COMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK

Transcription

1 COMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK MICHAEL STESSIN AND KEHE ZHU* ABSTRACT. Suppose ϕ is a holomorphic mapping from the polydisk D m into the polydisk D n, or from the polydisk D m into the unit ball B n, we consider the action of the associated composition operator C ϕ on Hardy and weighted Bergman spaces of D n or B n. We first find the optimal range spaces and then characterize compactness. As a special case, we show that if ϕ(z) = (ϕ 1 (z),, ϕ n (z)), z = (z 1,, z n ), is a holomorphic self-map of the polydisk D n, then C ϕ maps A p α(d n ) boundedly into A p β (Dn ), the weight β = n(2 + α) 2 is best possible, and the operator C ϕ : A p α(d n ) A p β (Dn ) the function [ n ] [ / n ] (1 z k 2 ) n (1 ϕ k (z) 2 ) k=1 k=1 tends to 0 as z approaches the full boundary of D n. This settles an outstanding problem concerning composition operators on the polydisk. 1. INTRODUCTION Let C be the complex plane and let C n be the n-dimensional complex Euclidean space. We use D = {z C : z < 1} to denote the open unit disk in C, and use D n = D D = {z = (z 1,, z n ) C n : z k < 1, 1 k n} to denote the polydisk in C n. Although we use the same symbol z to denote a point in C or a point in C n, the reader should have no trouble recognizing the meaning of z at every occurance Mathematics Subject Classification. 32A36 and 32A25. * Partially supported by a grant from the NSF. 1

2 2 MICHAEL STESSIN AND KEHE ZHU* Let H(D n ) denote the space of all holomorphic functions in D n. For any 0 < p < the Hardy space H p (D n ) consists of all f H(D n ) such that f p p = sup f(rζ) p dσ(ζ) <, 0<r<1 T n where T n = {ζ = (ζ 1,, ζ n ) C n : ζ k = 1, 1 k n} is the distinguished boundary of D n and dσ(ζ) = dζ 1 dζ n (2π) n is the normalized Haar measure on T n. It is well known that for every function f H p (D n ), the radial it f(ζ) = f(rζ) r 1 exists for almost every ζ T n. Furthermore, f p p = f(ζ) p dσ(ζ). T n See [16] for basic information about Hardy spaces of the polydisk. For 0 < p < and α > 1 the weighted Bergman space A p α(d n ) consists of all functions f H(D n ) such that f p p,α = f(z) p dv α (z) <, D n where dv α (z 1,, z n ) = da α (z 1 ) da α (z n ). Here da α (z) = (α + 1)(1 z 2 ) α da(z) is a weighted area measure on D with dx dy da(z) = π being normalized area measure on D. We say that a sequence {f k } in A p α or H p converges to 0 weakly if the sequence is bounded in norm and it converges to 0 uniformly on compact sets. We say that a bounded linear operator T from A p α or H p into some L p (X, dµ) is compact if T f k p dµ = 0 k X whenever {f k } converges to 0 weakly in A p α or H p.

3 COMPOSITION OPERATORS 3 Throughout the paper we fix two positive integers m and n. For any holomorphic mapping ϕ : D m D n, say ϕ(z 1,, z m ) = (ϕ 1 (z 1,, z m ),, ϕ n (z 1,, z m )), we consider the composition operator defined by C ϕ : H(D n ) H(D m ) C ϕ f = f ϕ, f H(D n ). The main results of our paper are Theorems A, B, and C below, together with their counterparts when ϕ maps the polydisk D m into the open unit ball B n in C n. Theorem A. For any p > 0 and α > 1 the composition operator C ϕ maps A p α(d n ) boundedly into A p β (Dm ), where β = n(2 + α) 2. Furthermore, the operator C ϕ : A p α(d n ) A p β (Dm ) n z D m k=1 (1 z 1 2 ) (1 z m 2 ) 1 ϕ k (z 1,, z m ) 2 = 0, where D m is the full topological boundary of D m. We also show that the constant β in Theorem A is the best possible. Theorem B. Suppose p > 0 and n > 1. Then the composition operator C ϕ maps H p (D n ) boundedly into A p n 2(D m ). Furthermore, the operator n z D m k=1 C ϕ : H p (D n ) A p n 2(D m ) (1 z 1 2 ) (1 z m 2 ) 1 ϕ k (z 1,, z m ) 2 = 0. Theorem C. Suppose p > 0 and ϕ : D m D is holomorphic, then the composition operator C ϕ maps H p (D) boundedly into H p (D m ). Furthermore, the operator C ϕ : H p (D) H p (D m ) z 1 N ϕ (z) 1 z 2 = 0, where N ϕ is a certain version of the Nevanlinna counting function of ϕ.

