AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/610237 610237 Research Article Distance from Bloch-Type Functions to the Analytic Space F(p,q,s) Yuan Cheng 1 http://orcid.org/0000-0001-9505-3815 Tong Cezhong 2 Sabatini Marco 1 Institute of Mathematics School of Science Tianjin University of Technology and Education, Tianjin 300222 China tute.edu.cn 2 Department of Mathematics School of Science, Hebei University of Technology, Tianjin 300401 China hebut.edu.cn 2014 1982014 2014 19 04 2014 09 07 2014 04 08 2014 19 8 2014 2014 Copyright © 2014 Cheng Yuan and Cezhong 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,s) can 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.

1. Introduction

Let D denote the unit disc {zC:|z|<1} of the complex plane C and let T={zC:|z|=1} be its boundary. As usual, H(D) denotes the space of all analytic functions on D.

Recall that, for 0<α<, the Bloch-type space Bα is the space of analytic functions on D satisfying (1)fBα=supzD(1-|z|2)α|f(z)|<. The little Bloch-type space Bα0 is the subspace of all fBα with (2)lim|z|1(1-|z|2)α|f(z)|=0. It is well known that Bα is a Banach space under the norm (3)fBα*=|f(0)|+fBα. In particular, when α=1, Bα becomes the classic Bloch space B, which is the maximal Möbius invariant Banach space that has a decent linear functional; see [1, 2] for more details on the Bloch spaces.

For aD, the involution of the unit disk is denoted by σa(z)=(a-z)/(1-a¯z). It is well known and easy to check that (4)1-|σa(z)|2=(1-|a|2)(1-|z|2)|1-a¯z|2=(1-|a|2)|σa(z)|.

Let 0<p<, -2<q<, 0s<, and -1<q+s<. The space F(p,q,s), introduced by Zhao in  and known as the general family of function spaces, is defined as the set of fH(D) for which (5)fF(p,q,s)p=supaDD|f(z)|p(1-|z|2)q(1-|σa(z)|2)sdA(z)<, where dA(z) is the normalized area measure on D. The space F0(p,q,s) consists of all fF(p,q,s) such that (6)lim|a|1D|f(z)|p(1-|z|2)q(1-|σa(z)|2)sdA(z)=0. 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 1<s<. The space F(p,p,0) is the classical Bergman space Lap(D), and F(p,p-2,0) is the classical Besov space Bp. The spaces F(2,0,s) are the Qs spaces, in particular, F(2,0,1)=BMOA, and the function space of bounded mean oscillation. See  for these basic facts.

For 0<s<, we say that a nonnegative Borel measure μ defined on D is an s-Carleson measure if (7)μCMs=supITμ(S(I))|I|s<, where the supremum ranges over all subarcs I of T, |I| denotes the arc length of I, and (8)S(I)={z=reiθD:1-|I|r<1,eiθI} is the Carleson square based on a subarc IT. We write CMs for the class of all s-Carleson measures. Moreover, μ is said to be a vanishing s-Carleson measure if (9)lim|I|0μ(S(I))|I|s=0.

For f an analytic function on D, we define (10)dμf(z)=|f(z)|p(1-|z|2)q+sdA(z). It was proved in  that fF(p,q,s) if and only if dμf is an s-Carleson measure and fF0(p,q,s) if and only if dμf is a vanishing s-Carleson measure.

Let XBα be an analytic function space. The distance from a Bloch-type function f to X is defined by (11)distBα(f,X)=infgXf-gBα.

The following result is obtained by Zhao in .

Theorem 1.

Suppose 1p<, 0<s1, and fB. The following two quantities are equivalent:

dist B(f,F(p,p-2,s));

inf{ɛ:χΩɛ(f)(1-|z|2)s-2 d A(z)isaCarlesonmeasure}  ,

where Ωɛ(f)={zD:|f(z)|(1-|z|2)ɛ} and χ denotes the characteristic function of a set.

When p=2 and s=1, 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 ). Also, similar type results can be found in . Specifically, distance from Bloch function to QK-type space is given in ; to the little Bloch space is obtained in , and to the Qp space of the ball is characterized in . All these spaces are Möbius invariant.

This paper is dedicated to characterize the distance from fB(q+2)/p to F(p,q,s), which extends Zhao’s result. The main result is following.

Theorem 2.

Suppose 1p<, 0<s1, -1<q+s<, and fB(2+q)/p. Then (12) dist B(q+2)/p(f,F(p,q,s))inf{ɛ>0:χΩ~ɛ(f)(z)(1-|z|2)s-2 dA(z)CMs}, where (13)Ω~ɛ(f)={zD:(1-|z|2)(q+2)/p|f(z)|ɛ}.

