Bergman spaces under maps of monomial type
aa r X i v : . [ m a t h . C V ] A p r BERGMAN SPACES UNDER MAPS OF MONOMIAL TYPE
ALEXANDER NAGEL AND MALABIKA PRAMANIK
Abstract.
For appropriate domains Ω , Ω we consider mappings Φ A : Ω → Ω of mono-mial type. We obtain an orthogonal decomposition of the Bergman space A (Ω ) intofinitely many closed subspaces indexed by characters of a finite Abelian group associated tothe mapping Φ A . We then show that each subspace is isomorphic to a weighted Bergmanspace on Ω . This leads to a formula for the Bergman kernel on Ω as a sum of weightedBergman kernels on Ω . Introduction
Let Ω , Ω ⊆ C n be open sets and let Φ A : Ω → Ω be a surjective holomorphic mapping ofmonomial type. In this paper we obtain a decomposition of weighted Bergman spaces on Ω associated to the mapping Φ A as well as relationships between the weighted Bergman kernelsof Ω and Ω . In this Introduction we begin by recalling the definitions of these concepts,and then state our main results.1.1. Bergman projections and kernels.
Let Ω be an open set in C n , n ≥ dV . Given a continuous weight function ω : Ω → (0 , ∞ ), denote by L (Ω , ω ) theHilbert space of (equivalence classes of) Lebesgue-measurable functions on Ω that are square-integrable with respect to the measure ω ( z ) dV ( z ). The closed subspace A (Ω , ω ) ⊆ L (Ω , ω )consisting of functions that are holomorphic on Ω is the corresponding weighted Bergmanspace . The orthogonal projection P ω Ω : L (Ω , ω ) → A (Ω , ω ) is the weighted Bergmanprojection . For f ∈ L (Ω , ω ) and z ∈ Ω the projection P ω Ω [ f ] is given by(1.1) P ω Ω [ f ]( z ) = Z Ω B Ω ( z , w ; ω ) f ( w ) ω ( w ) dV ( w ) . The integration kernel B Ω ( · , · ; ω ) : Ω × Ω → C is the weighted Bergman kernel . If { ψ j : j ≥ } is any complete orthonormal basis for A (Ω , ω ) then(1.2) B Ω ( z , w ; ω ) = ∞ X j =1 ψ j ( z ) ψ j ( w ) , Date : April 9, 2020.2010
Mathematics Subject Classification.
ALEXANDER NAGEL AND MALABIKA PRAMANIK where the series converges absolutely and uniformly on compact subsets of Ω × Ω. The valueof the Bergman kernel when z = w is the solution of an extremal problem:(1.3) B Ω ( z , z ; ω ) = sup n | h ( z ) | : h ∈ A (Ω , ω ) and || h || ,ω ≤ o , where || · || ,ω denotes the norm in L (Ω , ω ). It follows that(1.4) Ω ⊆ Ω = ⇒ B Ω ( z , z ; ω ) ≤ B Ω ( z , z ; ω )for all z ∈ Ω . See [14] for the basic facts about the Bergman kernel and projection. Weoften omit ω when ω ≡
1, in which case A (Ω) and B Ω are referred to respectively as the standard Bergman space and standard Bergman kernel of Ω.In this paper we are concerned with one aspect of the following general question: If Ω , Ω ⊆ C n are open and Φ : Ω → Ω is a surjective holomorphicmapping, how are weighted Bergman spaces on Ω related to those on Ω ? When ω ≡ → Ω is biholomorphic, the answer to the above question is well-known. Specifically, we have that R Ω f ( w ) dV ( w ) = R Ω f (cid:0) Φ( z ) (cid:1) | det J Φ( z ) | dV ( z ) for every f ∈ L (Ω ) where J Φ is the complex Jacobian matrix of Φ. Since det J Φ( z ) is nonvanishingand holomorphic, it follows that P Ω (cid:0)(cid:2) det J Φ (cid:3) · (cid:2) f ◦ Φ (cid:3)(cid:1) = (cid:2) det J Φ (cid:3) · (cid:2) P Ω f ◦ Φ (cid:3) , (1.5) Z Ω B Ω ( z , u ) f (cid:0) Φ( u ) (cid:1) det J Φ( u ) d u = det J Φ( z ) Z Ω B Ω (Φ( z ) , v ) f ( v ) d v . (1.6)Since v = Φ( w ) for a unique w ∈ Ω , it follows from (1.6) that(1.7) B Ω ( z , w ) = (cid:2) det J Φ( z ) (cid:3)(cid:2) B Ω (Φ( z ) , Φ( w )) (cid:3)(cid:2) det J Φ( w ) (cid:3) . For details see for example [14], Proposition 1.4.12, page 52.There has been considerable previous work which has resulted in generalizations of theformulas in (1.5), (1.6), and (1.7). In [2] and [3] Steven Bell generalized these results byshowing that equations (1.5) and (1.6) continue to hold if each Ω i is a bounded domain andΦ : Ω → Ω is a proper holomorphic mapping ( i.e. Φ − ( K ) ⊂ Ω is compact for eachcompact subset K ⊂ Ω ). Proper mappings are finite branched coverings. If Φ is an m -foldbranched covering let Ψ , . . . , Ψ m denote the m local inverses of Φ. In this case the identity(1.7) is replaced by the formula(1.8) m X i =1 B Ω (cid:0) z , Ψ i ( v ) (cid:1) det J Ψ i ( v ) = det J Φ( z ) B Ω (Φ( z ) , v ) , z ∈ Ω , v ∈ Ω . In another direction, Siqi Fu [11], using the Poisson summation formula, established similarformulas in certain cases of infinite covering maps from tube domains to Reinhardt domains.In Section 6.2 below we illustrate our results for the domainΩ = (cid:8) ( z , z ) ∈ C : 0 < | z | p < | z | q < (cid:9) , with p and q positive integers, which is a generalization of the Hartogs triangle Ω = (cid:8) ( z , z ) ∈ C : 0 < | z | < | z | < (cid:9) . We are grateful to Jeff McNeal and Debraj Chakrabarti for bringing our attention to recent work [9, 10, 4, 6, 5] related to this example. For example,Edholm and McNeal have studied the Bergman kernel for domains related to the “fat Hartogstriangle” Ω γ = (cid:8) ( z , z ) ∈ C : | z | γ < | z | < (cid:9) where γ > . In [9] they show that if γ is a positive integer then the Bergman kernel is bounded on L r (Ω γ )if and only if r ∈ ((2 γ + 2) / ( γ + 2) , (2 γ + 2) /γ ). In [10] they show that if γ is irrationalthen the Bergman kernel is bounded on L r (Ω γ ) only when r = 2. Chakrabarti, Edholm,and McNeal [4] study duality and approximation issues on these and more general boundedReinhardt domains. Chakrabarti, Konkel, Mainkar, and Miller [6] calculate the Bergmankernel for more general domains Ω k = (cid:8) ( z , . . . , z n ) ∈ C n : | z | k · · · | z n | k n < (cid:9) wherethe exponents k j are (possibly negative) integers. In these papers the authors decomposethe Bergman kernel for a domain into “sub-Bergman kernels” and these are related to thedecompositions we make. Chakrabarti and Edholm [5] study the relationship between the L r -mapping properties of the Bergman kernels on two domains one of which is the quotientof the other, under the action of a finite group of biholomorphic automorphisms.There is an extensive literature addressing the fundamental role of the Bergman projectionand its kernel in complex function theory. This paper is one in a series of work by the authors[16, 17, 18, 19, 20] dealing with estimates for the Bergman kernel in various domains. Ourresults in this paper are motivated by our interest in estimates for complex monomial balls ,discussed in Section 6 below. Our results and objectives are of a different nature than in theearlier work of Bell [2], [3], and are based on the algebraic structure of the mapping Φ A .1.2. Monomial mappings.
