JIN Jian-jun,LI Hua-bing,TANG Shu-an

(1.School of Mathematics Sciences,Hefei University of Technology,Xuancheng Campus,Xuancheng,242000,China;2.School of Mathematics Sciences,Guizhou Normal University,Guiyang,550001,China)

Abstract:In this paper,we study univalent functions f for which logf′belongs to the analytic Morrey spaces.By using the characterization of higher order derivatives of functions in analytic Morrey spaces,we establish some new descriptions for the analytic Morrey domains in terms of two kinds of generalized Schwarzian derivatives.

Key words:Analytic Morrey spaces;univalent functions;Schwarzian derivative;generalized Schwarzian derivatives;Carleson measure

§1.Introduction and Main Results

Let Δ={z:|z|<1}be the unit disk and S1be the unit circle in the complex plane C.Let H(Δ)denote the space of all analytic functions on Δ.

For p∈ (0,+∞),we say f∈ H(Δ)belongs to the Hardy space Hp(Δ)if

For any subarc I of S1,|I|denotes the normal length of I,i.e.,

For f∈H(Δ),let

be the average of f on I.For s∈ (0,1],f∈ H2(Δ),we say f belongs to the analytic Morrey space L2,s(Δ)if

Furthermore,we say f belongs to the little analytic Morrey space(Δ)if f ∈(Δ)and

It is clean that when s=1,the analytic Morrey(Δ)becomes BMOA,the space of analytic functions with bounded mean oscillation,and(Δ)becomes the VMOA space,see[9].The analytic Morrey spaces L2,s(Δ)were introduced and studied in[27]by Wu and Xie.Xiao and Xu[29]considered the composition operators between analytic Morrey spaces.Cascante,Fàbrega and Ortega studied the Corona theorem of analytic Morrey spaces in[6].More recently,Li,Liu and Lou studied the Volterra integral operator on analytic Morrey spaces[14].For more materials and recent results about these spaces,see Liu’s thesis[15].

For the subarc I of S1,its Carleson box S(I)is defined as

For s∈ (0,∞).Letμbe a positive measure on Δ,we sayμis a bounded s-Carleson measure if

Furthermore,μ is said to be a compact s-Carleson measure if‖μ‖C,s< ∞ and

For a∈Δ,we let

φa(z)is a M?bius transform of Δ.We have the following characterization for the s-Carleson measure(see[28]).

Theorem 1.1 For s∈ (0,∞),letμ be a positive measure on Δ.Then μ is a bounded s-Carleson measure if and only if

μis a compact s-Carleson measure if and only if Mμ<∞and

Wu and Xie proved in[27]that

Theorem 1.2 Let s∈ (0,1],f∈H(Δ).Then we have f∈(Δ)(resp.(Δ))if and only if|f′(z)|2(1-|z|2)dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ).

Let f be a univalent function on Δ,i.e.,f belongs to H(Δ)and is one-to-one.For a subspace X ? H(Δ),following Garnett and Marshall[10],we say that Ω =f(Δ)is an X domain if logf′∈ X.Many such domains have been characterized in the literature.In 1986,Astala and Gehring showed that?Ω = ?f(Δ)is a quasicircle,i.e.,f can be extended to a quasiconformal mapping in the whole complex plane,if and only if logf′belongs to the interior of the set of all mappings logf′under the the Bloch norm(see[1]).In a paper[19],Pommerenke proved that?Ω is an asymptotically conformal curve if and only if logf′belongs to the little Bloch space.

Here the little Bloch space,denoted by B0,is a closed subspace of the Bloch space B,which is defined as

where ‖ ·‖Bis called Bloch norm and B0is the class of all functions φ ∈ B satisfying lim

|z|→1-φ′(z)(1-|z|2)=0.

Recently certain domains have been characterized in terms of the Schwarzian derivative of univalent functions via Carleson measure conditions.The Schwarzian derivative S(f)(z),for a locally univalent function f on Δ,is defined as

Here N(f)(z)is called pre-Schwarzian derivative.

