Research Article Distance from Bloch-Type Functions to the Analytic Space

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1 Abstract and Applied Analysis, Article I 60237, 7 pages Research Article istance from Bloch-Type Functions to the Analytic Space F(p, q, s Cheng Yuan and Cezhong Tong 2 Institute of Mathematics, School of Science, Tianjin University of Technology and Education, Tianjin , China 2 epartment of Mathematics, School of Science, Hebei University of Technology, Tianjin 30040, China Correspondence should be addressed to Cezhong Tong; cezhongtong@hotmail.com Received 9 April 204; Revised 9 July 204; Accepted 4 August 204; Published 9 August 204 Academic Editor: Marco Sabatini Copyright 204 C. Yuan and C. Tong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The analytic space F(p,q,scan be embedded into a Bloch-type space. We establish a distance formula from Bloch-type functions to F(p,q,s, which generalizes the distance formula from Bloch functions to BMOA by Peter Jones, and to F(p, p 2, s by Zhao.. Introduction Let denote the unit disc {z C : z <}of the complex plane C and let T ={z C : z =}be its boundary. As usual, H( denotes the space of all analytic functions on. Recall that, for 0<α<, the Bloch-type space B α is the space of analytic functions on satisfying f B α = sup( z 2 α f (z z <. ( The little Bloch-type space B 0 α is the subspace of all f B α with lim ( z z 2 α f (z =0. (2 It is well known that B α is a Banach space under the norm f B α = f (0 + f B α. (3 In particular, when α =, B α becomes the classic Bloch space B, which is the maximal Möbius invariant Banach space that has a decent linear functional; see [, 2] formore details on the Bloch spaces. For a, the involution of the unit disk is denoted by σ a (z = (a z/( az. Itiswellknownandeasytocheck that σ a (z 2 = ( a 2 ( z 2 az 2 =( a 2 σ a (z. (4 Let 0<p<, 2<q<, 0 s<,and < q+s <.ThespaceF(p, q, s,introducedbyzhaoin[3]andknown as the general family of function spaces,is defined as theset of f H( for which f p F(p,q,s = sup f p (z a ( z 2 q ( σ a (z 2 s da (z <, (5 where da(z is the normalized area measure on.thespace F 0 (p, q, s consists of all f F(p,q,ssuch that lim a f p (z ( z 2 q ( σ a (z 2 s da (z =0. (6 For appropriate parameter values p, q, and s, F(p, q, s coincides with several classical function spaces. For instance, F(p, q, s = B (q+2/p if <s<.thespacef(p, p, 0 is the classical Bergman space L p a (, andf(p, p 2, 0 is the classical Besov space B p.thespacesf(2, 0, s are the Q s spaces, in particular, F(2, 0, = BMOA, and the function space of bounded mean oscillation. See[3 9] forthesebasic facts. For 0<s<, we say that a nonnegative Borel measure μ defined on is an s-carleson measure if μ CM s μ (S (I = sup I T I s <, (7

2 2 Abstract and Applied Analysis where the supremum ranges over all subarcs I of T, I denotes the arc length of I,and S (I = {z =re iθ : I r<, e iθ I} (8 is the Carleson square based on a subarc I T. Wewrite CM s for the class of all s-carleson measures. Moreover, μ is said to be a vanishings-carleson measure if μ (S (I lim I 0 I s =0. (9 For f an analytic function on, we define dμ f (z = f (z p ( z 2 q+s da (z. (0 It was proved in [3]thatf F(p,q,sif and only if dμ f is an s-carleson measure and f F 0 (p, q, s if and only if dμ f is a vanishing s-carleson measure. Let X B α be an analytic function space. The distance from a Bloch-type function f to X is defined by dist Bα (f, X = inf g X f g B α. ( The following result is obtained by Zhao in [9]. Theorem. Suppose p<, 0<s,andf B.The following two quantities are equivalent: ( dist B (f, F(p, p 2, s; (2 inf{ε : χ Ωε (f( z 2 s 2 da(z is a Carleson measure}, where Ω ε (f = {z : f (z ( z 2 ε}and χ denotes the characteristic function of a set. When p=2and s=, the above characterization is Peter Jone s distance formula from a Bloch function to BMOA (Peter Jone never published his result but a proof was provided in [0]. Also, similar type results can be found in [ 3]. Specifically, distance from Bloch function to Q K -type space is given in []; to the little Bloch space is obtained in [2], and to the Q p space of the ball is characterized in [3]. All these spaces are Möbius invariant. This paper is dedicated to characterize the distance from f B (q+2/p to F(p, q, s, which extends Zhao s result. The main result is following. Theorem 2. Suppose p<, 0<s, < q + s <, and f B (2+q/p.Then dist B(q+2/p (f, F (p, q, s where inf {ε>0:χ Ω ε (f (z ( z 2 s 2 da (z CM s }, (2 Ω ε (f = {z :( z 2 (q+2/p f (z ε}. (3 The strategy in this paper follows from Theorem 3..3 in [4]. The distance from a B α function to Campanato-Morrey space was given in [5] with similar idea. Notation. Throughout this paper, we only write U V(or V U for U cvfor a positive constant c, and moreover U Vfor both U Vand V U. 2. Preliminaries We begin with a lemma quoted from Lemma 3.. in [4]. Lemma 3. Let s, α (0,, andμ be nonnegative Radon measures on.then,μ CM s if and only if μ CM s,α = sup ( w 2 α w wz α+s dμ (z <. (4 According to Lemma 3 and the fact that f F(p,q,sif and only if dμ f is an s-carleson measure, we can easily get the following corollary. Corollary 4. Let f be an analytic function on. f F(p,q,s if and only if there exists an α>0such that f p F(p,q,s,α ( w 2 α = sup w wz α+s f p (z ( z 2 q+s da (z <. (5 We will also need the following standard result from [6]. Lemma 5. Suppose t> and c>0.then, for all z. ( w 2 t wz 2+t+c da (w ( z 2 c (6 The following lemma, quoted from Lemma in [9], is an extension of Lemma 5.See also[7]. Lemma 6. Suppose s> and r, t>0.ift<s+2<r,then ( w 2 s wz r wζ t da (w ( z 2 r s 2 ζz t. (7 Next, we see that F(p, q, s is contained in B (2+q/p.We thank Zhao for pointing out that the following result is firstly proved in [3].Here,wegiveanotherproofwithadifferent approach. Lemma 7. For p<, 2<q<,and0 s<, F(p, q, s B (2+q/p.Inparticular,ifs>,thenF(p, q, s = B (2+q/p.