In this paper, we consider mappings Φ A and functions F b of monomial type . If b = ( b , · · · , b n ) ∈ Z n , if A = { a j,k } is a non-singular n × n matrix withinteger entries, and if z = ( z , · · · , z n ) ∈ C n , we set(1.9) F b ( z ) := z b z b · · · z b n n , Φ A ( z ) := ( F a ( z ) , · · · , F a n ( z )) , where a j = ( a j, , · · · , a j,n ) denotes the j th row vector of A . If all the entries of the matrix A are non-negative integers, then Φ A is holomorphic on all of C n . If A has at least onenegative entry, then Φ A is holomorphic at z if and only if z ∈ C n \ H A where H A = n [ k =1 n z ∈ C n : z k = 0 , and there exists 1 ≤ j ≤ n such that a j,k < o . In particular, for any non-singular n × n matrix with arbitrary integer entries, the mappingΦ A is always holomorphic on C n ∗ := C n \ H , where H is the union of coordinate hyperplanes:(1.10) H := n z = ( z , · · · , z n ) ∈ C n : z z · · · z n = 0 o . For any integer-valued matrix A , the Jacobian of Φ A can be singular only at points in H .Basic properties of monomial type functions and mappings are presented in Section 2. ALEXANDER NAGEL AND MALABIKA PRAMANIK
The groups G A and b G A . We now introduce algebraic objects associated with mono-mial mappings. In this paper, all vectors in R n are considered row vectors, i.e., 1 × n matrices.Matrix multiplication is denoted by “ · ”. M n ( Z ) and M n ( R ) denote the spaces of n × n ma-trices with integer and real entries respectively. The transpose and inverse of a matrix M are denoted by M t and M − . The notation h· , ·i stands for the real inner product, i.e. if z = ( z , · · · , z n ) , w = ( w , · · · , w n ) ∈ C n then h z , w i := P nj =1 z j w j . Let e , . . . , e n denotethe standard basis elements of R n . Definition 1.1. (a) If A ∈ M n ( Z ) then C ( A ) := (cid:8) m · A t : m ∈ Z n (cid:9) denotes the Z -submodule of Z n generatedby the columns of A , and C ( A t ) := (cid:8) m · A : m ∈ Z n (cid:9) denotes the Z -submodule of Z n generated by the rows of A .(b) G A := Z n / C ( A ) and G A t := Z n / C ( A t ) denote the quotient groups; if m ∈ Z n then [ m ] denotes its equivalence class in G A and [[ m ]] denotes its equivalence class in G A t .(c) If [ m ] ∈ G A set (1.11) ξ j ([ m ]) := exp (cid:2) πi h m , e j · A − i (cid:3) and ξ ([ m ]) := (cid:0) ξ ([ m ]) , · · · , ξ n ([ m ]) (cid:1) . (d) If v = ( v , . . . , v n ) , w = ( w , . . . , w n ) ∈ C n then v ⊗ w = ( v w , . . . , v n w n ) denotes theHadamard vector product.(e) b G A denote the group of characters of G A , i.e. the set of group homomorphisms from G A to the unit circle T = (cid:8) z ∈ C : | z | = 1 (cid:9) , equipped with point-wise multiplication. Anelement of b G A is thus a map χ : G A → T such that χ ([ m ] + [ n ]) = χ ([ m ]) χ ([ n ]) .(f ) If b ∈ Z n the function χ b : G A → T given by χ b ([ m ]) := exp (cid:2) πi h m , b · A − i (cid:3) is acharacter of G A . In Section 2 we study the algebraic structure of C ( A ), G A , and b G A . We see that G A and b G A are finite abelian groups of order det( A ). We also show that the binary operation(1.12) ([ m ] , z ) → ξ ([ m ]) ⊗ z = (cid:16) e πi h m , e · A − i z , . . . , e πi h m , e n · A − i z n (cid:17) is a a faithful action of G A on C n ∗ .Note that if b , b ∈ Z n then the charcters χ b , χ b defined in part (f) of Definition 1.1 areequal if and only if b − b = n · A for some n ∈ Z n ; i.e. if and only if b − b ∈ C ( A t ).The correspondence [[ b ]] χ b therefore generates a mapping ϕ : G A t → b G A (1.13) ϕ ([[ b ]])([ m ]) := χ b ([ m ]) = exp (cid:2) πi h m , b · A − i (cid:3) . Lemma 3.5 below shows that ϕ defined in (1.13) is a group isomorphism. Thus the charactersof G A are parameterized by elements of the group G A t .1.4. Invariant domains and orthogonal decompositions.
An open set Ω ⊆ C n is saidto be invariant under the action of G A defined in equation (1.12) if for every z ∈ Ω ∗ = Ω ∩ C n ∗ and every m ∈ Z n , the point ξξξ ([ m ]) ⊗ z is also in Ω ∗ . A function f : Ω → C is said to be invariant under this group action if(1.14) f (cid:0) ξ ([ m ]) ⊗ z (cid:1) = f ( z ) for all z ∈ Ω ∗ and for all [ m ] ∈ G A . Suppose that Ω ⊆ C n is invariant under the action of G A . For each χ ∈ b G A and anyfunction f : Ω → C , define(1.15) Π χ [ f ]( z ) := 1 G A ) X [ m ] ∈ G A χ ([ m ]) f (cid:0) ξ ([ m ] ⊗ z ) (cid:1) for z ∈ Ω . In Section 4 we will show the following. • Each Π χ is a projection: Π χ = Π χ and f ( z ) = P χ ∈ b G A Π χ [ f ]( z ) for z ∈ Ω . • If b ∈ Z n and χ b is the character given in part (f) of Definition 1.1 then for all z ∈ Ω Π χ b [ f ] (cid:0) ξ ([ m ]) ⊗ z (cid:1) = χ b ([ m ]) − Π χ b [ f ]( z ) , F b ( ξ ([ m ]) ⊗ z ) = χ b ([ m ]) F b ( z ) . • The function Π χ b [ f ]( · ) F b ( · ) is invariant under the action of G A :(1.16) Π χ b [ f ] (cid:0) ξ ([ m ]) ⊗ z (cid:1) F b ( ξ ([ m ]) ⊗ z ) = Π χ b [ f ]( z ) F b ( z )These observations lead to the following orthogonal decompositions of the spaces L (Ω ; ω )and A (Ω ; ω ), parameterized by the characters of G A . Theorem 1.2.
Let A ∈ M n ( Z ) be non-singular, possibly with negative entries. Let Ω ⊂ C n be an open set and let ω : Ω → (0 , ∞ ) be continuous, both invariant under the action of G A . For each character χ ∈ b G A , let Π χ be the projection operator defined in equation (1.15) .Then the following conclusions hold.(a) The mapping Π χ acting on L (Ω ; ω ) or A (Ω ; ω ) is an orthogonal projection.(b) Denote by L χ (Ω ; ω ) := Π χ (cid:2) L (Ω ; ω ) (cid:3) and A χ (Ω ; ω ) := Π χ (cid:2) A (Ω ; ω ) (cid:3) the rangesof the projection Π χ . Then the following are direct sum decompositions into mutuallyorthogonal subspaces: L (Ω ; ω ) = M χ ∈ b G A L χ (Ω ; ω ) , A (Ω ; ω ) = M χ ∈ b G A A χ (Ω ; ω ) . The proof of Theorem 1.2 is given in Section 5.1.5.
Isomorphisms between Bergman spaces.
Let Ω , Ω ⊆ C n be open sets, let A ∈ M n ( Z ) be non-singular, and recall that H = (cid:8) ( z , . . . , z n ) ∈ C n : z j = 0 for some 1 ≤ j ≤ n (cid:9) .Since A is non-singular, it is easy to check that z ∈ H if and only if Φ A ( z ) ∈ H . We shallsuppose(1.17) Ω = Φ A (Ω ) and Φ A : Ω → Ω is holomorphicthough not necessarily biholomorphic. In particular this means that if Ω ∩ H = ∅ , then forevery j such that Ω ∩ { z ∈ C n : z j = 0 } 6 = ∅ , the j th column of A has only non-negativeinteger entries. On the other hand if Ω ∩ H = ∅ then any non-singular A ∈ M n ( Z ) generatesa holomorphic map Φ A : Ω → Ω , and in this case Ω ∩ H = ∅ as well. For i = 1 , ∗ i := Ω i \ H . It follows from (1.17) that Ω ∗ = Φ A (Ω ∗ ). We shall assume that Ω ∗ is invariantunder the action of G A defined in part (c) of Definition 1.1. Thus we assume(1.18) ξξξ [ m ] ⊗ z ∈ Ω ∗ whenever z ∈ Ω ∗ ; or equivalently Ω ∗ = Φ − A (Ω ∗ ) . ALEXANDER NAGEL AND MALABIKA PRAMANIK
Let ω j : Ω j → (0 , ∞ ), j = 1 ,
2, be positive, continuous weight functions such that(1.19) ω ( z ) = ω (cid:0) Φ A ( z ) (cid:1) , z ∈ Ω . In particular, this implies that the function ω is invariant under the group action of G A .The measure dµ = ω dV is then also invariant under this action, in a sense that will bemade precise in equation (4.2) in Section 4. Under these conditions there is an isomorphismbetween A (Ω ∗ , ω ) and a direct sum of weighted Bergman spaces on Ω ∗ . Theorem 1.3.