In 1991,Astala and Zinsmeister[2] first established a characterization of this type for the BMOA domain.Pau and Peláez[17]extended those results in[2]to the Qpdomains.P′erez-González and R¨atty¨a[18]obtained the analogous characterizations for the little Qpdomains.Very recently,Zorboska[34]considered more general Besov-type domains and Zhou[31]studied the QKdomains.More different characterizations for such domains can be found in[8],[22]and[13].

For the analytic Morrey domains,Wang and Xiao[26]proved that

Theorem 1.3 For s ∈ (0,1),let f be a univalent function on Δ.Then we have

(1)if logf′∈ L2,s(Δ)(resp.(Δ)),then logf′∈ B and|S(f)(z)|2(1-|z|2)3dxdy is a bounded s-Carleson measure on Δ (resp.a compact s-Carleson measure on Δ);

(2)if logf′∈ B0and|S(f)(z)|2(1-|z|2)3dxdy is a s-Carleson measure on Δ (resp.a compact s-Carleson measure on Δ),then logf′∈ L2,s(Δ)(resp.(Δ)).

More characterizations for analytic Morrey domains can be found in[12].

Remark 1.4 For s∈ (0,1),I ? S1,let f be a univalent function on Δ.Bishop and Jones proved in[3,Page 105]that there is a positive constant C such that

for all I?S1.It follows that

Then we see from Theorem 1.3 and(1.1)that

Proposition 1.5 For s ∈ (0,1),let f be a univalent function on Δ.Then we have

(1)logf′∈ L2,s(Δ)if and olny if|S(f)(z)|2(1-|z|2)3dxdy is a bounded s-Carleson measure on Δ;

(2)logf′∈ L20,s(Δ)if and only if|S(f)(z)|2(1-|z|2)3dxdy is a compact s-Carleson measure on Δ.

There are several kinds of generalizations of Schwarzian derivative in the literature.In this paper,we are concerned with the follwing two kinds of generalized Schwarzian derivatives.One type of higher order Schwarzian derivatives were introduced by Schippers in[21],which have nice invariance properties for the M?bius transformations like the classical Schwarzian derivative.

definition 1.6 For a locally univalent function f on Δ,the higher order Schwarzian derivatives σn(f)(z)is defined inductively as follows:

For any M?bius transformation g,we have

Using these invariance properties of σn(f)(z)for the M?bius transformations,Buss[5]proved that the higher Schwarzian operators induced by σn(f)(z)acting on functions of the disc induce analytic mappings of Teichmüller spaces.We recently proved some similar results for the subspaces of universal Teichmüller space,see[24],[25].

Another kinds of generalized Schwarzian derivatives were introduced by Chuaqui,Gr?hn and R¨atty¨a.This type of generalized Schwarzian derivatives have a natural connection to higher order linear differential equations(see[7]).

definition 1.7 For a locally univalent function f on Δ,let

The generalized Schwarzian derivatives Sn(f)(z)is defined as

Remark 1.8 It is clean that S1(f)(z)=N(f)(z)and S2(f)(z)=S(f)(z).It is not hard to see that each term in σn+1(f)(z)is a constant multiple of the corresponding term in Sn(f)(z).

In this paper,we will establish some new descriptions for the analytic Morrey domains in terms of higher order Schwarzian derivatives σn(f)(z)and generalized Schwarzian derivatives Sn(f)(z).These results enrich our understanding of the theory of analytic Morrey spaces and generalized Schwarzian derivatives.We obtain that

Theorem 1.9 For s∈ (0,1),integer n ≥ 3,let f be a univalent function on Δ.Then we have

(1)if logf′∈ L2,s(Δ)(resp.L20,s(Δ)),then logf′∈ B and|σn(f)(z)|2(1-|z|2)2n-3dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ).

