Derivative－free characterizations of a class of Möbius invariant spaces

HAN Jin－zhuang，ZHOU Ji－zhen

（1.School of Mathematics，Hulunbeier University，Inner Mongolia，Hulunbeier 021008，China；2.School of Sciences，Anhui University of Science and Technology，Huainan 232001，China）

On the other hand，by Fubini’s Theorem，we have

0 Introduction

Throughout this paper，the unit disc is denoted byD.For any givena∈D，let

be a M¨obius transformation of the unit disc.LetK：（0，∞］→ （0，∞］be a right－continuous and nondecreasing function.For 0＜p＜ ∞ ，the spaceQK（p，p－2）consists of all analytic functionsfonDsatisfying

wheredλ（z）＝（1－｜z｜2）－2dA（z）.Note thatdA（z）is the Eculidean area measure onDso thatA（D）＝1.Equipped with the norm‖f‖＝｜f（0）｜＋‖f‖K，p，the spaceQK（p，p－2）is a Banach space whenp≥1.We know that the spaceQK（p，p－2）is a subset of the Bloch space［1］.The basic properties ofQK（p，p－2）spaces can be founded in［1］.Ifp＝2，thenQK（p，p－2）spaces are justQKspaces［2－3］.IfK（0）＞0andp＞2，then the spaceQK（p，p－2）coincides with the analytic Besov space.

It is clear that the spaceQK（p，p－2）is Mobius invariant，i.e.，‖f◦φa‖K，p＝‖f‖K，p.The spaceQK（p，p－2）is nontrivial if and only if it contains all polynomials［4］，that is

Making a change of variables in the above integral and simplifying the result using polar coordinates，we obtain that the spaceQK（p，p－2）is nontrivial if and only if

In the paper，we may need two conditions onKas follows

and

where

The derivative－free characterizations ofQKspaces have been study in many papers［4－5］.In this paper，we study the derivative－free characterization inQK（p，p－2）spaces by applying the Bergman metric andp－mean oscillation.The technique is suggested by the theory of Bergman space.The result is important which enriches the theory ofQKtype spaces.It has important significance in application.

Furthermore，we suppose that the nondecreasing functionKsatisfiesK（2t）≈K（t），that is，K（2t）≺K（t）≺K（2t）.Note，K（2t）≺K（t）means that there exists a constantC（independent oft），such that

1 Derivative－Free characterizations of QK（p，p－2）spaces

In this section，we give some derivative－free characterizations ofQK（p，p－2）spaces.We now introduce some basic definitions.

Recall that the Bergman metric ofDis given by

For anyz∈Dandr＞0，the pseudo－hyperbolic disc is defined by

The pseudo－hyperbolic discD（z，r）is an Euclidean disc［5］.Let｜D（z，r）｜be the Euclidean area ofD（z，r）.Obviousy，｜D（z，r）｜≈ （1－｜z｜2）2aszapproaches the unit circle for any givenr＞0.It is well known that

Iffis an analytic function onD，we define the oscillation offatzin the Bergman metric as

Thep－mean oscillation offatzin the Bergman metric is defined by

where

andp＞0.Ifp＝2，we call it the mean oscillation offatzin the Bergman metric.See section 7.1 of［6］.Furthermore，we define

forp，r＞0.

The following result will be needed in the proof of the main theorem.

Lemma 1［7］LetKsatisfy the condition（3）.For any givenz∈D，then

Lemma 2［8］Let 0＜p＜ ∞，－1＜q＜ ∞ .Iffis analytic onD，then

Lemma 3LetKsatisfy the condition（3）and 0＜p＜ ∞ .Letfbe analytic onD，then

if and only if

ProofBy Lemma 2，we have

Note that

Make a change of variables，Then

Given a fixedr＞0，then

for allw∈D（z，r）.Proposition 4.3.8of［6］and（4）imply

On the other hand，by Fubini’s Theorem，we have

Lemma 1Gives

The proof is complete.

Theorem 1LetKsatisfy the condition（3）.Ifr＞0，0＜p＜∞ ，then the following conditions are equivalent for all analyticsfonD.

（1）The functionfbelongs toQK（p，p－2）spaces.

ProofWe make two change variablesu＝φa（z）andv＝φa（w）.Then

Lemma 3implies

Take the supremum overa∈Dboth side.This shows（1）⇔（2）.

By Proposition 4.3.8of［6］，we have the following inequality

Replacingfbyf－f（z），then

for all analyticsfandz，w∈D.This means

whenw∈D（z，r）.Ifw∈D（z，r），thenD（w，r）⊂D（z，2r）and｜D（z，2r）｜≈｜D（w，r）｜.By（5）we have

This gives

We have showed（2）⇒ （3）.Ifw，u∈D（z，r），then

So we have

This gives

Then we obtain

This shows（3）⇒（4）.

We apply（6）to get the following inequality

This proves（4）⇒（5）.

Letfbe analytic onDwithf（0）＝0.Denote bySinceis subharmonic forp＞0，we have

Replacingfbyf◦φz－f（z），we obtain

Making a change of variables，we have

So（5）implies（1）.The proof is complete.

Denote by∂Dthe unit circle.For a subarcI⊂∂D，letθbe the midpoint ofIand denote

where｜I｜denotes the length ofI.If｜I｜＞1，we setS（I）＝D.S（I）is called the Carleson box.A positive measure dμis said to be aK－Carleson measure onDprovided

IfKsatisfies the conditions（2）and（3），then dμis aK－Carleson measure if and only if

by Theorem 3.1of［3］.By Theorem 1，we have the following result.

Corollary 1LetKsatisfy the conditions（2）and（3）.Ifr＞0，0＜p＜ ∞ ，then the following conditions are equivalent for all analyticsfonD.

（1）The functionfbelongs toQK（p，p－2）spaces.

（3）（ωr（f）（z））pdλ（z）is aK－Carleson measure.

（4）（MOp，r（f）（z））pdλ（z）is aK－Carleson measure.

（5）（Op，r（f）（z））pdλ（z）is aK－Carleson measure.

2 Derivative－Free characterizations of QK，0（p，p－2）spaces

For 0＜p＜ ∞ ，the spaceQK，0（p，p－2）consists of all analytic functionsfonDsatisfying

A positive measure dμis said to be a vanishingK－Carleson measure onDprovided

IfKsatisfies the conditions（2）and（3），then dμis a vanishingK－Carleson meansure if and only if

by Theorem 3.1of［3］.

Carefully checking the proof of Theorem 1and Corollary 1.We see that the little version of Theorem 1and Corollary 1hold as well，from which we obtain the following results.

Theorem 2LetKsatisfy the condition（3）.Ifr＞0，0＜p＜∞ ，then the following conditions are equivalent for all analyticsfonD.

Corollary 2LetKsatisfy the conditions（2）and（3）.Ifr＞0，0＜p＜ ∞ ，then the following conditions are equivalent for all analyticsfonD.

（1）The functionfbelongs toQK，0（p，p－2）spaces.

（3）（ωr（f）（z））pdλ（z）is a vanishingK－Carleson measure.

（4）（MOp，r（f）（z））pdλ（z）is a vanishingK－Carleson measure.

（5）（Op，r（f）（z））pdλ（z）is a vanishingK－Carleson measure.

We leave the details of Theorem 2and Corollary 2to the interested reader.

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