Bull. Korean Math. Soc. 35 (1998), No. 2, pp. 293{299
A PROBLEM FOR ANALYTIC FUNCTIONS OF BOUNDED AND VANISHING MEAN OSCILLATION Hasi Wulan Abstract. In this note we consider some characterizations of an-
alytic functions of bounded and vanishing mean oscillation on the unit disk in C and answer a question about them in the negative.
1. Introduction Let D = fz : jzj < 1g be the unit disk in the complex plane, z = x + iy, and denote by dxdy the usual area measure on D. For w 2 D, let the Mobius transformation 'w : D ! D be de ned by 'w (z) = 1w,,wzz ; w 2 D: For 0 < r < 1, let (w; r) = fz 2 D : j'w (z)j < rg be the pseudohyperbolic disk with center w and radius r. The space BMOA ("Bounded Mean Oscillation", see [1]) is the set of all analytic functions on D for which kf kBMOA < 1, where Z 2 1=2 i 2 kf kBMOA = sup jf ('w (e )) , f (w)j d : w2D
0
Contained in BMOA is the subspace V MOA ("Vanishing Mean Oscillation"), the set of all analytic functions f on D for which Z 2 lim jf ('w (ei )) , f (w)j2d = 0: jwj!1, 0
Received May 17, 1997. 1991 Mathematics Subject Classi cation: 30D45. Key words and phrases: BMOA function, V MOA function, -Carleson measure.
Hasi Wulan
It is well-known that for a function f analytic on D we have (see [3]) ZZ (1.1) f 2 BMOA () sup jf 0(z)j2(1 , j'w (z)j2 )dxdy < 1 w2D
and (see [5]) (1.2) f 2 V MOA () jwlim j!1,
D
ZZ D
jf 0(z)j2 (1 , j'w (z)j2 )dxdy = 0:
The Bloch space B is the set of all analytic functions f on D for which kf kB = supz2D jf 0 (z)j(1 , jzj2 ) < 1, and the little Bloch space B0 is contained in B for which limjzj!1, jf 0(z)j(1 ,jzj2 ) = 0. For 0 < p < 1 and 1 < < 1, we know that (see [6])
f 2 B () sup
ZZ
w2D
and
f 2 B0 () jwlim j!1,
D
jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 ) dxdy < 1
ZZ D
jf 0(z)jp (1 , jzj2)p,2 (1 , j'w (z)j2) dxdy = 0:
The Bloch space B and the space BMOA share many analogous properties, as do the little Bloch space B0 and the space V MOA. Motivated by these facts and the observation of the equivalents (1.1) and (1.2) for BMOA and V MOA, respectively, Stroetho asked in [6] the following: Question. Let f be an analytic function on D and 0 < p < 1. Are the following statements true? ZZ f 2 BMOA () sup jf 0(z)jp (1 ,jzj2 )p,2 (1 ,j'w (z)j2 )dxdy < 1: w2D
f 2 V MOA () jwlim j!1,
D
ZZ D
jf 0(z)jp (1,jzj2)p,2 (1,j'w (z)j2)dxdy = 0:
Choa and Miao settled the question above in the negative, respectivly, that is: 294
A problem for analytic functions of bounded and vanishing mean oscillation
Theorem A ([2],[4]).
(i) If 0 < p < 2, then there exists an analytic function f 2 BMOA such that ZZ sup jf 0 (z)jp(1 , jzj2 )p,2(1 , j'w (z)j2 )dxdy = 1: w2D
D
(ii) If 0 < p < 2, then there exists an analytic function f 2 V MOA such that ZZ lim jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 )dxdy 6= 0: jwj!1, D
Moreover, Stroetho proved in [6] the following result: Theorem B. Let f be an analytic function f on D and 0 < p < 1; 0 < < 1. Then RR (i) supw2D D jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 ) dxdy < 1 =)
f 2 BMOA; RR (ii) limjwj!1 D jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2) dxdy = 0 =) f 2 V MOA.
However, Theorem B gave only sucient conditions for an analytic function f to belong to the spaces BMOA and V MOA. A natural question then arises: for 0 < p < 1; 0 < < 1, are the conditions (i) and (ii) in Theorem B necessary for f to belong to BMOA and V MOA, respectively? In this paper we answer this question in the negative. Our result is the following: Theorem. Let 0 < p < 1 and 0 < < 1. (A) There exists an analytic function f 2 BMOA such that ZZ sup jf 0(z)jp (1 , jzj2 )p,2(1 , j'w (z)j2 ) dxdy = 1: w2D
D
(B) There exists an analytic function f 2 V MOA such that ZZ lim jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 ) dxdy 6= 0: jwj!1, D
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Hasi Wulan
2. Lemma Before embarking into the proof of Theorem, let us state the de nition of -Carleson measure and its some equivalent conditions, which will be used in our proof. For a subarc I @ D, where @ D is the boundary of D, let
S (I ) = fz 2 D : 1 , jI j jzj < 1; z=jzj 2 I g: If jI j 1 then we put S (I ) = D. For 0 < < 1, we say that a positive measure de ned on D is an -Carleson measure if supf(S (I ))=jI j : I @ Dg < 1: If = 1, we get the classical Carleson measure (see [3]): Lemma [7]. Let be a positive measure and > 1. Then the following statements are equivalent: (a) is an -Carleson measure. (b) for 0 < r < 1, there exists constant C such that
((w; r)) C (1 , jwj) ; w 2 D: (c)
ZZ 1 , jwj2 d(z) < 1: sup w2D D j1 , wz j2
Remark. Lemma above is not true for the case 0 < 1.
