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A note on Qp domains
Accepted Manuscript A note on Q p domains Jianjun Jin PII: DOI: Reference:
S0022-247X(13)00419-8 http://dx.doi.org/10.1016/j.jmaa.2013.05.002 YJMAA 17586
To appear in:
Journal of Mathematical Analysis and Applications
Received date: 23 February 2013 Please cite this article as: J. Jin, A note on Q p domains, J. Math. Anal. Appl. (2013), http://dx.doi.org/10.1016/j.jmaa.2013.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A NOTE ON Qp DOMAINS JIANJUN JIN Abstract. In this paper, we first give a new characterization for Qp,0 domain, 0 < p < 1, in terms of quasiconformal extension of univalent function. As application, using this characterization, we obtain two more characterizations of these domains by means of some kernel functions introduced recently by Hu and Shen [9].
1. Introduction Let ∆ = {z : |z| < 1} denote the unit disk in the extended complex plane ˆ = C ∪ {∞}, ∆∗ = C ˆ \ ∆ be the exterior of ∆ and T = ∂∆ = ∂∆∗ be the unit C circle. Let H(∆) denote the space of all analytic functions on ∆. The Bloch space B is the class of all functions f ∈ H(∆) satisfy kf kB = sup |f ′ (z)|(1 − |z|2 ) < ∞. z∈∆
The little Bloch space B0 is defined by
B0 = {f ∈ H(∆) : lim |f ′ (z)|(1 − |z|2 ) = 0}. |z|→1−
wz ¯ For z, w ∈ ∆, we let g(z, w) = log | 1− z−w | be the Green’s function of ∆ with pole at w. For −1 < p < ∞, we say that f belongs to Qp space, if f ∈ H(∆) and ZZ kf k2Qp = sup |f ′ (z)|2 g p (z, w) dxdy < ∞. w∈∆
∆
Moreover, we say that f belongs to little Qp space, denoted by Qp,0 , if ZZ lim |f ′ (z)|2 g p (z, w) dxdy = 0. |w|→1−
∆
Qp spaces were introduced by Aulaskari, Xiao and Zhao in [5] and they have been much studied in recent years. It is known that Q0 is the classical Dirichlet space, Q1 is BM OA and Q1,0 is V M OA. The space BM OA consists of those functions f in Hardy space H 2 whose boundary values have bounded mean oscillation on the unit circle. The space V M OA is the closure of the polynomials in the BM OA norm(see [7]). It is proved in [3] that Qp is the Bloch space if 1 < p < ∞, and in [10] that Qp contains only constants if −1 < p < 0. For 0 < p < 1, Qp is a proper subspace of BM OA and Qp,0 is a proper subspace of V M OA. 2000 Mathematics Subject Classification. Primary 30H25; Secondary 30C62. Key words and phrases. Quasiconformal extension of univalent function; Qp spaces; Carleson measure. 1
2
by
JIANJUN JIN
Let I denote the subarc of T. The Carleson sector S(I), based on I, is defined
S(I) = {z = reiθ : 1 − |I| ≤ r < 1, eiθ ∈ I}, R 1 |dζ|. where |I| is the normalized length of I, i.e. |I| = 2π I For 0 < p < ∞, a nonnegative measure λ on ∆ is called a bounded p-Carleson measure if there exists a positive constant C such that, for any subarc I of T, λ(S(I)) ≤ C|I|p .
We say that λ is a compact p-Carleson measure, if in addition λ(S(I)) = o(|I|p ),
|I| → 0.
When p = 1, we get the standard definition of the original Carleson measure. For a nonnegative measure λ on ∆∗ , replacing S(I) in the above definition by the following Carleson sector S∗ (I) = {z = reiθ : 1 < r ≤ 1 + |I|, eiθ ∈ I},
we similarly obtain the definition of bounded(compact) p-Carleson measure on ∆∗ . In [4], Aulaskari, Stegenga and Xiao characterized functions in Qp spaces by means of above modified Carleson measure. They proved that Theorem A. Let f ∈ H(∆). For 0 < p < ∞, we have (A1 ) f ∈ Qp (∆) if and only if dλ = |f ′ (z)|2 (1 − |z|2 )p dxdy is a bounded pCarleson measure on ∆. (A2 ) f ∈ Qp,0 (∆) if and only if dλ = |f ′ (z)|2 (1 − |z|2 )p dxdy is a compact pCarleson measure on ∆. For more information about these spaces, we refer the readers to Xiao’s recent monographs [15] and [16]. Let f be a univalent function on ∆. For a subspace X ⊂ H(∆), adopting the terminology from [8], we say that Ω = f (∆) is an X domain if log f ′ ∈ X. There have been some results in the characterization of such type domains. Astala and Gehring [1] proved that ∂Ω = ∂f (∆) is a quasicircle, i.e. f has a quasiconformal extension to the whole plane, if and only if log f ′ belongs to the Bloch norm interior of the set of all mappings log f ′ , where f is a univalent function on ∆. In [13], Pommerenke obtained that ∂Ω is an asymptotically conformal curve if and only if log f ′ belongs to little Bloch space B0 . Recently, some domains have been characterized in terms of Schwarzian derivative via Carleson measure conditions. Here, the Schwarzian derivative, for a locally univalent function f , is defined by 1 Sf (z) = Nf′ (z) − Nf2 (z), 2
Nf (z) = (log f ′ )′ (z) =
f ′′ (z) . f ′ (z)
In 1991, Astala and Zinsmeister [2] gave the following description of BM OA domain. Theorem B. Let f be a univalent function on ∆. Then we have the following equivalent statements. (B1 ) log f ′ ∈ BM OA(∆). (B2 ) |Sf (z)|2 (1 − |z|2 )3 dxdy is a Carleson measure on ∆. More recently, Pau and Pel´ aez [11] proved the following extension of Theorem B. It should be pointed out that this result has been predicted in [15, Page 23] by Xiao, and part of it has been obtained there.
