Analytic Besov spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND Analytic 157, 3 18-336 (1991) APPLICATIONS Besov Spaces KEHE ZHU* Department of Mathematics, State University Albany, New York 12222 of New York, Submitted by C. Foias Received June 5, 1989 1. INTRODUCTION Let D be the open unit disk in C and dA the normalized area measure on D. For 1 < p < +co, the analytic Besov space BP consists of analytic functionsfon D such that (1 - Izl’)f’(z) is in LP(D, &), where d&z) d4z) = (, _ lz12)2 is the Mobius invariant measure on D. The Besov space B, consists of analytic functions f on D such that f(z) =CT=? anql,(z) with C,‘=“l [anI < +co, where 1, ED and VA(Z) For 1 <p< zz A--2 1-Xz’ ZED. +co, we write llfllB,= lI(l - Iz12)f’(Z)IlU(di)~ When p = 1, we write llfll B,= hf { y M: f(z) = F anlni..(z))’ n=l n=l Note that B, is also called the Bloch space. I( jl B, is a complete norm on B,. When 1 < p d +co, 11iI4 is a complete semi-norm on BP. For all 1 d p < +co, 11/IBPis Mobius invariant in the sense that IIf0 cp]IBP= llfll 4 for all f~ B, and cpE Aut(D), the Mobius group of D. * Research supported by the National Science Foundation. 318 0022-247X/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. 319 ANALYTIC BESOV SPACES The advantage of the above definition for B, is that I/ IIB, is clearly Mobius invariant. The disadvantage of this definition is that it makes B, appear isolated from the other spacesB,. It follows from Theorem A below that for 1 < p < + 00 an analytic function f(z) on D is in B, if and only if (1 - IzI *)*f”(z) is in LP(D, &). Thus we see that B, is indeed compatible with the other B,‘s. In particular, we see that f is in B, if and only if s D \f"(Z)I dA(z)< +a. This characterization for B, can be found in [2], and we will use it occasionally. To describe the main results of the paper, we need to use the Bergman metric of D, which is given by 1 B(z, w)=px 1 + kPz(wl 1 _ ,ql(w),’ z, WED. For any z E D and Y> 0, let E(z, r) = {w E D: /?(z, w) < Y} and IE(z, r)j be the (normalized) area of E(z, r). If f is analytic in D, we define the oscillation off at z in the Bergman metric as o,(.f)(z)=suP{If(z)-f(W)I: wEE(z, y,). We’ll also consider the mean oscillation off in the Bergman metric. Let ^ 1 fJz) = lqz, r)l I E(;,r)f(w) dA(w), ZED. The mean oscillation off at z in the Bergman metric is then defined to be MQ,(f ^ 1 )(z) = ,E(z, r), sE(=,r)If(w) -f,(z)1 dA(w). For f analytic in D, we’ll let 1f(E(z, r))l denote the (normalized) area of the image of E(z, Y) under$ The main results of the paper are Theorems A, B, C, and D. A. Suppose 1 d p < +CO, m 2 2, P is the Bergman projection on D, and f is analytic in D, then the following are equivalent: THEOREM (1) f+; (2) f E PLP(D, di,); (3) (1 - IZ/2)“‘f(n’)(z)ELP(D, dA). 