Báo cáo hóa học: "NORM EQUIVALENCE AND COMPOSITION OPERATORS BETWEEN BLOCH/LIPSCHITZ SPACES OF THE BALL" potx

CLAHANE AND STEVO STEVI ´C Received 11 October 2005; Revised 30 January 2006; Accepted 12 February 2006 For p > 0, letᏮpBn and ᏸpBn denote, respectively, the p-Bloch and holomorphic p-Li

BETWEEN BLOCH/LIPSCHITZ SPACES OF THE BALL

DANA D CLAHANE AND STEVO STEVI ´C

Received 11 October 2005; Revised 30 January 2006; Accepted 12 February 2006

For p > 0, letᏮp(Bn) and ᏸp(Bn) denote, respectively, the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ballBninCn It is known thatᏮp(Bn) andᏸ1− p(Bn) are equal as sets when p ∈(0, 1) We prove that these spaces are additionally norm-equivalent, thus extending known results forn =1 and the polydisk As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operatorCφfromᏸp(Bn) toᏸq(Bn)

Copyright © 2006 D D Clahane and S Stevi´c This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Background and terminology

Letn ∈ N, and suppose thatDis a domain inCn Denote the linear space of complex-valued, holomorphic functions onDbyᏴ(D) Ifᐄ is a linear subspace of Ᏼ(D) and

φ : D → Dis holomorphic, then one can define the linear operatorCφ:ᐄ→Ᏼ(D) by

Cφ(f ) = f ◦ φ for all f ∈ᐄ.Cφ is called the composition operator induced by φ.

The problem of relating properties of symbolsφ and operators such as Cφ that are induced by these symbols is of fundamental importance in concrete operator theory However, efforts to obtain characterizations of self-maps that induce bounded compo-sition operators on many function spaces have not yielded completely satisfactory results

in the several-variable case, leaving a wealth of basic open problems

In this paper, we try to make progress toward the goal of characterizing the holo-morphic self-maps of the open unit ball Bn in Cn that induce bounded composition operators between holomorphic p-Lipschitz spacesᏸp(Bn) for 0< p < 1 by translating

the problem to (1− p)-Bloch spaces Ꮾ1− p(Bn) via an auxiliary Hardy/Littlewood-type norm-equivalence result of potential independent interest This method was also used in [7] forB 1and in [3] for the unit polydiskΔn

The function-theoretic characterization of analytic self-maps ofB 1that induce

bound-ed composition operators onᏸp(B 1) for 0< p < 1 is due to Madigan [7], and the case of

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 61018, Pages 1 11

DOI 10.1155/JIA/2006/61018

Δnwas handled in a joint paper by the present authors with Zhou [3], in which a full characterization of the holomorphic self-mapsφ ofΔn that induce bounded composi-tion operators betweenᏸp(Δn) andᏸq(Δn), and, more generally, between Bloch spaces

Ꮾp(Δn) andᏮq(Δn), is obtained forp, q ∈(0, 1), along with analogous characterizations

of compact composition operators between these spaces

Although our main results concerning composition operators,Theorem 2.1and Cor-ollary 2.2, are not full characterizations, they do generalize Madigan’s result for the disk to

Bn; on the other hand, we obtain a complete Hardy-Littlewood norm-equivalence result forp-Bloch and (1 − p)-Lipschitz spaces ofBnfor alln ∈ N This norm-equivalence result should lead to an eventual extension toBnof the characterizations of bounded composi-tion operators established onB 1in [7] and onΔnin [3]

Most of our several complex variables notation is adopted from [8] Ifz =( 1, , z n) andw =(w1, , wn) are points inCn, then we define a complex inner product by z, ω  =

n

k =1zk wk¯ and put| z |:= z, z  We callBn:= { z ∈ C n:| z | < 1 } the (open) unit ball of

Cn

Letp ∈(0,∞) Thep-Bloch spaceᏮp(Bn) consists of all f ∈Ᏼ(Bn) with the property that there is anM ≥0 such that

b( f , z, p) : =1− | z |2 p ∇ f (z)  ≤ M ∀ z ∈ B n. (1.1)

