)|2(1-lzl2)adA(z)/7t .(5.10) r—bl- a [D In (5.10) let N —> oo . Since f satisfies (5.7) the integral at the right of (5.10) tends to 0 and (5.8) follows. III Theorem 5.3 : Let 0 < r < 1. For an analytic function f on D the following quantities are equivalent: (B) sup Mnfo <0}1 -f(h)). heD Proof: Fix 0 < r < 1. Let f be analytic on D. By Lemma 5.2 1/2 (1-r2) M(r,f - f(0)) s JEII If'(z)|2(1-IzI2)2d4(z)/7t D Combining the above inequality with Lemma 5.1 we get M(nf- f(0)) 5 £- ||f - f(O)“ 2 (1’7 La Let h e D. Applying the above inequality to f o (,p}. - f (h) we get 86 _ 2 M(r.f°¢l-f(h))s J— "focal-f(hflle. (5.11) (1‘7 a and with the help of Theorem 1.1 it follows that quantity (B) is less than or equal to a constant times the Bloch norm of f. To show the converse, note that If ' (O) I r S r? (r, f ) , so that (l-lhl2)|f'(h)| s .1;'1)7(r,fo (pl -f(h)), (5.12) which implies that the Bloch norm of f is less than or equal quantity (B). C] As usual the equivalences of the previous theorem carry over to the little Bloch space, and we have: Theorem 5.4 : Let 0 < r < 1. For an analytic function f on D the following statements are equivalent: (a) f6 130; (b) M(r,fo (pl1 - f(h)) -) 0 as Ihl—>1'. Proof: Fix 0 < r < 1. Let f be analytic on D. It follows immediately from (5.12) that (b) implies (a). The converse follows from (5.11) and Theorem 1.2. CI We now wish to investigate the spaces BM 0A and VMOA. In view of Theorem 1.7, comparison of the two equivalences 87 1/2 IlfllSB = sup Ilf'(z)l2 (1-1a (2)12)2 dA(z)/7t , heD ID A and ”SM 1/2 IlfIIBMOA p IIIf'(z)I2(1-I¢A(z)|2) dA(z)/7t , heD ID leads to the following question: Question : Let 0 < p < co and let f be an analytic function on D. Are the following true? (i) f eBMOA 4:) sup If'(z)Ip(1-lz|2)p'2(1-|(pl(z)|2) dA(z)/7t < co? heD ID (ii) f e VMOA 4:) I |f'(z)|p(1-|z|2)p'2(l -|(pl(z)l2) dA(z)/7t —) 0 as lhl-—)1' ? D We do not know an answer for the above question. The classical results of Littlewood and Paley ([22], Theorems 5 and 6, page 54) and a change of variables give the following implications for an analytic function f on D: (l)For 0fe moa. D 88 (Il)For 2Sp o as 121—91' ID As mentioned above, these implications follow from Littlewood and Paley‘s theorems, but we can also prove them directly, using Theorem 1.7: Proof: (1) Let 0 < p S 2. If for an analytic function f on D sup IIf'(z)I”(1-IzI2)”‘2<1-Icpl(z)|2) «14(2):: < .., heD D then it follows from Theorem 1.7 that f e D. Since we have I If ' (z) 12 (1 — I (pl(z) 12) dA(z)/7t D .<_ Ilfllgp Ilf'(z)|p(1-Iz|2)p'2(1-I(pl(z)l2) dA(z)/7t , D both implications in (I) follow. 89 (I!) Let 2 S p < oo . Now make use of the inequality I If'(z)|p(l-Iz|2)p'2(1 -I(pl(z)|2) dA(z)/7t [D P-2 , 2 2 S Ilfll13 I If (z)I (l-lqpl(z)| ) dA(z)/7t , D and since BMOA C 13, the implications in ([1) follow immediately. [I What we can prove is the following theorem. Theorem 5.5 : Let 0 < p < oo , 0 < O'< 1. Then for an analyticfitnction f on D we have the following two implications: (i) sup If’(z)lp(l-I2I2)p'2(l-Itp (2)12)“ dA(z)/7t < oo heD ID '1 implies that f e BMOA ; (ii) I |f’(z)lp(1-|zl2)p'2(1-|(pl(z)l7‘)° dA(z)/7t -) 0 as 121—51' ID implies that f e VMOA . The proof of Theorem 5.5 makes use of the following weighted Garcia-norm equivalences for the Bloch space and the little Bloch space. 90 Lemma 5.6 : Let -1 < a < co and 0 < p < oo . Then for an analytic function f on D we have: (i) Ilfll =sup 13 hraD (‘I l/p |f<¢1(2))-f(h)|p (I-Izlz)“ dA(z)/a ) ; D (ii) f e :80 (=> [I |f((pl(z))-f(h)lp (14212)"l dA(z)/7t —) 0 as lhI—al' I. ID Proof: Take -1 < a < co and 0