320 KEHE ZHU Moreover, if 1 < p < +co, then (4) jr& (If(z)-f(w)l”/ll -Z~14)~4Z)~~(~)< THEOREM B. following If r > 0, 1 < p < +oo, and ,f is analytic are equivalent: (1) f ~4,; (2) MO,(f) E LV, (3) w(f) (4) If(E(z, r))l”‘E THEOREM C. +a. in D, then the d2); E LP(Q dA); LP(Q d2). Suppose f(z) = CT:0 a,z” is analytic in D, then we have (1) For any 16 p < +co, there is a constant C, > 0 (independent off) such that IanId C, Ilf /IBpn--lip, (2) n = 1, 2, ... . There is a constant C > 0 such that THEOREM D. For any 1 < p < +oo, there exists a constant C, > 0 such that If(z)-f(w)1 GCp Ilf IIBpP(Z,w)“y for all f E B, and z, w E D, where /I is the Bergman metric and l/p + l/q = 1. We note that the above results are well-known for the case p = +co. See [ 1, 3,4]. Actually, this case serves as our main motivation for the paper. Theorem A will be proved in the next section by a number of lemmas some of which are of independent interest. We prove Theorem B in Section 3, where several other geometric characterizations for the analytic Besov spaces are also obtained. Sections 4 and 5 are devoted to the proof of Theorems C and D, respectively. The author thanks the referee for a very careful reading of the manuscript and several insightful comments and suggestions. This work was done while the author was at the University of Waterloo. 321 ANALYTIC BESOV SPACES 2. ANALYTIC CHARACTERIZATIONS Recall that the Bergman kernel of D is the function K(z, w) = 1 (1 -zG) 2’ z, WED. The Bergman projection is the operator P defined by P!(z) = j f(w) &, w) dA(w), ZED. D For any LED, let 1 - /q* k,(z) = (1 _ Xz)2. It is easy to see that k,(z) = -q;,(z), thus we have the following change of variable formula jD.bCPAZ) dA(z)= jDS(4lk,&)12 dA(z). Since each cp).is an involution, we also have This section is devoted to the proof of Theorem A. We break the proof into several lemmas some of which are of independent interest. LEMMA 1. Iff Proof: Fix any nonnegative integer n and z E D, then is a polynomial, ix0 then f E PLP(D, d%)for all 1 ,< p 6 +oo. (m+ l)z* jD (1 - 1~1’)~w’?V’dA(w) =(n+l)z” j~(l-lw~*)‘l~~~“dA(w) 2 =(n+2)(n+3)Z”’ 322 KEHE ZHU It follows that each monomial is in PLP(D, &) for all 1 < p < +oo. Since the Bergman projection is linear, we see that each polynomial is in PLP(D, dA) for all 1 <p< +co. 1 LEMMA 2. Suppose m 3 l,f~ . . = f”“‘(O) = 0, then Proof L’(D, dA) is analytic, andf(0) =f’(O) = Since f(0) = f’0) = . . = f”“)(O) = 0, the integral (1- Iw12)“f’“‘(w)dA(w) g(z)=$S, $yl-zw)2 converges for all z E D and defines an analytic function g(z) in D. Taking derivatives n times inside the integral gives SincefE L’(D, dA), we have (1 - Iz12)“f’“‘(z) E L’(D, dA) [S]. By 7.1.4 of [6] (and an application of dominated convergence), we have ZED. g’“‘(z) = f’“‘(z), For any 0 <k B m - 1, Taylor expansion shows that (1 - Iw12)“f’“‘(w) g’*‘(o)=i!!$!?j dA(w)=o -m-k W D Thus we have g’“‘(z) =f’“‘(z) and gCk’(0)=f’“‘(O) for all 0 d k <m - 1. Hence f(z) = g(z), completing the proof of Lemma 2. 