The strategy in this paper follows from Theorem  3.1.3 in . The distance from a Bα function to Campanato-Morrey space was given in  with similar idea.

Notation. Throughout this paper, we only write UV (or VU) for UcV for a positive constant c, and moreover UV for both UV and VU.

2. Preliminaries

We begin with a lemma quoted from Lemma  3.1.1 in .

Lemma 3.

Let s, α(0,), and μ be nonnegative Radon measures on D. Then, μCMs if and only if (14)μCMs,α=supwDD(1-|w|2)α|1-w¯z|α+s dμ(z)<.

According to Lemma 3 and the fact that fF(p,q,s) if and only if dμf is an s-Carleson measure, we can easily get the following corollary.

Corollary 4.

Let f be an analytic function on D. fF(p,q,s) if and only if there exists an α>0 such that (15)fF(p,q,s),αp=supwDD(1-|w|2)α|1-w¯z|α+s|f(z)|p(1-|z|2)q+s dA(z)<.

We will also need the following standard result from .

Lemma 5.

Suppose t>-1 and c>0. Then, (16)D(1-|w|2)t|1-w¯z|2+t+c dA(w)1(1-|z|2)c for all zD.

The following lemma, quoted from Lemma  1 in , is an extension of Lemma 5. See also .

Lemma 6.

Suppose s>-1 and r, t>0. If t<s+2<r, then (17)D(1-|w|2)s|1-w¯z|r|1-w¯ζ|t dA(w)1(1-|z|2)r-s-2|1-ζ¯z|t.

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 . Here, we give another proof with a different approach.

Lemma 7.

For 1p<, -2<q<, and 0s<, F(p,q,s)B(2+q)/p. In particular, if s>1, then F(p,q,s)=B(2+q)/p.

Proof.

We can use the reproducing formula for f to get that (18)f(z)=CD(1-|w|2)b-1f(w)(1-w¯z)b+1dA(w) for some constant C, where b is a real number greater than 1+(q+s)/p; see, for example, [14, page 55].

Let 0<α<2+q. If p>1, denote p=p/(p-1); it follows from the Hölder’s inequality and (15) that (19)|f(z)|D(1-|w|2)(q+s)/p(1-|z|2)α/p|f(w)||1-w¯z|(s+α)/p×(1-|w|2)b-1-(q+s)/pdA(w)(1-|z|2)α/p|1-w¯z|b+1-(s+α)/p(D(1-|z|2)α|1-w¯z|s+α|f(w)|p(1-|w|2)q+sdA(w))1/p×(D(1-|w|2)p(b-1-(q+s)/p)dA(w)(1-|z|2)p(α/p)|1-w¯z|p(b+1-(s+α)/p))1/pfF(p,q,s),α(1-|z|2)α/p×(D(1-|w|2)p(b-1-(q+s)/p)dA(w)|1-w¯z|p(b+1-(s+α)/p))1/pfF(p,q,s),α1(1-|z|2)α/p×(1(1-|z|2)(2-α+q)/(p-1))1/p=fF(p,q,s),α1(1-|z|2)(2+q)/p. Apparently, we have used Lemma 5 in the last inequality. This gives that F(p,q,s)B(q+2)/p when 1<p<.

If p=1, then (20)(1-|z|2)2+q|f(z)|D(1-|w|2)q+s(1-|z|2)α|f(w)||1-w¯z|α+s×(1-|w|2)b-1-q-sdA(w)(1-|z|2)α-2-q|1-w¯z|b+1-s-αD|f(w)|(1-|w|2)q+s(1-|z|2)α|1-w¯z|α+sdA(w)×supwD(1-|w|2)b-1-q-s(1-|z|2)2+q-α|1-w¯z|b+1-α-sfF(p,q,s),αsupwD(1-|w|2)b-1-q-s(1-|z|2)2+q-α|1-w¯z|b+1-α-s. Recall that b>1+q+s and 0<α<2+q. We can easily use (4) to check that (21)supz,wD(1-|w|2)b-1-q-s(1-|z|2)2+q-α|1-w¯z|b+1-α-s1. Thus, F(p,q,s)B(q+2)/p when p=1.