3. The proof of Theorem
(A) For 0 < p < 2, by (i) in Theorem A there exsits f 2 BMOA such that ZZ (3.1) sup jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 )dxdy = 1; w2D
D
296
A problem for analytic functions of bounded and vanishing mean oscillation
it follows that for 0 < < 1 ZZ (3.2) sup jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2) dxdy = 1: w2D
D
By Holder's inequality, for 0 < p < 2 and 0 < < 1 we get ZZ jf 0 (z)jp (1 , jzj2 )p,2(1 , j'w (z)j2 )dxdy D ZZ p=2 0 2 2 jf (z)j (1 , j'w (z)j ) dxdy D ZZ (2,p)=2 2,p 2 ,2 2 2,p (1 , jzj ) (1 , j'w (z)j ) dxdy D ZZ p=2 0 2 2 = jf (z)j (1 , j'w (z)j ) dxdy D (3.3) !(2,p)=2 1 , jwj2 22,,pp ZZ 2,p 2 ,2+ 2,p (1 , jzj ) dxdy : j1 , wzj2 D 2,p
Since the di erential form (1 ,jzj2 ),2+ 2,p dxdy is 22,,pp -Carleson measure and 22,,pp > 1, by Lemma we have 1 , jwj2 22,,pp ZZ 2 , p (3.4) sup (1 , jzj2 ),2+ 2,p j1 , wzj2 dxdy = M 6= 0: w2D D Therefore, by (3.3) and (3.4) we obtain ZZ jf 0(z)jp (1 , jzj2 )p,2 (1 , j'w (z)j2 )dxdy D ZZ p=2 0 0 2 2 (3.5) M sup jf (z)j (1 , j'w (z)j ) dxdy : w2D
D
Thus, by (3.1) and (3.5) we have ZZ (3.6) sup jf 0(z)j2 (1 , j'w (z)j2) dxdy = 1: w2D
D
297
Hasi Wulan
By (3.2) and (3.6) Theorem is valid for cases 0 < p 2 and 0 < < 1. Now we consider the cases 2 < p < 1 and 0 < < 1. Let q = 1 + p2 . By (i) in Theorem A there exsits g 2 BMOA such that ZZ sup jg0(z)jq (1 , jzj2)q,2(1 , j'w (z)j2)dxdy = 1: w2D
D
By Holder's inequality we have ZZ 1 = sup jg0(z)jq (1 , jzj2 )q,2(1 , j'w (z)j2 )dxdy w2D ZZD = sup jg0(z)j1+ p2 (1 , jzj2 ) p2 ,1 (1 , j'w (z)j2) p +1, p dxdy w2D D ZZ 1=p 0 2+p 2 p 2 sup jg (z)j (1 , jzj ) (1 , j'w (z)j ) dxdy (3.7)
w2D
sup
D
ZZ
w2D
D
(1 , jzj
2
p, ),2 (1 , j'w (z)j2) p,1 dxdy
(p,1)=p
:
Since g 2 BMOA B; we set supz2D jg0(z)j(1 , jzj2 ) = K . Hence ZZ 1 = sup jg0(z)jq (1 , jzj2 )q,2(1 , j'w (z)j2 )dxdy w2D D ZZ 1=p 0 2+p 2 p 2 sup jg (z)j (1 , jzj ) (1 , j'w (z)j ) dxdy w2D D ZZ (p,1)=p p, 2 ,2 2 p,1 sup (1 , jzj ) (1 , j'w (z)j ) dxdy w2D D ZZ 1=p 2=p 0 p 2 p,2 2 K sup jg (z)j (1 , jzj ) (1 , j'w (z)j ) dxdy w2D D !(p,1)=p pp,,1 ZZ 2 p , 1 , j w j sup (1 , jzj2 ),2+ p,1 j1 , wzj2 dxdy (3.8)
w2D
K 0 sup w2D
D
ZZ D
jg0(z)jp (1 , jzj2)p,2 (1 , j'w (z)j2) dxdy 298
1=p
A problem for analytic functions of bounded and vanishing mean oscillation p,
since (1 , jzj2 ),2+ p,1 dxdy is pp,,1 -Carleson measure and pp,,1 > 1. Therefore, from (3.8) we know that there exsits g 2 BMOA such that ZZ sup jg0(z)jp (1 , jzj2)p,2 (1 , j'w (z)j2) dxdy = 1 w2D
D
for 2 < p < 1; 0 < < 1. This shows that (A) holds for all cases. Similar to the proof of (A), we can get (B) by (ii) in Theorem A. Thus the proof of Theorem is complete.
References [1] A. Baernstein, Analytic functions of bounded mean oscillation, Aspects of Contemporary Complex Analysis, Academic Press, New York, 1979, pp. 3{36. [2] J. S. Choa, Note on the space BMOA, Canad. Math. Bull. 35 (1992), 40{45. [3] J. Garnett, Bounded analytic functions, Academic Press, New York, 1981. [4] J. Miao, A property of analytic functions with Hadamard gaps, Bull. Austral. Math. Soc. 45 (1992), 105{112. [5] S. C. Power, Vanishing Carleson measuer, Bull. London Math. Soc. 12 (1980), 207-210. [6] K. Stroetho , Besov-type characterizations for the Bloch spaces, Bull. Austral. Math. Soc. 39 (1989), 405{420. [7] H. Wulan, Carleson measure and the derivatives of functions in BMO, J. Inner Mongolia Normal University 2 (1993), 1{9. Department of Mathematics, Inner Mongolia Normal University, Hohhot 010022, People's Republic of China
Current address:
Department of Mathematics, University of Joensuu, P. O. Box 111, FIN80101 Joensuu, Finland E-mail :
[email protected].
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