(2)if logf′∈ B0and|σn(f)(z)|2(1-|z|2)2n-3dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ),then logf′∈ L2,s(Δ)(resp.L20,s(Δ)).

Theorem 1.10 For s∈ (0,1),n ∈ N,let f be a univalent function on Δ.Then we have

(1)if logf′∈ L2,s(Δ)(resp.L20,s(Δ)),then logf′∈ B and|Sn(f)(z)|2(1-|z|2)2n-1dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ).

(2)if logf′∈ B0and|Sn(f)(z)|2(1-|z|2)2n-1dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ),then logf′∈ L2,s(Δ)(resp.L20,s(Δ)).

Remark 1.11 Both Theorem 1.9 and 1.10 are extensions of Theorem 1.3.From Theorem 1.9 and 1.10,we see that

Corollary 1.12 For s∈(0,1),integer n≥ 3,let f be a univalent function on Δ with logf′∈ B0.Then logf′∈ L2,s(Δ)(resp.L20,s(Δ))if and only if|σn(f)(z)|2(1-|z|2)2n-3dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ).

Corollary 1.13 For s∈ (0,1),n ∈ N,let f be a univalent function on Δ with logf′∈ B0.Then logf′∈ L2,s(Δ)(resp.L20,s(Δ))if and only if|Sn(f)(z)|2(1-|z|2)2n-1dxdy is a bounded s-Carleson measure on Δ(resp.a compact s-Carleson measure on Δ).

To prove Theorem 1.9 and 1.10,we need the following characterization of higher order derivatives of functions belonging to the analytic Morrey spaces.

Theorem 1.14 Let s∈ (0,1],n ∈ N.Then for an analytic function f on Δ,the following statements are equivalent.

(1)f∈L2,s(Δ);

(2)

(3)|f(n)(z)|2(1-|z|2)2n-1dxdy is a bounded s-Carleson measure on Δ.

Moreover,we have the following equivalent statements.

(4)f∈L2,s0(Δ);

(5)

(6)|f(n)(z)|2(1-|z|2)2n-1dxdy is a compact s-Carleson measure on Δ.

Remark 1.15 Theorem 1.15 was obtained and proved by Wu and Xie in[27].We will present a different proof for this result in the next section.It should be pointed out that the main reason for giving a new proof of Theorem 1.14 is that,as we shall see in Section 3,to prove Theorem 1.9 and 1.10,we not only need Theorem 1.14,but also need some arguments given in our direct proof of Theorem 1.14 in Section 2.

In what follows,we will denote the set of all bounded s-Carleson measures and compact s-Carleson measures on Δ by CMs(Δ)and CMs,0(Δ),respectively.C,C1,C2,···will denote some universal positive constants that might change from one line to another,while C(·),C1(·),C2(·),···will denote some positive constants that depend on the parameters in the brackets.

§2.The n-th Derivative of the Analytic Morrey Spaces

n this section,we characterize the higher order derivatives of functions belonging to the analytic Morrey spaces and present a direct proof of Theorem 1.14.We first recall some wellknown lemmas.

Lemma 2.1 Let α ∈ (0,∞),φ ∈ H(Δ).Then

Remark 2.2 A proof of Lemma 2.1 can be found in[3].

Lemma 2.3 Let α ＞ -1,γ,β ＞ 0,γ+β-α ＞ 2,if β < α+2<γ,then,for a,b∈Δ,we have

Remark 2.4 In particular,when b=a,we have

See[30]for a proof of Lemma 2.3.

For τ＞ 0,the Bloch-type space Bτis defined as

The little Bloch-type space Bτ,0is the class of all functions φ ∈ Bτsatisfying

When τ=1,Bτis the Bloch space B and Bτ,0is the little Bloch space B0.It was proved in[32]that

Lemma 2.5 Let τ＞ 0,n ∈ N.Then,for an analytic function φ on Δ,the following statements are equivalent.

(1)φ∈Bτ;

(2)

Moreover,we have the following equivalent statements.