Now, suppose s>1 and let fB(q+2)/p, then (22)|f(z)|(1-|z|2)(q+2)/pf(q+2)/p< for all zD. It follows that (23)fF(p,q,s)p=supaDD|f(z)|p(1-|z|2)q+s(1-|a|2|1-a¯z|2)sdA(z)=supaDD|f(z)|p(1-|z|2)q+2×(1-|z|2)s-2(1-|a|2|1-a¯z|2)sdA(z)fB(q+2)/ppsupaD(1-|a|2)sD(1-|z|2)s-2|1-a¯z|2sdA(z)fB(q+2)/pp. 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 Ta,b as (24)Ta,bf(z)=D(1-|w|2)b-1|1-w¯z|a+bf(w)dA(w)zD.

The following lemma is similar to Theorem  2.5 in . Indeed, Qiu and Wu proved the case 1<p<. Specially, the p=2 case is just Lemma  3.1.2 in .

Lemma 8.

Assume 0<s1, 1p<, and α>-1. Let b>(α+1)/p, let a>1-(α+1)/p, and let f be Lebesgue measurable on D. If |f(z)|p(1-|z|2)α dA(z) belongs to CMs, then |Ta,bf(z)|p(1-|z|2)p(a-1)+α dA(z) also belongs to CMs.

Proof.

We firstly prove the case p=1 and then sketch the outline argument of the case 1<p< modified from  for the completeness.

When p=1, according to Lemma 3, it is sufficient to show that (25)supaDD(1-|a|2)x|1-a¯z|x+s|Ta,bf(z)|(1-|z|2)a-1+αdA(z)< for some x>0. That is to show (26)supaDD(1-|a|2)x|1-a¯z|x+s|D(1-|w|2)b-1f(w)|1-w¯z|a+bdA(w)|×(1-|z|2)a-1+αdA(z) is finite. By Fubini’s theorem, it is enough to verify that (27)supaDD(1-|a|2)xD(1-|z|2)a-1+αdA(z)|1-w¯z|a+b|1-a¯z|x+s×|f(w)|(1-|w|2)b-1dA(w) is finite.

Choosing x such that x+s<a+1+α, we can use Lemma 6 to control the last integral by (28)supaDD(1-|a|2)x|1-a¯w|x+s|f(w)|(1-|w|2)αdA(w). Since |f(z)|(1-|z|2)αdA(z) is an s-Carleson measure, we can complete the proof by using Lemma 3 again.

When 1<p<, we need to verify that (29)1|I|sS(I)|Ta,bf(z)|p(1-|z|2)p(a-1)+αdA(z)1

holds for any arc IT. In order to make this estimate, let NI, be the biggest integer satisfying NI-log2|I|, and let In, n=0,1,2,,NI, denotes the arcs on T with the same center as I and length 2n|I|, and INI+1 is just T. We can control and decompose the integral as (30)S(I)|Ta,bf(z)|p(1-|z|2)p(a-1)+αdA(z)S(I)(S(I1)(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+bhhhhhhhhhh×|f(w)|dA(w)S(I1)(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+b)pdA(z)+S(I)(DS(I1)(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+bhhhhhhhhhhhh×|f(w)|dA(w)DS(I1)(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+b)pdA(z)=Int1+Int2.

In order to estimate Int1, we define the linear operator B:Lp(D)Lp(D) as (31)B(f)(z)=DK(z,w)f(w)dA(w), where (32)K(z,w)=(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+b. If we choose a test function g(z)=(1-|z|2)-1/pp, then Schur’s lemma combines with Lemma 5 implying that (33)DK(w,z)gp(w)dA(w)gp(z),DK(w,z)gp(z)dA(z)gp(w). Hence, B is a bounded operator. Letting h(w)=|f(w)|(1-|w|2)α/pχS(I1)(w), then hLp(D) with (34)hLpp=S(I1)|f(w)|p(1-|w|2)αdA(w)|I|s. Thus, (35)Int1D|B(h)(z)|pdA(z)=B(h)LpphLpp|I|s.

To handle Int2, first note that, for n=0,1,,NI, if zS(I) and wS(In+1)S(In), then |1-w¯z|2n|I|. Further, it is easy to check that, for any fixed β>-1, (36)S(In)(1-|w|2)βdA(w)(2n|I|)β+2,n=0,1,,NI. Now, splitting DS(I1) as (37)n=1NIS(In+1)S(In)=n=1NIS~n+1, we have (38)Int2S(I)|n=1NIS~n+1(1-|w|2)b-1|f(w)||1-w¯z|a+bdA(w)|phhh×(1-|z|2)p(a-1)+αdA(z)|I|p(a-1)+α+2×(n=1NI1(2n|I|)a+bhhh×S(In+1)(1-|w|2)b-1|f(w)|dA(w)n=1NI)p. Recall that |f(z)|p(1-|z|2)αdA(z)CMs. It follows from Hölder’s inequality that (39)S(In+1)(1-|w|2)b-1|f(w)|dA(w)|In+1|s/p·(2n+1|I|)b-1-α/p+2/p.