4 4 MICHAEL STESSIN AND KEHE ZHU* The case m = 1 in Theorems A and B was considered in [21]. Our proofs here are, in fact, reductions to this case. Composition operators acting on spaces of analytic functions in the unit disk have been well understood. In particular, every analytic self-map ϕ of the unit disk induces a bounded composition operator on H p (D) and on A p α(d). Furthermore, the operator C ϕ is compact on A p α(d) if and only if 1 z 2 z 1 1 ϕ(z) = 0; 2 and the operator C ϕ is compact on H p (D) if and only if z 1 N ϕ (z) 1 z 2 = 0. The monographs [5], [20] and [23] contain more information about composition operators on the unit disk. One-dimensional results for other spaces of holomorphic functions in the unit disk can be found in [1], [15] [3] and others. Composition operators acting on spaces of holomorphic functions of several variables have been known to behave in much more complicated ways, see [5],[9],[10],[12],[14],[22]. Although we have determined here the optiomal range spaces for the action of C ϕ on Hardy and Bergman spaces of D n when ϕ is a holomorphic self-maps of the polydisk D n, the corresponding problem for the unit ball is still open. The papers [12], [4], [13], [9] and [10] made partial progress on this problem. 2. PRELIMINARIES ON CARLESON MEASURES An important tool in the study of composition operators on holomorphic function spaces is the notion of Carleson measures. We will talk about Carleson measures for Hardy spaces as well as Carleson measures for Bergman spaces. Let Ω denote either the polydisk D n or the unit ball B n. A positive Borel measure on Ω is called a Carleson measure for the Hardy space H p (Ω) if there exists a positive constant C > 0 such that f(z) p dµ(z) C f p p Ω for all f H p (Ω), where f p is the norm of f in H p (Ω). We say that µ is a vanishing Carleson measure for the Hardy space H p (Ω) if f k (z) p dµ(z) = 0 k Ω whenever {f k } is a sequence in H p (Ω) that converges to 0 weakly.

5 COMPOSITION OPERATORS 5 Lemma 1. For a positive Borel measure µ on Ω the following two conditions are equivalent. (a) µ is a Carleson measure for H p (Ω) for some p > 0. (b) µ is a Carleson measure for H p (Ω) for every p > 0. An analogous characterization also holds for vanishing Carleson measures for H p (Ω). Proof. See [5] and [24] for the case Ω = B n. See [2] and [8] for the case Ω = D n. Similarly, a positive Borel measure µ on Ω is called a Carleson measure for the Bergman space A p α(ω) if there exists a constant C > 0 such that f(z) p dµ(z) C f(z) p dv α (z) Ω for all f A p α(ω). And µ is called a vanishing Carleson measure for A p α(ω) if f k (z) p dµ(z) = 0 k Ω whenever {f k } is a sequence in A p α(ω) that converges to 0 weakly. Lemma 2. Suppose α > 1 and µ is a positive Borel measure on Ω. Then the following two conditions are equivalent. (a) µ is a Carleson measure for A p α(ω) for some p > 0. (b) µ is a Carleson measure for A p α(ω) for every p > 0. Ω Proof. See [5] and [25]. 3. THE ACTION OF C ϕ ON BERGMAN SPACES In this section we study the action of C ϕ on Bergman spaces when ϕ is a holomorphic mapping from D m into D n. Theorem 3. For any p > 0 and α > 1 the operator C ϕ maps A p α(d n ) boundedly into A p β (Dm ), where β = n(2 + α) 2. Proof. It is easy to check that biholomorphic maps of D n induce bounded composition operators on Bergman and Hardy spaces of D n. Therefore, by an obvious change of variables if necessary, we may assume that ϕ(0) = 0. We also assume that m > 1; the case m = 1 was considered in [21]. Let S 2m 1 as usual stands for the unit sphere in C m and CP k for the k-dimensional complex projective space. Each l = [ζ 1,..., ζ m ] CP m 1 with m j=1 ζ j 2 = 1 intersects D m by a one-dimensional disk D l of radius