(3)φ ∈ Bτ,0;

(4)(1-|z|2)τ+n-1|φ(n)(z)|→ 0, |z|→ 1-1.

Next,we will prove Theorem 1.14 by a similar way as in[3].

[Proof of Theorem 1.14] First,it is easy to see that(2)is equivalent to(3)by Theorem 1.1.We will show(1)is equivalent to(2).We first show(1)(2).In view of Theorem 1.1 and 1.2,we see that(1)(2)holds for n=1.Suppose now that(1)(2)holds for some fixed n ∈ N.Takingin Lemma 2.1.A simple computation yields that

Then we get from the first inequality of Lemma 2.1 that

Let

Then

It follows from(2.1)that

On the other hand,we have

Here we used the following simple estimate,for all a∈Δ,

We know that L2,s(Δ)?,see[29].Let τ=.Then,by using Lemma 2.5,we obtain that

where we used Lemma 2.3 with

Consequently,combining(2.2)-(2.4),we see that Λf(a,n)< ∞ implies Λf(a,n+1)< ∞.Thus,by induction on n,(1)=?(2)holds for all n∈N.

We now prove(2)=?(1).Assume that(2)holds,we first show f∈For λ ∈ Δ,0

It is known that

for z ∈ D(λ,r),see[33].

By subharmonicity,we have that for an analytic function g on Δ,

For a∈ Δ,taking g(w)=f(n)?φa(w)in(2.6)and by a change of variable,we obtain

Noting that

Then we get that

It follows that

In view of(2.5),we see that

for z∈D(a,r).Hence we get from(2.7)and(2.8)that

On the other hand,noting that φa? φa(z)=z and D(a,r)= φa(Δr),then we see that

for z∈D(a,r).Thus,

Therefore,(2.9)and(2.10)yield that

Then we see from Lemma 2.5 that f∈

We now proceed to show(2)=?(1).By Theorem 1.1 and 1.2,we see that(2)=?(1)holds for n=1.Assume that

Noting that f∈Bτ,τ=,we have

and

where we used Lemma 2.3 with α =s,β =

From(2.11),(2.12),(2.13)and(2.4),we see that Λf(a,n+1)< ∞ implies Λf(a,n)< ∞.Then we obtain that(2)1)holds for all n∈N,by induction on n.This proves(1)(3).Similarly,by checking the proof above,we can show that(4)(5)(6).The proof of Theorem 1.14 is complete.

§3.Proofs of the Main Results

In this section,we prove Theorem 1.9 and 1.10.We first present the proof of Theorem 1.9.The main ideas of the proof are from[11].In the rest of this section,for simplicity,we write σn,N,and ρ to denote σn(f)(z),N(f)(z),and 1-|z|2,respectively.In order to prove Theorem 1.9,we need the following lemma,which has appeared in[11].For the convenience of the reader,we include a proof for this lemma.

Lemma 3.1 Let f be a univalent function on Δ.Then,for integers n ≥ 3,k≥ 0,we have

Moreover,if logf′∈ B0,then

Proof The case k=0 of(3.1)is a basic result about univalent fucntions(see[20]).Then,in view of Lemma 2.5 with τ=1,we see that(3.1)holds for any k ≥ 0.If logf′∈ B0,by again Lemma 2.5 with τ=1,we can obtain that(3.3)holds for any k ≥ 0.

Recall σ3=N′-,we know from Leibniz’s formula for higher derivatives that

Then we get from

and(3.1)that(3.2)holds for n=3,k≥0.Note that the case n=3,k=0 has been known(see[16]).

Now,for fixed n≥3,we suppose that,for any k∈N∪{0},

By the definition of σn+1,we have

It follows from(3.1),(3.6)and(3.7)that(3.2)holds for σfor any k∈N∪{0}.Then,by induction on n,we know that(3.2)holds for any n≥3,k≥0.If logf′∈B0,using(3.3)and by induction,we can similarly show that(3.4)is true for any n≥3,k≥0.The lemma is proved.