-1. Let s' denote the conjugate index of s, i.e., s' = s/(s - 1). An application of H61der's inequality gives that I'f( -1, we have that the integral at the left of (5.13) is finite, in fact it is equal to l/(sa + 1). It follows that 1 sa+1 PS a Up 1/ (I who) -f(2)|p(1-|z|2)°‘ «(2)/u) s( )psllfwl'fw'i ' D (5.14) 91 To obtain an inequality in the other direction choose a number q e (1,00) large enough so that q >a + 1. By Hblder's inequality I my» - f(l)!” dA(zyrc n) = I vole» 4(4)!” (1 4212)“ a 4212)” dam/7r n) W m , s (Imago) - f(h)Ip (1 -12 12)“ dA(z)/7t ) I (14212)“‘1/4 dA(z)/7t ID ID Because q - 1 > a , we have -aq'/q = —a/(q - 1) > -1, and thus the integral at the right of the last inequality is finite, in fact it is equal to (q - 1)/(q - l - a). It follows that PM a 1/P - ( -1)/ "fog-f(hHIL .<.( q 1 )4 p(IIf< 2. Let n be an integer, n > 2. Then q = n/(n - l) S 2. Let g be an analytic function on D for which g(0) = 0. An application of Hblder's inequality and (5.4) gives 92 1/‘2 II g H qS Ilgll 2S (ZI |g'(z)|2 (1-Izl?) dA(zYn’) . (5.16) H H [D Let f be analytic on D, and assume for the moment that f (0) = 0. Apply (5.16) to the function g = f ” ' 1. This yields 1) .<_ 2(n- 1)2 I If(z)|2("'2) lf'(z)I2 (1-1212) dA(z)/7t . "qu: H ID Writing [3 = n + 0- 2, and using Hiilder's inequality with index n/2, which has conjugate index n/(n - 2), we get Ilfll2(:-1)S 2 (n- 1)2 I If'(z)l2(1 421523“ If(z)|2("’2)(1 4215145“ dA(z)/7t H [D 2/n S2(n-1)2 Ilf'(z)l" (1423)” d4(z)/7t x D l-2/n ".22 x Iva)?” (1.1.7.12)"’2 dA(z)/7t . (5.17) D Now let he D, and in (5.17) replace f by fo 19,1. - f(h). Then (5.17) becomes llfo ol-flh) "3:42 93 2/n s 2(n -1)2 I |f'((ol(z))|" Iol’(z)|" (1.1217‘)’3 dA(z)/7r x D l-2/n n_.2g x Ilf((pl(z)) - f(h)l2" (14212)“2 dA(z)/7t .(5.18) ID Making use of identity (0.20) and the change-of-variable formula (0.22a) we see that the first integral at the right hand side of (5.18) is equal to ¢Q)|2 IIf (4(2))I" (+i— ) (1-lzl2)fl|¢l'(z)|2dA(z)/7t = ~|z|2 D = I (pm/1(3):" (l-Itpl(z)|2)n'2 (14212)" Iago)? dA(z)/7t ID = I lf'(w)|" (1-lez)”'2 (1-|¢l(w)l7')° dA(w)/7t . D Since o< 1 we have that the exponent of (1 - lz I2) in the second integral at the right hand side of (5.18) is bigger than -1. By Lemma 5.6 there exists a constant C such that the second integral at the right of (5.18) is less than or equal C II f IIB. It follows that (n-l)< Ilfotpl- f(h)"; S 2/n s 2(n-1)2C1-2/n Ilfllg-M (I If'(z)In (14212)”2 (1-|(pA(z)l2)" dA(z)/7t ID (5.19) 94 An important observation to make is that the conditions in statements ( i) and (ii) imply that f 6 EB (by Theorem 1.7), so both statements (1') and (ii) follow at once from inequality (5.19). The general case is easily reduced to the previous case, again making use of the Bloch norm of f. Let 0 < p < oo . Choose an integer n > 2 such that n 2 p. Then we have I |f’(z)|n (1 .1212)”2 (1 -|(pl(z)F)° dA(z)/7t [D s Ilfll;-p I lf'(z)|p (1-1212)”‘2 (1-|(pl(z)I2)° dA(z)/7t . ID This completes the proof of this theorem. [3 The following assertion, which is implicit in the results of V.V. Peller ([26], Theorem 2' on page 454), is an immediate consequence of the above theorem. Corollary 5.7 : Let 1 < p < co . If f is an analytic fimction on D for which I If'(z)|p (l-lzl2)p'2 dA(z)/71: < oo , (5.20) D then f e VMOA. Hong Oh Kim proved that a Blaschke product satisfying (5.20) must be a finite Blaschke product ([19], Theorem 1.1 on page 176). A simple proof is provided by the 95 previous corollary, since it is easy to see that VMOA cannot contain Blaschke products with infinitely many zeros in D. (In fact if b is a Blaschke product and 11 e D is a zero for b, then II b 0 (PA - b (A) "”2 = II b o (p; IIH2 = 1. Thus b is not contained in VMOA if it has infinitely many zeros in D.) In [19] Hon g Oh Kim also proved the following result: Iffe fBand J lf'(z)|p (14212)“ dA(z)/7t < .. , (5.21) [D for 0 p. Put [3 = a + n - p. As in the proof of Theorem 5.5 we have 2/n Ilfll x:-1)_<_ 2(n-1)2 Ilf'(z)|" (1-lzl2)fidA(z)/1t