Let Ω and Ω be open sets in C n satisfying assumptions (1.17) and (1.18) .Let ω and ω be continuous weight functions satisfying (1.19) . Let b ∈ Z n and let χ = ϕ ([[ b ]]) be the character of G A defined (1.13) so that χ ([ m ]) = exp (cid:2) πi h m , b · A − i (cid:3) . Let Π χ be themapping defined in (1.15) .(a) If f : Ω ∗ → C is any function, there exists a unique function T b [ f ] : Ω ∗ → C so that T b [ f ] (cid:0) Φ A ( z ) (cid:1) = Π χ [ f ]( z ) F b ( z ) for all z ∈ Ω ∗ .(b) If g : Ω ∗ → C is any function and if f ( z ) = g ◦ Φ A ( z ) F − b ( z ) , then Π χ [ f ] = f and T b [ f ] = g .(c) If f is holomorphic on Ω ∗ then T b [ f ] is holomorphic on Ω ∗ .(d) Let c = c ( b ) := ( − b ) · A − − and let η b ( w ) := det( A ) − | F c ( w ) | ω ( w ) . Then forevery f ∈ L (Ω ; ω )(1.20) Z Ω | Π χ [ f ]( z ) | ω ( z ) dV ( z ) = Z Ω | T b [ f ]( w ) | η b ( w ) dV ( w ) . (e) If L χ (Ω ; ω ) = Π χ (cid:2) L (Ω ; ω ) (cid:3) and A χ (Ω ; ω ) = Π χ (cid:2) A (Ω ; ω ) (cid:3) , the mappings (1.21) T b : L χ (Ω , ω ) → L (Ω , η b ) and T b : A χ (Ω ∗ , ω ) → A (Ω ∗ , η b ) are isometric isomorphisms of Hilbert spaces.(f ) For each χ ∈ b G A choose b χ ∈ Z n with ϕ ([[ b χ ]]) = χ . Then there is an isomorphism (1.22) A (Ω ∗ , ω ) ∼ = M χ ∈ b G A A (Ω ∗ , η b χ ) and an identity of Bergman kernels: for z , w ∈ Ω ∗ , (1.23) B Ω ∗ ( z , w ; ω ) = X χ ∈ b G A F − b χ ◦ Φ A ( z ) B Ω ∗ (cid:0) Φ A ( z ) , Φ A ( w ); η b χ (cid:1) F − b χ ◦ Φ A ( w ) . In particular, (1.24) B Ω ∗ ( z , z ; ω ) = X χ ∈ b G A (cid:12)(cid:12) F − b χ ◦ Φ A ( z ) (cid:12)(cid:12) B Ω ∗ (cid:0) Φ A ( z ) , Φ A ( z ); η b χ (cid:1) . Theorem 1.3 is proved in Section 5.Since Ω ∗ ⊆ Ω , the extremal characterization (1.3) gives the inequality B Ω ∗ ( z , z ; ω ) ≥ B Ω ( z , z ; ω ). Combining this with (1.24), we get Corollary 1.4.
Under the same hypotheses as Theorem 1.3, we have for z ∈ Ω ∗ , (1.25) B Ω ( z , z ; ω ) ≤ X χ ∈ b G A (cid:12)(cid:12) F − b χ ◦ Φ A ( z ) (cid:12)(cid:12) B Ω ∗ (cid:0) Φ A ( z ) , Φ A ( z ); η b χ (cid:1) . Remarks:
1. In part (f) of Theorem 1.3, the choice of b χ ∈ Z n such that ϕ ([[ b χ ]]) = χ is not unique.Different choices lead to different choices of c ( b χ ) and η b χ as given in part (d), andhence lead to different spaces A (Ω , η b χ ). Thus (1.22) can be viewed as a family ofdecompositions for A (Ω ∗ , ω ) rather than a single one.2. In Theorem 1.3, it is important to note that the isomorphism between the two spaces L (Ω , ω ) and L (Ω , η b ) does not in general lead to an isomorphism between the cor-responding Bergman spaces A (Ω i , · ), but does lead to an isomorphism of the Bergmanspaces A (Ω ∗ i , · ) for the axes-deleted domains. Indeed the key point in part (e) of Theo-rem 1.3 is that the mapping T b : A χ (Ω ∗ , ω ) → A (Ω ∗ , η b ) is onto, whereas a priori themapping T b : A χ (Ω , ω ) → A (Ω ∗ , η b ) need not be onto. For example, suppose thatΩ = (cid:8) z = ( z , z ) ∈ C : | z z | < , | z | < (cid:9) and Φ A ( z , z ) = ( z z , z ) . Then G A is trivial (hence so is b G A ), andΩ = Φ A (Ω ) = (cid:8) w = ( w , w ) : | w | < , | w | < (cid:9) is the unit polydisk in C . Let us now choose the weight function ω ( w ) = | w | andthe holomorphic function g ( w ) = w /w on Ω \ H . Set b = , so that c = (0 , − η b ( w ) = | w | dV ( w ). We observe that g ∈ A (Ω ∗ , η b ). However, g does not lie in T b (cid:0) A χ (Ω , ω ) (cid:1) where χ is the identity character. This is because any f ∈ A χ (Ω , ω )with T b [ f ] = g must satisfy f ( z ) = z /z on Ω \ H . Such a function f does not admit aholomorphic extension to the origin.1.6. Bergman kernel estimates.
The Bergman kernel identity (1.23) involves the axes-deleted domains Ω ∗ and Ω ∗ rather than the original domains Ω and Ω . Also the upperbound in Corollary 1.4 is not sharp in general. In this section we state a result that forcertain choices of domain-weight pairs (Ω , ω ), an identity like (1.23) holds for Ω and Ω ,and the inequality in (1.25) is an equality. We begin by specifying the type of weights forwhich such results will hold. Definition 1.5.
Let Ω ⊆ C n be open and ω : Ω → (0 , ∞ ) a continuous weight function.(a) ω is said to be of monomial type if there exists µµµ = ( µ , . . . , µ n ) ∈ R n and a continuousfunction ϑ : Ω → (0 , ∞ ) such that (1.26) ω ( z ) = | F µµµ ( z ) | ϑ ( z ) , and inf (cid:8) ϑ ( z ) : z ∈ Ω (cid:9) > . (b) We call a monomial-type weight function ω admissible if (1.27) µ j < / for each index ≤ j ≤ n such that Ω ∩ (cid:8) z ∈ C n : z j = 0 (cid:9) = ∅ . For example, the weight function ω ≡ corresponding to the standard Bergman space isadmissible. Proposition 1.6. If ω : Ω → [0 , ∞ ) is an admissible weight function of monomial type on Ω , then A (Ω , ω ) = A (Ω ∗ , ω ) . We then have the following Bergman kernel identities for B Ω and B Ω . Theorem 1.7.