3 Abstract and Applied Analysis 3 Proof. We can use the reproducing formula for f to get that f ( w 2 b f (w (z =C da (w (8 ( wz b+ for some constant C, whereb is a real number greater than + (q + s/p;see,forexample,[4,page55]. Let 0<α<2+q.Ifp>, denote p =p/(p ;it follows from the Hölder s inequalityand (5that f (z ( w 2 (q+s/p ( z 2 α/p f (w wz (s+α/p ( w 2 b (q+s/p da (w ( z 2 α/p wz b+ (s+α/p ( z 2 α /p ( wz s+α f (w p ( w 2 q+s da(w ( f F(p,q,s,α ( z 2 α/p ( ( w 2 p (b (q+s/p da(w ( z 2 p (α/p wz p (b+ (s+α/p ( w 2 p (b (q+s/p da(w wz p (b+ (s+α/p f F(p,q,s,α ( z 2 α/p ( ( z 2 (2 α+q/(p /p /p /p = f F(p,q,s,α. ( z 2 (2+q/p (9 Apparently, we have used Lemma 5 in the last inequality. This gives that F(p, q, s B (q+2/p when <p<. If p=,then ( z 2 2+q f (z ( w 2 q+s ( z 2 α f (w wz α+s ( w 2 b q s da (w ( z 2 α 2 q wz b+ s α f (w ( w 2 q+s ( z 2 α α+s da (w wz ( w 2 b q s ( z 2 2+q α sup w wz b+ α s f F(p,q,s,α sup ( w 2 b q s ( z 2 2+q α. w wz b+ α s (20 Recall that b>+q+sand 0<α<2+q.Wecaneasilyuse (4tocheckthat ( w 2 b q s ( z 2 2+q α sup. (2 z,w wz b+ α s Thus, F(p, q, s B (q+2/p when p=. Now, suppose s>and let f B (q+2/p,then f (z ( z 2 (q+2/p f (q+2/p < (22 for all z.itfollowsthat f p F(p,q,s = sup f p (z a ( z 2 q+s ( a 2 s az 2 da (z = sup f p (z a ( z 2 q+2 ( z 2 s 2 ( a 2 s az 2 da (z f p B (q+2/p sup( a 2 s a f p B (q+2/p. ( z 2 s 2 az 2s da (z (23 Again, the above inequality follows from Lemma 5. This completes the proof. Our strategy relies on an integral operator preserving the s-carleson measures. For a, b > 0, we define the integral operator T a,b as ( w 2 b T a,b f (z = wz f (w da (w z. (24 a+b The following lemma is similar to Theorem 2.5 in [8]. Indeed,QiuandWuprovedthecase<p<.Specially, the p=2case is just Lemma 3..2 in [4]. Lemma 8. Assume 0<s, p<,andα>.let b > (α + /p, leta > (α + /p, andletf be Lebesgue measurable on.if f(z p ( z 2 α da(z belongs to CM s, then T a,b f(z p ( z 2 p(a +α da(z also belongs to CM s.