x ” ID l-2/n L23 x ( J |f(z)?" (14211)"‘2 dA(z)/7r . (5.22) ID Now estimate the first integral at the right of (5.22) as follows 96 I lf'(z)|" (1-12155 dA(z)/7t s Ilfllg'p J lf’(z)|p (14212)“ «(2)/7: < co . ID ID The exponent (n - 2/3)/(n - 2) = (2p - 2a - n)/(n - 2) in the second integral at the right of (5.22) is easily seen to be greater than -1, so that also this integral is finite (as a consequence of Lemma 5.6). Thus f e H", for arbitrary integers n > p. Hence f e H‘? finm10 0, choose an r e (O, 1) such that lf’(z)l(1-lzl2)1' D(AJ) for some r e (O, l). The following theorem should be compared with these results. Theorem 5.9 .° For an analytic fitnction f on D we have: 1 (i) feBMOA 4:) sup (Ilf’lsz/rt )dr Oas Ill—>1' . o D(l ,r) 101 Proof: Let f be an analytic function on D. Using characteristic functions we have .2 _ , 2 I If I dA/n' .. Jlf (z)I xix/mm dA(z)/7t , D(A,r) D thus 1 l r 2 _ ' 2 J( I If I dA/n‘ )dr — I If (z)I J‘xlX/Lr)(z) d; dA(z)/7t 0 D(AJ) D o I |f'(z) :2 (1 -|(pl(z)l) dA(z)/7t , D and both (i) and (ii) follow at once. [I The previous theorem can be used to give yet another proof of Pommerenke's result ([29], Satz 1) which states that for an analytic function f on D which is one-to-one, containment in $3, or in SEQ , implies that the function already belongs to BMOA , or VMOA , respectively. Proof: Suppose that f e B is one-to-one. Let ,1 e D. Since for every 2 e D(AJ) |f(z) — f(2)| s Ilfllg log 1—1: , we have the inclusion 102 {f(l) - f(l):z 6 0011)} C D(O, "fliglog T37). and it follows that 2 2 J lf’IZdA/rt = I{f(z) -f(l):z eD(/1,r)}l s IlfllEB (log-l—fr-) . (5.27) D(lr) Thus 1 1 2 2 1 2 I JIf’ldA/n drs IIfIISB J(log-1-_—r—)dr<°°, 0 D(lm) 0 and by (i) of Theorem 5.9 we have that f e BMOA . If f e 130 , then by Theorem 1.2 we have for each r e (O, 1) Ilf’lsz/It —+ 0 as 121—)1’, D(l,r) so that by (5.27) and the Lebesgue Dominated Convergence Theorem we have that l J I If’lsz/zr dr —) 0 as I’ll-91', O D(lJ) thus, by Theorem 5.9, f e VMOA . El Chapter 6 In this chapter we briefly discuss cyclic vectors in the little Bloch space. We generalize a theorem of Anderson, Clunie and Pommerenke and obtain a result very similar to one of Brown and Shields in the context of Dirichlet spaces. First some notation and a definition. In order to be able to compare our result for the little Bloch space with a result of Brown and Shields for the Dirichlet space we will give a general definition for a cyclic vector. Let 8 be a Banach space of analytic functions on D which contains the polynomials as a dense subset, which is invariant under multiplication by the function z , and for which all the point evaluations are bounded linear functionals on 8. For a function f e 8 let [f 18 denote the closure of the set { pf: p is a polynomial } in the Banach space 8. Definition : A fiinction f e 8 is called a W or M if [flg = 8 . The little Bloch space $30 furnishes an example of such a Banach space 6. That $30 is invariant under multiplication by the function z is easy to see, and that the polynomials form a dense subset of EEO is proved in [2],Theorem 2.1. In general, it is easy to show that a cyclic vector in 8 has no zeros in D (see, for example, [35], Proposition 4). In the case of the little Bloch space Anderson, Clunie and Pommerenke proved the following result [2], Theorem 3.8: For fe $30 the condition inf[lf(z)l : z e D} >0 implies that f is cyclicfor ‘30. In Corollary 6.4 we will extend their result and prove: 103 104 Iffe fBo,ge fBonH”, |f(z)l2lg(z)l in D,andifg2is cyclicfor £80,212“ f is cyclic for $30 . We are actually able to prove a result not just for cyclic vectors in 130 but one that gives an inclusion relation for the sets [f ]9ointroduced above. This will be the content of Theorem 6.3. The Dirichlet space D = { f e H (D) : f ' 6 L02} is another example of a Banach space 6 of analytic functions on D. Our Corollary 6.4 should be compared with the following result of Brown and Shields [10], Theorem 1: Iffe D,ge DnH‘”, lf(z)|2lg(z)| in D, andzfgzis cyclicfor D,then f is cyclicfor D. In [10], Proposition 11, Brown and Shields proved also that: Iff, g e D (WI-l”, andszg is cyclicfor D, then both f and g are cyclicfor D. This is also true for bounded functions in the little Bloch space. In Theorem 6.5 we will give an inclusion relation for the sets [f 190 introduced above. As a corollary we get: If f, g a $80 n H °°, and if f g is cyclic for 130 ,then both f and g are cyclic for $80. In the proofs of Theorems 6.2 and 6.5 we will need to use the following two lemmas. Recall that for an analytic function g on D, and for O < t < l the dilate g, of g is defined by the equation gt (2) = g (tz) (z e D). 105 Lemma 6.1 : Let g e 130. Then: sup (1-Izl)|g'(z)l log I-IIZI —> O as t—~) 1'. (6.1) ZED t l'lZI Proof: Since g,’ (z) = t g' (tz), and (1 - t I z I) lg' (t z)l S H g "56 we have the inequality l-tlzl l-IzI l-tlzl - ' < — _ (1 Izl)lgt(z)llog(1.lzl)_||g||‘la 1-tlzl log( 1-lzl) . (6.2) Take 0 < r < l . It is elementary to show that l-lzl l-tlzl l-r l-tr ."ff’s‘, l-tIzl log( 1-121) ‘ l-tr [03(14) ' (6‘3) It follows from (6.2) and (6.3) that l-tlzl l-Izl max (1 -Izl) Igt'(z)l log( IzI_<.r )—90as t—>1-. (6.4) Now let£>0begiven. Sincege 1'30,we can chooseanre (0, 1) such that (l-lwl)Ig'(w)l<£ wheneverr2 0 be given. Since he 130 , II h, - h "SD -—> 0 as t —> 1'. With (6.6) it follows that we can choose a t e (O, 1) for which f II(—-1)ht|ISB<£ and Ilht-hIIB<£. (6.7) ft The function h, /f, is analytic in a neighborhood of D, hence we can find a sequence of polynomials ( p" ) (of course depending on the t that we picked) such that the functions pn - h, If, and their derivatives converge to 0 uniformly on D. We claim that in fact I! IIf(pn--f7’)llEB —)0asn——)oo. (6.8) 107 To prove this claim, write gn = p" - ht/ft . Then by the choice of the sequence of polynomials gn —) 0 and gn' —> 0 uniformly on D as n —>oo . Using the product rule for differentiation we see that (l-lzl2)IEdz—(fgn)l _<_ (1-1212) |f'(z)Ilgn(z)| + (l-lzl2)lf(z)llg”l(z)l . (6.9) Again using the inequality x log (l/x) < 1 for O < x < 1, it follows from 1 |f(z) -f(0)| 5 log (m) IlfllSB , that (1-1212) |f(z)-f©)l s 2IIfIIflB, which combined with (6.9) gives that Ilfgn "13 s Ilfllia llgnllm + (2|lfllI3+ |f(0)l) llgn'llw . (6.10) Now, since both II gn II“, and II gn' II“, tend to O as n —-) oo, our claim (6.8) follows immediately from (6.10). We are now ready to finish the proof. By (6.8) there is a polynomial p such that h Ilf(p - 7:.)"9 < e. (6.11) 108 By the triangle inequality llf -h H sllf( -1)" +|l(£-1)h II +||h -h II p ‘B ’7 ft 53 j; r so I :3 ’ so that (6.7) and (6.11) imply that II f p - h IIB < 38 . We conclude that h e [ f 196 which implies that [h 1136 C [“536 . El Theorem 6.3 : Let f, g 6 $30 , such that I f (z) | 2 I g (z)I (z e D), and suppose that f is nonvanishing and that g is bounded. Then I gzkao c mafia. Proof: Let f, ge SBO , such that lf(z)|2 lg (z)I (z e D), and suppose that f is nonvanishing and that g is bounded. It is easy to see that gze T30 . So by Lemma 6.2 it suffices to show that (6.6) holds with h = g2. As in the proof of Theorem 2.1 note that - I | |f(z) - ft(z)l S Ilfll:B log(-1T_—t—IZZT) . (6.12) It is elementary to check that d f 2 _ f(l)-f,(2) , f'(Z) -ft'(2) 2 a— ((z ' DE!) - 12(2) 8‘03) 8, (Z) 1’ 12(2) 8, (2) 0(2) - 13(2))ft' (z) 2 - 2 g, (z) . i; (z) Using that If, (2) I 2 lg, (z) I (z e D) it follows that 109 s 2|f(z) -ft(z)llgt'(z)l + |f’(z) -ft'(z)l|| gllw + d f 2 Islam“) + |f(z) -ft(z)|Ift'(z)I . (6.13) Using the definition of the Bloch norm it