Let (Ω j , ω j ) , j = 1 , be as in Theorem 1.3. ALEXANDER NAGEL AND MALABIKA PRAMANIK (a) Suppose that A (Ω , ω ) = A (Ω ∗ , ω ) . Then the identities (1.23) and (1.24) hold, with B Ω ∗ on the left side replaced by B Ω . In particular, this is the case whenever ω isadmissible of monomial type and satisfies (1.19) .(b) Suppose that ω is a weight function of monomial type on Ω , not necessarily admissible.Then for every χ ∈ b G A , there exists a choice b χ ∈ Z n such that ϕ ([[ b χ ]]) = χ and suchthat the weight function η b χ is admissible of monomial type on Ω . For such choices theidentities (1.23) and (1.24) hold, with B Ω ∗ on the right side of those relations replacedby B Ω .(c) Suppose that both ω and ω are weight functions of monomial type obeying (1.19) , andthat ω is admissible. Then for each χ ∈ b G A there exist b χ ∈ Z n such that B Ω ( z , w ; ω ) = X χ ∈ b G A F − b χ ◦ Φ A ( z ) B Ω (cid:0) Φ A ( z ) , Φ A ( w ); η b χ (cid:1) F − b χ ◦ Φ A ( w )(1.28) B Ω ( z , z ; ω ) = X χ ∈ b G A (cid:12)(cid:12) F − b χ ◦ Φ A ( z ) (cid:12)(cid:12) B Ω (cid:0) Φ A ( z ) , Φ A ( z ); η b χ (cid:1) . (1.29) In particular, the relations (1.28) and (1.29) hold when ω ≡ , i.e., for the standardBergman space on Ω . Proposition 1.6 and Theorem 1.7 are proved in Section 5.
Remark:
It is important to note the distinction between Theorem 1.3 (d) and Theorem 1.7(b) and (c). The identities (1.23) and (1.24) hold for the axes-deleted domains Ω ∗ and Ω ∗ equipped with arbitrary continuous weight functions ω and ω obeying (1.19), and theseidentities remain valid for any choice of b χ ∈ Z n obeying ϕ ([[ b χ ]] = χ . In contrast, therelations (1.28) and (1.29) are true for the original domains Ω and Ω and for certainchoices of b χ , provided the associated weights are of appropriate monomial type.1.7. A simple example.
Before developing the general theory we consider a very simple example of our main results.Let Ω = Ω = D = (cid:8) z ∈ C : | z | < (cid:9) , and let Φ : D → D be the proper mapping Φ( z ) = z .The standard Bergman kernel and projection for the unit disk are given by B D ( z, w ) = 1 π (1 − zw ) − and P D [ f ]( z ) = 1 π Z D f ( w )(1 − zw ) dV ( w ) . It follows from Bell’s work that for this example, equations (1.5) and (1.8) become f ∈ L ( D ) and g ( z ) = 2 zf ( z ) = ⇒ P D [ g ]( z ) = 2 z P D [ f ]( z ) , √ w B D ( z, √ w ) − √ w B D ( z, −√ w ) = 2 z B D ( z , w ) . Our approach is to decompose h ∈ A ( D ) into even and odd functions, and then identify thecorresponding subspaces of A ( D ) with certain weighted Bergman spaces. If h ∈ A ( D ) setΠ e [ h ]( z ) = 12 (cid:0) h ( z ) + h ( − z ) (cid:1) , A e ( D ) = (cid:8) h ∈ A ( D ) : h ( z ) = h ( − z ) (cid:9) , Π o [ h ]( z ) = 12 (cid:0) h ( z ) − h ( − z ) (cid:1) , A o ( D ) = (cid:8) h ∈ A ( D ) : h ( z ) = − h ( − z ) (cid:9) . We see that A e ( D ) and A o ( D ) are closed complementary orthogonal subspaces of A ( D ),and hence A ( D ) = A e ( D ) ⊕ A o ( D ) with || h || = || Π e h || + || Π o h || where || · || denotes thenorm in L ( D ). Next if h ∈ A ( D ) there are unique holomorphic functions π e [ h ] and π o [ h ]on D so that Π e [ h ]( z ) = π e [ h ]( z ) and Π o [ h ]( z ) = z π o [ h ]( z ). Since Z D f ( w ) dV ( w ) = 2 Z D f ( z ) | z | dV ( z ) , it follows that || Π e [ h ] || = 12 Z D | π e [ h ]( z ) | | z | − dV ( z ) and || Π o [ h ] || = 12 Z D | π o [ h ]( z ) | dV ( z ) . Thus if we introduce weight functions ζ e ( w ) = | w | − and ζ o ( w ) ≡ on D , the mappings π e : A e ( D ) → A ( D ; ζ e dV ) and π o : A o ( D ) → A ( D ; ζ o dV )are isometric isomorphisms. In particular if π = ( π e , π o ) we have the following relationbetween Bergman projections and Bergman kernels:(1.30) π ◦ P D = (cid:0) P ζ e D ◦ π e , P ζ o D ◦ π o (cid:1) ,B D ( z, w ) = B D ( z , w ; ζ e ) + z w B D ( z , w ; ζ o ) . In the notation of Theorems 1.2 and 1.3, Φ A ( z ) = z , G A ∼ = b G A ∼ = {− , } . Following theprescription of Theorem 1.3 (d), we find that c = ( b = − , − if b = 0 , and hence η b ( z ) = ( = ζ o if b = − , | z | − = ζ e if b = 0 . Thus equation (1.28) shows that B D ( z, w ) = zwB D ( z , w ; η − ) + B D ( z , w ; η ), which is(1.30). 2. Functions and mappings of monomial type
We collect here basic facts concerning the functions and maps of the form (1.9). Set(2.1) O n := (cid:8) ( t , . . . , t n ) ∈ R n : t j > ≤ j ≤ n (cid:9) , C n ∗ := (cid:8) ( z , . . . , z n ) ∈ C n : n Y j =1 z j = 0 (cid:9) = C n \ H . Thus O n is the positive octant in R n and C n ∗ is C n with complex coordinate planes deleted.We denote by = (1 , . . . , ∈ Z n the vector with 1 in every entry. The vector e k =(0 , . . . , , . . . , ∈ Z n is the unit vector with 1 in the k th entry and zeros elsewhere. If a = ( a , · · · , a n ) ∈ R n then F a ( t ) = F a ( t , . . . , t n ) = t a t a · · · t a n n is a function of monomial-type and F a : O n → (0 , ∞ ). If each a j ∈ Z then F a extends to a holomorphic function on C n ∗ .If also each a j ≥ F a extends to a holomorphic function on C n . For { a , . . . , a n } ⊂ R n ,let A ∈ M n ( R ) be the matrix whose j th row vector is a j . Then Φ A ( t ) = ( F a ( t ) , · · · , F a n ( t )) is mapping of monomial-type corresponding to A , and Φ A : O n → O n . If A ∈ M n ( Z ) thenΦ A is a holomorphic mapping from C n ∗ to itself. If all the entries of A are non-negative thenΦ A is a holomorphic mapping from C n to itself. Let J Φ A ( t ) = det (cid:16) ∂F a j ∂t k (cid:17) ( t ) denote theJacobian matrix. For the proof of the following, see Lemma 4.2 in [16]. Proposition 2.1.
Let t ∈ O n .(a) If b j ∈ R n , c j ∈ R , and a = P kj =1 c j b j then F a ( t ) = Q kj =1 F b j ( t ) c j ;(b) If A , B ∈ M n ( R ) and a ∈ R n then F a · A ( t ) = F a (cid:0) Φ A ( t ) (cid:1) and Φ A · B ( t ) = Φ A (cid:0) Φ B ( t ) (cid:1) ;(c) Let A ∈ M n ( R ) and b = · A − ∈ R n . Then J Φ A ( t ) = det( A ) F b ( t ) .(d) If A ∈ M n ( R ) is invertible then Φ A : O n → O n is a diffeomorphism and Φ − A = Φ A − . The identities in (a), (b), and (c) continue to hold for t = z ∈ C n ∗ , and also for t = z ∈ C n provided the vectors and matrices have non-negative integer entries. If A ∈ M n ( Z ) is invert-ible and | det( A ) | 6 = 1 then Φ A : C n ∗ → C n ∗ is not one-to-one, and so is not biholomorphic.However, we have the following replacement. Proposition 2.2.
