Báo cáo toán học: "Weighted Composition Operators between Different Weighted Bergman Spaces in Polydiscs " pot

Necessary and sufficient conditions are established for the weighted composition operatorψCϕinduced byϕz and ψz to be bounded or compact between different weighted Bergman spaces in poly

Vietnam Journal of Mathematics 34:3 (2006) 255–264

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Weighted Composition Operators

Between Different Weighted Bergman

Spaces in Polydiscs

Li Songxiao*

Department of Math., Shantou University,515063, Shantou, Guangdong, China

and Department of Mathematics, Jiaying University,

514015, Meizhou, Guangdong, China

Received April 04, 2004 Revised September 18, 2005

Abstract. Let Dn be the unit polydiscs of C n, ϕ(z)=(ϕ1(z), ,ϕn(z)) be a holomor-phic self-map ofDn andψ(z)a holomorphic function on Dn Necessary and sufficient conditions are established for the weighted composition operatorψCϕinduced byϕ(z)

and ψ(z) to be bounded or compact between different weighted Bergman spaces in polydiscs

2000 Mathematics Subject Classification: 47B38, 32A36

Keywords: Bergman space, polydiscs, weighted composition operator.

1 Introduction

We adopt the notation described in [4-6] Denote by D n the unit polydisc in

Cn , by T n the distinguished boundary of D n , by A p α (D n) the weighted Bergman

spaces of order p with weightsQn

i=1 (1−|z i|2)α , α > −1 We use m nto denote the

By σ n we shall denote the volume measure on D n given by σ n (D n) = 1, and

by σ n,α we shall denote the weighted measure on D n given by σ n,α=Qn

i=1(1 −

|z i|2)α σ n If R is a rectangle on T n , then S(R) denote the corona associated to

∗

R In particular, if R = I1× I2× · · · × I n ⊂ T n , with I i being the intervals on

T n of length δ i and centered at e i(θ0i+δi/2) for i = 1, · · ·, n, then S(R) is given by

S(I i ) = {re iθ ∈ D : 1 − δ i < r < 1, θ i0< θ < θ i0+ δ i }.

For α > −1, 0 < p < ∞, recall that the weighted Bergman space A p

α (D n) consists of all holomorphic functions on the polydisc satisfying the condition

kf k p Ap

Z

Dn

|f (z)| p

n

Y

i=1

(1 − |z i|2)α dσ n,α < +∞.

Denoted by H(D n ) the class of all holomorphic functions with domain D n

Let ϕ be a holomorphic self-map of D n , the composition operator C ϕinduced by

ψC ϕ induced by ψ and ϕ is defined by

for z in D n and f ∈ H(D n)

It is interesting to characterize the composition operator on various analytic function spaces The book [2] contains plenty of information It is well known that composition operator is bounded on the Hardy space and the Bergman space

in the unit disc This result does not carry over to the case of several complex variables Singh and Sharma has showed in [7] that not every holomorphic

map from D n to D n induces a composition operator on H p (D n) For example,

ϕ(z1, z2) = (z1, z2) does not induce a bounded composition operator on H2(D n)

Jafari studied the composition operator on Bergman spaces A p

α (D n) in [6] His results can be stated as follows

Theorem A Let 1 < p < ∞, α > −1 and let ϕ be a holomorphic self-map

α (D n ) if and only if µ is an (compact) α Carleson measure.

Theorem B Let 1 < p < ∞ and α > −1 and let ϕ be a holomorphic self-map

(i) C ϕ is a bounded composition operator on A p

α (D n ) if and only if

sup

z0∈Dn

Z

Dn

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z0)i ϕ i|2

i2+α

dσ n,α ≤ M < ∞.

(ii) C ϕ is a compact composition operator on A p

α (D n ) if and only if

lim sup

z0∈Dn

Z

Dn

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z0)i ϕ i|2

i2+α

In this paper, we study the weighted composition operators between differ-ent weighted Bergman spaces in polydiscs Some measure characterizations and

function theoretic characterizations are given for the boundedness and compact-ness of the weighted composition operators

Throughout the remainder of this paper C will denote a positive constant,

the exact value of which may vary from one appearance to the next

2 Measure Characterization of Weighted Composition Operators

In this section, we give the measure characterization of weighted composition operators between different weighted Bergman spaces For this purpose, we should need some lemmas which will be stated as follows

Definition 1 A finite, nonnegative, Borel measure µ on D n is said to be a

η − α Carleson measure if

µ1(S(R)) ≤ C

n

Y

i=1

δ i 2+α

µ is said to be a compact η − α Carleson measure if

lim

δi →0 sup

θ∈Tn

µ1(S(R))

Qn i=1 δ i 2+α = 0.