follows from (6.12) and (6.13) that f 2 , l-tlzl "(f7 - l)gl ":3 s 4IIfII,B sue?) (1-121) lg: (z)I 108(7751') + , l-tIzI + Ilgllm IIf-ftllSB + 2IIfIISJB szietg (1-121) If: (z)I log( 1-121) . (6.14) Now, by Lemma 6.1 the first and the third term at the right of inequality (6.14) tend to 0 as we take the limit where t -—> 1'. Since f e 130 also || f - f, IIB -> O , and our claim that f 2 - "(J-f- l)gt "513—, O as t—)1 follows immediately. D The following corollary is an immediate consequence of Theorem 6.3 and the definition of a cyclic vector. Corollary 6.4 : Let f, g e 130 , such that If (2) I 2 lg (z)I (z e D), and suppose that g is bounded and g2 is cyclic for-Bo. Then f is cyclic for $30. Theorem 6.5 : Let f, g e 130 n H °°, and suppose that f is nonvanishing. Then [fglgo c mm. 1 1 0 Proof: Take f, g e EEO r) H °°, and suppose that f is nonvanishing. It is easy to see that then their product h = f g is in $0 . By Lemma 6.2 it suffices to show that the function h satisfies (6.6). It follows from d f d 3((3- - 1)}1‘) = lg((f‘ftlgfll IA lf'(z) - ft' (z)I IIgIIw + |f(z) -ft(z)I Igt'(z)l, and inequality (6.12) that f Il(7--1)htlltB s IIf-ftllfB |Ig||°°+ l , l-tIzl + lefll‘BSzlég(1-|2|)|gt(2)llog(1"2|). (6.15) Both terms at the right of inequality (6.15) tend to 0 as t —> 1' (that the second term tends to zero follows from Lemma 6.1). Thus f - Il(17.1)h‘ll,13 —> 0 as t—)1, t and by Lemma 6.2 we are done. E] The following corollary is an immediate consequence of Theorem 6.5 and the definition of a cyclic vector. 111 Corollary 6.6 : If f, g e 580 n H °°, and if f g is cyclic for 380 , then both f and g are cyclic for I30. Chapter 7 In this chapter we consider Hankel operators with integrable symbol. The Hankel operators that we study are defined by projecting onto the orthogonal complement of the Bergman space. We first prove that these Hankel operators transform in a unitarily equivalent way if the symbol is replaced by one of its Mbbius transforms. We then restrict our attention to Hankel operators with conjugate analytic symbol, and show that Sheldon Axler's results [6], Theorems 6 and 7, hold if the operator norm of the Hankel operator is obtained by putting a weighted [JD-norm on both its domain and its range. Recall that for 0 < p < co the Bergman space Lap is defined as the space of analytic functions f: D —> (I: such that 1/p Ilfll = I lf(z)|p «(2)/a < a. . La” D For 1 S p < co the Bergman space Lap is a Banach space. The Bergman space L02 is a Hilbert space; it is a closed subspace of the Hilbert space L2(D,dA/7r) with inner product given by = 1 mm dA(z)/a. ID for f, g e L2(D,dA/7t). Point evaluation is a bounded linear functional on the Hilbert space L02, thus for every II. 6 D there exists a unique function k A 6 La2 such that 112 113 f(l) = ,forallfe L02. These functions 1‘2 (11 e D) are called the reproducing kernels for L02. It is easy to verify that for every 1 e D the reproducing kernel k1 is given by the formula k (2): A ,forzeD. (l-llz) Because of the reproducing property of k1 we have < k}. , k}. > = 163(4) . Using the above formula for 1‘1. it follows at once that 2 l "k H 2: ——-—2—2. La (l-Ill) Let P denote the orthogonal projection of L2(D,dA/7t) onto L02. In view of the above formula for the reproducing kernels it is easy to see that for g e L2(D,dA/7t) we have the following formula for its projection P (g): 8 (W) (P(g» La and U2 . (L a ) —-) (La ) such that U1(H°°) C H°° and UZOHfoqpl = Hfo U1. 1 1 5 Proof: Take fe L1(ID,dA/a) and g e H°°.Let1'le ID. By (7.2) we have for z 6 ID my» «to, (w» 2 (H v (g)>(z) = I s (w) dA(wya . (7.3) f ° t D (1 - It? 2 ) In (7.3) make the substitution u = ”(w). Making use of identity (0.17) we have 1 (1-12.12)2_ (147202 (1.121152 (1.6/10m)2 Il-Zul4 (l-iz)2(l-a¢l(z))2 Ir-iul“ = (14,112)2 1 (1512)2 (1-E¢,(z))2(1-71u)2 so that change-of-variable formula (0.22a) transforms (7.3) into 22 -| | f((P(Z))-f(ll) (H (3))(2) = (1 .1 )2 fl 2 ——l—; swam» dA(u)/7r f°"’t (1-22) D (1461(2)) (1-2u) ( ( )) - (u) =(1-I).