Now, an easy computation gives that (40)Int2(n=1NI2-n(a-1+(α+2-s)/p))p|I|s|I|s, since a>1-(α+1)/p and 0<s1. This completes the proof.

3. Proof of the Main Result Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>.

For fB(q+2)/p, it is easy to establish the following formula (see, e.g., [19, (1.1)] or [14, page 55]. Notice that it is a special case of the α-order derivative of f, as α=0 in , which holds for all holomorphic f on D). Consider (41)f(z)=f(0)+D(1-|w|2)(q+2)/pf(w)w¯(1-w¯z)1+(q+2)/pdA(w)zD. Define, for each ɛ>0, (42)f1(z)=f(0)+Ω~ɛ(f)(1-|w|2)(q+2)/pf(w)w¯(1-w¯z)1+(q+2)/pdA(w),f2(z)=DΩ~ɛ(f)(1-|w|2)(q+2)/pf(w)w¯(1-w¯z)1+(q+2)/pdA(w). Then, (43)|f1(z)|fB(q+2)/pDχΩ~ɛ(f)(w)|1-w¯z|2+(q+2)/pdA(w)=fB(q+2)/pD(1-|w|2)2/p|1-w¯z|2+(q+2)/phhhhhhhhhhhhhhhh×χΩ~ɛ(f)(w)(1-|w|2)2/pdA(w). Write (44)g(w)=χΩ~ɛ(f)(w)(1-|w|2)2/p. Then, (45)|g(w)|p(1-|w|2)sdA(w)=χΩ~ɛ(f)(w)(1-|w|2)s-2dA(w).

So, if (46)χΩ~ɛ(f)(z)(1-|z|2)s-2dA(z) is in CMs, Lemma 8 implies that (47)|f1(z)|p(1-|z|2)q+sdA(z)CMs. By Corollary 4, f1F(p,q,s). Meanwhile, recall that, for wDΩ~ɛ(f) and (1-|w|2)(q+2)/p|f(w)|<ɛ, we can use Lemma 5 to obtain (48)|f2(z)|DΩ~ɛ(f)(1-|w|2)(q+2)/p|f(w)||1-w¯z|2+(q+2)/pdA(w)<ɛD1|1-w¯z|2+(q+2)/pdA(w)ɛ(1-|z|2)(2+q)/p. This means that (49)(1-|z|2)(2+q)/p|f2(z)|ɛ. To summarize the above argument, we have f=f1+f2, f1F(p,q,s) (by (47)), and f2B(2+q)/p (by (49)), and χΩ~ɛ(f)(z)(1-|z|2)s-2dA(z) is an s-Carleson measure for each ɛ>0. Thus, (50)distB(2+q)/p(f,F(p,q,s))inf{ɛ>0:χΩ~ɛ(f)(z)(1-|z|2)s-2dA(z)CMs}.

In order to prove the other direction of the inequality, we assume that ɛ0 equals the right-hand quantity of the last inequality and (51)distB(2+q)/p(f,F(p,q,s))<ɛ0. We only consider the case ɛ0>0. Then, there exists an ɛ1 such that (52)0<ɛ1<ɛ0,distB(2+q)/p(f,F(p,q,s))<ɛ1. Hence, by definition, we can find a function hF(p,q,s) such that (53)f-hB(2+q)/p<ɛ1. Now, for any ɛ(ɛ1,ɛ0), we have that (54)χΩ~ɛ(f)(z)(1-|z|2)s-2dA(z) is not in CMs. But, according to (53), we get (55)(1-|z|2)(2+q)/p|h(z)|>(1-|z|2)(2+q)/p|f(z)|-ɛ1iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiizD, and so (56)χΩ~ɛ(f)(z)χΩ~ɛ-ɛ1(h)(z)zD. This implies that (57)χΩ~ɛ-ɛ1(h)(z)(1-|z|2)s-2dA(z) does not belong to CMs. But, it follows from (13) that Ω~ɛ-ɛ1(h)={zD:(1-|z|2)(q+2)/p|h(z)|ɛ-ɛ1}. Therefore, (58)χΩ~ɛ-ɛ1(h)(z)(1-|z|2)s-2dA(z)=χΩ~ɛ-ɛ1(h)(z)(1-|z|2)q+s(1-|z|2)q+2dA(z)|h(z)|p(ɛ-ɛ1)p(1-|z|2)q+sχΩ~ɛ-ɛ1(h)(z)dA(z)1(ɛ-ɛ1)p|h(z)|p(1-|z|2)q+sdA(z). Since hF(p,q,s), Corollary 4 implies that (59)|h(z)|p(1-|z|2)q+sdA(z) is in CMs. This means that (60)(ɛ-ɛ1)pχΩ~ɛ-ɛ1(h)(z)(1-|z|2)s-2dA(z) is in CMs, and so is χΩ~ɛ-ɛ1(h)(z)(1-|z|2)s-2dA(z). This contradicts (57). Thus, we must have (61)distB(2+q)/p(f,F(p,q,s))ɛ0 as required.