Let A ∈ M n ( Z ) be non-singular.(a) Φ A : C n ∗ → C n ∗ is a proper holomorphic mapping.(b) If Φ A ( z ) = w ∈ C n ∗ there is a neighbourhood U z of z in C n ∗ so that Φ A : U z → Φ A ( U z ) isa biholomorphic mapping.Proof. If A ∈ M n ( Z ) is invertible, then the rows a j of A form a basis for R n . Solving thelinear system Ax = e k using Cramer’s rule, we have e k = det( A ) − P nj =1 b j,k a j where each b j,k ∈ Z . It follows from part (a) of Proposition 2.1 that(2.2) z det( A ) k = F e k ( z ) det( A ) = n Y j =1 F a j ( z ) b j,k . Suppose now that K is a compact subset of C n ∗ . For part (a), we need to show that Φ − A ( K ) = { z ∈ C n ∗ : Φ A ( z ) ∈ K } ⊂ C n ∗ is compact, i.e., closed and bounded. That the latter setis closed follows easily from the fact that K is closed and Φ A is continuous. To provethat Φ − A ( K ) is bounded, we observe that there exist positive numbers ǫ < N such that K ⊂ (cid:8) w ∈ C n : ǫ ≤ | w k | ≤ N, ≤ k ≤ n (cid:9) . Thus, if w = Φ A ( z ) = (cid:0) F a ( z ) , . . . , F a n ( z ) (cid:1) ∈ K then we have that ǫ ≤ | F a j ( z ) | ≤ N for 1 ≤ j ≤ n . It follows from (2.2) that each | z k | is bounded and bounded away from zero by constants depending on ǫ , N , the integers b j,k ,and det( A ). This implies that Φ − A ( K ) is compact, proving (a). Part (b) follows from theholomorphic inverse function theorem since J Φ A ( z ) = 0 for all z ∈ C n ∗ . (cid:3) The explicit nature of Φ A allows us to describe the pre-image of any point in C n ∗ . To thisend, and for any z ∈ C n ∗ , let us write its polar form z = (cid:0) r e πiθ , . . . , r n e πiθ n (cid:1) = r ⊗ exp[2 πiθθθ ] , where r = ( r , . . . , r n ) ∈ O n and exp[ v ] = ( e v , · · · , e v n ) for any row vector v . Thus r isuniquely determined and θ = ( θ , . . . , θ n ) ∈ R n is determined up to translation by an elementof Z n . Lemma 2.3.
Let A ∈ M n ( Z ) be non-singular and let w = ρ ⊗ exp[2 πi φ ] ∈ C n ∗ .(a) Then there exists z ∈ C n ∗ such that Φ A ( z ) = w .(b) If z , z ∈ C n ∗ are given by the polar forms z = r ⊗ exp[2 πi θ ] and z = r ⊗ exp[2 πi θ ] ,then Φ A ( z ) = Φ A ( z ) if and only if r = r and ( θ − θ ) · A t ∈ Z n .(c) The inverse image of w under Φ A can be identified with the group G A via the one-to-oneand onto mapping G A ∋ [ m ] −→ ξ ([ m ]) ⊗ z , defined in (1.12) . In particular, for every w ∈ C n ∗ , the cardinality of Φ − A ( w ) is the same, and equals G A ) .Proof. If z = r e πi θ then Φ A ( z ) = (cid:0) F a ( r ) e πi h a , θ i , . . . , F a n ( r ) e πi h a n , θ i (cid:1) , and soΦ A ( z ) = w ⇐⇒ (cid:0) F a ( r ) e πi h a , θ i , . . . , F a n ( r ) e πi h a n , θ i (cid:1) = (cid:0) ρ e πiφ , . . . , ρ n e πiφ n (cid:1) ⇐⇒ Φ A ( r ) = ρ and θ · A t = φ + m for some m ∈ Z n ⇐⇒ Φ A ( r ) = ρ and θ = φ · ( A − ) t + m · ( A − ) t for some m ∈ Z n . Now Φ A : O n → O n is invertible by Proposition 2.1, part (d). For the rest of this proof, letus denote by Φ − A ( ρρρ ) the unique pre-image of ρρρ in O n . Set z = Φ − A ( ρ ) ⊗ exp (cid:2) πi φ · ( A − ) t (cid:3) .It follows that Φ A ( z ) = w , proving (a). Next, if Φ A ( r ⊗ exp[2 πi θ ]) = Φ A ( r ⊗ exp[2 πi θ ])then Φ A ( r ) = Φ A ( r ) and ( θ − θ ) · A t = m for some m ∈ Z n . Since Φ A is invertible on O n it follows that r = r , proving (b). Finally, for part (c) the computation above showsthat the inverse image of w under Φ A is contained in the set of all points of the form z m = Φ − A ( ρ ) ⊗ exp (cid:2) πi φ · ( A − ) t + 2 πi m · ( A − ) t (cid:3) , m ∈ Z n . But z m = z m if and only if ( m − m ) · ( A − ) t ∈ Z n , which means that m − m ∈ C ( A ).Thus the map m ∈ Z n z m lifts naturally to [ m ] ∈ G A z m = z [ m ] . Moreover, acomparison of the expression above with the definition of ξ in (1.12) shows that z [ m ] = z ⊗ exp (cid:2) πi m · ( A t ) − (cid:3) = z ⊗ ξ ([ m ]) . This completes the proof. (cid:3) The group G A To describe the structure of the group G A we use a normal form for integer matrices, some-times called the Smith normal form . We state this in the lemma below; a proof of it canbe found in [1, Chapter 12, Theorem 4.3], [15, Chapter 1, Theorem 11.3], or [13, Chapter 3,Theorem 5].
Lemma 3.1.
Let A ∈ M n ( Z ) be non-singular.(a) There exist S , T , Λ ∈ M n ( Z ) with | det( S ) | = | det( T ) | = 1 and Λ a diagonal matrixsuch that (3.1) S · A · T = Λ . (b) The diagonal entries λ , . . . , λ n of Λ satisfy ≤ λ ≤ λ ≤ · · · ≤ λ n , and each λ j divides λ j +1 for ≤ j ≤ n − . They are called the invariant factors of A . Remark 3.2.
Since | det( S ) | = | det( T ) | = 1 , it follows from Cramer’s rule that the inversematrices S − , T − ∈ M n ( Z ) ; i.e. they also have integer entries. Structure of G A . If λ , . . . , λ n are the invariant factors of a non-singular matrix A ∈ M n ( Z ), the next lemma shows that G A is isomorphic to L nj =1 Z /λ j Z . In order to define theisomorphism, we set(3.2) s = max { ℓ : 1 ≤ ℓ ≤ n, λ ℓ = 1 } , with the convention that s = 0 if λ >
1. Then(3.3) n M j =1 Z /λ j Z = n M j = s +1 Z /λ j Z . Let S ∈ M n ( Z ) be the matrix from Lemma 3.1. Then define b ι : Z n → L nj =1 Z /λ j Z by(3.4) b ι ( m ) = (cid:0) π ( h e , m · S t i ) , . . . , π n ( h e n , m · S t i (cid:1) , where π j : Z → Z /λ j Z is the group homomorphism which sends each integer m ∈ Z to itsequivalence class [ m ] λ j ∈ Z /λ j Z . If s ≥
1, then π j ≡ j ≤ s . It is easy to see that b ι is agroup homomorphism. Lemma 3.3.