|2)k(z) I f (p12 f (l-Illzflc (u) (go (p )(u) dA(u)/7t l - 2 l l D (1' it $107)) 2 = (1-1165(2) Hf((1 -1111 u, (g o ¢l))(¢l(2)). Thus we have 2 Hf°¢l(g) = (1 -121 ”A Hf((1-|l|2)kl(g° ripe (p, . (7.4) 116 Define the operator U : L2(D,dA/7t) —9LZ(D,dA/7z) by U (g) = (1 - 12.12) it,1 (g o 6,1), for g e L2(ID,dA/a). Since (1 - 1M2) k1 = - (01' , we have for g e L2(D,dA/tr) 2 2 IIU(g)II 2 = I 1(goo )(z)|2l¢ '(z)l2dA(z)/7t = Ilgll 2 , L (ID.dA/7r) D “I “I L (lD.dA/2r) so that U is well-defined. For g , h e L2(D,dA/7t) we have < U (g) ,h ) = I (l -IAI2)kl(z)g WAC?» m dA(z)/7t . D In the above integral make the substitution u = (p112) . We get < 061),}: > = j (1 -IAI2)k2( h(/a . D Now using the identity (0.18) it is easy to verify that . 2 _ k2(¢2(u)) l =1 g(u) (1-1212)k2(u)h(tp2(u))dA(u)/a = (g ,U(h) >. D Hence U is a self-adjoint operator on L2(D,dA/7t). Take g e L2(D,dA/rt) and put h = U (g). Differentiating the identity ¢A(¢l(z)) = 2 we see that for each 2 e D 2 — (l—IAIZ) kl(z) k22((p2(z)) — 1 , so that _ _ 22 = U (h)(2) — (1 W ) k2(2) k2(¢2(2)) 8(2) 8 (2) , and thus U c U = l . Hence U is a unitary operator on L2(D,dA/7t). Observe that U (L22) c L22, U (H°°) c H°°, and U ( (L22)i) c (1.22%. The first two of these inclusions are obvious fiom the definition of U. The last inclusion follows from the first since the operator U is self-adjoint. Let U1 : La2 —> La2 and U2 : (1.22)i —> (L22)i be the restrictions of U to L22 and (L22)i respectively. Then both U1 and U2 are unitary operators and U1(H°°) C H°°. We claim that UZOHf°¢1 = Hf 0 U1. Let g e H°°, then it follows from (7.4) that H20 226;) = (1-111612 (H, o U,)(g)o (p2 . so that 118 2 (U20 22)(g) = (1--|/1|)k/1 ( (p (g)° (p2) f° 2 fo = (14111sz2 (k2o (p2) (H, o U,)(g> = (Hf o Ul)(g) . and our claim is verified. This completes the proof of Theorem 7.1. E] In order to state a corollary of the above theorem we need to introduce more notation. For a linear operator S : L 222 -) (L222) i, densely defined on H°°, let II S II denote the operator norm of S obtained by putting the Lz-norm on both the domain and the range of S, i.e., |IS|I=sup{I|S(g)I| 2 :geHwardIIgII 51}. L (D,dA/7r) L 2 a Let L(Laz, (L02) i) denote the set of all bounded linear operator T : La2 -) (L222) l, densely defined on H°°. For T e L(Laz, (L222) J-), define its singular numbers sn(T) by sn(T) = inf { II T - FM :17 e L (150241.29) has rankatmostn ]. for n e INO. Note that so(T) = II T II. For 0 < p < co the Schatten-von Neumann class cp is defined to be the set of all bounded linear operators T : L22 —> (L22) i, densely defined on H°°, for which co l/p _ p o, IITII P- Erna) < . C n=0 119 Let C °° denote the set of all bounded linear operators T : L 222 -> (L 02) i, densely defined on H°°, which are compact. Then clearly C P C C°° for 0 < p < 00. Take f in L1(D,dA/7t) and suppose that l e D. Let U1 and U2 be the unitary operators of Theorem 7.1. If for an n e INO operator Fe L(L22, (L02) i) has rank at most n , then also U20 F c U 1'1 has rank at most n . Since U1 and U2 are unitary Operators it follows that -1 ”Hfowl- FII = IIHf- U2°F° U1 II, which implies that for each n e IN 0 sn(Hf o 4’21) = sn(Hf) . Thus we get the following corollary. Corollary 7.2 .