4 4 Abstract and Applied Analysis Proof. We firstly prove the case p=and then sketch the outline argument of the case <p< modified from [8] for the completeness. When p =, according to Lemma 3, itissufficientto show that ( a 2 x sup a az x+s T a,bf (z ( z 2 a +α da (z < (25 ( w 2 b ( z 2 (a +α/p + ( S(I \S(I wz a+b = Int + Int 2. f (w da (w pda (z (30 for some x>0.thatistoshow ( a 2 x sup a az x+s ( z 2 a +α da (z ( w 2 b f (w wz a+b da (w is finite. By Fubini s theorem, it is enough to verify that sup ( a 2 x ( z 2 a +α da (z a wz a+b az x+s f (w ( w 2 b da (w (26 (27 In order to estimate Int, we define the linear operator B: L p ( L p ( as where B (f (z = K (z, w f (w da (w, (3 K (z, w = ( w 2 b ( z 2 (a +α/p wz a+b. (32 If we choose a test function g(z = ( z 2 /pp,thenschur s lemma combines with Lemma 5 implying that is finite. Choosing x such that x+s < a++α,wecanuselemma 6 to control the last integral by ( a 2 x sup a aw x+s f (w ( w 2 α da (w. (28 Since f(z ( z 2 α da(z is an s-carleson measure, we can complete the proof by using Lemma 3 again. When <p<, we need to verify that I s T a,bf (z p S(I ( z 2 p(a +α da (z (29 holds for any arc I T. In order to make this estimate, let, be the biggest integer satisfying log 2 I, and let I n, n=0,,2,...,, denotes the arcs on T with the same center as I and length 2 n I, andi NI + is just T. Wecancontroland decompose the integral as T a,bf (z p S(I ( z 2 p(a +α da (z ( w 2 b ( z 2 (a +α/p ( S(I S(I wz a+b K (w, z g p (w da (w g p (z, K (w, z g p (z da (z g p (w. (33 Hence, B is a bounded operator. Letting h(w = f(w ( w 2 α/p χ S(I (w,thenh L p ( with Thus, h p L p = f (w p ( w 2 α da (w I s. (34 S(I Int B (h(z p da (z = B (h p L h p p L I s. p (35 To handle Int 2,firstnotethat,forn = 0,,...,,ifz S(I and w S(I n+ \S(I n,then wz 2 n I.Further,it is easy to check that, for any fixed β>, ( w 2 β da (w (2 n I β+2, n=0,,...,. S(I n (36 Now, splitting \S(I as f (w da (w pda (z S(I n+ \S(I n = S n+, (37

5 Abstract and Applied Analysis 5 we have Then, Int 2 S(I S n+ ( w 2 b f (w p da(w wz a+b ( z 2 p(a +α da (z I p(a +α+2 (38 f (z f B (q+2/p = f B (q+2/p χ Ω ε (f (w da (w 2+(q+2/p wz ( w 2 2/p wz 2+(q+2/p (43 ( (2 n I a+b χ Ω ε (f (w da (w. ( w 2 2/p ( w 2 b f (w p. da(w S(I n+ Recall that f(z p ( z 2 α da(z CM s.itfollowsfrom Hölder s inequality that Write Then, χ Ω ε (f (w g (w =. ( w 2 2/p (44 ( w 2 b f (w da (w S(I n+ I n+ s/p (2 n+ I b α/p+2/p. Now, an easy computation gives that (39 g (w p ( w 2 s da (w =χ Ω ε (f (w ( w 2 s 2 da (w. (45 So, if χ Ω ε (f (z ( z 2 s 2 da (z (46 is in CM s, Lemma 8 implies that Int 2 ( 2 n(a +(α+2 s/p p I s I s, (40 since a > (α + /p and 0<s.Thiscompletesthe proof. 3. Proof of the Main Result Proof of Theorem 2. For f B (q+2/p,itiseasytoestablish the following formula (see, e.g., [9, (.] or [4, page 55]. Notice that it is a special case of the α-order derivative of f, as α=0in [4], which holds for all holomorphic f on. Consider ( w 2 (q+2/p f (w f (z =f(0 + da (w z. w( wz +(q+2/p (4 efine, for each ε>0, f (z =f(0 + Ω ε (f ( w 2 (q+2/p f (w w( wz +(q+2/p da (w, ( w 2 (q+2/p f (w f 2 (z = da (w. \ Ω ε (f w( wz +(q+2/p (42 f (z p ( z 2 q+s da (z CM s. (47 By Corollary 4, f F(p,q,s. Meanwhile, recall that, for w \ Ω ε (f and ( w 2 (q+2/p f (w < ε,wecanuselemma 5 to obtain f 2 (z \ Ω ε (f <ε This means that ( w 2 (q+2/p f (w da (w wz 2+(q+2/p da (w 2+(q+2/p wz ε ( z 2 (2+q/p. (48 ( z 2 (2+q/p f 2 (z ε. (49 To summarize the above argument, we have f = f +f 2, f F(p,q,s (by (47, and f 2 B (2+q/p (by (49, and χ Ω ε (f (z( z 2 s 2 da(z is an s-carleson measure for each ε>0.thus, dist B(2+q/p (f,f(p,q,s inf {ε > 0 : χ Ω ε (f (z ( z 2 s 2 da (z CM s }. (50

