# Distance from Bloch-Type Functions to the Analytic SpaceF(p,q,s)

Authors: Cheng Yuan; Cezhong Tong
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The analytic spaceF(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.


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## Body

## 1. Introduction

LetD denote the unit disc {z∈C:|z|<1} of the complex plane C and let T={z∈C:|z|=1} be its boundary. As usual, H(D) denotes the space of all analytic functions on D.Recall that, for0<α<∞, the Bloch-type space Bα is the space of analytic functions on D satisfying
(1)∥f∥Bα=sup⁡z∈D(1-|z|2)α|f′(z)|<∞.
The little Bloch-type space Bα0 is the subspace of all f∈Bα 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)∥f∥Bα*=|f(0)|+∥f∥Bα.
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.Fora∈D, 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)|.Let0<p<∞, -2<q<∞, 0≤s<∞, and -1<q+s<∞. The space F(p,q,s), introduced by Zhao in [3] and known as thegeneral family of function spaces, is defined as the set of f∈H(D) for which
(5)∥f∥F(p,q,s)p=sup⁡a∈D∫D‍|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 f∈F(p,q,s) such that
(6)lim⁡|a|→1∫D‍|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 ofbounded mean oscillation. See [3–9] for these basic facts.For0<s<∞, we say that a nonnegative Borel measure μ defined on D is an s-Carleson measure if
(7)∥μ∥CMs=sup⁡I⊂Tμ(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=reiθ∈D:1-|I|≤r<1,eiθ∈I}
is the Carleson square based on a subarc I⊆T. 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.Forf an analytic function on D, we define
(10)dμf(z)=|f′(z)|p(1-|z|2)q+sdA(z).
It was proved in [3] that f∈F(p,q,s) if and only if dμf is an s-Carleson measure and f∈F0(p,q,s) if and only if dμf is a vanishing s-Carleson measure.LetX⊂Bα be an analytic function space. The distance from a Bloch-type function f to X is defined by
(11)distBα(f,X)=inf⁡g∈X∥f-g∥Bα.The following result is obtained by Zhao in [9].Theorem 1.
Suppose1≤p<∞, 0<s≤1, and f∈B. The following two quantities are equivalent: (1)
distB(f,F(p,p-2,s));(2)
inf⁡{ɛ:χΩɛ(f)(1-|z|2)s-2dA(z)isaCarlesonmeasure},whereΩɛ(f)={z∈D:|f′(z)|(1-|z|2)≥ɛ} and χ denotes the characteristic function of a set.Whenp=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 [10]). Also, similar type results can be found in [11–13]. Specifically, distance from Bloch function to QK-type space is given in [11]; to the little Bloch space is obtained in [12], and to the Qp space of the ball is characterized in [13]. All these spaces are Möbius invariant.This paper is dedicated to characterize the distance fromf∈B(q+2)/p to F(p,q,s), which extends Zhao’s result. The main result is following.Theorem 2.
Suppose1≤p<∞, 0<s≤1, -1<q+s<∞, and f∈B(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-2dA(z)∈CMs},
where
(13)Ω~ɛ(f)={z∈D:(1-|z|2)(q+2)/p|f′(z)|≥ɛ}.The strategy in this paper follows from Theorem3.1.3 in [14]. The distance from a Bα function to Campanato-Morrey space was given in [15] with similar idea.Notation. Throughout this paper, we only write U≲V (or V≳U) for U≤cV for a positive constant c, and moreover U≈V for both U≲V and V≲U.