- Let fe L1(ID,dA/tt), and 0 < p 5 co . If er cp, then for each A e D Before we proceed note that equation (7.4) can be used to obtain a formula for H2091) . Since (1 - 1,112)2 191(k)1 W1) = 1, it follows from (7.4) that Hfo¢2(k2) = k2 Hf(1)°(p2 = "i (fotp2 - HUMP/.2). Replacing f by f 0 (p3 we get the formula 120 Hf(k2)=(f-P(f°¢2)o¢2)k2- (7.5) Let 1 < p < co and -l < a < p - 1. For a Lebesgue measurable function g on D let the weighted LP-norm of g be defined by Mo IIgII = J'lg(z)lp(l-Iz|2)adA(z)/7r P-0 D For f eLa1 think of H f as an operator from H°° to the class of all functions on D. The operator norm II H f “p,a of H f is obtained by putting the weighted lP-norm II . "22202 on both the domain and the range of H f , i.e., II -II = I - : °° I II s . Hf p,a sup{IHfgllp’a geH andlg pg 1} Thus II H f "2,0 coincides with our notation II H ;- I| used before Corollary 7.2. In [6] Sheldon Axler showed that the operator norm II H f II and the Bloch norm II f “13 are equivalent. In the following theorem we extend this result to the operator norms II H; IIM . Theorem 7.3 : Let 1

'/2 the improper integral in the above inequality is finite, and (7.6c) follows. This completes the proof of this lemma. CI The next lemma will play a crucial role in the proof of Theorem 7.3, where it will be used twice. Lemma 7.6 : Let 0 < a < l . Then there exists a finite positive constant C (depending on a) such that for every analytic function f on D and for all z e D : I WW) "“22“ .4. w...) s __C__ IIfIISB . (7.9) [D 11-sz (1-le2)a (1-Izl2)a Proof: Take 0 < a < 1. Let f be an analytic function on D. Fix a point z e D. In the integral at the left of (7.9) make the change of variables A = (pl (w). We get I f(W)-f(Z)| 1 U ml2 (1-1.9.730 dA(w) = 125 1 |f<¢2(/1)- f(2)| 1 - —— —— dA(l) . (14212)“ D 11-12120“) (l-Itl’i" Since ll f 0 (p2 "$13 = II f I133 it suffices to show that there exists a finite positive constant C such that for every analytic function f on D and for all z e D : I |f(h) -f(0)| 1 (1A). II II . . D 11-712120”) (l-IAIZ)“ ()5 C f9 (710) Fix f e I3 , and let ze D. Using that for every he D, |f(h) -f(0)| s Ilfll log , 9 l-l/ll the integral at the left of (7.10) is less than or equal to IlfllSB I log( 1 ) 1 —1— dA(t). D 1"“ Il-izIm‘a) (l-Illz)“ so it suffices to show that sup log (4) 1 -—1— dA(A) < oo. (7.11) zeD ID l-IM '1-iz|2(1'a) (141.30 It is easy to see that the integrals in (7.11) depend only on the modulus p = I z I of z in the disk D, so we have to show that there exists a finite positive constant C such that 126 for all p e [0, l) we have 1 1t Irlog(—1—)—-1—-( 1 d0)dr so. (7.12) l-f 2)a i9 20-0) 0 (1” 4, ll-rpe | Distinguish the following three cases. Case (a): '/2 <0: <1. ApplyingLemma7.5with]3 = l - a, sothat O<[3 <'/2 , we have 1 i9l2tl-a) d6 5 c ((1-rp)2“'1+ 1) s zc, -fl ll-rpe and (7.12) follows immediately. Case (b): a = '/2. Then Lemma 7.5 gives us that 1 d05C(l+log1 )SC(1+log-l—), i9l2(1-a) l-rp 1" 4, ll-rpe from which (7.12) follows easily. Case(c): O-‘—-(I 1, and, s 0 (1'r2)a 4, ll-rpe'elw'a) o . (7.13b) 161D ID I1_1W|2(a'P+2) As in the proof of Lemma 7.6 we have for p = I I'll I (l - I M770! dA(W)/7r = [D Il-iwlm'p”) 1 It = I r (1_,2)a (I 1 do ) dr (7.14) 0 7r |12rpei9lla-p-t-2) Put [3 = a - p + 2 . Then it follows from a < p - 1 that [3 < 1. It is however not ruled out that ,6 is negative. We will first show that statement (7.13a) holds. Distinguish the following four cases. Case 1: '/z < fi< 1. Then by (7.6c) 129 so that the integral in (7.14) is less than or equal to l C Ir (1- r2)a——— dr C 22'3'1J‘r(1-r2)a'2fi+1dr (1- r12)B 0 IA 5 C2254 1 s 3?— 2(a-2fi+2) P-l 9 since a - 25 + 2 = 2(p - 1) - a >p -1, and 213 -1< 1. This proves (7.13a) for this case. Case 2: B = '/2 . Then by (7.6b) we have 1 d9.<_ C(1+log )5 C(1+lo — “9'23 MP 3 1r,) 7, Il-rpe and it follows that the integral in (7. 