A NOTE ON Qp DOMAINS
3
Theorem C. Let 0 < p < ∞ and f be a univalent function on ∆. Then we have the following equivalent statements. (C1 ) log f ′ ∈ Qp (∆). (C2 ) |Sf (z)|2 (1 − |z|2 )2+p dxdy is a bounded p-Carleson measure on ∆. For the little Qp domains, P´erez-Gonz´ alez and R¨ atty¨a [12] obtained the following result. Theorem D. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (D1 ) log f ′ ∈ Qp,0 (∆). (D2 ) |Sf (z)|2 (1 − |z|2 )2+p dxdy is a compact p-Carleson measure on ∆. In particular, when p = 1, we have log f ′ ∈ V M OA(∆) if and only if |Sf (z)|2 (1− 2 3 |z| ) dxdy is a compact Carleson measure on ∆. Supposing in addition in Theorem B that ∂Ω = ∂f (∆) is a quasicircle, Astala and Zinsmeister [2, Theorem 4] also obtained the following equivalent condition in terms of quasiconformal extension of univalent function. That is (B3 ) f can be extended to a quasiconformal mapping in the whole plane whose 2 ∗ complex dilatation µ induces a Carleson measure |µ(z)| |z|2 −1 dxdy on ∆ . Very recently, for V M OA domain, Shen and Wei [14] proved the following Theorem E. Let f be a bounded univalent function on ∆, ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (E1 ) log f ′ ∈ V M OA(∆). (E2 ) |Sf (z)|2 (1 − |z|2 )3 dxdy is a compact Carleson measure on ∆. (E3 ) f can be extended to a quasiconformal extension in the whole plane whose 2 ∗ complex dilatation µ induces a compact Carleson measure |µ(z)| |z|2 −1 dxdy on ∆ . Now, It is natural to ask whether there exist analogous characterizations for Qp domains in terms of quasiconformal extension of univalent function. In this note, for the little Qp domain, 0 < p ≤ 1, we prove the following result, which is an extension of Theorem E. Theorem F. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (F1 ) log f ′ ∈ Qp,0 (∆). (F2 ) |Sf (z)|2 (1 − |z|2 )2+p is a compact p-Carleson measure on ∆. (F3 ) f can be extended to a quasiconformal extension in the whole plane whose 2 ∗ complex dilatation µ induces a compact p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ . This paper is organized as follows. In section 2, we present the proof of Theorem F. In section 3, we will use Theorem F to give more two characterizations of little Qp domains by means of some kernel functions introduced by Hu and Shen [9]. 2. Proof of Theorem F In this section, we give the proof of Theorem F. In what follows, C will denote some positive constant which may vary from line to line. Proof of Theorem F. We only need to show that F2 ⇔ F3 . F3 ⇒ F2 . Following Astala and Zinsmeister, we prove the equivalent form of this implication. We assume that f is a univalent function on C\∆ with the following
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JIANJUN JIN
series expansion at infinity b1 + ··· , w and f can be extended to a qusiconformal mapping in the whole plane. |µf (z)|2 We will show that, if (1−|z| 2 )2−p dxdy is a compact p-Carleson measure on ∆, 2 2 2+p then |Sf (z)| (|z| − 1) dxdy is a compact p-Carleson measure on ∆∗ . By the representation theorem of quasiconformal mappings we have the following identity. ZZ ZZ ¯ (w) ∂f 2 ¯ ∂f (w)dudv = lim ζ (2.1) dudv 2 ζ→∞ ∆ (ζ − w) ∆ f (w) = w +
= −π lim ζ 2 (f ′ (ζ) − 1) ζ→∞
= πb1 = − For z ∈ ∆∗ , let
π lim ζ 4 Sf (ζ). 6 ζ→∞
1 + wz . w + z¯ It is clear that α is an automorphism of ∆∗ mapping ∞ to z. Let α(w) =
γ(w) = −
(|z|2 − 1)f ′ (z) , w − f (z)
we see that g = γ ◦ f ◦ α has a series expansion at infinity as follows. g(w) = w + b0 +
b1 + ··· . w
It follows from Sg (w) = Sf ◦α (w) = (Sf ◦ α)(w)[α′ (w)]2 that
Sf (z) = lim Sf ◦α (ζ)[α′ (ζ)]−2 = (|z|2 − 1)−2 lim ζ 4 Sg (ζ), ζ→∞
ζ→∞
then, in view of (2.1), we obtain
ZZ 6 Sf (z) = − (|z|2 − 1)−2 µg (z)∂g(z) dxdy π Z Z∆ 6 µf ◦α (z)∂g(z) dxdy. = − (|z|2 − 1)−2 π ∆
Hence, by Cauchy’s inequality, we have ZZ ZZ 2 2 2+p 2 p−2 2 |Sf (z)| (|z| − 1) ≤ C(|z| − 1) |∂g(z)|2 dxdy. |µf ◦α (z)| dxdy ∆
∆
Let k = kµg k∞ = kµf k∞ , and Jg be the Jacobian of g, then, by Koebe area theorem, we have ZZ ZZ 1 J (z) dxdy |∂g(z)|2 dxdy = 2 g ∆ 1 − |µg (z)| ∆ ZZ 1 ≤ Jg (z) dxdy 1 − k2 ∆ π . ≤ 1 − k2
A NOTE ON Qp DOMAINS
5
For the simplicity, we will write µ(z) instead of µf (z). Therefore, after a change of variable, we obtain ZZ |µ(ζ)|2 dξdη. |Sf (z)|2 (|z|2 − 1)2+p ≤ C(|z|2 − 1)p 4 ∆ |ζ − z|
Then, for the subarc I of T, the measure dλ = |Sf (z)|2 (|z|2 − 1)2+p dxdy satisfies Z Z ZZ |µ(ζ)|2 (|z|2 − 1)p (2.2) λ(S∗ (I)) ≤ C dξdη dxdy. 4 S∗ (I) ∆ |ζ − z| Next, we estimate the integral Z Z ZZ E := (|z|2 − 1)p S∗ (I)
∆
|µ(ζ)|2 dξdη |ζ − z|4
dxdy.
We may assume that |I| < 41 . We use zI to denote the central point of I, aI to denote the arc whose arc length is a|I| with the same center as I and set ! ZZ ZZ |µ(ζ)|2 2 p (|z| − 1) E = (2.3) dξdη dxdy 4 ∆\S(4I) |ζ − z| S∗ (I) ! ZZ ZZ 2 |µ(ζ)| dξdη dxdy + (|z|2 − 1)p 4 S∗ (I) S(4I) |ζ − z| =
E1 + E2 .
Estimate of E1 . When ζ ∈ ∆ \ S(4I), we see that |ζ − z| ≥ C|ζ − zI |, for any z ∈ S∗ (I). Then we have ! ZZ ZZ |µ(ζ)|2 2 p dξdη dxdy (|z| − 1) E1 ≤ C 4 ∆\S(4I) |ζ − zI | S∗ (I) Z 1+|I| ZZ |µ(ζ)|2 ≤ C|I| r(r2 − 1)p dr dξdη 4 ∆\S(4I) |ζ − zI | 1 ZZ |µ(ζ)|2 dξdη. ≤ C|I|2+p 4 ∆\S(4I) |ζ − zI | Let N be the smallest positive integer n such that 2n |I| ≥ 1. We set 2N I = T, 1 1 S(2N I) = ∆. It is clear that [log2 |I| ] ≤ N ≤ [log2 |I| ] + 1. For any positive integer n ∈ [2, N − 1], we have |ζ − zI |4 ≥ C(1 − |ζ|2 )2−p (2n |I|)2+p ,
It follows that E1
≤
C|I|2+p
≤
C|I|2+p
=
N −1 Z Z X
S(2n+1 I)\S(2n I)
n=2
N −1 X n=2
C
N −1 X n=2
1
1 n (2 |I|)2+p ZZ
(2n )2+p
ZZ
ζ ∈ S(2n+1 I) \ S(2n I).