Ꮾp(Bn) is a Banach space with norm f Ꮾpgiven by

 f Ꮾp =f (0)+ sup

z ∈B n

The little p-Bloch spaceᏮp

0(Bn) is defined as the closed subspace ofᏮp(Bn) consisting of the functions that satisfy

lim

z → ∂Bn



1− | z |2 p ∇ f (z)  =0. (1.3)

Forp ∈(0, 1),ᏸp(Bn ) denotes the holomorphic p-Lipschitz space which is the set of all

f ∈Ᏼ(Bn) such that for someC > 0,

f (z) − f (w)  ≤ C | z − w | p for everyz, w ∈ B n. (1.4) These functions extend continuously toBn(cf [3, Lemma 4.4]) Therefore, ifA(Bn) is the ball algebra [8, Chapter 6], then

ᏸp

Bn

=Lipp

Bn

∩ A

Bn

where Lipp(Bn) is the set of all f :Bn → Csatisfying (1.4) for someC > 0 and all z ∈ B n

ᏸp(Bn) is endowed with a complete norm · ᏸpthat is given by

 f ᏸp =f (0)+ sup

z = w:z,w ∈B

 f (z) − f (w)

| z − w | p



In (1.4) and (1.6),BnandBnare interchangeable, since functions inᏸp(Bn) extend con-tinuously toBn The supremum above is called the Lipschitz constant for f As in [8, page 13],σ represents the unique rotation-invariant positive Borel measure on ∂Bnfor which

σ(∂Bn)=1, and for f ∈ L1(σ), C[ f ] denotes Cauchy integral of f onBn(see [8, page 38])

Letu ∈ ∂Bnand f ∈Ᏼ(Bn ) The directional derivative of f at z ∈ B nin the direction

ofu ∈ ∂Bnis given by

Du f (z) = lim

λ →0,λ ∈C

f (z + λu) − f (z)

Observe that

We define the partial differential operators Djas in [8, Chapter 1] The radial derivative operator [8, page 103] inCnwill be denoted byRand is linear LetU =u1,u2, , u n

be an orthonormal basis for the Hilbert spaceCnwith its usual Euclidean structure We define a gradient operator∇UonᏴ(D) with respect toUby

∇Uf (z) =D u1 f (z), D u2 f (z), , D u n f (z)

and we denote∇Uby∇whenUis the typically ordered standard basis forCn

Letx and y be two positive variable quantities We write x  y (and say that x and y are comparable) if and only if x/ y is bounded above and below.

2 Main results on composition operators

Our norm-equivalence result (Theorem 3.5) ties our results concerning Cφ between

p-Lipschitz spaces ofBnto the following result for general Bloch spaces

Theorem 2.1 Let p, q ∈(0,∞ ), and suppose that φ :Bn → B n is holomorphic Then the following statements hold.

(A) If there is an M ≥ 0 such that for all z ∈ B n and j ∈ {1, , n } ,



1− | z |2 q

1−φ(z) 2 p ∇ φ j(z)  ≤ M, (2.1)

thenCφ is bounded fromᏮp(Bn ) (resp.,Ꮾp

0(Bn )) toᏮq(Bn ).

(B) IfCφ is bounded fromᏮp(Bn ) (resp.,Ꮾp

0(Bn )) toᏮq(Bn ), then there is an M  ≥0

such that for all z ∈ B n and u ∈ ∂Bn,



1− | z |2 q

1− φ(z), u 2 p ∇ φ(z), u M  (2.2)

Theorem 2.1above andCorollary 2.2 below for 0< p = q < 1 appear in [2, Chapter 4] It should be pointed out thatTheorem 2.1, part (A) is similar to a statement that is proved in [11]; furthermore, [11] contains a result that is in the same direction as part (B)

ofTheorem 2.1and that is proven using different testing functions Unlike [11], however, the present paper addresses composition operators betweenᏸp(Bn) andᏸq(Bn) and the coincidence and norm equivalence ofᏮ1− p(Bn) andᏸp(Bn), respectively