Remark When η = 1, the definition of Carleson measures for polydiscs is due

to Chang(see [1])

Modifying the proof of Theorem 2.5 in [5], we get the following lemma

Lemma 1 Suppose that 1 < p < ∞, α > −1, η ≥ 1 Let I be the identity

α (D n ) into L ηp (D n , µ) Then I is a bounded operator if and only if µ is an η − α Carleson measure.

α (D n) then

n Z

Dn

|f | ηp dµo1

≤ C

Z

Dn

if and only if

µ1(S(R)) ≤ C

n

Y

j=1

For this purpose, suppose that (1) holds for all f ∈ A p α (D n) Define

f (z) =

n

Y

j=1

(1 − α j z j)−(α+2)/p ,

where α j = (1 − δ j )e i(θ0j+δj/2)

It is easy to see that f (z) ∈ A p α (D n) In addition,

since on S(R),

|f (z)| ηp > 2 −η(α+2)

n

Y

j=1

δ j −η(α+2) ,

we have

n Z

Dn

|f (z)| ηp dµo1

≥n Z

S(R)

|f (z)| ηp dµo1

≥ 2−(α+2)

n

Y

j=1

δ j −(α+2) µ1(S(R)). (3)

Then the result follows from (1) and (3)

Conversely, suppose that (2) holds for all rectangles in T n Fix z ∈ D nand

let 1 − |z j|2= δ j , consider a polydisc W z centered at z and of radius δ j /2 in the

z j coordinate If R = I1× × I n is the rectangle on T n with I j centered at

z j /|z j | and |I j | = 2δ j , then W z ⊂ S(R) (see [5]) Therefore for any f ∈ A p

α (D n),

by the sub mean value property for |f |, we get (see [5])

|f (z)| ≤ Qn C

j=1 δ j α+2

Z

S(R)

|f |dσ n,α

Since σ n,α (S(R)) = CQn

j=1 δ j α+2, then

|f (z)| ≤ C

σ n,α (S(R))

Z

S(R)

Now define

M (f ) = sup

R

1

σ n,α (S(R))

Z

S(R)

|f |dσ n,α

We get

We will show that there exists a constant C independent of s such that

Given (6), since M is a sublinear operator of type (∞, ∞), it is obvious that

kM (f )k∞ ≤ kf k∞ If we define 1p = θ,1q = θ η , 0 < θ < 1, i.e q = pη, by the

Marcinkiewicz interpolation theorem we obtain that

n Z

Dn

|M (f )| ηp dµo1

≤ C

Z

Dn

|f | p dσ n,α (7)

Combining (5) and (7) we get

n Z

Dn

|f (z)| ηp dµo1

≤ Cn Z

Dn

|M (f )(z)| ηp dµo1

≤ C

Z

Dn

|f (z)| p dσ n,α

This prove (1)

To complete the proof we need to show that if µ1(S(R)) ≤ Cσ n,α (S(R)), then (6) holds Let R z = I1× × I n denote rectangle on T n with I j denoting

intervals centered at z j /|z j | and of radius (1 − |I j |)/2 Let S z denote the corona

associated with R z Note that z ∈ S z, define

Z

Sz

|f |dσ n,α > s( + σ n,α (S z))o

It is easily to check that the following equality holds

Λ = {z ∈ D n : M (f ) > s} = [

>0

A  s ,

i.e µ(Λ) = lim →0 µ(A  s ) Furthermore, if z ∈ A  s and S z are disjoint for the

different z ∈ Λ, then by (8) we have

z∈Λ

( + σ n,α (S z )) <X

z∈Λ

Z

Sz

|f |dσ n,α ≤ kf k 1,α

Hence

z∈Λ

( + σ n,α (S z )) ≤ kf k 1,α (9)

Consider the last inequality (9), it shows that there are only finitely many z ∈

s so that their corresponding S z are disjoint From these extract the points,

are multiplied by five in each coordinate then the resulting sets cover A 

s This follows from covering lemma Write the S z associated with these points as

S1, S2, , S l Since A 

s⊂Sl k=1 5S k , S k are pairwise disjoint,

µ(A  s) ≤ 5n

l

X

k=1

(see [5]) Also, by hypothesis

combining (9), (10) and (11), we get

µ(A  s) ≤ 5n

l

X

k=1

µ(S k ) ≤ C

l

X

k=1

( + σ n,α (S k))η ≤ C(s−1kf k 1,α)η

Letting  tend to zero we obtain

Using Lemma 1, we give a characterization of the boundedness of the weighted

composition operator ψC ϕ : A p α (D n ) → A ηp β (D n)

Theorem 1 Suppose that 1 < p < ∞, β, α > −1, η ≥ 1 Let ϕ be a

µ(E) = ν(ϕ−1(E))(E ⊂ D n ) Then ψC ϕ : A p α (D n ) → A ηp β (D n ) is bounded if and only if µ is a η − α Carleson measure.