|µ(ζ)|2 dξdη |ζ − zI |4
S(2n+1 I)\S(2n I)
S(2n+1 I)\S(2n I)
|µ(ζ)|2 dξdη (1 − |ζ|2 )2−p
|µ(ζ)|2 dξdη. (1 − |ζ|2 )2−p
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JIANJUN JIN 2
|µ(ζ)| Since (1−|ζ| 2 )2−p dξdη is a compact p-Carleson measure on ∆, this means that, for any ε > 0, there exists a positive constant δ < 1 such that when |J| ≤ δ, we have ZZ 1 |µ(ζ)|2 β(J) := dξdη < ε. 2 2−p |J|p S(J) (1 − |ζ| )
We note that there is a positive constant C such that β(J) < C, for all subarcs J of T. δ Let N0 = [log2 |I| ], we see that |2n+1 I| = 2n+1 |I| ≤ δ, for any n < N0 . Then, for ε > 0, we have ZZ N −1 X 1 |µ(ζ)|2 E1 ≤ C dξdη 2 2−p (2n )2+p S(2n+1 I) (1 − |ζ| ) n=2 =
C
N −1 X n=2
≤
C|I|p
≤
p
β(2n+1 I)(2n+1 |I|)p (2n )2+p
N −1 X n=2
C|I|
β(2n+1 I) 4n
N −1 X
n=N0
NX 0 −1 C ε + n 4n 4 n=2
!
.
Noting N0 , N → ∞ as |I| → 0, we see from above inequality that
(2.4)
E1 = o(|I|p ),
|I| → 0.
Estimate of E2 . When ζ ∈ S(4I), we first obtain ZZ (|z|2 − 1)p dxdy 3+p S∗ (I) |ζ − z| Z |I| Z |I| tp dtds ≤C 3+p [(t + 1 − |ζ|)2 + s2 ] 2 0 0 Z |I| Z |I| tp dtds ≤C (t + s + 1 − |ζ|)3+p 0 0 C . ≤ 1 − |ζ|2
In view of the fact that for ζ ∈ S(4I), z ∈ S∗ (I), it holds that |ζ − z|1−p ≥ C(1 − |ζ|2 )1−p , we get that ZZ ZZ |µ(ζ)|2 (|z|2 − 1)p dxdy E2 = dξdη 3+p |ζ − z|1−p S(4I) |ζ − z| S∗ (I) ! ZZ ZZ (|z|2 − 1)p |µ(ζ)|2 dxdy dξdη ≤ C 2 1−p 3+p S∗ (I) |ζ − z| S(4I) (1 − |ζ| ) ZZ |µ(ζ)|2 ≤ C dξdη. 2 2−p S(4I) (1 − |ζ| ) It follows that
(2.5)
E2 ≤ C|I|p β(4I) = o(|I|p ),
|I| → 0.
A NOTE ON Qp DOMAINS
7
Then it is easy to see from (2.2)-(2.5) that |Sf (z)|2 (|z|2 − 1)2+p dxdy is a compact p-Carleson measure on ∆∗ . This proves that F3 implies F2 . F2 ⇒ F3 . If |Sf (z)|2 (1 − |z|2 )2+p dxdy is a compact p-Carleson measure on ∆, this means that log f ′ ∈ Qp,0 (∆), then log f ′ belongs to little Bloch space. By a result of Becker and Pommerenke [6], we get that there is a positive constant R > 1 such that f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation µ satisfies 1 1 1 |µ(z)| = |Nf ( )|(1 − 2 ) , z¯ |z| |z|
1 < |z| < R.
Then, for z ∈ {z : 1 < |z| < R}, we have (2.6)
|Nf ( z1¯ )|2 (|z|2 − 1)p 1 |µ(z)|2 = ≤ |Nf ( )|2 (|z|2 − 1)p . 2 2−p (|z| − 1) |z|6 z¯
On the other hand, in view of Theorem A, we know that |Nf (z)|2 (1−|z|2)p dxdy is compact p-Carleson measure on ∆. It is easy to see that |Nf ( 1z¯ )|2 (|z|2 −1)p dxdy is a 2
compact p-Carleson measure on ∆∗ . Then, by (2.6), we conclude that (|z||µ(z)| 2 −1)2−p dxdy ∗ is a compact p-Carleson measure on ∆ . This completes the proof of Theorem F. 3. Another two characterizations of Qp,0 domains
In this section, we give more two new characterizations of Qp,0 domains in terms of some kernel functions introduced by Hu and Shen [9]. We first review some basic definitions and results from Teichm¨ uller theory and also some lemmas from [9] and [14]. Let h be a sense preserving self-homeomorphism h of the unit circle T. we say h is quasisymmetric if there exists a constant C(h) such that |h(2I)| ≤ C(h)|h(I)| for any subarc I ⊂ T with |I| ≤ 12 , where 2I is the subarc with same center as I but with double length and | · | denotes the normalized Lebesgue measure. Let f be a univalent function on ∆ with ∂f (∆) is a Jordan curve in the extended ˆ and g be a Riemann mapping from ∆∗ onto C ˆ − f (∆). we call complex plane C −1 h = f ◦ g is the conformal welding corresponding to f . It is well known that h is quasisymmetric if and only if f has a quasiconformal extension to the whole plane. For a quasisymmetric homeomorphism h, Hu and Shen [9] introduced two kernel functions as follows. Z h(w) 1 dw, (ζ, z) ∈ ∆ × ∆, φh (ζ, z) = 2πi T (1 − ζw)2 (1 − zh(w)) 1 ψh (ζ, z) = 2πi
Z
T
h(w) dw, (ζ − w)2 (1 − zh(w))
(ζ, z) ∈ ∆ × ∆.