1 LEMMA 3. rf 16 p < + a3 and f E B,, then f E PLP(D, dl,). By Lemma 1, we may as well assume that f(0) =.f’(O) = . . . = Proof f ‘4’(O)= 0. By Lemma 2, we have f = Pq with cp(z)= Cl- Iz12)f’(4 z ZED. If 1 < p d +co and f E B,, then cpE LP(D, dA) and hencef E PLP(D, dA). Iff EBi, then f(Z)= y anV2n(z) I, = 1 323 ANALYTIC BESOV SPACES for a sequence {An} in D and C,:=“, la,/ < +co. It is easy to compute that f”(Z) = -2 :cr an/in (:_;;$ II= I By 1.4.10 of [6], there is a constant C>O such that jD(1- lz12)2 If”(z)1 4z)=s,If”(Z)1 d‘e) 62 +f I% Cl- IW~J: ,ly;,3 n ,z = 1 II=1 By Lemma 2, we have S = PCJJwith q(z) = (1 - Izl*)*fw 22* in L’(D, dll). This completes the proof of Lemma 3. 1 LEMMA 4. The operator T defined by Tf(z) = (1 - 1~1~) jDfIl”~z$;’ is bounded on LP(D, d;l) for all 1 < p < + CO. Proof: l<p< By 1.4.10 of [6], T is bounded on L”(D). +oc and l/p+l/q=l. Let 1 El=-+- 2q 3 P’ 1 &l &*=-+- Assume that 2 4’ then we have E, + c2= 3, qE2> 2, and pc, > 3. By Hiilder’s inequality, we have 324 KEHE ZHU Since qc2 > 2, 1.4.10of [6] implies that there is a constant C, > 0 such that c dA(w) ID d 11 -zzw/qc2 (1 _ YIP )2):yF-2 for all ZE D. Since E, + .s2= 3 and l/p + l/q = 1, it is easy to see that 1 - (qE2- 2)/q = F, -2/p. It follows that ITf(z)l”< C,(l - Iz12)P@ IS( p dA(w) sD 11- ZWlPC’ for all z ED. Applying Fubini’s theorem, we get jD IV(z)lp WI 6 C, j”DIf( p dA(w) i, “l;!?;;:;4 dA(z). Since pcl - 4 > - 1, 1.4.10 of [6] implies that there exists another constant C, > 0 such that (1 - IzJ2)PQ-4 c2 D I 1 - ZWlP”’ dA(z)6 (1 -jut\‘)’ 1 for all w E D. This implies that j I?‘f(z)lp4z)~C,C, D jD If(w)l”d4w), completing the proof of Lemma 4. 1 COROLLARY. Proof. If 1 < p d + cc and f E PLP(D, dA), then f E BP. Suppose cpE LP(D, dll) and f = Pep, then for any z E D, we have Differentiating under the integral sign and then multiplying the result by (1 - lzl*), we get By Lemma 4, we have (1 - lzl*)f)(z)~ When p = 1, we have f”(z) = 6 LP(D, dE,) for all 1 <p< !:,,l;y;;;, dA(w) +co. 325 ANALYTIC BESOV SPACES and =6 j Icp(w)l dA(w)< +a. I D We have proved the equivalence of (I) and (2) in Theorem A. LEMMA 5. Suppose a > 1 and T, is the operator defined by Tnf(z) = (I- Iz12YjD :,(~z;I::,,. Then T, is bounded on L,(D, dA) for all 1 6 p d +CO. Proof: c, Ilfli,. By 1.4.10 of [6], there is a constant C, >O such that IlT,/I, d On the other hand, 1.4.10 of [6] gives another constant C, > 0 such that sD ITxf(z)l W)d I, If(w)1 dA(w) jD (l;+;r;22dA(z) d c2 If( sD (1 Mw) - (,i2)’ = C2 jD If(w)ld~(w). By interpolation [S], T, is bounded on LP(D, d;l) for all 1 < p < + CO. 1 Remark. The referee pointed out that Cl- lz12Y If(w)I dA(w) IT,f(z)l d t1 _ lzlIm-~ .rD ll-zWl3 . Thus Lemma 5 follows from the proof of Lemma 4 in the case 1 < p < +a, and hence interpolation is not needed for the proof of Lemma 5. COROLLARY. Suppose 1 6 p 6 + co and m 3 2, then f E PLP(D, dA) if and only [f (1 - /z12)*f (M)(z) E LP(D, dA). ProoJ The “if” part follows from Lemmas 1 and 2. To seethe “only if” part, let f = Pep with q E LP(D, d;l). Then differentiating under the integral sign gives The desired result now follows from the above lemma. m 326 KEHE ZHU This