6 6 Abstract and Applied Analysis In order to prove the other direction of the inequality, we assume that ε 0 equals the right-hand quantity of the last inequality and dist B(2+q/p (f, F (p, q, s < ε 0. (5 We only consider the case ε 0 >0. Then, there exists an ε such that 0<ε <ε 0, dist B(2+q/p (f, F (p, q, s < ε. (52 Hence,bydefinition,wecanfindafunctionh F(p,q,s such that f h B (2+q/p <ε. (53 Now, for any ε (ε,ε 0,wehavethat χ Ω ε (f (z ( z 2 s 2 da (z (54 is not in CM s. But, according to (53, we get ( z 2 (2+q/p h (z >( z 2 (2+q/p f (z ε and so This implies that z, (55 χ Ω ε (f (z χ Ω ε ε (h (z z. (56 χ Ω ε ε (h (z ( z 2 s 2 da (z (57 does not belong to CM s.but,itfollowsfrom(3 that Ω ε ε (h = {z :( z 2 (q+2/p h (z ε ε }. Therefore, χ Ω ε ε (h (z ( z 2 s 2 da (z =χ Ω ε ε (h (z ( z 2 q+s da (z ( z 2 q+2 h p (z (ε ε p ( z 2 q+s χ Ω ε ε (h (z da (z (ε ε p h p (z ( z 2 q+s da (z. Since h F(p,q,s, Corollary4 implies that (58 h p (z ( z 2 q+s da (z (59 is in CM s.thismeansthat (ε ε p χ Ω ε ε (h (z ( z 2 s 2 da (z (60 is in CM s,andsoisχ Ω ε ε (h (z( z 2 s 2 da(z. This contradicts (57. Thus, we must have dist B(2+q/p (f, F (p, q, s ε 0 (6 as required. Remark 9. Theorem 2 characterizes the closure of F(p, q, s in the B (q+2/p norm. That is, for f B (q+2/p, f is in the closure of F(p, q, s in the B (q+2/p norm if and only if, for every ε>0, Ω ε (f S(I ( z 2 s 2 da (z I s (62 for any Carleson square S(I. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors would like to thank the referee for her/his helpful comments and suggestions which improved this paper. Cheng Yuan is supported by NSFC of China and Tianjin Advanced Education evelopment Fund 20005; Cezhong Tong is supported by the National Natural Science Foundation of China (Grant nos and 7087 and Natural Science Foundation of Hebei Province (Grant no. A References [] R. Timoney, Bloch functions in several complex variables. I, The Bulletin of the London Mathematical Society, vol.2,no.4, pp , 980. [2] R. Timoney, Bloch functions in several variables, Journal für die Reine und Angewandte Mathematik,vol.39,pp. 22,980. [3] R. H. Zhao, On a general family of function spaces, Annales Academiæ Scientiarum Fennicæ Mathematica issertationes, vol. 05, pp. 56, 996. [4] R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, in Complex Analysis and Its Applications, Pitman ResearchNotesinMathematics305,pp.36 46,Longman Scientific & Technical, Harlow, UK, 994. [5] R.Aulaskari,.A.Stegenga,andJ.Xiao, Somesubclassesof BMOA and their characterization in terms of Carleson measures, The Rocky Mountain Mathematics,vol.26, no. 2,pp ,996. [6] R. Aulaskari, J. Xiao, and R. H. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, vol. 5, no. 2, pp. 0 2, 995. [7] J. Rättyä, n-th derivative characterizations, mean growth of derivatives and F(p, q, s, Bulletin of the Australian Mathematical Society,vol.68,pp ,2003. [8] R. H. Zhao, On logarithmic Carleson measures, Acta Scientiarum Mathematicarum,vol.69,no.3-4,pp ,2003. [9] R. H. Zhao, istances from Bloch functions to some Möbius invariant spaces, Annales Academiæ Scientiarum Fennicæ Mathematica,vol.33,pp ,2008. [0] P. G. Ghatage and. C. Zheng, Analytic functions of bounded mean oscillation and the Bloch space, Integral Equations and Operator Theory,vol.7,no.4,pp.50 55,993. [] Z. Lou and W. Chen, istances from Bloch functions to QKtype spaces, Integral Equations and Operator Theory,vol.67,no. 2, pp. 7 8, 200.

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