They defined the following two operators on A2 induced by φh and ψh , respectively. ZZ 1 − φh (ζ, z¯)ψ(z) dxdy, ψ ∈ A2 , ζ ∈ ∆, Th ψ(ζ) = π ∆ ZZ 1 ψh (ζ, z¯)ψ(z) dxdy, ψ ∈ A2 , ζ ∈ ∆, Th+ ψ(ζ) = π ∆
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JIANJUN JIN
where A2 is the Hilbert space consists of all analytic functions φ on ∆ with the inner product and norm ZZ 1 1 φ(w)ψ(w) dudv, kφk = hφ, φi 2 < ∞. hφ, ψi = π ∆
It was proved in [9] that both Th− and Th+ are bounded operators from A2 into itself. For a univalent function f on ∆, define the Grunsky kernel as U (f, ζ, z) =
and set
1 f ′ (ζ)f ′ (z) − , [f (ζ) − f (z)]2 (ζ − z)2
φh (z) = U (f, z) =
1 ZZ
1 π
2
∆
|φh (ζ, z)|2 dξdη
∆
|U (f, ζ, z)|2 dξdη
π
ZZ
(ζ, z) ∈ ∆ × ∆, z ∈ ∆,
,
21
,
z ∈ ∆,
Shen and Wei [14] proved that Lemma G. Let f be a univalent function on ∆ and h = f −1 ◦ g be the corresponding quasisymmetric conformal welding. Then we have U (f, z) ≤ φh (¯ z ) ≤ kTh+ kU (f, z),
z ∈ ∆.
Lemma H. Let f be a univalent function on ∆, h = f −1 ◦ g be the corresponding quasisymmetric conformal welding, and υ be the Beltrami coefficient of a quasiconformal extension of h−1 to ∆. Then we have ZZ 1 1 |υ(w)|2 (1 − |z|2 )2 dudv. |Sf (z)|2 ≤ U 2 (f, z) ≤ φ2h (¯ z) ≤ (3.1) 2 36 π ¯w|4 ∆ 1 − |υ(w)| |1 − z Now, we can state the main result of this section. Theorem I. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and h = f −1 ◦ g be the corresponding quasisymmetric conformal welding. Then each of the following two statements is equivalent to those in Theorem F. (F 4) φ2h (¯ z )(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. 2 (F 5) U (f, z)(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. In order to prove our theorem, we also need the following lemma(see [17]). Lemma J. Let s > −1, r, t > 0, and r + t − s > 2. If t < s + 2 < r, then we have ZZ (1 − |w|2 )s C dudv ≤ . r |1 − z w| t 2 )r−s−2 |1 − ζ z |1 − ζ w| ¯ ¯ (1 − |ζ| ¯|t ∆ Proof of Theorem I. First we note that Lemma G implies F4 ⇔ F5 and the first inequality of (3.1) implies F5 ⇒ F2 . It remains to prove F3 ⇒ F4 . We assume that f can be extended to a quasiconformal extension in the whole plane whose complex 2 ∗ dilatation µ induces a compact p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ . Then we know that
|µ( z1¯ )|2 (1−|z|2 )2−p dxdy −1
is a compact p-Carleson measure on ∆.
Noting that h = f ◦ g, it is clear that f˜ = g −1 ◦ f is a quasiconformal extension of h−1 to ∆∗ and f˜ has the same complex dilatation µ as f . Then, considering the
A NOTE ON Qp DOMAINS
9
mapping fˆ = j ◦ f˜ ◦ j, where j(z) = z1¯ , we see that fˆ is a quasiconformal extension of h−1 to ∆ and its complex dilatation v satisfies |υ(z)| = |µ( 1z¯ )|. Therefore |υ(z)|2 (1−|z|2 )2−p dxdy
is a compact p-Carleson measure on ∆. Then, by Lemma H, we have p ZZ 1 − |ζ|2 φ2h (w)(1 ¯ − |w|2 )p dudv lim 2 ¯ |ζ|→1− |1 − ζw| ∆ p ZZ ZZ 1 1 1 − |ζ|2 |υ(z)|2 2 p dxdy lim − ≤ (1 − |w| ) dudv 2 |1 − z w| 2 ¯ π |ζ|→1 1 − |υ(z)| ¯4 |1 − ζw| ∆ ∆ p ZZ 1 − |ζ|2 |υ(z)|2 ≤ C lim dxdy ¯ 2 (1 − |z|2 )2−p |ζ|→1− |1 − ζz| ∆ ZZ (1 − |w|2 )p |1 − ζ z¯|2p (1 − |z|2 )2−p dudv. × |1 − ζ w| ¯ 2p |1 − z w| ¯4 ∆
Using Lemma J with r = 4, t = 2p, s = p and noticing the following characterization of compact p-Carleson measure, we see that φ2h (¯ z )(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. A nonnegative measure λ on ∆ is a compact p-Carleson measure if and only if (see [4]) ZZ
1 − |ζ|2 ¯ 2 − |ζ|→1 |1 − ζz| ∆ This finishes the proof of Theorem I. lim
p
dλ(z) = 0.
4. Remarks Checking the proof of Theorem F, we may obtain the following theorem, which is a generalization of Theorem 1 in [2]. Theorem K. Let 0 < p ≤ 1 and f be a univalent function on ∆. If f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation 2 ∗ ′ µ induces a p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ , then log f ∈ Qp (∆). We don’t know whether the converse of Theorem K is also true. We end this paper with the following Conjecture L. Let 0 < p < 1, f be a univalent function on ∆ and ∂Ω = ∂f (∆) be a quasicircle. If log f ′ ∈ Qp (∆), then f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation µ induces a p-Carleson measure |µ(z)|2 ∗ (|z|2 −1)2−p dxdy on ∆ . Acknowledgment The author would like to thank Prof. Yuliang Shen for his encouragement and the anonymous referee for his/her helpful comments and suggestions. References 1. K. Astala, F. Gehring, Injectivity, the BMO norm and the universal Teichmller space, J. Anal. Math., 46(1986), pp. 16-57. 2. K. Astala, M. Zinsmeister, Teichm¨ uller spaces and BMOA, Math. Ann., 289(1991), pp. 613625.