It is natural to consider the application of corresponding “little-oh” arguments to obtain a compactness result analogous toTheorem 2.1, in which “bounded” is replaced

by “compact” and the limit of the left-hand side of each inequality in the statement is taken as | φ(z) | →1−, with inequality replaced by equality to 0 However, in the case, thatp ∈(0, 1),Ꮾp(Bn) is the same as and norm-equivalent toᏸ1− p(Bn), whose compact composition operators are known (by a result due to J H Shapiro) to be generated pre-cisely by holomorphic self-mapsφ ofBnwith supremum norm strictly less than 1 (see [4, Chapter 4])

The following corollary follows from Theorems2.1and3.5and extends the main result

of [7]

Corollary 2.2 Let p, q ∈ (0, 1), and suppose that φ :Bn → B n is holomorphic Then the following statements hold.

(A) If there is an M ≥ 0 such that



1− | z |2  1− q

1−φ(z) 2  1− p ∇ φj(z)  ≤ M, (2.3)

for all j ∈ {1, 2 , n } and z ∈ B n, thenCφ is a bounded operator fromᏸp(Bn ) to

ᏸq(Bn ).

(B) IfCφ is a bounded operator fromᏸp(Bn ) toᏸq(Bn ), then there is an M  ≥ 0 such that for all z ∈ B n and u ∈ ∂Bn,



1− | z |2  1− q

1− φ(z), u 2  1− p ∇ φ(z), u M  (2.4) Choosingn =1,p = q ∈(0, 1), andu =1 inCorollary 2.2leads to the following result, which appears in [7]

Theorem 2.3 Let 0 < p < 1, and suppose that φ is an analytic self-map ofB 1 ThenCφ is bounded onᏸp(B 1) if and only if

sup

z ∈B1



1− | z |2

1−φ(z)1− pφ (z)< ∞ . (2.5)

3 Norm equivalence ofᏸp(Bn) andᏮ1− p(Bn) for 0< p < 1

To generalizeTheorem 2.3toBn, we needTheorem 3.5, which is the ball analogue of the following result for the disk [7, Lemma 2] The first statement inTheorem 3.1can be

derived from a classical theorem of Hardy/Littlewood forn =1 (see [6], [5, page 74], and [4, page 176])

Theorem 3.1 Let 0 < p < 1 If f :B 1→ C is analytic, then f ∈ᏸp(B 1) if and only if

f (z)  = O



1

1− | z |2

 1− p

Furthermore, the Lipschitz constant of f and the quantity

sup

z ∈B1



1− | z |2  1− pf (z) (3.2)

are comparable as f varies throughᏸp(B 1).

We remark that the polydisk version ofTheorem 3.1is stated and proved in [3] How-ever, the argument used there cannot be applied toBn, so we need a different approach for that domain We will proceed by listing some lemmas, which together essentially form the norm-equivalence result,Theorem 3.5

For 0< p < 1, we define · RᏮ1− ponᏸp(Bn) by

 f R

Ꮾ 1− p =f (0)+ sup

z ∈B n



1− | z |2  1− p(Rf )(z). (3.3)

It can be shown by subsequent applications of Lemmas3.2and3.3below that · ᏾Ꮾ1− pis

a norm onᏸp(Bn)

We start with the following lemma

Lemma 3.2 Suppose that 0 < p < 1 Thenᏸp(Bn)⊂Ꮾ1− p Furthermore, there is a C p ≥0

such that for all f ∈ᏸp(Bn ),

 f R

Proof The proof of the first statement is standard and left to the reader Since functions

inᏸp(Bn) extend continuously toBn, they are automatically inL1(σ) [8, Remark, page 107] and since the quotients of these functions and theirᏸp-norms satisfy [8, equation (1), page 107], the second statement is obtained from [8, Theorem 6.4.9]  The following lemma is also a portion ofTheorem 3.5