For b ι as in (3.4) , the following conclusions hold.(a) b ι (cid:0) C ( A ) (cid:1) = ; hence b ι induces a group homomorphism ι : G A → L nj =1 Z /λ j Z .(b) The homomorphism ι is an isomorphism and so G A ) = | Q nj =1 λ j | = | det( A ) | .(c) G A is generated by the ( n − s ) elements (cid:8)(cid:2) e s +1 · ( S t ) − (cid:3) , . . . , (cid:2) e n · ( S t ) − (cid:3)(cid:9) .Proof. If n ∈ C ( A ) then n = m · A t for some m ∈ Z n . It follows from the Smith normalform (3.1) for A that n · S t = m · A t · S t = m · ( T − ) t · Λ · ( S − ) t · S t = m · ( T − ) t · Λ . Since m · ( T − ) t ∈ Z n it follows that the k th entry of n · S t is an integer multiple of λ k andso π k ( h e k , n · S t ) = [0] λ k , and this establishes (a).Now suppose that n ∈ Z n and that ι ([ n ]) = . Then π j ( h e j , n · S t i ) = [0] λ j ∈ Z /λ j Z so thereexists k = ( k , · · · , k n ) ∈ Z n such that n · S t = ( k λ , · · · , k n λ n ) = k · Λ . Thus n = k · Λ · ( S t ) − = k · Λ t · ( S t ) − = k · T t · A t ∈ C ( A ) , which means that [ n ] = [ ] ∈ G A , and so ι is one-to-one. Next, fix any ([ k ] λ , · · · , [ k n ] λ n ) ∈ L nℓ =1 Z /λ ℓ Z . Set k = ( k , · · · , k n ) and n = k · ( S t ) − ∈ Z n . Then ι ([ n ]) = b ι ( n ) = (cid:0) π ( h e , k · ( S t ) − · S t i ) , . . . , π n ( h e n , k · ( S t ) − · S t i ) (cid:1) = (cid:0) π ( h e , k i ) , . . . , π n ( h e n , k i ) (cid:1) = ([ k ] λ , · · · , [ k n ] λ n ) , which shows that ι is surjective, proving (b).It remains to prove part (c). For λ j >
1, the group Z /λ j Z under addition is generatedby [1] λ j . In view of the identification (3.3), the group L nj =1 Z /λ j Z is generated by the elements { f k : s + 1 ≤ k ≤ n } , with f k = (cid:0) , · · · , , [0] λ s +1 , . . . , [1] λ k , . . . , [0] λ n (cid:1) , where all thecomponents are zero except the k th , which is one. But ι (cid:0)(cid:2) e k · ( S t ) − (cid:3)(cid:1) = f k since (cid:2) e j · ( S t ) − (cid:3) λ k = π k (cid:0) h e k , e j · ( S t ) − · S t i (cid:1) = π k (cid:0) h e k , e j i (cid:1) = ( [1] λ j if k = j [0] λ j if k = j , completing the proof. (cid:3) Corollary 3.4. If A ∈ M n ( Z ) is non-singular and w ∈ C n ∗ then the cardinality of the inverseimage Φ − A ( { w } ) is | det( A ) | .Proof. This follows by combining Lemma 2.3 (c) and Lemma 3.3 (b). (cid:3)
The characters of G A . A character of a group G is a group homomorphism χ : G → T = { z ∈ C : | z | = 1 } . Theset of all characters is denoted by b G and is a group under point-wise multiplication. Let usrecall that if m ∈ Z n , then [ m ] denotes its equivalence class in G A = Z n / C ( A ) and [[ m ]]denotes its equivalence class in G A t = Z n / C ( A t ). Lemma 3.5.
Let A ∈ M n ( Z ) be non-singular.(a) If χ is a character of G A there exists b ∈ Z n so that (3.5) χ ([ m ]) = exp (cid:2) πi h m , b · A − i (cid:3) for every [ m ] ∈ G A . Conversely, every b ∈ Z n defines a character in this way.(b) Two elements b , b ∈ Z n define the same character χ if and only if b − b = n · A forsome n ∈ Z n ; i.e. b − b ∈ C ( A t ) .(c) If [[ b ]] ∈ G A t define ϕ ([[ b ]])([ m ]) := exp (cid:2) πi h m , b · A − i (cid:3) . Then ϕ : G A t → b G A is agroup isomorphism.Proof. It is easy to check that every character b χ of Z n is given by b χ ( m ) = exp (cid:2) πi h m , θ i (cid:3) for a unique θ = ( θ , . . . , θ n ) ∈ T n . If χ is a character of G A = Z n / C ( A ), it lifts to a character b χ of Z n and so b χ ( m ) =exp (cid:2) πi h m , θ χ i (cid:3) for a unique θ χ ∈ T n . Moreover b χ must be the identity on C ( A ), hence(3.6) exp (cid:2) πi h m · A t , θ χ i (cid:3) = 1 for all m ∈ Z n . Thus θ χ · A ∈ Z n and so θ χ = b · A − for some b ∈ Z n . On the other hand, if b ∈ Z n then m −→ exp (cid:2) πi h m , b · A − i (cid:3) is clearly a character of Z n which is the identity on thesubgroup m ∈ C ( A ). This proves (a).If b , b ∈ Z n define the same character of G A then exp (cid:2) πi h m , ( b − b ) · A − i (cid:3) = 1 for all m ∈ Z n , and so ( b − b ) · A − = p ∈ Z n , and this proves (b). Assertion (c) then followsfrom (a) and (b). (cid:3) We now define certain special characters of G A . Let ϕ : G A t → b G A be the isomorphismfrom part (c) of Lemma 3.5. For 1 ≤ k ≤ n define(3.7) ξ k ([ m ]) = ϕ ([[ e k ]])([ m ]) = exp (cid:2) πi h m , e k · A − i (cid:3) ∈ T . We note that this agrees with the definition of ξ k in (1.11). Proposition 3.6.
The characters ξ , . . . , ξ n generate the dual group b G A .Proof. By part (a) of Lemma 3.5, every character χ of G A is of the form χ = ϕ ([[ b ]]) forsome b ∈ Z n . We can write b = P nk =1 b j e k ∈ Z n , from which it follows that χ = ϕ ([[ b ]]) = n Y k =1 ϕ ([[ e k ]]) b k = n Y k =1 ξ b k k . The last equation shows that { ξ , · · · , ξ n } is a set of generators. (cid:3) The action of G A . Given the characters ξ k defined in (3.7) let us recall the definitionof ξ in (1.11). We use ξ to define an action of the group G A on the set C n via the relation(c). A group action is faithful if for any two distinct elements of the group, there exist someelement of the set that produces distinct images under the action. Lemma 3.7.
The following conclusions hold.(a) For any [ m ] , [ m ] ∈ G A and any z ∈ C n ∗ ,(i) ξ ([ m + m ]) ⊗ z = ξ ([ m ]) ⊗ (cid:0) ξ ([ m ]) ⊗ z (cid:1) ;(ii) if ξ ([ m ]) ⊗ z = ξ ([ m ]) ⊗ z for some z ∈ C n ∗ then [ m ] = [ m ] .In particular, the mapping ([ m ] , z ) → ξ ([ m ]) ⊗ z is a faithful action of G A on C n ∗ .(b) Φ A (cid:0) ξ ([ m ]) ⊗ z (cid:1) = Φ A ( z ) for all [ m ] ∈ G A and all z ∈ C n ∗ .(c) If z , z ∈ C n ∗ and Φ A ( z ) = Φ A ( z ) then there exists [ m ] ∈ G A such that ξ [ m ] ⊗ z = z .(d) If b ∈ Z n then F b (cid:0) ξ ([ m ]) (cid:1) = exp (cid:2) πi h m , b · A − i (cid:3) .Proof. The property in (i) of part (a) follows from the fact that each ξ k is a character of G A .If ξ [ m ] ⊗ z = ξ [ m ] ⊗ z for some z ∈ C n ∗ then ξ ([ m ]) = ξ ([ m ]) since all the componentsof z are non-zero. Hence for 1 ≤ k ≤ n ,exp (cid:2) πi (cid:10) m , e k · A − (cid:11)(cid:3) = exp (cid:2) πi (cid:10) m , e k · A − (cid:11)(cid:3) , orexp (cid:2) πi (cid:10) e k , ( m − m )( A t ) − (cid:11)(cid:3) = 1 , so ( m − m ) · ( A t ) − ∈ Z n . But this says that m − m ∈ C ( A ) and so [ m ] = [ m ],completing the proof of (a).Part (b) follows from Lemma 2.3 (c), with z = z and w = Φ A ( z ) in the notation of thatlemma.Next, let z ℓ = r ℓ ⊗ exp[2 πi θ ℓ ] ∈ C n ∗ for ℓ = 1 ,
2. From Lemma 2.3 (b) we know thatΦ A ( z ) = Φ A ( z ) if and only if r = r and θ = θ + m · ( A t ) − for some m ∈ Z n . But this is true if and only if z = ξ ([ m ]) ⊗ z , establishing (c). Finally, for part (d) we have F b (cid:0) ξ ([ m ]) (cid:1) = n Y j =1 exp (cid:2) πib j h m , e j · A − i (cid:3) = exp h πi (cid:10) m , n X j =1 b j e j · A − (cid:11)i = exp (cid:2) πi (cid:10) m , b · A − (cid:11)(cid:3) . (cid:3) Group actions and characters
In this section we recall some basic facts about group characters and group actions for anarbitrary finite abelian group G . Let G ) denote the cardinality of G and let e denote theidentity element. The following orthogonality relations are well-known; see for example [7,Corollary 4.2] for a proof. Proposition 4.1.