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3. R. Aulaskari, P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex Analysis and Its Applications, Pitman Research Notes in Mathematics, vol. 305, pp. 136-146, Pitman, London, 1994. 4. R. Aulaskari, D. Stegenga, J. Xiao, Some subclasses of BMOA and their characterization in terms of Carleson measure, Rocky Mountain J. Math., 26(2)(1996), pp. 485-506. 5. R. Aulaskari, J. Xiao, R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15(1995), pp. 101-121. ¨ 6. J. Becker, C. Pommenerke, Uber die quasikonforme Fortsetzung schlichter Funktionen, Math. Z., 161(1978), pp. 69-80. 7. J. Garnett, Bounded Analytic Functions, Graduate Texts in Mathematics, vol. 236, Springer, Berlin, 2007. 8. J. Garnett, D. Marshall, Harmonic Measure, New Math. Monogr., vol. 2, Cambridge Univ. Press, Cambridge, 2005. 9. Y. Hu, Y. Shen, On quasisymmetric homeomorphisms, Israel J. Math., (191)2012, pp. 209-226. 10. A. Nicolau, J. Xiao, Bounded Functions in M¨ obius Invariant Dirichlet Spaces, J. Funct. Anal., 150 (1997), pp. 383-425. 11. J. Pau, J. Pel´ aez, Logarithms of the derivative of univalent functions in Qp spaces, J. Math. Anal. Appl., 350(1)(2009), pp. 184-194. 12. F. P´ erez-Gonz´ alez, J. R¨ atty¨ a, Dirichlet and VMOA domains via Schwarzian derivative, J. Math. Anal. Appl., 359(2)(2009), pp. 543-546. 13. C. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann., 236(1978), pp. 199-208. 14. Y. Shen, H. Wei, Universal Teichm¨ uller space and BMO, Adv. Math., 234(2013), pp. 129-148. 15. J. Xiao, Holomorphic Q Classes, Lecture Notes in Mathematics, vol. 1767, Springer, Berlin (2001). 16. J. Xiao, Geometric Q Functions, Frontiers in Mathematics, Birkh¨ auser, Basel (2006). 17. R. Zhao, Distances from Bloch functions to some M¨ obius invariant spaces , Ann. Acad. Sci. Fenn. Math., 33(2008), pp. 303-313. Department of Mathematics, Soochow University, Suzhou 215006, People’s Republic of China E-mail address: [email protected]
S0022-247X(13)00419-8 http://dx.doi.org/10.1016/j.jmaa.2013.05.002 YJMAA 17586
To appear in:
Journal of Mathematical Analysis and Applications
Received date: 23 February 2013 Please cite this article as: J. Jin, A note on Q p domains, J. Math. Anal. Appl. (2013), http://dx.doi.org/10.1016/j.jmaa.2013.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A NOTE ON Qp DOMAINS JIANJUN JIN Abstract. In this paper, we first give a new characterization for Qp,0 domain, 0 < p < 1, in terms of quasiconformal extension of univalent function. As application, using this characterization, we obtain two more characterizations of these domains by means of some kernel functions introduced recently by Hu and Shen [9].
1. Introduction Let ∆ = {z : |z| < 1} denote the unit disk in the extended complex plane ˆ = C ∪ {∞}, ∆∗ = C ˆ \ ∆ be the exterior of ∆ and T = ∂∆ = ∂∆∗ be the unit C circle. Let H(∆) denote the space of all analytic functions on ∆. The Bloch space B is the class of all functions f ∈ H(∆) satisfy kf kB = sup |f ′ (z)|(1 − |z|2 ) < ∞. z∈∆
The little Bloch space B0 is defined by
B0 = {f ∈ H(∆) : lim |f ′ (z)|(1 − |z|2 ) = 0}. |z|→1−
wz ¯ For z, w ∈ ∆, we let g(z, w) = log | 1− z−w | be the Green’s function of ∆ with pole at w. For −1 < p < ∞, we say that f belongs to Qp space, if f ∈ H(∆) and ZZ kf k2Qp = sup |f ′ (z)|2 g p (z, w) dxdy < ∞. w∈∆
∆
Moreover, we say that f belongs to little Qp space, denoted by Qp,0 , if ZZ lim |f ′ (z)|2 g p (z, w) dxdy = 0. |w|→1−
∆
Qp spaces were introduced by Aulaskari, Xiao and Zhao in [5] and they have been much studied in recent years. It is known that Q0 is the classical Dirichlet space, Q1 is BM OA and Q1,0 is V M OA. The space BM OA consists of those functions f in Hardy space H 2 whose boundary values have bounded mean oscillation on the unit circle. The space V M OA is the closure of the polynomials in the BM OA norm(see [7]). It is proved in [3] that Qp is the Bloch space if 1 < p < ∞, and in [10] that Qp contains only constants if −1 < p < 0. For 0 < p < 1, Qp is a proper subspace of BM OA and Qp,0 is a proper subspace of V M OA. 2000 Mathematics Subject Classification. Primary 30H25; Secondary 30C62. Key words and phrases. Quasiconformal extension of univalent function; Qp spaces; Carleson measure. 1
2
by
JIANJUN JIN
Let I denote the subarc of T. The Carleson sector S(I), based on I, is defined
S(I) = {z = reiθ : 1 − |I| ≤ r < 1, eiθ ∈ I}, R 1 |dζ|. where |I| is the normalized length of I, i.e. |I| = 2π I For 0 < p < ∞, a nonnegative measure λ on ∆ is called a bounded p-Carleson measure if there exists a positive constant C such that, for any subarc I of T, λ(S(I)) ≤ C|I|p .
We say that λ is a compact p-Carleson measure, if in addition λ(S(I)) = o(|I|p ),
|I| → 0.
When p = 1, we get the standard definition of the original Carleson measure. For a nonnegative measure λ on ∆∗ , replacing S(I) in the above definition by the following Carleson sector S∗ (I) = {z = reiθ : 1 < r ≤ 1 + |I|, eiθ ∈ I},
we similarly obtain the definition of bounded(compact) p-Carleson measure on ∆∗ . In [4], Aulaskari, Stegenga and Xiao characterized functions in Qp spaces by means of above modified Carleson measure. They proved that Theorem A. Let f ∈ H(∆). For 0 < p < ∞, we have (A1 ) f ∈ Qp (∆) if and only if dλ = |f ′ (z)|2 (1 − |z|2 )p dxdy is a bounded pCarleson measure on ∆. (A2 ) f ∈ Qp,0 (∆) if and only if dλ = |f ′ (z)|2 (1 − |z|2 )p dxdy is a compact pCarleson measure on ∆. For more information about these spaces, we refer the readers to Xiao’s recent monographs [15] and [16]. Let f be a univalent function on ∆. For a subspace X ⊂ H(∆), adopting the terminology from [8], we say that Ω = f (∆) is an X domain if log f ′ ∈ X. There have been some results in the characterization of such type domains. Astala and Gehring [1] proved that ∂Ω = ∂f (∆) is a quasicircle, i.e. f has a quasiconformal extension to the whole plane, if and only if log f ′ belongs to the Bloch norm interior of the set of all mappings log f ′ , where f is a univalent function on ∆. In [13], Pommerenke obtained that ∂Ω is an asymptotically conformal curve if and only if log f ′ belongs to little Bloch space B0 . Recently, some domains have been characterized in terms of Schwarzian derivative via Carleson measure conditions. Here, the Schwarzian derivative, for a locally univalent function f , is defined by 1 Sf (z) = Nf′ (z) − Nf2 (z), 2
Nf (z) = (log f ′ )′ (z) =
f ′′ (z) . f ′ (z)
In 1991, Astala and Zinsmeister [2] gave the following description of BM OA domain. Theorem B. Let f be a univalent function on ∆. Then we have the following equivalent statements. (B1 ) log f ′ ∈ BM OA(∆). (B2 ) |Sf (z)|2 (1 − |z|2 )3 dxdy is a Carleson measure on ∆. More recently, Pau and Pel´ aez [11] proved the following extension of Theorem B. It should be pointed out that this result has been predicted in [15, Page 23] by Xiao, and part of it has been obtained there.