Lemma 3.3 If p ∈ (0, 1), thenᏮ1− p(Bn)⊂ᏸp(Bn ), and

 f ᏸp ≤2 + 2p −1 

 f Ꮾ1− p ∀ f ∈Ꮾ1− p

Bn



Proof Suppose that f ∈Ꮾ1− p(Bn) If f =0, thenf ∈ᏸp(Bn) trivially, so assume hence-forward that f =0 A well-known result [8, Chapter 6] applied to f /  f Ꮾ 1− pimplies that

for allz, w ∈ Bn,

1

 f Ꮾ 1− p

f (z) − f (w)  ≤ 1 + 2p −1 

from which the first statement of the lemma follows Moreover,

 f ᏸp =f (0)+ sup

z,w ∈B n:z = w

f (z) − f (w)

| z − w | p

≤f (0)+

1 + 2p −1

 f Ꮾ 1− p ≤2 + 2p −1

 f Ꮾ 1− p

(3.7)

 The following fact also constitutes a part ofTheorem 3.5

Lemma 3.4 Let p > 0 Then f ∈Ꮾp(Bn ) if and only if there is an M ≥ 0 such that for all

z ∈ B n, |(Rf )(z) |(1− | z |2)p ≤ M If p ∈ (0, 1], then there is a Cp ≥ 0 such that  f Ꮾp ≤

Cp  f RᏮp for all f ∈Ꮾp(Bn ).

Proof For a proof of the first statement, see [10, Proposition 1] To prove the second statement, we use the weighted Bergman projection Pswith kernelKs and the mapLs

defined onP s[L ∞(B n)] by



L s g

(z) =(s + 1) −1 

1− | z |2 

(n + s + 1)g(z) + (Rg)(z)

∀ z ∈ B n, (3.8)

wheres ∈ Csatisfies Res > −1 (see [1]) By [1, Corollary 13], we have thatPs ◦ Lsis the identity onᏮ1(Bn) for all such values ofs In particular, P0◦ L0is the identity onᏮp(Bn), since this set is contained inᏮ1(Bn) Note that the assumptionp ∈(0, 1] is used here

We then obtain that there is aC ≥0 such that for allz ∈ B nand f ∈Ꮾ1(Bn),

f (z) =P0◦ L0



(f )(z) = C



Bn



1− | w |2 

K0(z, w)

(n + 1) f (w) +Rf (w)

dV (w).

(3.9) Hence, there is aC  ≥0 such that for all f ∈Ꮾp(Bn) andz ∈ B n,

∇ f (z)  ≤ C 



Bn



1− | w |2∇ K0(z, w)f (w)dV (w)

+C 



Bn



1− | w |2∇ K0(z, w)Rf (w)dV (w). (3.10)

Letε ∈(1− p, 1) Subsequent applications of the above inequality (see [9, Lemma 2] and [8, Theorem 1.4.10]) imply that there are nonnegative constantsC andC such that

for allz ∈ Bnand f ∈Ꮾp(Bn), the following chain of inequalities holds:

∇ f (z)  ≤ C 



Bn



1− | w |2 

| w |

1−  z, w n+2f (w)dV (w)

+C 



Bn



1− | w |2 

| w |

1−  z, w n+2Rf (w)dV (w)

≤ C   f R

Ꮾp



Bn



1− | w |2 ε

1−  z, w n+2 dV (w)

+C   f R

Ꮾp



Bn



1− | w |2  1− p

1−  z, w n+2 dV (w)

≤ C   f RᏮp

1



1− | z |1− ε+C

 f RᏮp

1



1− | z |p .