Let G be a finite abelian group. Suppose that χ , χ ∈ b G and g ∈ G .Then X g ∈ G χ ( g ) χ ( g ) = ( G ) if χ = χ , if χ = χ and X χ ∈ b G χ ( g ) = ( G ) if g = e , if g = e. Now suppose that G has an action ρ on a set X ; i.e. ρ is a group homomorphism from G tothe group of permutations of X . If g ∈ G the action of ρ ( g ) on x ∈ X is denoted by ρ ( g ) · x .The action ρ is faithful if ρ ( g ) = ρ ( g ) implies g = g . If F : X → C and χ ∈ b G set(4.1) Π χ [ F ]( x ) := F χ ( x ) = 1 G ) X g ∈ G χ ( g ) F ( ρ ( g ) · x ) . Note that in the context of this paper, the above definition of Π χ is the same as the onespecified in (1.15). A positive measure µ on X is said to be invariant under the action ρ of G if for all g ∈ G and all f ∈ L ( X, dµ ) the following relation holds:(4.2) Z X f ( x ) dµ ( x ) = Z X f ( ρ ( g ) · x ) dµ ( x ) . Proposition 4.2.
Suppose that G is a finite abelian group, equipped with an action ρ on aset X , which in turn supports a positive measure µ that is invariant under ρ . Let F : X → C .(a) F ( x ) = P χ ∈ b G F χ ( x ) .(b) If h ∈ G then F χ ( ρ ( h ) · x ) = χ ( h ) F χ ( x ) = χ ( h ) − F χ ( x ) .(c) If G : X → C and G ( ρ ( h ) · x ) = χ ( h ) − G ( x ) for all h ∈ G then Π χ [ G ] = G .(d) If F ∈ L ( X, dµ ) and χ , χ ∈ b G then Z X F χ ( x ) F χ ( x ) dµ ( x ) = ( || F χ || L ( X,dµ ) if χ = χ , if χ = χ .(e) If χ , χ ∈ b G then Π χ (cid:2) Π χ [ F ] (cid:3) ( x ) = ( Π χ [ F ]( x ) if χ = χ , if χ = χ . Proof.
We have P χ ∈ b G F χ ( x ) = G ) P g ∈ G h P χ ∈ b G χ ( g ) i F ( ρ ( g ) · x ) = F ( x ) by Proposition4.1, which gives (a). Next, using a reparametrization g ∈ G gh in the sum, we arrive atthe relation F χ ( ρ ( h ) · x ) = G ) − X g ∈ G χ ( g ) F (cid:0) ρ ( g ) · ρ ( h ) · x ) = G ) − X g ∈ G χ ( g ) F (cid:0) ρ ( gh ) · x )= G ) − X g ∈ G χ ( gh − ) F (cid:0) ρ ( g ) · x ) = χ ( h ) G ) − X g ∈ G χ ( g ) F (cid:0) ρ ( g ) · x ) , which gives (b). A similar argument gives (c). If χ = χ then the defining property (4.2)of an invariant measure yields Z X F χ ( x ) F χ ( x ) dµ ( x ) = G ) − Z X X g,h ∈ G χ ( g ) χ ( h ) F ( ρ ( g ) · x ) F ( ρ ( h ) · x ) dµ ( x )= G ) − Z X X g,h ∈ G χ ( g ) χ ( h ) F ( ρ ( gh − ) · x ) F ( x ) dµ ( x )= G ) − Z X X g,h ∈ G χ ( gh ) χ ( h ) F ( ρ ( g ) · x ) F ( x ) dµ ( x )= G ) − Z X X g ∈ G χ ( g ) h X h ∈ G χ ( h ) χ ( h ) i F ( ρ ( g ) · x ) F ( x ) dµ ( x ) = 0by Proposition 4.1. FinallyΠ χ (cid:2) Π χ [ F ] (cid:3) ( x ) = Π χ h G ) − X g ∈ G χ ( g ) F ( ρ ( g ) · x ) i = G ) − X g ∈ G χ ( g ) X h ∈ G χ ( h ) F ( ρ ( g ) · ρ ( h ) · x )= G ) − X g,h ∈ G χ ( h ) χ ( g ) F ( ρ ( gh ) · x )= G ) − X g,h ∈ G χ ( h ) χ ( gh − ) F ( ρ ( g ) · x )= G ) − X g ∈ G χ ( g ) h X h ∈ G χ ( h ) χ ( h ) i F ( ρ ( g ) · x ) , and (d) then follows from Proposition 4.1. (cid:3) Corollary 4.3. If F ∈ L ( X, dµ ) , then the norm of F decomposes as follows: || F || L ( X,dµ ) = X χ ∈ b G || F χ || L ( X,dµ ) . Now let S ( X ) be a vector space of functions on X that is invariant under the action ρ of G ,i.e., if F ∈ S ( X ), then for every g ∈ G , the function given by x ∈ X F ( ρ ( g ) · ) is also in S ( X ). In this case, it follows from the definition (4.1) of Π χ and Proposition 4.1 that thelinear operator Π χ maps S ( X ) into S ( X ). For each χ ∈ b G set(4.3) S ( X ) χ := Π χ (cid:2) S ( X ) (cid:3) = (cid:8) F ∈ S ( X ) : F = Π χ [ F ] (cid:9) ⊂ S ( X ) . The following is then an easy consequence of Proposition 4.2.
Corollary 4.4.
For each χ ∈ b G , the mapping Π χ : S ( X ) → S ( X ) is a linear map. Moreover,(a) S ( X ) χ ∩ S ( X ) χ = { } if χ = χ .(b) S ( X ) = L χ ∈ b G S ( X ) χ .(c) If S ( X ) ⊂ L ( X, dµ ) then the subspaces {S ( X ) χ : χ ∈ b G } are mutually orthogonal.In particular, if S ( X ) ⊂ L ( X, dµ ) , then Π χ acting on S ( X ) is an orthogonal projection, inwhich case the direct sum in (b) gives an orthogonal decomposition of S ( X ) into mutuallyorthogonal subspaces. Proofs of theorems 1.2, 1.3, and 1.7
Proof of Theorem 1.2.