6 6 MICHAEL STESSIN AND KEHE ZHU* 1 r l =. We remark that 1 r max{ ζ j :j=1,...,m} l m. For l CP m 1 let us denote by τ l (w; α) the following function in the unit disk D m τ l (w; α) = (1 rl 2 ζ j w 2 ) α. We obviously have j=1 (1) τ l (w; α) (1 w 2 ) α It is easily seen that the A p α-norm on D m is equivalent to (2) ( ( = CP m 1 S 2m 1 ( ) ) 1/p f(r l wζ) p τ l (w; α)da(w) ω m 1 (l) D ( ) 1/p f(r l wζ) p τ [ζ] (w; α)da(w) dσ(ζ)), D where ω m 1 is the Fubini-Study volume form on CP m 1, dσ is the measure on S 2m 1 invarian under the action of the unitary group and l = [ζ 1,..., ζ m ]. For ζ S 2m 1 we denote by ψ ζ : D D m the mapping given by ψ ζ (w) = r l wζ, 1 where as above r l =. Then g max{ ζ j :j=1,...,m} ζ = ϕ ψ ζ maps the unit disk D into D n and sends the origin in C to the origin in C n. By Theorem 4 of [21], there exists a constant C > 0, independent of ζ, such that for every function f A p α(d n ) we have f(g ζ (w)) p da β (w) C f(z) p dv α (z). D D n Since dσ is a probability measure, the result follows from (1) and (2). Theorem 4. Suppose p > 0, α > 1, and β = n(2 + α) 2. Then the operator n (3) z D m k=1 C ϕ : A p α(d n ) A p β (Dm ) (1 z 1 2 ) (1 z m 2 ) 1 ϕ k (z 1,, z m ) 2 = 0.

7 COMPOSITION OPERATORS 7 Proof. Fix α, β, and ϕ. We define the pull-back measure µ in a standard way: for any Borel set E in D n µ(e) = v β (ϕ 1 (E)). Then µ is a positive Borel measure on D n and it follows from standard measure theory that we have the following change of variables formula: f(ϕ) p dv β = f p dµ. D m D n Therefore, the composition operator C ϕ : A p α(d n ) A p β (Dm ) the measure µ is a vanishing Carleson measure for the Bergman space A p α(d n ). According to Lemma 2, the condition that µ is a vanishing Carleson measure for A p α(d n ) is independent of p. Therefore, we may assume p = 2 without loss of generality. Also, any biholomorphic mapping of D n induces an invertible composition operator on A p α(b n ). The same goes for Hardy spaces as well. Therefore, we may assume ϕ(0) = 0 without loss of generality. For the rest of the proof we assume that p = 2, ϕ(0) = 0, and m > 1. The case m = 1 was considered in [21]. We can consider the adjoint operator C ϕ : A 2 β(d m ) A 2 α(d n ). It is easy to see that, for the normalized reproducing kernels of A 2 β (Dm ), m (1 z k 2 ) (2+β)/2 k z (w) = (1 w k z k ), 2+β we have D C k z 2 da α = k=1 [ n k=1 ] 2+α (1 z 1 2 ) (1 z m 2 ). 1 ϕ k (z 1,, z m ) 2 As z = (z 1,, z m ) approaches the full topological boundary D m of D m, the functions k z converges to 0 weakly in A 2 β (Dm ). If C ϕ : A 2 α(d n ) A 2 β (Dm ) is compact, then Cϕ : A 2 β (Dm ) A 2 α(d n ) is compact, and so condition (3) must hold. Conversely, assume we have condition (3). We will use the notation introduced in the proof of Theorem 3. Let us remark that for every ζ S 2m 1 (3) implies n 1 w 2 (4) w 1 1 g ζ,j (w) = 0, 2 j=1