(3.11)

It follows that for all f ∈Ꮾp(Bn) andz ∈ B n,



1− | z |2 p ∇ f (z)  ≤2p+1 C   f R

The second statement in the lemma now follows from the above statement and an

Next, we state and prove this section’s main result, the analogue ofTheorem 3.1for

Bn We emphasize that while the statement of equality in the theorem is known and can

be obtained, for example, from [12], the norm-equivalence portion requires additional work that includes the previous lemmas and the proof below Furthermore, neither this result nor its proof has appeared previously in any literature that is known to the authors, though it seems to be part of the folklore The proof of this rather fundamental theorem seems to be nontrivial and worthy of recording

Theorem 3.5 If 0 < p < 1, thenᏮ1− p(Bn)=ᏸp(Bn ); furthermore,

 f Ꮾ 1− p  f RᏮ1− p  f ᏸp

as f varies throughᏸp(Bn ).

Proof The first statement is known, sinceᏸp(Bn)= A(Bn)∩Lipα(Bn) (see [8, Chapter 6]), which is set theoretically equal toᏮ1− p(Bn) (see [10]) ByLemma 3.4, it follows that there is a Cp ≥0 such that for all f ∈ᏸp(Bn),  f Ꮾ 1− p ≤ Cp  f RᏮ1− p It follows from Lemma 3.2that there is aC  p ≥0 such that for all f ∈ᏸp(Bn), f Ꮾ 1− p ≤ C p  f R

Ꮾ 1− p ≤

CpC  p  f ᏸp, which is less than or equal toCpC  p(2 + 2p −1) f Ꮾ 1− p byLemma 3.3 The

4 Proof of Theorem 2.1

In the proof ofTheorem 2.1, part (B), we will use part of the following lemma, which is obtained by straightforward estimates involving (1.8) (see [2, Chapter 4])

Lemma 4.1 Let f ∈Ᏼ(D), whereDis an open subset of Cn , and suppose thatUis an orthonormal basis forCn Then for all z ∈ D ,

We are now ready to proveTheorem 2.1

Proof of Theorem 2.1 (A) Suppose that for some M ≥0,



1− | z |2 q

1−φ(z) 2 p ∇ φ j(z)  ≤ M ∀ z ∈ B n, ∈ {1, 2, , n } (4.2)

Ifz ∈ B nandF(z) =(1− | z |2)q |∇(Cφ f )(z) |, then we have that

F(z) =1− | z |2 q



 n

i =1

Di(f ◦ φ)(z) 2

≤1− | z |2 qn

i =1

Di(f ◦ φ)(z)

≤1− | z |2 q

n n



j =1

∇ f

φ(z) ∇ φj(z)

= n ∇ f

φ(z)1−φ(z) 2 p 

1− | z |2 q

1−φ(z) 2 p

n



j =1

∇ φ j(z)

≤ n sup

w ∈B n



∇ f (w)1− | w |2 p n

j =1



1− | z |2 q

1−φ(z) 2 p ∇ φ j(z)  ≤ n  f Ꮾp nM,

(4.3)

by inequality (4.2) It follows thatCφ f Ꮾq ≤(C + n2M)  f Ꮾp for every f ∈Ꮾp(Bn), thus completing the proof ofTheorem 2.1, part (A)

(B) We proceed by modifying the argument given in [4, pages 187-188] forn =1 For

a ∈ B n, define f a:Bn → Cto be the function that vanishes at 0 and is an antiderivative of

ψa:Bn → Cgiven byψa(t) =(1− at)¯ − p Letw ∈ B nandu ∈ ∂Bn DefineFw,u:Bn → Cby

Fw,u(z) = f  w,u 

Defineφ u:Bn → B1 byφ u(z) =  φ(z), u  Letu(1):= u, and choose u(2),u(3), , u(n) so thatU = { u(1),u(2),u(3), , u(n) }is an orthonormal basis forCn For allz ∈ B nand j ∈ {2, 3, , n }, we have that

D u(j) F w,u(z) =lim

λ →0

F w,u

z + λu(j)

− F w,u(z) λ

=lim

λ →0

f  w,u(1)

z + λu(j),u(1) − f  w,u(1)

z, u(1)

(4.5)

On the other hand, for everyz ∈ B n,

Du(1)Fw,u(z) =lim

λ →0

Fw,u(z + λu) − Fw,u(z)

λ

=lim

λ →0

f  w,u 

 z, u +λ

− f  w,u 

 z, u 

 z, u .