completes the proof of the equivalence of (l), (2), and (3) in Theorem A. LEMMA 6. Zf 1~ p < +m and f is analytic in D, then f E B, ij’ and only if is D D ProoJ If(z)-f(w)l” )l-nq4 dA(z) dA(w) < +m. If g E L’(D, dA) is analytic, then Taylor expansion shows that g’(O)= 2 jD $g(w) - g(O))Mw). By Holder’s inequality, we have Ig’(O)1’ 6 2’ jD Ig(w)- g(O)1 pWw). Replace g by f 0cp=,then Using Fubini’s theorem and a chage of variable, we get s D (1 -l4’)” If'(z)l"d4z) =2p!*,!“D ‘f’;‘~+-;;“p dA(z) d/l(w). This proves the “if” part of the lemma. On the other hand, by Theorem A of [S], there is a constant C> 0 (depending only on p) such that is DD If(z)-f(w)lpd~(Z)dA(w) ll-z+14 =~DW)~D If~cp,(w)-ff(z)lPdNw) 327 ANALYTIC BESOV SPACES Since (focp,)’ (w)=f’((~Aw)) d(w) and (1 - 14’) Id(w)I = 1- Iv,(w)l*, we have ss If(z) -f(w)l p &(=) dA(w) D * (1-zW14 4 z1JD (1 GCJD I~z(~~~)l’YIf’(cP,(w))l” dA(w) = c j di(z) j (1 - lwl’)” If’(W)IP Ikl(W)12dA(w) D D = c jD Cl- Iwl’)” If’(~~)l”~4W) I, lly~;14 = c jD (1- [WI’)” If’(w)“&(w). This proves the “only if” part of Lemma 6. 1 This completes the proof of Theorem A. We remark that the following are equivalent norms on B, for 1 < p < +a: (1) lIfllBp+ If(O (2) llfll = I.m)l + If’(O)+ ... + If’“- “(O)l + lltl- Iz12~‘~f~“~~z~llLP(di)~ Ilfll =inf(IIgll,~,,:f=~g); c4) /tf /I = If(O)I + &, SD(If(z) -fcw)l "/I 1 - zw14) &4(z) &l(w)]? (3) 3. GEOMETRIC CHARACTERIZATIONS Recall that p is the Bergman metric on D and for z E D, r > 0, E(z, r) = {w E D: fi(z, w) < r}. If f is a function on D, then the oscillation off metric is o,(.f)(z)=supfIf(z)-f(w)l: at z in the Bergman wEE(z, r)>, and the mean oscillation off at z in the Bergman metric is 1 1 M~r(fNz) = ,E(z, r), sE(;,r)If(w)-.Ixz)l dA(W)> 328 KEHE ZHU where?Jz) is the averaging function defined by For any open set E c D and any bounded function dist(f, H”(E))=sup(I f(z)-g(z)l:zEE, f on E, we write gEH”(E)}. The main purpose of this section is to prove Theorem B. In the process of the proof, we’ll obtain several other characterizations of the analytic Besov spaces.We need the following well-known estimates (see [4] for example): (1) There is a constant C > 0 (depending only on r) such that 1 C I& C r)l 6 IUw)12 6 IE(z, ).), for all z E D and w E E(z, Y). (2) There is a constant C > 0 (depending only on Y) such that lfcz,l6--j IE(z, r)l ,q;,r) IfCw)' dA(w) for all analytic functions f and z E D. We state the main results of the section as THEOREM 7. If r > 0, 1~ p < +CXI, and f is analytic in D, then the following are equivalent (note that in some casesfunctions are ident$ed with their values at z): (1) fEBp; (2) MO,(f) E LV, (3) (4) If(E(z, r))11i2ELPP, di); C&i,r) If’(w)12 d4w)l”2 E LPP, d2); SE(w)If "(w)I dA(w) E LP(D, dA); diWIEcz,,,, H”(E(z, r))) E LP(D, dA); o,(f)ELVX dA); IE(z, rj1-l jEcz,rjIf(w)-f(z)I dA(w)ELP(D,dA). (5) (6) (7) (8) dJ%); ProoJ The theorem will be proved in the following orders: (1) o (2) o (7)--V); (l)-(5); and (l)-=-(3)-(3)-(4)-(6). ANALYTIC