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 {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, for 0<α<∞, the Bloch-type space Bα is the space of analytic functions on D satisfying
(1)∥f∥Bα=supz∈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.
For a∈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)|.
Let 0<p<∞, -2<q<∞, 0≤s<∞, and -1<q+s<∞. The space F(p,q,s), introduced by Zhao in [3] and known as the general family of function spaces, is defined as the set of f∈H(D) for which
(5)∥f∥F(p,q,s)p=supa∈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 of bounded mean oscillation. See [3–9] 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)∥μ∥CMs=supI⊂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.
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 [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.
Let X⊂Bα be an analytic function space. The distance from a Bloch-type function f to X is defined by
(11)distBα(f,X)=infg∈X∥f-g∥Bα.
The following result is obtained by Zhao in [9].
Theorem 1.
Suppose 1≤p<∞, 0<s≤1, and f∈B. The following two quantities are equivalent:
where Ωɛ(f)={z∈D:|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 [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 from f∈B(q+2)/p to F(p,q,s), which extends Zhao’s result. The main result is following.
Theorem 2.
Suppose 1≤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-2
dA(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 Theorem 3.1.3 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 Lemma 3.1.1 in [14].
Lemma 3.
Let s, α∈(0,∞), and μ be nonnegative Radon measures on D. Then, μ∈CMs if and only if
(14)∥μ∥CMs,α=supw∈D∫D(1-|w|2)α|1-w¯z|α+s
dμ(z)<∞.
According to Lemma 3 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.
Let f 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=supw∈D∫D(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 [16].
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 z∈D.
The following lemma, quoted from Lemma 1 in [9], is an extension of Lemma 5. See also [17].
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 [3]. Here, we give another proof with a different approach.
Lemma 7.
For 1≤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 for f′ 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].
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/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<∞.
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-α≲∫D|f′(w)|(1-|w|2)q+s(1-|z|2)α|1-w¯z|α+sdA(w)×supw∈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),αsupw∈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)supz,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, suppose s>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=supa∈D∫D|f′(z)|p(1-|z|2)q+s(1-|a|2|1-a¯z|2)sdA(z)=supa∈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)/ppsupa∈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 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)∀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.
Assume 0<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)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 [18] for the completeness.
When p=1, according to Lemma 3, it is sufficient to show that
(25)supa∈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)supa∈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)supa∈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.
Choosing x such that x+s<a+1+α, we can use Lemma 6 to control the last integral by
(28)supa∈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.
When 1<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 arc I⊂T. 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)(∫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 handle Int2, 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 ResultProof of Theorem 2.
For f∈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 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 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.
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 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).
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