α (D n ) → A ηp β (D n ) is bounded, then there exists a constant C

such that

kψf ◦ ϕk Aηp

β ≤ Ckf k Ap

α

for all f ∈ A p

α (D n), i.e

n Z

Dn

|ψf ◦ ϕ| ηp dσ n,β (z)o1

≤ C

Z

Dn

|f | p dσ n,α (z).

By the definition of µ, we have (see [3, p 163])

Z

Dn

|ψf ◦ ϕ| ηp dσ n,β=

Z

Dn

|f | ηp dµ.

Hence

n Z

Dn

|f | ηp dµo1

≤ C

Z

Dn

|f | p dσ n,α (z).

The assertion follows from Lemma 1

Conversely, suppose that µ is an η − α Carleson measure By Lemma 1, there exists a constant C such that

n Z

Dn

|f | ηp dµo1

≤ C

Z

Dn

|f | p dσ n,α (z)

for all f ∈ A p

α (D n ) By the definition of µ, kψf ◦ ϕk Aηp

β ≤ Ckf k Ap

α Hence

ψC ϕ : A p α (D n ) → A ηp β (D n) is bounded We are done 

To characterize the compactness of weighted composition operators between different weighted Bergman spaces, we will need the following lemma, whose proof is an easy modification of that Proposition 3.11 in [2], we omit the proof

Lemma 2 Suppose that 1 < p < ∞, β, α > −1, η ≥ 1 Let ϕ be a holomorphic

α (D n ), kψf k ◦ ϕk ηp ηp,β → 0 as

k → ∞.

Theorem 2 Suppose that 1 < p < ∞, β, α > −1, η ≥ 1 Let ϕ be a

α (D n ) → A ηp β (D n ) is compact if and only if µ is a compact η − α Carleson measure.

α (D n ) → A ηp β (D n) is compact Let

f δ (z1, z2, , z n) =

n

Yδ β−(α+2)/p i (1 − η i z i)β ,

where 0 < δ j < 1, β > (α + 2)/p and η j = (1 − δ j )e i(θ0j+δj/2) These functions

are bounded in A p

α (D n ), and tend to zero weakly as δ i → 0 Since on regions

|f δ (z1, z2, , z n)|p >

n

Y

i=1

δ i (β−(α+2)/p)p

(2β δ β i)p =

n

Y

i=1

1

δ α+2 i 2pβ ,

then we get

Z

Dn

|ψf δ ◦ ϕ| ηp dσ n,β

≥ Z

Dn

i=1 δ i η(α+2)2ηpβ , where (δ) → 0 as δ i → 0 for some i Hence µ1(S) ≤ (δ)2 nβpQn

i=1 δ i α+2, i.e

lim

δi→0 sup

θ∈Tn

µ1(S(R))

Qn i=1 δ i α+2 = 0.

Therefore µ is a compact η − α Carleson measure.

Conversely, suppose that µ is a compact η − α Carleson measure, then for every  > 0, there is δ such that

sup

θ∈Tn

µ1(S(R))

Qn i=1 δ α+2 i ≤  for all δ i < δ Let f k ⊂ A p α (D n) converge uniformly to 0 on each compact subsets

of D n It just only need to show that kψf k ◦ ϕk ηp,β→ 0 We have

kψf k ◦ ϕk ηp ηp,β=

Z

Dn

|ψf k ◦ ϕ| ηp dσ n,β=

Z

Dn

|f k|ηp dµ

= Z

Dn\(1−δ)Dn

|f k|ηp dµ +

Z

(1−δ)Dn

|f k|ηp dµ

= I1+ I2 Write µ = µ1+ µ2, where µ1 is the restriction of µ to (1 − δ)D n and µ2lies on

the complement of this set in D n Then, since µ2≤ µ, we get

supµ2(S(R))

t n(α+2) ≤ supµ(S(R))

t n(α+2) , where the supremums are extended over all θ ∈ T n and for all 0 < t < δ Then

it is clearly that µ2is a compact η − α Carleson measure Hence

Z

Dn\(1−δ)Dn

t ηn(α+2) kf kkηp

Apα≤ Ckf kkηp

Apα.