A NOTE ON Qp DOMAINS
3
Theorem C. Let 0 < p < ∞ and f be a univalent function on ∆. Then we have the following equivalent statements. (C1 ) log f ′ ∈ Qp (∆). (C2 ) |Sf (z)|2 (1 − |z|2 )2+p dxdy is a bounded p-Carleson measure on ∆. For the little Qp domains, P´erez-Gonz´ alez and R¨ atty¨a [12] obtained the following result. Theorem D. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (D1 ) log f ′ ∈ Qp,0 (∆). (D2 ) |Sf (z)|2 (1 − |z|2 )2+p dxdy is a compact p-Carleson measure on ∆. In particular, when p = 1, we have log f ′ ∈ V M OA(∆) if and only if |Sf (z)|2 (1− 2 3 |z| ) dxdy is a compact Carleson measure on ∆. Supposing in addition in Theorem B that ∂Ω = ∂f (∆) is a quasicircle, Astala and Zinsmeister [2, Theorem 4] also obtained the following equivalent condition in terms of quasiconformal extension of univalent function. That is (B3 ) f can be extended to a quasiconformal mapping in the whole plane whose 2 ∗ complex dilatation µ induces a Carleson measure |µ(z)| |z|2 −1 dxdy on ∆ . Very recently, for V M OA domain, Shen and Wei [14] proved the following Theorem E. Let f be a bounded univalent function on ∆, ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (E1 ) log f ′ ∈ V M OA(∆). (E2 ) |Sf (z)|2 (1 − |z|2 )3 dxdy is a compact Carleson measure on ∆. (E3 ) f can be extended to a quasiconformal extension in the whole plane whose 2 ∗ complex dilatation µ induces a compact Carleson measure |µ(z)| |z|2 −1 dxdy on ∆ . Now, It is natural to ask whether there exist analogous characterizations for Qp domains in terms of quasiconformal extension of univalent function. In this note, for the little Qp domain, 0 < p ≤ 1, we prove the following result, which is an extension of Theorem E. Theorem F. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and ∂Ω = ∂f (∆) be a closed Jordan curve. Then we have the following equivalent statements. (F1 ) log f ′ ∈ Qp,0 (∆). (F2 ) |Sf (z)|2 (1 − |z|2 )2+p is a compact p-Carleson measure on ∆. (F3 ) f can be extended to a quasiconformal extension in the whole plane whose 2 ∗ complex dilatation µ induces a compact p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ . This paper is organized as follows. In section 2, we present the proof of Theorem F. In section 3, we will use Theorem F to give more two characterizations of little Qp domains by means of some kernel functions introduced by Hu and Shen [9]. 2. Proof of Theorem F In this section, we give the proof of Theorem F. In what follows, C will denote some positive constant which may vary from line to line. Proof of Theorem F. We only need to show that F2 ⇔ F3 . F3 ⇒ F2 . Following Astala and Zinsmeister, we prove the equivalent form of this implication. We assume that f is a univalent function on C\∆ with the following
4
JIANJUN JIN
series expansion at infinity b1 + ··· , w and f can be extended to a qusiconformal mapping in the whole plane. |µf (z)|2 We will show that, if (1−|z| 2 )2−p dxdy is a compact p-Carleson measure on ∆, 2 2 2+p then |Sf (z)| (|z| − 1) dxdy is a compact p-Carleson measure on ∆∗ . By the representation theorem of quasiconformal mappings we have the following identity. ZZ ZZ ¯ (w) ∂f 2 ¯ ∂f (w)dudv = lim ζ (2.1) dudv 2 ζ→∞ ∆ (ζ − w) ∆ f (w) = w +
= −π lim ζ 2 (f ′ (ζ) − 1) ζ→∞
= πb1 = − For z ∈ ∆∗ , let
π lim ζ 4 Sf (ζ). 6 ζ→∞
1 + wz . w + z¯ It is clear that α is an automorphism of ∆∗ mapping ∞ to z. Let α(w) =
γ(w) = −
(|z|2 − 1)f ′ (z) , w − f (z)
we see that g = γ ◦ f ◦ α has a series expansion at infinity as follows. g(w) = w + b0 +
b1 + ··· . w
It follows from Sg (w) = Sf ◦α (w) = (Sf ◦ α)(w)[α′ (w)]2 that
Sf (z) = lim Sf ◦α (ζ)[α′ (ζ)]−2 = (|z|2 − 1)−2 lim ζ 4 Sg (ζ), ζ→∞
ζ→∞
then, in view of (2.1), we obtain
ZZ 6 Sf (z) = − (|z|2 − 1)−2 µg (z)∂g(z) dxdy π Z Z∆ 6 µf ◦α (z)∂g(z) dxdy. = − (|z|2 − 1)−2 π ∆
Hence, by Cauchy’s inequality, we have ZZ ZZ 2 2 2+p 2 p−2 2 |Sf (z)| (|z| − 1) ≤ C(|z| − 1) |∂g(z)|2 dxdy. |µf ◦α (z)| dxdy ∆
∆
Let k = kµg k∞ = kµf k∞ , and Jg be the Jacobian of g, then, by Koebe area theorem, we have ZZ ZZ 1 J (z) dxdy |∂g(z)|2 dxdy = 2 g ∆ 1 − |µg (z)| ∆ ZZ 1 ≤ Jg (z) dxdy 1 − k2 ∆ π . ≤ 1 − k2
A NOTE ON Qp DOMAINS
5
For the simplicity, we will write µ(z) instead of µf (z). Therefore, after a change of variable, we obtain ZZ |µ(ζ)|2 dξdη. |Sf (z)|2 (|z|2 − 1)2+p ≤ C(|z|2 − 1)p 4 ∆ |ζ − z|
Then, for the subarc I of T, the measure dλ = |Sf (z)|2 (|z|2 − 1)2+p dxdy satisfies Z Z ZZ |µ(ζ)|2 (|z|2 − 1)p (2.2) λ(S∗ (I)) ≤ C dξdη dxdy. 4 S∗ (I) ∆ |ζ − z| Next, we estimate the integral Z Z ZZ E := (|z|2 − 1)p S∗ (I)
∆
|µ(ζ)|2 dξdη |ζ − z|4
dxdy.
We may assume that |I| < 41 . We use zI to denote the central point of I, aI to denote the arc whose arc length is a|I| with the same center as I and set ! ZZ ZZ |µ(ζ)|2 2 p (|z| − 1) E = (2.3) dξdη dxdy 4 ∆\S(4I) |ζ − z| S∗ (I) ! ZZ ZZ 2 |µ(ζ)| dξdη dxdy + (|z|2 − 1)p 4 S∗ (I) S(4I) |ζ − z| =
E1 + E2 .