BESOV SPACES 329 We begin with the inequality Replace .f by f - f{z), then for all analytic f and z, WED. If w E E(z, v), then E(w, r) c E(z, 2r) and there is another constant C2 > 0 such that It follows that for all z E D. Choose a constant C > 0 such that C2 G c Ik,(N IJW, WI 2 for all z E D and w E E(z, 2r), Since each lk,(w)l’dA(w) measure, we have is a probability This proves the implication (1) + (7) by Lemma 6 (and a change of variable). The implication (7) => (8) is obvious from the observation that 409:157'2-3 330 KEHE ZHLJ That (8) implies (2) follows from the inequality =- 2 Im 4 I I ‘q:,r) If(w) -mi ~40 By Taylor expansion and the symmetry of E(0, Y), we have s’(O)=Cj E(O. r) a4 d-0) for all analytic function g in D, where C > 0 is some constant independent of g (but dependent on r). Replace g by g - A and crack in the absolute values, then I g(w) - 4 Mw) for all analytic g and all complex numbers A. Replace g by fo cp; and A by .ih), then (1- lz12)V-WI G c j w, r) If%04 -m d4w). Changing the variable in the above integral and using the fact that cp,VW, r)) = JW, 4, we get (1 - Izl’) If’(Z)I Q c j IS(w) -“M E(;,r) Choose another constant C’>O such that c Ikwl’~ C’ ,E(z, r)l for all z ED and w E E(z, Y) then (1 - IA21 I.fWl G C’MWfk) Ik(w)l’ Ww). ANALYTIC BESOV SPACES 331 for all analytic f and z E D. This proves that (2) implies (l), and we have completed the proof of the equivalences of (l), (2), (7) and (8). Next we prove the equivalence of (1) and (5). By a change of variable, ) lf”(W)l dA(w) = I,(o,r) If”(cpAw))l lk(w)12 dA(w). sE(-,r Since E(0, Y) is a Euclidean disk with center 0 and radius less than 1 (depending on Y only), there is a constant C> 0 such that 1 ,< C( 1 - Iwl’)’ for all M’E E(0, r). Thus s I.f”(~~)l dA(w) < c j If”(P4w))l (1 - 142)2 lkb412 Mw) E(:.r) E(0.r) =Cj uo.r) If”(cP,(w))l (1 - I4&4’)‘~~(w) 6 c sD If”(cPAw))l Cl- I~‘04’)‘~~(w) = c [ If”(W)\ (1 - Iwl’)2 ~k;(w)~‘liA(w). JD then (1 - I~l~)~f”(w) is bounded in D, which implies that If .fEB,,, jE(z,rj If”(w)1 &I(w) is bounded in z since each lk,(w)12d,4(w) is a probably measure. Moreover, if 1 d p < +cc, then Holder’s inequality implies that D If”(W)lP (1 - lw12)2pIk,(W)l’dA(~~), and Fubini’s theorem gives ~dE.(z)~Cp~D(l-~M;‘)~~lf”(w)~“drl(w). I.f”(W)l &l(w) 1 Therefore, (1) implies (5) by Theorem A. On the other hand, we can find a constant C > 0 (depending on r only) such that C If “(ZII 6 lE(z, sE(-,r,I.f”(W)l dA(w) for ail z E D and all analytic J: Since lE(z, r)l - (1 - 1~1~)~for any fixed r > 0, we can find another constant C’ > 0 such that I f”(w)l dA(w). E(;,r) (1 - \z12)2I.f”(Z)l d C’ 1 Thus (5) implies (1) by Theorem A. 332 KEHE ZHU Finally we prove the equivalences of (1 ), (3), (4), and (6). That (4) implies (3) is clear, because IE(Z,rjlf’(w)12 &t(w) is the area of f(E(z, r)) counting multiplicity. By the proof for the case p = +co in [3], there are constants C, > 0 and C’, > 0 such that (1 - lzl’) If’(z)1 <Cl dist(flE(z,r),H”(E(z, r)))GC2 