Because {f k } converges uniformly to 0 on (1 − δ)D n , I2can be made arbitrarily

small by choosing large k Since  is arbitrary, we have ψC ϕ f k → 0 in A ηp β (D n),

that is ψC ϕ : A p (D n ) → A ηp (D n) is compact This completes the proof 

3 Function Theoretic Characterization of Weighted Composition Operators

In this section, we give some function theoretic characterizations of weighted composition operators For this purpose, we should first modify the Proposition

8 and Proposition 12 of [6] and give the following lemma

Lemma 3 Let µ be a nonnegative, Borel measure on D n Then

(i) µ is an η − α Carleson measure if and only if

sup

z0∈Dn

Z

Dn

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0 i z i)|2

i(2+α)η

(ii) µ is a compact η − α Carleson measure if and only if

lim sup

z0∈Dn

Z

Dn

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0i z i)|2

i(2+α)η

Proof Suppose that (12) holds, we show that µ is an η − α Carleson measure.

Note that

R = {(e iθ1, · · ·, e iθn) ∈ T n : |θ i − (θ0)i | < δ i}

and

S = S(R) = {(r1e iθ1, · · ·, r n e iθn) ∈ D n : 1 − δ i < r i < 1, |θ i − (θ0)i | < δ i }.

Hence if z0 = 0, (12) implies that µ(D n ) ≤ M < ∞ Therefore we can assume that δ i <14 for all i Take(z0)i= 1 −δi

2e i(θ0 ) i

, then for all z ∈ S, we have

n

Y

i=1

C (1 − |(z0)i|2)2+α ≤

n

Y

i=1

 1 − |(z0)i|2

|1 − (z0)i z i|2

2+α

So

µ1(S) =n Z

S

= C

n

Y

i=1

(1 − |(z0)i|2)2+αn Z

S

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0i z i)|2

i(2+α)η

≤ CM1

n

Y

i=1

δ i 2+α

Hence µ is an η − α Carleson measure.

Conversely, suppose that µ is an η − α Carleson measure Let z0 ∈ D n, if

||z0|| ≤ 1, it is obviously that (12) holds, since the integrand can be bounded

uniformly Also, if |(z0)i | <34, the term corresponding to this i in the integrand

in (12) can be bounded So let us suppose |(z0)i | > 3 for all i, and let

E k =n

i

|z i− (z0 ) i

|(z0 ) i ||

1 − |(z0)i| < 2 ko

Note that if z ∈ E1, then

n

Y

i=1

C (1 − |(z0)i|2)2+α ≥

n

Y

i=1

 1 − |(z0)i|2

|1 − (z0)i z i|2

2+α

and for k ≥ 2 if z ∈ E k − E k−1, then

n

Y

i=1

C (1 − |(z0)i|2)2+α ≥

n

Y

i=1

 1 − |(z0)i|2

|1 − (z0)i z i|2

2+α

Since µ is an η − α Carleson measure, we have

Z

Dn

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0 i z i)|2

i(2+α)η dµ

≤

Z

E1

+

∞

X

k=2

Z

Ek−Ek−1

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0i z i)|2

i(2+α)η dµ

≤ C

∞

X

k=2

Qn i=1 (1 − |(z0)i|)(2+α)η

≤ C

∞

X

k=2

Qn i=1 δ i (2+α)η

≤ C < ∞.

This completes the proof of (i) With the same manner of (i) and the proof of Proposition 8 (ii) in [6] we can give the proof of (ii), we omit the details 

Theorem 3 Suppose that 1 < p < ∞, β, α > −1, η ≥ 1 Let ϕ be a holomorphic

(i) ψC ϕ : A p

α (D n ) → A ηp β (D n ) is bounded if and only if

sup

z0∈Dn

Z

Dn

|ψ(z)| ηp

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0 i ϕ i)|2

i(2+α)η

(ii) ψC ϕ : A p

α (D n ) → A ηp β (D n ) is compact if and only if

lim sup

z0∈Dn

Z

Dn

|ψ(z)| ηp

n

Y

i=1

h 1 − |(z0)i|2

|1 − (z 0i ϕ i)|2

i(2+α)η

is a bounded or compact η − α Carleson measure Then by Lemma 3 we get the

1 S Y A Chang, Carleson measure on the bi-disc, Ann Math 109 (1979) 613–

620

2 C C Cowen and B D MacCluer, Composition operators on Spaces of Analytic Functions, CRC Press, Boea Raton, 1996.

3 P R Halmos, Measure Theory, Springer–Verlag, New York, 1974.

4 F Jafari, Composition Operators in Polydisc, Dissertation, University of

Wiscon-sin, Madison, 1989

5 F Jafari, Carleson measures in Hardy and weighted Bergman spaces of polydisc,

Proc Amer Math Soc 112 (1991) 771–781.

6 F Jafari, On bounded and compact composition operators in polydiscs, Canad.

J Math 5 (1990) 869–889.

7 R K Singh and S D Sharma, Composition operators and several complex

vari-ables, Bull Aust Math Soc 23 (1981) 237–247.