(4.6)

From (4.5) and (4.6), it follows that

∇UF w,u(z)  =  ψ  w,u 

 z, u   = 1−  w, u  z, u − p

We observe that the quantity above is bounded whenu is fixed This fact andLemma 4.1 together imply thatFw,u ∈Ꮾp

0(Bn) Also, we have

F w,u(0)= f  w,u 

0,u = f  w,u (0)=0. (4.8)

Furthermore, byLemma 4.1, we have that

sup

z ∈B n



1− | z |2 p ∇ Fw,u(z)  =sup

z ∈B n



1− | z |2 p ∇UFw,u(z)

=sup

z ∈B n



1− | z |2 p1−  w, u  z, u − p

Note that

1−  w, u  z, u − p

≤1− | z |− p ≤ 2p

1− | z |2 p (4.10)

It follows that the quantity (4.9) is less than or equal to 2p Hence,F w,u ∈Ꮾp(Bn) for everyw ∈ B nandu ∈ ∂Bn; moreover, the set

F w,u

Ꮾp:u ∈ ∂Bn,w ∈ B n

(4.11)

is bounded This fact and the hypothesis together imply that there existC and M ≥0 such that for everyw ∈ B nandu ∈ ∂Bn,

Fw,u ◦ φ

Ꮾq ≤ CFw,u

Therefore, we obtain that

sup

u ∈ ∂B ,z,w ∈B



∇f  w,u  ◦ φu

(z)1− | z |2 q

Now for eachj ∈ {1, 2, , n }, we have that

Dj

f  w,u  ◦ φu

(z) = f   w,u 

 φ(z), u D j

φ(z), u

=1−  w, u  φ(z), u − p D j

It follows that

∇f  w,u  ◦ φ u

(z) =1−  w, u  φ(z), u − p ∇ φ(z), u (4.15) Using (4.15), we can rewrite (4.13) as

sup

u ∈ ∂Bn,z,w ∈B n



1− | z |2 q

1−  w, u 

φ(z), u p ∇ φ(z), u CM. (4.16)

In particular, we have that

sup

u ∈ ∂Bn,z ∈B n



1− | z |2 q

1−  φ(z), u 2p ∇ φ(z), u CM, (4.17) from which the statement ofTheorem 2.1, part (B), follows 

By restricting the values ofu, one obtains various necessary conditions for

bounded-ness ofC φfrom part (B) ofTheorem 2.1 Two of such conditions are listed inCorollary 4.2 below We point out that the boundedness of quantity (4.18) below whenCφis bounded fromᏮp(Bn) toᏮq(Bn) is a result given by Zhou in [11]

Corollary 4.2 Let p, q > 0 IfCφ is a bounded operator fromᏮp(Bn ) (resp.,Ꮾp

0(Bn )) to

Ꮾq(Bn ), then there is an M ≥ 0 such that the following statements hold.

(i) For all z ∈ B n with φ(z) = 0,



1− | z |2 q

1−φ(z) 2 p

J φ(z) T · φ(z)

(ii) For all z ∈ B n and j ∈ {1, 2, , n } ,



1− | z |2 q

1−φ j(z) 2 p ∇ φj(z)  ≤ M. (4.19)

Proof Putting u : = φ(z)/ | φ(z) |inTheorem 2.1, part (B), one obtains that quantity (4.18)

is no larger than someM  ≥0 for allz ∈ Bn such thatφ(z) =0 Successively replacing

u ∈ ∂BninTheorem 2.1, part (B), by the typically ordered standard basis elementsej of

Cnfor j =1, 2, , n, we see that the left side of inequality (4.19) is no larger than some

be obtained, for example, from [12], the norm -equivalence portion requires additional work that includes the previous lemmas and. ..

 f R

Proof The proof of the first statement is standard and left to the reader Since functions

inᏸp(Bn)