Remark 9.

Theorem 2 characterizes the closure of F(p,q,s) in the B(q+2)/p norm. That is, for fB(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 ɛ>0, (62)Ω~ɛ(f)S(I)(1-|z|2)s-2dA(z)|I|s 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 11226086 of China and Tianjin Advanced Education Development Fund 20111005; Cezhong Tong is supported by the National Natural Science Foundation of China (Grant nos. 11301132 and 11171087) and Natural Science Foundation of Hebei Province (Grant no. A2013202265).

Timoney R. Bloch functions in several complex variables. I The Bulletin of the London Mathematical Society 1980 12 4 241 267 10.1112/blms/12.4.241 MR576974 Timoney R. Bloch functions in several variables Journal für die Reine und Angewandte Mathematik 1980 319 1 22 Zhao R. H. On a general family of function spaces Annales Academiæ Scientiarum Fennicæ Mathematica Dissertationes 1996 105 1 56 Aulaskari R. Lappan P. Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal Complex Analysis and Its Applications 1994 Harlow, UK Longman Scientific & Technical 136 146 Pitman Research Notes in Mathematics 305 Aulaskari R. Stegenga D. A. Xiao J. Some subclasses of BMOA and their characterization in terms of Carleson measures The Rocky Mountain Journal of Mathematics 1996 26 2 485 506 10.1216/rmjm/1181072070 MR1406492 2-s2.0-0030516589 Aulaskari R. Xiao J. Zhao R. H. On subspaces and subsets of BMOA and UBC Analysis 1995 15 2 101 121 10.1524/anly.1995.15.2.101 MR1344246 Rättyä J. n-th derivative characterizations, mean growth of derivatives and F(p, q, s) Bulletin of the Australian Mathematical Society 2003 68 405 421 Zhao R. H. On logarithmic Carleson measures Acta Scientiarum Mathematicarum 2003 69 3-4 605 618 MR2034196 Zhao R. H. Distances from Bloch functions to some Möbius invariant spaces Annales Academiæ Scientiarum Fennicæ Mathematica 2008 33 303 313 Ghatage P. G. Zheng D. C. Analytic functions of bounded mean oscillation and the Bloch space Integral Equations and Operator Theory 1993 17 4 501 515 10.1007/BF01200391 MR1243993 ZBL0796.46011 2-s2.0-33748325045 Lou Z. Chen W. Distances from Bloch functions to QK-type spaces Integral Equations and Operator Theory 2010 67 2 171 181 10.1007/s00020-010-1762-2 MR2650769 2-s2.0-77953006602 Tjani M. Distance of a BLOch function to the little BLOch space Bulletin of the Australian Mathematical Society 2006 74 1 101 119 10.1017/S0004972700047493 MR2248810 ZBL1101.30035 2-s2.0-33748298745 Xu W. Distances from Bloch functions to some Möbius invariant function spaces in the unit ball of Cn Journal of Function Spaces and Applications 2009 7 91 104 Xiao J. Geometric Qp Functions 2006 Basel, Switzerland Birkhäauser Frontiers in Mathematics MR2257688 Xiao J. Yuan C. Analytic campanato spaces and their compositions Indiana University Mathematics Journal, preprint Zhu K. Operator Theory in Function Spaces 2007 Providence, RI, USA American Mathematical Society 10.1090/surv/138 MR2311536 Ortega J. M. Fàbrega J. Pointwise multipliers and corona type decomposition in BMOA Annales de l'institut Fourier 1996 46 1 111 137 10.5802/aif.1509 MR1385513 2-s2.0-0040834421 Qiu L. Wu Z. s-Carleson measures and function spaces Report Series 2007 12 University of Joensuu, Department of Physics and Mathematics Arcozzi N. Blasi D. Pau J. Interpolating sequences on analytic besov type spaces Indiana University Mathematics Journal 2009 58 3 1281 1318 10.1512/iumj.2009.58.3589 ZBL1213.30069 2-s2.0-69549083462