## 2. Preliminaries

We begin with a lemma quoted from Lemma3.1.1 in [14].Lemma 3.
Lets, α∈(0,∞), and μ be nonnegative Radon measures on D. Then, μ∈CMs if and only if
(14)∥μ∥CMs,α=sup⁡w∈D∫D‍(1-|w|2)α|1-w¯z|α+sdμ(z)<∞.According to Lemma3 and the fact that f∈F(p,q,s) if and only if dμf is an s-Carleson measure, we can easily get the following corollary.Corollary 4.
Letf be an analytic function on D. f∈F(p,q,s) if and only if there exists an α>0 such that
(15)∥f∥F(p,q,s),αp=sup⁡w∈D∫D‍(1-|w|2)α|1-w¯z|α+s|f′(z)|p(1-|z|2)q+sdA(z)<∞.We will also need the following standard result from [16].Lemma 5.
Supposet>-1 and c>0. Then,
(16)∫D‍(1-|w|2)t|1-w¯z|2+t+cdA(w)≈1(1-|z|2)c
for all z∈D.The following lemma, quoted from Lemma  1 in [9], is an extension of Lemma 5. See also [17].Lemma 6.
Supposes>-1 and r, t>0. If t<s+2<r, then
(17)∫D‍(1-|w|2)s|1-w¯z|r|1-w¯ζ|tdA(w)≲1(1-|z|2)r-s-2|1-ζ¯z|t.Next, we see thatF(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, we give another proof with a different approach.Lemma 7.
For1≤p<∞, -2<q<∞, and 0≤s<∞, 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 forf′ to get that
(18)f′(z)=C∫D‍(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].
Let0<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/p′≲∥f∥F(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/p′≲∥f∥F(p,q,s),α1(1-|z|2)α/p×(1(1-|z|2)(2-α+q)/(p-1))1/p′=∥f∥F(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<∞.
Ifp=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-α≲∫D‍|f′(w)|(1-|w|2)q+s(1-|z|2)α|1-w¯z|α+sdA(w)×sup⁡w∈D(1-|w|2)b-1-q-s(1-|z|2)2+q-α|1-w¯z|b+1-α-s≲∥f∥F(p,q,s),αsup⁡w∈D(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)sup⁡z,w∈D(1-|w|2)b-1-q-s(1-|z|2)2+q-α|1-w¯z|b+1-α-s≲1.
Thus, F(p,q,s)⊂B(q+2)/p when p=1.
Now, supposes>1 and let f∈B(q+2)/p, then
(22)|f′(z)|(1-|z|2)(q+2)/p≤∥f∥(q+2)/p<∞
for all z∈D. It follows that
(23)∥f∥F(p,q,s)p=sup⁡a∈D∫D‍|f′(z)|p(1-|z|2)q+s(1-|a|2|1-a¯z|2)sdA(z)=sup⁡a∈D∫D‍|f′(z)|p(1-|z|2)q+2×(1-|z|2)s-2(1-|a|2|1-a¯z|2)sdA(z)≲∥f∥B(q+2)/ppsup⁡a∈D(1-|a|2)s∫D‍(1-|z|2)s-2|1-a¯z|2sdA(z)≈∥f∥B(q+2)/pp.
Again, the above inequality follows from Lemma 5. This completes the proof.Our strategy relies on an integral operator preserving thes-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)∀z∈D.The following lemma is similar to Theorem  2.5 in [18]. Indeed, Qiu and Wu proved the case 1<p<∞. Specially, the p=2 case is just Lemma  3.1.2 in [14].Lemma 8.
Assume0<s≤1, 1≤p<∞, 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)αdA(z) belongs to CMs, then |Ta,bf(z)|p(1-|z|2)p(a-1)+αdA(z) also belongs to CMs.Proof.
We firstly prove the casep=1 and then sketch the outline argument of the case 1<p<∞ modified from [18] for the completeness.
Whenp=1, according to Lemma 3, it is sufficient to show that
(25)sup⁡a∈D∫D‍(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)sup⁡a∈D∫D‍(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)sup⁡a∈D∫D‍(1-|a|2)x∫D‍(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.
Choosingx such that x+s<a+1+α, we can use Lemma 6 to control the last integral by
(28)sup⁡a∈D∫D‍(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.
When1<p<∞, we need to verify that
(29)1|I|s∫S(I)‍|Ta,bf(z)|p(1-|z|2)p(a-1)+αdA(z)≲1
holds for any arcI⊂T. In order to make this estimate, let NI, be the biggest integer satisfying NI≤-log⁡2|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)‍(∫D∖S(I1)‍(1-|w|2)b-1(1-|z|2)(a-1)+α/p|1-w¯z|a+bhhhhhhhhhhhh×|f(w)|dA(w)∫D∖S(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 h∈Lp(D) with
(34)∥h∥Lpp=∫S(I1)‍|f(w)|p(1-|w|2)αdA(w)≲|I|s.
Thus,
(35)Int1≲∫D‍|B(h)(z)|pdA(z)=∥B(h)∥Lpp≲∥h∥Lpp≲|I|s.
To handleInt2, first note that, for n=0,1,…,NI, if z∈S(I) and w∈S(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 D∖S(I1) as
(37)⋃n=1NI‍S(In+1)∖S(In)=⋃n=1NI‍S~n+1,
we have
(38)Int2≲∫S(I)‍|∑n=1NI‍∫S~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<s≤1. This completes the proof.


## 3. Proof of the Main Result

Proof of Theorem2.
Forf∈B(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 [14], 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)∀z∈D.
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)=∫D∖Ω~ɛ(f)‍(1-|w|2)(q+2)/pf′(w)w¯(1-w¯z)1+(q+2)/pdA(w).
Then,
(43)|f1′(z)|≲∥f∥B(q+2)/p∫D‍χΩ~ɛ(f)(w)|1-w¯z|2+(q+2)/pdA(w)=∥f∥B(q+2)/p∫D‍(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, f1∈F(p,q,s). Meanwhile, recall that, for w∈D∖Ω~ɛ(f) and (1-|w|2)(q+2)/p|f′(w)|<ɛ, we can use Lemma 5 to obtain
(48)|f2′(z)|≤∫D∖Ω~ɛ(f)‍(1-|w|2)(q+2)/p|f′(w)||1-w¯z|2+(q+2)/pdA(w)<ɛ∫D‍1|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, f1∈F(p,q,s) (by (47)), and f2∈B(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 thatɛ0 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 h∈F(p,q,s) such that
(53)∥f-h∥B(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)|-ɛ1iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀z∈D,
and so
(56)χΩ~ɛ(f)(z)≤χΩ~ɛ-ɛ1(h)(z)∀z∈D.
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)={z∈D:(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 h∈F(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.
Theorem2 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 ɛ>0,
(62)∫Ω~ɛ(f)∩S(I)‍(1-|z|2)s-2dA(z)≲|I|s
for any Carleson square S(I).



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