Estimate of E1 . When ζ ∈ ∆ \ S(4I), we see that |ζ − z| ≥ C|ζ − zI |, for any z ∈ S∗ (I). Then we have ! ZZ ZZ |µ(ζ)|2 2 p dξdη dxdy (|z| − 1) E1 ≤ C 4 ∆\S(4I) |ζ − zI | S∗ (I) Z 1+|I| ZZ |µ(ζ)|2 ≤ C|I| r(r2 − 1)p dr dξdη 4 ∆\S(4I) |ζ − zI | 1 ZZ |µ(ζ)|2 dξdη. ≤ C|I|2+p 4 ∆\S(4I) |ζ − zI | Let N be the smallest positive integer n such that 2n |I| ≥ 1. We set 2N I = T, 1 1 S(2N I) = ∆. It is clear that [log2 |I| ] ≤ N ≤ [log2 |I| ] + 1. For any positive integer n ∈ [2, N − 1], we have |ζ − zI |4 ≥ C(1 − |ζ|2 )2−p (2n |I|)2+p ,
It follows that E1
≤
C|I|2+p
≤
C|I|2+p
=
N −1 Z Z X
S(2n+1 I)\S(2n I)
n=2
N −1 X n=2
C
N −1 X n=2
1
1 n (2 |I|)2+p ZZ
(2n )2+p
ZZ
ζ ∈ S(2n+1 I) \ S(2n I).
|µ(ζ)|2 dξdη |ζ − zI |4
S(2n+1 I)\S(2n I)
S(2n+1 I)\S(2n I)
|µ(ζ)|2 dξdη (1 − |ζ|2 )2−p
|µ(ζ)|2 dξdη. (1 − |ζ|2 )2−p
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JIANJUN JIN 2
|µ(ζ)| Since (1−|ζ| 2 )2−p dξdη is a compact p-Carleson measure on ∆, this means that, for any ε > 0, there exists a positive constant δ < 1 such that when |J| ≤ δ, we have ZZ 1 |µ(ζ)|2 β(J) := dξdη < ε. 2 2−p |J|p S(J) (1 − |ζ| )
We note that there is a positive constant C such that β(J) < C, for all subarcs J of T. δ Let N0 = [log2 |I| ], we see that |2n+1 I| = 2n+1 |I| ≤ δ, for any n < N0 . Then, for ε > 0, we have ZZ N −1 X 1 |µ(ζ)|2 E1 ≤ C dξdη 2 2−p (2n )2+p S(2n+1 I) (1 − |ζ| ) n=2 =
C
N −1 X n=2
≤
C|I|p
≤
p
β(2n+1 I)(2n+1 |I|)p (2n )2+p
N −1 X n=2
C|I|
β(2n+1 I) 4n
N −1 X
n=N0
NX 0 −1 C ε + n 4n 4 n=2
!
.
Noting N0 , N → ∞ as |I| → 0, we see from above inequality that
(2.4)
E1 = o(|I|p ),
|I| → 0.
Estimate of E2 . When ζ ∈ S(4I), we first obtain ZZ (|z|2 − 1)p dxdy 3+p S∗ (I) |ζ − z| Z |I| Z |I| tp dtds ≤C 3+p [(t + 1 − |ζ|)2 + s2 ] 2 0 0 Z |I| Z |I| tp dtds ≤C (t + s + 1 − |ζ|)3+p 0 0 C . ≤ 1 − |ζ|2
In view of the fact that for ζ ∈ S(4I), z ∈ S∗ (I), it holds that |ζ − z|1−p ≥ C(1 − |ζ|2 )1−p , we get that ZZ ZZ |µ(ζ)|2 (|z|2 − 1)p dxdy E2 = dξdη 3+p |ζ − z|1−p S(4I) |ζ − z| S∗ (I) ! ZZ ZZ (|z|2 − 1)p |µ(ζ)|2 dxdy dξdη ≤ C 2 1−p 3+p S∗ (I) |ζ − z| S(4I) (1 − |ζ| ) ZZ |µ(ζ)|2 ≤ C dξdη. 2 2−p S(4I) (1 − |ζ| ) It follows that
(2.5)
E2 ≤ C|I|p β(4I) = o(|I|p ),
|I| → 0.
A NOTE ON Qp DOMAINS
7
Then it is easy to see from (2.2)-(2.5) that |Sf (z)|2 (|z|2 − 1)2+p dxdy is a compact p-Carleson measure on ∆∗ . This proves that F3 implies F2 . F2 ⇒ F3 . If |Sf (z)|2 (1 − |z|2 )2+p dxdy is a compact p-Carleson measure on ∆, this means that log f ′ ∈ Qp,0 (∆), then log f ′ belongs to little Bloch space. By a result of Becker and Pommerenke [6], we get that there is a positive constant R > 1 such that f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation µ satisfies 1 1 1 |µ(z)| = |Nf ( )|(1 − 2 ) , z¯ |z| |z|
1 < |z| < R.
Then, for z ∈ {z : 1 < |z| < R}, we have (2.6)
|Nf ( z1¯ )|2 (|z|2 − 1)p 1 |µ(z)|2 = ≤ |Nf ( )|2 (|z|2 − 1)p . 2 2−p (|z| − 1) |z|6 z¯
On the other hand, in view of Theorem A, we know that |Nf (z)|2 (1−|z|2)p dxdy is compact p-Carleson measure on ∆. It is easy to see that |Nf ( 1z¯ )|2 (|z|2 −1)p dxdy is a 2
compact p-Carleson measure on ∆∗ . Then, by (2.6), we conclude that (|z||µ(z)| 2 −1)2−p dxdy ∗ is a compact p-Carleson measure on ∆ . This completes the proof of Theorem F. 3. Another two characterizations of Qp,0 domains
In this section, we give more two new characterizations of Qp,0 domains in terms of some kernel functions introduced by Hu and Shen [9]. We first review some basic definitions and results from Teichm¨ uller theory and also some lemmas from [9] and [14]. Let h be a sense preserving self-homeomorphism h of the unit circle T. we say h is quasisymmetric if there exists a constant C(h) such that |h(2I)| ≤ C(h)|h(I)| for any subarc I ⊂ T with |I| ≤ 12 , where 2I is the subarc with same center as I but with double length and | · | denotes the normalized Lebesgue measure. Let f be a univalent function on ∆ with ∂f (∆) is a Jordan curve in the extended ˆ and g be a Riemann mapping from ∆∗ onto C ˆ − f (∆). we call complex plane C −1 h = f ◦ g is the conformal welding corresponding to f . It is well known that h is quasisymmetric if and only if f has a quasiconformal extension to the whole plane. For a quasisymmetric homeomorphism h, Hu and Shen [9] introduced two kernel functions as follows. Z h(w) 1 dw, (ζ, z) ∈ ∆ × ∆, φh (ζ, z) = 2πi T (1 − ζw)2 (1 − zh(w)) 1 ψh (ζ, z) = 2πi
Z
T
h(w) dw, (ζ − w)2 (1 − zh(w))
(ζ, z) ∈ ∆ × ∆.