8 8 MICHAEL STESSIN AND KEHE ZHU* where g ζ = (g ζ,1,..., g ζ,n ) = (ϕ 1 (r l wζ),..., ϕ n (r l wζ)). Indeed, if ζ j < max{ ζ k : 1 k n} then 1 w 2 < 1 rl 2 wζ j 2, and therefore the product in (4) is dominated by the product in (3) computed at ψ ζ (w) = (r l wζ 1,..., r l wζ m ). Since ψ ζ (w) approaches the topological boundary of D m as w 1, (4) follows. Let {f k } be a sequence in A 2 α(d n ) that converges to 0 weakly and consider the functions F k (ζ) = f k (g ζ (w) 2 da β (w). D As it was mentioned in the proof of Theorem 3 f k ϕ 2 2,β= f k ϕ 2 dv β D ( m ) f k (g ζ (w)) 2 τ [ζ] (w; β)da(w) S 2m 1 D dσ(ζ) F k (ζ)dσ(ζ), S 2m 1 where the last inequality follows from (1). Since g ζ (0) = 0 for all ζ S 2m 1, by Theorem 4 of [21], there exists a constant C > 0, independent of k and independent of ζ, such that F k (ζ) C. On the other hand, it follows from condition (4) and Theorem 4 of [21] that F k(ζ) = 0. k This together with dominated convergence gives k D n f k (ϕ) 2 dv β = 0. This shows that C ϕ : A 2 α(d n ) A 2 β (Dm ) is compact whenever condition (3) is satisfied. Remark Our proofs of Theorems 3 and 4 show that similar results hold for holomorphic mappings of an arbitrary bounded symmetric domain in C m into D n. We conclude the section with an example which shows that the constant β in Theorem 3 is the best possible. Let ϕ : D m D n be defined by ϕ(z 1,..., z m ) = (z 1, z 1,..., z 1 ). Denote by M the subspace of functions in A 2 α(d n ) vanishing on the diagonal, M = {f A 2 α(d n ) : f(z, z,...z) = 0, z D}. It follows from [21] that the restriction of functions in A 2 α(d n ) to the diagonal is a bounded invertible operator from A 2 α(d n ) M onto A 2 β (D), where

9 COMPOSITION OPERATORS 9 β = n(α + 2) 2. Define ψ : D m D by ψ(z 1,..., z m ) = z 1. It is easy to check that for every γ > 1 the operator C ψ : A 2 γ(d) A 2 γ(d m ) is an isometry. If we denote by the same letter the diagonal mapping of the unit disk D into D n, then C ϕ = C ψ C. This shows that the constant β = n(α + 2) 2 is the best possible in Theorem THE ACTION OF C ϕ ON HARDY SPACES In this section we study the action of C ϕ on Hardy spaces when ϕ is a holomorphic map from D m into D n. Theorem 5. Suppose p > 0 and ϕ is a holomorphic map from D m into D n, where m 1 and n > 1. Then the composition operator maps H p (D n ) boundedly into A p n 2(D m ). Furthermore, the operator n z D m k=1 C ϕ : H p (D n ) A p n 2(D m ) (1 z 1 2 ) (1 z m 2 ) (1 ϕ k (z 1,, z m ) 2 = 0. Proof. The proof is similar to that of Theorems 3 and 4. We leave the details to the interested reader. When n = 1, it is clear that the space A p n 2(D m ) must be replaced by the Hardy space H p (D m ). To characterize compactness, we need to introduce a version of the Nevannlina counting function for holomorphic functions on D m. Recall that if f is an analytic function on the unit disk, the Nevanlinna counting function N f is defined by (5) N f (z) = { log 1 w : f(w) = z }, z f(0). The well-known Littlewood s inequality states that if f maps D into D and z f(0), then 1 f(0)z N f (z) log z f(0) ;