If(Q, ~))l”~. Thus we have (4)=> (3)9 (6) 5 (1). It remains to show that (1) implies (4). First we note that l/2 1 IfW12dA(w) < JE(z, r)ySup{If’(w)l: If w E E(z, v), then E(w, r) c E(z, 2r) and IE(w, r)l WEE(Z, Y)]. is comparable to I%, r)l, thus If’(w)l 9 ,,E: r), s,,,.,, Is’(u)I dA(u) <- c2 If’(U)1 Mu) IEk r)l sE(z.Zr) for all w E E(z, Y). It follows that I) [i EC:, If’ba2 dA(w) 1’2f c, (E(z, r)l Ii2 E(-2 , If’(~)l dA(u) I% r)l s -, r 1 for all z E D. Since IE(z, r)/ ‘j2 is comparable to (1 - IWI’) and jE(z, r)] ’ is comparable to Ik,(w)l’ for all ZE D and w E E(z, r), we can find a constant C > 0 such that d C i (lD 1~1~)If’(w)1 Ik(w)12 MM)) for all z E D. The implication (1) + (4) now follows easily from an application of the Cauchy-Schwarz inequality (note that lkz(w)12 d(w) is a probability measure for each z E D) and Fubini’s theorem. This completes the proof of Theorem 7. 1 ANALYTIC 333 BESOV SPACES 4. TAYLOR COEFFICIENTSOF FUNCTIONS IN B, In this section, we estimate the growth of the Taylor coefficients of functions in the analytic Besov spacesB,. The main result of the section is Theorem C in the Introduction. For convenience of reference, we restate the theorem as THEOREM 8. Suppose f(z) = Cz=x0 a,,z” is analytic in D, then we have (1) For all 1 <p< +a, there is a constant C,>O n = 1, 2, ... . I4 GCp lIfllBpn-“p~ (2) with There is a constant C > 0 such that +cr 14 6 C IlfllB,. ,t=0 Proof: Since the Bloch space B, contains all the other Besov spaces, we may as well assume that f E B,. In this case, we have a,z=(n+l)~~(l-l;i2)f’(~)in~1dA(z), n = 1, 2, ... . a,,=f(n+l)J n = 2, 3, ... . (l-~z~2)2f”(z)~‘*-2dA(z), D It follows from the first equation above and Holder’s inequality that if 1 <p< +co, then Ia,/ <(n+ 1) jD (1 - lz12) If’(z)/ Izlnp’ (l- 1~1’)~dl(z) l/Y <(n+ 1) IIf II4 Izl(n-‘)q (1 - IzI*)~~d(z) j i D I for all n = 1, 2, .... where l/p + l/q = 1. Using polar coordinates and r-functions, we can easily get By Stirling’s formula, Q(n-l)q+2) T((n+l)q+l)-n24 1 (n-t +co). 334 KEHE ZHU Therefore we can find a constant C > 0 such that ?Y bnl GC IIfIlBJ~+ 1) p& ( n = 1, 2, ... 3 > Since l/p + l/q = 1, we have 2 - l/q = 1 + l/p. It follows that there is a constant C,>O with bnl Q cp IlfllEp-‘i” for all n=l,2,.... This proves (1) for l<p< +co. The case p= +cc is trivial from the first formula for a,. Moreover, it is clear that C, = 2. To prove the case p = 1 in (1 ), we use the second formula for a, : a,=&+l) j (1 -~z~2)2,f”(Z)Zn~*dA(Z), n = 2, 3, ... . D Elementary calculus shows that the maximum of (n + 1) Iz\~-~ (1 - 1~1’)~ in D is (n+ 1) (s)(np2)‘2 (5) which is clearly less than 16/n. Since jn If”(z)1 &t(z) is dominated by /If IIB, (see the proof of Lemma 3), we can find a constant C > 0 such that (a , <c IlfllB, n --. n for all n b 2. The case n = 1 is easy (point evaluations of derivatives of functions are bounded linear functionals on the analytic spaces). This proves (1). Part (2) also follows from the second formula for a,: un=; (n+ 1) JD (1 - lz~2)2f”(z)Z~~* &t(z). We have nF* l%l <;j: (I- lz12)’ If”(Z)l dA(z) +cx(n+ 1) 14n-2 n=2 G 2 sD If”(Z)l dA(z) ANALYTIC BESOV SPACES 335 f IIB,.Since la01+ la,] is dominated by I/f IIB,, the which is dominated by 11 desired result in (2) now follows. This finishes the proof of Theorem 8. 