They defined the following two operators on A2 induced by φh and ψh , respectively. ZZ 1 − φh (ζ, z¯)ψ(z) dxdy, ψ ∈ A2 , ζ ∈ ∆, Th ψ(ζ) = π ∆ ZZ 1 ψh (ζ, z¯)ψ(z) dxdy, ψ ∈ A2 , ζ ∈ ∆, Th+ ψ(ζ) = π ∆
8
JIANJUN JIN
where A2 is the Hilbert space consists of all analytic functions φ on ∆ with the inner product and norm ZZ 1 1 φ(w)ψ(w) dudv, kφk = hφ, φi 2 < ∞. hφ, ψi = π ∆
It was proved in [9] that both Th− and Th+ are bounded operators from A2 into itself. For a univalent function f on ∆, define the Grunsky kernel as U (f, ζ, z) =
and set
1 f ′ (ζ)f ′ (z) − , [f (ζ) − f (z)]2 (ζ − z)2
φh (z) = U (f, z) =
1 ZZ
1 π
2
∆
|φh (ζ, z)|2 dξdη
∆
|U (f, ζ, z)|2 dξdη
π
ZZ
(ζ, z) ∈ ∆ × ∆, z ∈ ∆,
,
21
,
z ∈ ∆,
Shen and Wei [14] proved that Lemma G. Let f be a univalent function on ∆ and h = f −1 ◦ g be the corresponding quasisymmetric conformal welding. Then we have U (f, z) ≤ φh (¯ z ) ≤ kTh+ kU (f, z),
z ∈ ∆.
Lemma H. Let f be a univalent function on ∆, h = f −1 ◦ g be the corresponding quasisymmetric conformal welding, and υ be the Beltrami coefficient of a quasiconformal extension of h−1 to ∆. Then we have ZZ 1 1 |υ(w)|2 (1 − |z|2 )2 dudv. |Sf (z)|2 ≤ U 2 (f, z) ≤ φ2h (¯ z) ≤ (3.1) 2 36 π ¯w|4 ∆ 1 − |υ(w)| |1 − z Now, we can state the main result of this section. Theorem I. Let 0 < p ≤ 1, f be a bounded univalent function on ∆, and h = f −1 ◦ g be the corresponding quasisymmetric conformal welding. Then each of the following two statements is equivalent to those in Theorem F. (F 4) φ2h (¯ z )(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. 2 (F 5) U (f, z)(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. In order to prove our theorem, we also need the following lemma(see [17]). Lemma J. Let s > −1, r, t > 0, and r + t − s > 2. If t < s + 2 < r, then we have ZZ (1 − |w|2 )s C dudv ≤ . r |1 − z w| t 2 )r−s−2 |1 − ζ z |1 − ζ w| ¯ ¯ (1 − |ζ| ¯|t ∆ Proof of Theorem I. First we note that Lemma G implies F4 ⇔ F5 and the first inequality of (3.1) implies F5 ⇒ F2 . It remains to prove F3 ⇒ F4 . We assume that f can be extended to a quasiconformal extension in the whole plane whose complex 2 ∗ dilatation µ induces a compact p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ . Then we know that
|µ( z1¯ )|2 (1−|z|2 )2−p dxdy −1
is a compact p-Carleson measure on ∆.
Noting that h = f ◦ g, it is clear that f˜ = g −1 ◦ f is a quasiconformal extension of h−1 to ∆∗ and f˜ has the same complex dilatation µ as f . Then, considering the
A NOTE ON Qp DOMAINS
9
mapping fˆ = j ◦ f˜ ◦ j, where j(z) = z1¯ , we see that fˆ is a quasiconformal extension of h−1 to ∆ and its complex dilatation v satisfies |υ(z)| = |µ( 1z¯ )|. Therefore |υ(z)|2 (1−|z|2 )2−p dxdy
is a compact p-Carleson measure on ∆. Then, by Lemma H, we have p ZZ 1 − |ζ|2 φ2h (w)(1 ¯ − |w|2 )p dudv lim 2 ¯ |ζ|→1− |1 − ζw| ∆ p ZZ ZZ 1 1 1 − |ζ|2 |υ(z)|2 2 p dxdy lim − ≤ (1 − |w| ) dudv 2 |1 − z w| 2 ¯ π |ζ|→1 1 − |υ(z)| ¯4 |1 − ζw| ∆ ∆ p ZZ 1 − |ζ|2 |υ(z)|2 ≤ C lim dxdy ¯ 2 (1 − |z|2 )2−p |ζ|→1− |1 − ζz| ∆ ZZ (1 − |w|2 )p |1 − ζ z¯|2p (1 − |z|2 )2−p dudv. × |1 − ζ w| ¯ 2p |1 − z w| ¯4 ∆
Using Lemma J with r = 4, t = 2p, s = p and noticing the following characterization of compact p-Carleson measure, we see that φ2h (¯ z )(1 − |z|2 )p dxdy is a compact p-Carleson measure on ∆. A nonnegative measure λ on ∆ is a compact p-Carleson measure if and only if (see [4]) ZZ
1 − |ζ|2 ¯ 2 − |ζ|→1 |1 − ζz| ∆ This finishes the proof of Theorem I. lim
p
dλ(z) = 0.
4. Remarks Checking the proof of Theorem F, we may obtain the following theorem, which is a generalization of Theorem 1 in [2]. Theorem K. Let 0 < p ≤ 1 and f be a univalent function on ∆. If f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation 2 ∗ ′ µ induces a p-Carleson measure (|z||µ(z)| 2 −1)2−p dxdy on ∆ , then log f ∈ Qp (∆). We don’t know whether the converse of Theorem K is also true. We end this paper with the following Conjecture L. Let 0 < p < 1, f be a univalent function on ∆ and ∂Ω = ∂f (∆) be a quasicircle. If log f ′ ∈ Qp (∆), then f can be extended to a quasiconformal mapping in the whole plane whose complex dilatation µ induces a p-Carleson measure |µ(z)|2 ∗ (|z|2 −1)2−p dxdy on ∆ . Acknowledgment The author would like to thank Prof. Yuliang Shen for his encouragement and the anonymous referee for his/her helpful comments and suggestions. References 1. K. Astala, F. Gehring, Injectivity, the BMO norm and the universal Teichmller space, J. Anal. Math., 46(1986), pp. 16-57. 2. K. Astala, M. Zinsmeister, Teichm¨ uller spaces and BMOA, Math. Ann., 289(1991), pp. 613625.
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