10 10 MICHAEL STESSIN AND KEHE ZHU* see [20], page 187, or [5], page 33 for details. In particular, if f(0) = 0, then N f (z) log 1 z, z 0. If f is a holomorphic function from D m into D, where m > 1, we consider slice functions of the form where f ζ (w) = f(wζ), w D, ζ = (1, ζ 2,, ζ m ) T m. Each f ζ is then an analytic self-map of the unit disk. We now define the Nevanlinna counting function of f as follows. (6) N f (z) = N fζ (z) dσ(ζ 2,, ζ m ), z D. T m 1 It follows directly from the one-dimensional Littlewood inequality that 1 f(0)z (7) N f (z) log z f(0), z D {f(0)}. We can now settle the case n = 1. Theorem 6. Suppose p > 0 and ϕ : D m D is holomorphic. Then the composition operator C ϕ maps H p (D) boundedly into H p (D m ). Moreover, the operator C ϕ : H p (D) H p (D m ) N ϕ (z) (8) z 1 1 z = 0. 2 Proof. Once again, we may assume that p = 2 and ϕ(0) = 0. For any f H 2 (D) we first use Fubini s theorem to write f(ϕ) f(0) 2 dσ = dσ(ζ) f(ϕ ζ (η) f(0) 2 dη T m T m 1 T 2π. We then apply the Littlewood-Paley formula (see [6], page 236) to the inner integral above. The result is f(ϕ) f(0) 2 dσ = dσ(ζ) f (ϕ ζ (w) 2 ϕ ζ(w) 2 log 1 T m T m 1 D w da(w).

11 COMPOSITION OPERATORS 11 Applying a well-known change of variables formula (see Proposition of [23] for example) to the inner integral above, we obtain f(ϕ) f(0) 2 dσ = dσ(ζ) f (w) 2 N ϕζ (w) da(w). T m T m 1 D By Fubini s theorem again, we have (9) f(ϕ) f(0) 2 dσ = f (w) 2 N ϕ (w) da(w). T m D It follows from (7) and the Littlewood-Paley identity again that f(ϕ) f(0) H 2 (D m ) f f(0) H 2 (D) for all f H 2 (D). This proves that the composition operator C ϕ is a contraction from H 2 (D) into H 2 (D m ). By equation (9) and the Littlewood-Paley identity, the operator the measure C ϕ : H 2 (D) H 2 (D m ) N ϕ (z)da(z) is a vanishing Carleson measure for the Bergman space H(D) L 2 (D, log(1/ z ) da(z)). Since log(1/ z ) is comparable to 1 z 2 as z 1, we conclude that the compactness of the operator C ϕ : H 2 (D) H 2 (D m ) is equivalent to the condition that the measure dµ(z) = N ϕ (z)da(z) is a vanishing Carleson measure for the Bergman space A 2 1(D). It follows from Lemma 1 of [11] and a standard argument (see Exercise 4 on page 124 of [23]) that the above measure µ is a vanishing Carleson measure for A 2 1(D) if and only if the condition in (8) holds. This completes the proof of the theorem. 5. THE CASE WHEN ϕ MAPS D m INTO B n In this section we consider the open unit ball in C n, Here z = (z 1,, z n ) and B n = {z C n : z < 1}. z = z z n 2. For the above z and w = (w 1,, w n ) we also write z, w = z 1 w z n w n.