Now,we begin to prove Theorem 1.9.Firstly,we prove the part(1)of Theorem 1.9.Since f is univalent on Δ,we know that logf′∈ B.Assume that logf′∈ L2,s(Δ).We will prove that|σn|2ρ2n-3dxdy ∈ CMs(Δ)for any n ≥ 3,by induction.

According to Theorem 1.14,we see that,for any k∈N∪{0},

For k∈N,by using(3.5)and the following inequality

we get from(3.8)that

Then it follows form(3.8)that

It is clean that|σ(k)3|2ρ2k+3dxdy ∈ CMs(Δ)also holds for k=0.

Therefore,for fixed n≥3,we can now suppose that

for any k∈N∪{0}.Noting(3.7),using again the inequality(3.9),we get

Consequently,by(3.1)and(3.10),we see that,for any k∈N∪{0},

Hence,by induction on n,we see that|σn|2ρ2n-3dxdy ∈ CMs(Δ)for any n ≥ 3.Similarly,we can show that if logf′∈ L20,s(Δ),then|σn|2ρ2n-3dxdy ∈ CMs,0(Δ)for any n ≥ 3.This proves the part(1)of Theorem 1.9.

Secondly,we prove the part(2)of Theorem 1.9.Assume that logf′∈ B0.Noting that Theorem 1.3 and Theorem 1.2,it is enough to show that|σn|2ρ2n-3dxdy ∈ CMs(Δ)(n ≥ 4)implies|N|2ρdxdy ∈ CMs(Δ)(resp.|σn|2ρ2n-3dxdy ∈ CMs,0(Δ)(n ≥ 4)implies|N|2ρdxdy ∈CMs,0(Δ)).

Suppose now that|σn+1|2ρ2n-1dxdy ∈ CMs(Δ),n ≥ 3,that is

We will show|N|2ρdxdy ∈ CMs(Δ).In view of Theorem 1.14,it is equivalent to show

Noting that

By again Leibniz’s formula for higher derivatives,we get that

and for 2≤j≤n-2,

It follows that

Then we have from(3.9)that

Hence we obtain

We may assume that f is univalent on={z:|z|≤1},i.e.,f is analytic onand oneto-one,since if not,we can first consider the dilatation function fr(z)=f(rz),and then take r→ 1-to get the desired results.Then we have N2ρdxdy∈ CMs(Δ).In view of Theorem 1.14 and its proof given above,we see that there exists a positive constant C2(n)such that

for all 0 ≤ i≤ n-2.We now estimate(a,n),(a,n),(a,n).Note that logf′∈ B0,by Lemma 3.1,we see that,for any ∈＞ 0,there exist a constant δ∈∈ (0,1)such that

for all 3≤i≤n,0≤k≤n,0≤j≤n.

Then we get from(3.13)that

for all 0≤i≤n-3.Similarly,we have

for all 2≤j≤n-2,0≤i≤n-j-1,and

By Lemma 3.1,we obtain,for 0≤i≤n-3,

Similarly,we have,for 2≤j≤n-2,0≤i≤n-j-1,

and

Then we get from(3.14)-(3.19)that

It follows from(3.12)that

That is

we see that

which means that logf′∈ L2,s(Δ).This proves that if logf′∈ B0and|σn|2ρ2n-3dxdy ∈CMs(Δ)(n ≥ 3),then logf′∈ L2,s(Δ).Similarly,checking the proof above,we can show that logf′∈ B0and|σn|2ρ2n-3dxdy ∈ CMs,0(Δ)(n ≥ 3)implies logf′∈ L2,s0(Δ).This proves Theorem 1.9.

Finally,we note that each term in σn+1(f)(z)is a constant multiple of the corresponding term in Sn(f)(z).By checking the proof of Theorem 1.9 carefully,we can similarly show that Theorem 1.10 also holds.The proofs of Theorem 1.9 and 1.10 are finished.