1 Remark. The above result for p = + so is well-known, see [l] example. The casep = 1 in (1) is also well-known, see [2]. for 5. GROWTH OF FUNCTIONS IN BP Finally in this section we prove Theorem D, a Lipschitz type estimate for functions in the analytic Besov spaces. THEOREM 9. For any 1 6 p < +CD, there exists a constant C, > 0 such that If(z)-f(w)1 GC, Ilf IIB,,P(ZT wry for all f E B, and z, w E D, where /? is the Bergman metric and l/p + l/q = 1. Proof: If p = + co, then q = 1 and the desired result is well-known (with C, = l), see [3] for example. When p = 1, then q = +cc and the desired result is obvious by the definition of the norm ((f jlB,. So we now assume 1 < p < +GO. Since both II 1)4 and /I are Mobious invariant, it suffices to prove that there is a constant C, such that If(z)-f(O)1 G c, II.fIIBJm z)“y for all f E B, and all z E D. By Taylor expansion, we have for all f E B, and z E D. It is easy to find a constant C, > 0 (see Lemma 15 of [7]) such that If(z)-f(O)1 ss, (1 -lwl’) If'(w)l ,w, ,1 -zu’12 d c, s (lD dA(w) Iw12) If’(w)1 dA(w) II -zzw(’ = C, jD Cl- b12) If'(w)1 'I,_'!;~'d&v,. 336 KEHE ZHU Applying HGlder’s inequality, we get By 1.4.10 of [6], s (l- lw12Y2 dA(w)-~log D (l-zW12Y 2 1+ IZI --/?(O, z) 1 - lzl (lzl + 1 -). Thus there are constants 6 E (0, 1) and C2 > 0 such that If(z) -“m)l G c2 IIf IIBPPa ZP for allfEBp and 6 < Iz/ < 1. The case IzI < 6 can be deduced from Theorem C. In fact, if f(z) = CcT0 u,zn, then Theorem C gives a constant C3 > 0 such that If(z)-f(O)1 6 y I4 l4”6C, IlfllB,, y lZl”n-l’P ?I=1 n=l fC3 llfllB&b& Ilf llBpIzl + Ilf IIBpB(O~ z)l’y Ilf 113lW& for all f~ B, and IzI < 6. This completes the proof of Theorem 9. [ REFERENCES 1. J. M. ANDERSON, J. CLUNIE, AND CH. POMMERENKE, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. 2. J. ARAZY, S. FISHER, AND J. PEETRE, Mobius invariant function spaces, J. Reine Angew. Math. 363 (1985) llc-145. 3. S. AXLER, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315-332. 4. S. AXLER, Bergman spaces and their operators, in “Surveys of Some Recent Results in Operator Theory” (J. B. Conway and B. B. Morrel, Eds.), pp. l-50, Research Notes in Mathematics, Vol. 1, Pitman, London, 1988. 5. J. BERG AND J. LBFSTR~M, “Interpolation Spaces,” Grundlehren Math. Wiss., Vol. 223, Springer-Verlag, New York/Berlin, 1976. 6. W. RUDIN, “Function Theory on the Unit Ball of C”,” Springer-Verlag, New York/Berlin, 1980. 7. K. H. ZHU, Duality and Hankel operators on the Bergman spaces of bounded symmetric domains, J. Funct. Anal. 81 (1988), 26&278. 8. K. H. ZHU, The Bergman spaces, the Bloch space, and Gleason’s problem, Trans. Amer. Math. Sot. 309 (1988), 253-268.