12 12 MICHAEL STESSIN AND KEHE ZHU* Let H(B n ) denote the space of all holomorphic functions in B n. Throughout this section we fix a holomorphic map ϕ from the polydisk D m into the unit ball B n and consider the composition operator C ϕ : H(B n ) H(D m ). More specifically, we are going to study the action of C ϕ on Hardy and weighted Bergman spaces of the unit ball B n. Recall that for p > 0 the Hardy space H p (B n ) consists of all functions f H(B n ) such that f p p = sup f(rζ) p dσ(ζ <, 0<r<1 S n where dσ is the normalized Lebesgue measure on the unit sphere S n = B n = {ζ C n : ζ = 1}. It is well known that for every function f H p (B n ), the radial it f(ζ) = f(rζ) r 1 exists for almost all ζ S n. Moreover, f p p = f(ζ) p dσ(ζ). S n See [17] and [24] for more information about Hardy spaces of the unit ball. Similarly, for p > 0 and α > 1 we consider the weighted Bergman space A p α(b n ) consisting of functions f H(B n ) such that f p p,α = f(z) p dv α (z) <, B n where dv α (z) = c α (1 z 2 ) α dv(z). Here dv is Lebesgue measure on B n and c α is normalizing constant such that v α (B n ) = 1. Theorem 7. For any p > 0 and α > 1 the composition operator C ϕ maps A p α(b n ) boundedly into A p β (Dm ), where β = n + α 1. Furthermore, the operator C ϕ : A p α(b n ) A p β (Dn ) (1 z 1 2 ) (1 z m 2 ) = 0. z D m 1 ϕ(z 1,, z m ) 2 Proof. This is similar to the proof of Theorems 3 and 4.

13 COMPOSITION OPERATORS 13 Theorem 8. Suppose p > 0 and n > 1. Then the composition operator maps H p (B n ) boundedly into A p β (Dm ), where β = n 2. Moreover, the operator C ϕ : H p (B n ) A p β (Dm ) (1 z 1 2 ) (1 z m 2 ) = 0. z D m 1 ϕ(z 1,, z m ) 2 Proof. Once again, this is similar to the proof of Theorems 3 and FURTHER RESULTS AND REMARKS It was shown in [9] that if ϕ : B n D is holomorphic, then the composition operator C ϕ maps H p (D) boundedly into H p (B n ), and the range space H p (B n ) is best possible. Using the same arguments from the proof of Theorem 6, we can prove the following little oh version. Theorem 9. If p > 0 and ϕ : B n D is holomorphic, then the operator C ϕ : H p (D) H p (B n ) z 1 N ϕ (z) 1 z 2 = 0. Here the Nevanlinna counting function of ϕ : B n D is defined as (10) N ϕ (z) = N ϕζ (z) dσ(ζ), z D, S n where for ζ S n, ϕ ζ (z) = ϕ(zζ), z D, is the standard slice function induced by ϕ. We mention a general definition of the Nevanlinna counting function used in the value distribution theory for functions of several complex variables, and point out the relationship between this and our earlier definitions. Our references for this discussion are [7] and [18]. Let A be a k-dimensional analytic set in C n and 0 A. Define r dt (11) N(A, r) = ω k, 0 t A B t where r > 0, B t = tb n is a ball of radius t, and ω is the Fubini-Study form, namely, ω = dd c log z 2.