14) is less than or equal 1 CJr(1-r2)a(1+log-1-1_-;-)dr -1, and (7.13a) follows. Case 3:0 < fl < '/2 . Then by (7.6a) d0 3 C ((1-rp)"23+ 1) s C ((1-r)1‘2’3+ 1) , _,,I1-rpe I25 130 so that the integral in (7.14) is less than or equal 1 C Jr (1-r2)a ((1-r)1‘2”+ 1) dr 5 C ( 1 + 1 0 2(05-2fi+2) 2(a+1) < C(p+a) _ 2(p-1)(a+1>' since a - 25 + 2 > p - 1, and it follows that (7.13a) holds. Case 4: [3 S 0. Then the trivial estimate 1. d6 s 31, -,, |1-rpe'9l2fi 225 shows that the integral in (7.14) is bounded uniformly in p e [0, 1). This completes the proof of statement (7.13a). To show that statement (7.13b) holds we need to consider only two cases. Case 1: 0 <fi < 1. Then the trivial inequality 1 1 Il-iwlzfi 22’“ implies that statement (7.13b) is true. Case 2: B S 0. Now using the inequalities 131 1 1 225 —-:—— 2 —- 2 -—-——’ ll-Awlz’3 (14wa (1411112)?” we get 2 a I meyx 2 22’3 I (1—le )“‘2’3 dA(w)/7t . 1D Il-lwl [D Since a1wehavethata-2fi+1=2p- a-3>p-2>-1, sothat the last integral converges to a positive number. Thus statement (7.13b) is proved and the proof of this lemma is complete. C] Proof of Theorem 7.3: Let 1 O such thatpy e (O,p - 1) n (a, (1+1); then clearly y< 1. Writing p' for the conjugate index of p, i.e., p' = p/(p - 1), it follows immediately from O 05+ 2 we have that (1 - | u|)2P‘ “ 2 (1 - |u|)"““2 and it follows from the above inequality that there is a number 6 > 0 independent of A e D [in fact, we can take 5: min {(1 - r), (1 - r )‘M 1}], such that for all z e D(Lr) lkl(z)lp (1 -lz|2)a 2 6 . (7.18) (1 -I,1I2)2”'°‘ Combining (7.17) and (7.18) we have I I(Hf kl)(z)lp (1 - |z|2)a dA(z)/7r 2 5 I |f(z) -f(2.)lpdA(z)/7t, ow) (1 41' ‘0‘ D(l,r) which, together with formula (0.21) for the normalized Lebesgue area of a pseudo-hyperbolic disk, gives us that 135 i i I J’ |f(z)-f(l)|pdA(z)/7t s D( ,r) D(lr) s —:— (1-IAI2)2P'“'ZJ |(H-f- kAXz)lp (l-lzl2)a dA(z)/7r r 6 D(JLJ) l 2p-a-2 P s r—zg (1-IAI2) IIkalllp’a (7.19) Now, making use of Lemma 7.7 and the definition of the operator norm ll H fll p. a we have lleklllpas Ilellpa C , P- P. (1_IM2)@-a-2 and it follows from (7.19) that 1 J' |f(z)-f(2)|pdA(z)/7r 5 523—11771; , lD(2.,r)| D(lLr) r 5 ' and by Theorem 1.1 there exists a finite positive constant C ' such that IIfIISB s C In? Ilp'a, completing the proof of this theorem. E1 Proof of Theorem 7.4 : Let1 O weakly in If” as I ll —) 1'. That the set {g(oa'l/P' : g e L°°(D, dA/n)] is dense in I?” follows easily from the fact that L°°(D, dA/rt) is dense in [P'(D, dA/It). Since {nlz 2. e D] is norm-bounded in 11"“ it suffices to show that (n; , gwa’l/P') -) O as Ill --> 1', for all g e L°°(D, dA/7r). Fix a g e L°°(D, dA/n). Noting that wa'l/P' (pa = ma ,p we have the estimate '“P' < I J I I = II II Ilk II 720 |(k’1 .8 (Ga )l _ | gllm k1. org/p dA/fl g 0° 1 Imp . (. ) ID 137 We would like to estimate the norm ll ’9. "1.0: ,p , but Lemma 7 .7 does not apply to this norm. The idea is to use Lemma 7.7 with an index slightly bigger than 1, but not too big, so that the necessary estimates work out. It is easy to see that we can choose a number q such that l < q < p and -l < qa /p < q - 1. Now, by Lemma 7.7 there exists a finite positive constant C’ such that for every 1 e D "kill a/ so 1 . M p (1—IAI2)2'°"P'2"1 By Holder's inequality ll k)."1,a/p 5 ll 1‘1 “(ma/p , so that we have |(kl ,gwa'l’P'w s C' II gllw 1 . (7.21) (1-|l|2)2' a/p ~2/q By Lemma 7.7 there is a finite positive constant C such that for every ,1 e D ——1— s C (1-I2I2)2'°"P'2”’ . (7.22) Ilk II 1. p,a Combining (7.21) and (7.22) we get 2 2 |(nl,gwa-1/P')IS CC' llg II” (142.12)" P, which implies that (n,1 , gwa'l/P') —> o as I AI -+ 1', and the claim is proved. Now, since H 7 is a compact Operator and n; -> O weakly in LP” as l/ll —-) 1‘, we must have ll H ;- n1 Ilp'a -) O as I ll —> 1'. It follows from (7.18) and Lemma 7.7 that 138 l lD(lJ) I J |f(z)-f(2)I”dA(z)/zt s -C_ IIH— n I” , D(lLr) therefore we have 1 lD(ll,r)l J |f(z) -f(l)|pd4(z)/7t —) O as Ill—) 1-, D(lr) and by Theorem 1.2 it follows that f e 130. 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