14 14 MICHAEL STESSIN AND KEHE ZHU* It is well known that (12) N(A, r) = à N(A l, r)ω k, where à is the projectivazation of A and N(A l, r) is the counting function of the divisor A l, in the one-dimesnsional complex line l CP n 1 defined in a regular way as the logarithmic average given by (5). If f is a holomorphic function from C n to C and if z f(0), we define (13) N f (r, z) = N(A(f, z), r) = N(A(f, z) l, r)ω n 1, where Ã(f,z) A(f, z) = {w B n : f(w) = z} is an (n 1)-dimensional analytic set in C n. If f is a holomorphic function from B n into D, the Nevanlinna counting function of f can be defined as N f (z) = N f (1, z). Since ω n 1 is the volume form for the Fubini-Study metric, we can express this version of the counting function as an integral mean of sectional one-dimensional counting functions over the unit sphere. Therefore, this definition of the Nevanlinna counting function coincides with our early definition given in (10) in the case of the unit ball. In the case of the unit polydisk, the measure used to define the Hardy spaces H p lives on the torus T n which has measure 0 in the full boundary of D n. In this case, the above definition based on projectivization is not appropriate for the study of composition operators on Hardy spaces, and we must resort to the direct integral form given in (6). We also mention that our results on compactness of composition operators on Hardy spaces can be strengthened to yield Shapiro s formula for the essential norm. Theorem 10. If ϕ is a holomorphic function from Ω into D, where Ω is either the polydisk or the unit ball, then the essential norm of the operator is given by the formula C ϕ : H 2 (D) H 2 (Ω) C ϕ e = sup z 1 N ϕ (z) log 1. z Proof. This follows from our proof of Theorem 6 and Shapiro s arguments in [19].

15 COMPOSITION OPERATORS 15 Similar to the one-dimensional case, inner functions in either D n or B n can be described in terms of Nevanlinna counting functions. The following result can be obtained as an adaptation of the corresponding theorem in [19] to the case of functions of several variables. Theorem 11. Let Ω be either D n or B n. A holomorphic mapping ϕ : Ω D is inner if and only if 1 ϕ(0)z N ϕ (z) = log z ϕ(0) for almost all z in D \ 0. Finally, as was pointed out in the introduction, we mention that it is still an open problem to determine the optimal range space for the action of composition operators C ϕ on a Hardy or weighted Bergman space when ϕ is a holomorphic self-map of the unit ball. We consider this a fundamental issue in the theory of composition operators. REFERENCES [1] P. Bourdon, J. Cima and A. Matheson, Compact composition operators on BMOA, Pacific J. Math. 351 (1999), [2] S.-Y. Chang, Carleson measure on the bi-disc, Ann. Math. 109 (1979), [3] J. Cima and A. Matheson, Weakly compact composition operators on VMO, Rocky Mountain J. Math. 2002, no. 3, [4] J. Cima and P. Mercer, Composition operators between Bergman spaces on convex domains, J. Operator Theory 33 (1995), [5] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, [6] J. Garnett, Bounded Analytic Functions, Academic Press, New York, [7] Ph. Griffiths and J. King, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), [8] F. Jafari, Carleson measures in Hardy and weighted Bergman spaces of polydiscs, Proc. Amer. Math. Soc. 112 (1991), [9] H. Koo and W. Smith, Composition operators between spaces of functions of several variables, preprint, [10] H. Koo and W. Smith, Composition operators induced by smooth self-mappings of the unit ball in C n, preprint, [11] D. Lueking and K. Zhu, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (1992), [12] B. MacCluer, Compact composition operators on H p (B n ), Michigan J. Math. 32 (1985), [13] B. MacCluer and P. Mercer, Composition operators between Hardy and weighted Bergman spaces on convex domains in C N, Proc. Amer. Math. Soc. 123 (1995), [14] B. MacCluer and J. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canadian J. Math. 38 (1986),

16 16 MICHAEL STESSIN AND KEHE ZHU* [15] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), [16] W. Rudin, Function Theory in Polydiscs, Benjamin, New York, [17] W. Rudin, Function Theory in the Unit Ball of C n, Springer, New York, [18] B.V. Shabat, Distribution of Values of Holomorphic Mappings, American Mathematical Society, Providence, [19] J. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), [20] J. Shapiro, Composition operators and classical function theory, Spriger-Verlag, New York, [21] M. Stessin and K. Zhu, Composition operators on embedded disks, preprint, [22] W.R.Wogen, The smooth mapping which preserve the Hardy space H 2 B, Operator Theory: Advances Appl. 35 (1988), [23] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, [24] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, [25] K. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), DEPARTMENT OF MATHEMATICS, SUNY, ALBANY, NY 12222, USA address: stessin@math.albany.edu, kzhu@math.albany.edu