According to Lemma 3.7 the mapping ([ m ] , z ) → ξ ([ m ]) ⊗ z from equation (1.12) defines agroup action ρ of G A on C n ∗ . Set X = Ω ∩ C n ∗ = Ω \ H . By assumption, Ω is invariantunder this action; hence it follows from the definition of domain invariance on page 4 that([ m ] , z ) → ξ ([ m ]) ⊗ z generates a group action of G A on X as well. Further, if ω is acontinuous non-negative weight function that is invariant under ρ according to the definitionon page 4, then the measure dµ ( z ) = ω ( z ) dV ( z ) is also an invariant measure on X , in thesense of (4.2).We now choose two ρ -invariant vector spaces of functions. The first is S ( X ) = L (Ω \ H , dµ ),which is isomorphic to L (Ω , ω ), since H has Lebesgue measure zero. The second choice of S ( X ) is the subspace of L (Ω , ω ) consisting of holomorphic functions on Ω \ H that admita holomorphic extension to Ω ∩ H . Note that the latter space is isomorphic to the weightedBergman space A (Ω , ω ). The results of Section 4 apply for these choices of S ( X ). Parts(a) and (b) of Theorem 1.2 then follow immediately from Corollary 4.4.5.2. Proof of Theorem 1.3.Part (a):
The following two identities follow respectively from part (d) of Lemma 3.7 andpart (b) of Proposition 4.2: for every [ m ] ∈ G A , F b ( ξ ([ m ]) ⊗ z ) = χ ([ m ]) F b ( z ) and Π χ [ f ]( ξ ([ m ]) ⊗ z ) = χ ([ m ]) − Π χ [ f ]( z ) . Combining these two observations we find that F b ( ξ ([ m ]) ⊗ z )Π χ [ f ]( ξ ([ m ]) ⊗ z ) = F b ( z ) Π χ [ f ]( z ) , so Π χ [ g ] F b is invariant under the action of G A . Thus, the function f b = T b [ f ] defined by(5.1) f b ( w ) = F b ( z )Π χ [ f ]( z ) for any z ∈ Φ − A ( w ) , w ∈ Ω ∗ is well-defined as a function on Ω ∗ . (cid:3) Part (c): If f is holomorphic on Ω ∗ , then so is Π χ [ f ]. Any function F b is holomorphic on C n ∗ , and hence on Ω ∗ . The function f b is a composition of their product F b ( · )Π χ [ f ]( · ) withΦ − A . We have shown in Proposition 2.2(b) that Φ A : C n ∗ → C n ∗ is locally a biholomorphicmapping. Thus f b is holomorphic on Ω ∗ . (cid:3) Part (b):
Next, for any g : Ω ∗ → C , let us set f ( z ) = g (cid:0) Φ A ( z ) (cid:1) F − b ( z ). We know fromLemma 3.7(c) and (d) that Φ A ( ξ ([ m ]) ⊗ z ) = Φ A ( z ) and F − b (cid:0) ξ ([ m ]) (cid:1) = χ ([ m ]) − . Substi-tuting these into the expression (1.15) for Π χ [ f ], we arrive atΠ χ [ f ]( z ) = 1 G A ) X [ m ] ∈ G A χ ([ m ]) f ( ξ [ m ] ⊗ z )= 1 G A ) X [ m ] ∈ G A χ (cid:0) [ m ]) g ◦ Φ A ( ξ ([ m ]) ⊗ z (cid:1) F − b (cid:0) ξ ([ m ]) ⊗ z (cid:1) = 1 G A ) X [ m ] ∈ G A χ (cid:0) [ m ]) g (cid:0) Φ A ( z ) (cid:1) χ ([ m ]) − F − b ( z ) = f ( z ) . The same relation also yields the second claim; namely, for every w = Φ A ( z ) ∈ Ω , T b [ f ]( w ) = Π χ [ f ]( z ) F b ( z ) = f ◦ Φ A ( z ) F b ( z ) = g ( w ) . (cid:3) Part (d):
For any integrable function G : Ω → C , the change of variables w = Φ A ( z ) gives(5.2) Z Ω G ( w ) dV ( w ) = det( A ) − Z Ω G (cid:0) Φ A ( z ) (cid:1) | J Φ A ( z ) | dV ( z ) , since Φ A is det( A )-to-one. Now ω (cid:0) Φ A ( z ) (cid:1) = ω ( z ), and(5.3) | J Φ A ( z ) | = det( A ) | F · A − ( z ) | , | F c (cid:0) Φ A ( z ) (cid:1) | = | F c · A ( z ) | . Given any f ∈ L (Ω , ω ), there exists according to part (a) a unique function f b = T b [ f ] :Ω → C so that g b (cid:0) Φ A ( z ) (cid:1) = Π χ [ f ]( z ) F b ( z ). Applying the change of variable formula (5.2)with G ( w ) = | T b [ f ]( w ) | η b ( w ) and invoking the relations (5.3), we obtain Z Ω | T b [ f ]( w ) | η b ( w ) dV ( w ) = det( A ) − Z Ω | T b [ f ]( w ) | | F c ( w ) | ω ( w ) dV ( w )= det( A ) − Z Ω | Π χ [ f ]( z ) | | F b ( z ) | | F c (cid:0) Φ A ( z ) (cid:1) | | J Φ A ( z ) | ω (Φ A ( z )) dV ( z )= Z Ω | Π χ [ f ]( z ) | | F c · A + b ( z ) | | F · A − ( z ) | ω ( z ) dV ( z )= Z Ω | Π χ [ f ]( z ) | | F c · A + b + · A − ( z ) | ω ( z ) dV ( z )= Z Ω | Π χ [ f ]( z ) | ω ( z ) dV ( z ) . This completes the proof of (d). (cid:3) Part (e):
The definition (5.1) of T b shows that it is linear. Let us first show that T b isinjective, on L χ (Ω , ω ), and hence on A χ (Ω ∗ , ω ). If T b [ f ] ≡ L χ (Ω , ω ), then thedefinition (4.3) dictates that Π χ [ f ] = f . It follows then from the norm identity (1.20) inpart (d) that T b [ f ] ≡ L (Ω , η b ) = ⇒ Π χ [ f ] = f ≡ L (Ω , ω ) , proving injectivity.Surjectivity of T b on L (Ω , ω ) and on A (Ω ∗ , η b ) follows from parts (b) and (d). Given g ∈ L (Ω , η b ), the function f defined in part (b) lies in L χ (Ω , ω ) and obeys T b [ f ] = g . If g ∈ A (Ω ∗ , η b ), the same function f is also in A χ (Ω ∗ , ω ), by virtue of part (c). The identity(1.20) in part (d) shows that T b preserves norms, and hence is an isometry. This proves thetwo isomorphisms claimed in (1.21). (cid:3) Part (f):
Since Ω ∗ and ω are both invariant under the group action (c), the space A (Ω ∗ , ω )admits the direct sum decomposition ensured by Theorem 1.2. Combining this with part(e), we see that A (Ω ∗ , ω ) = M(cid:8) A χ (Ω ∗ , ω ) : χ ∈ b G A (cid:9) ∼ = M(cid:8) A χ (Ω ∗ , η b χ ) : χ ∈ b G A , ϕ ([[ b χ ]] = χ (cid:9) . It remains to the prove the Bergman kernel identities (1.22) and (1.24). For each χ ∈ b G A let (cid:8) ψ χ ( · ; ω ) , ψ χ ( · ; ω ) , . . . (cid:9) be a complete orthonormal basis for A χ (Ω ∗ , ω ). Since A (Ω ∗ , ω ) = M χ ∈ b G A A χ (Ω ∗ , ω )is an orthogonal decomposition, the set of functions (cid:8) ψ χk ( · ; ω ) : χ ∈ b G A , k ≥ (cid:9) is a com-plete orthonormal basis for A (Ω ∗ , ω ). In view of (1.2), this yields the following expressionfor the Bergman kernel: B Ω ∗ ( z , w ; ω ) = X χ ∈ b G A ∞ X k =1 ψ χk ( z ; ω ) ψ χk ( w ; ω ) , z , w ∈ Ω ∗ . For each χ ∈ b G A , let us choose b χ ∈ Z n so that ϕ ([[ b χ ]]) = χ . Then according to (e), thereare functions θ χk ∈ A (Ω ∗ , η b χ ), k = 1 , , . . . , so that T b χ [ ψ χk ( · ; ω )] = θ χk ( · ; η b χ ); in otherwords,(5.4) θ χk (cid:0) Φ A ( z ); η b χ (cid:1) = ψ χk ( z ; ω ) F b χ ( z ) . Since T b is norm-preserving, and hence angle-preserving, for all b ∈ Z n , it follows that forevery χ ∈ b G A , the set of functions { θ χk ( · ; η b χ ) : k ≥ } is a complete orthonormal basis for A (Ω ∗ ; η b χ ). By (1.2), this again implies(5.5) B Ω ∗ ( ξ , ζ ; η b χ ) = ∞ X k =1 θ χk ( ξ ; η b χ ) θ χk ( ζ ; η b χ ) , ξ , ζ ∈ Ω ∗ . Combining (5.4) and (5.5) gives B Ω ∗ ( z , w ; ω ) = X χ ∈ b G A ∞ X k =1 ψ χk ( z ; ω ) ψ χk ( w ; ω )= X χ ∈ b G A F − b χ ( z ) h ∞ X k =1 (cid:0) ψ χk ( z ) F b χ ( z ) (cid:1)(cid:0) ψ χk ( w ) F b χ ( w ) (cid:1)i F − b χ ( w )= X χ ∈ b G A F − b χ ( z ) h ∞ X k =1 θ χk (cid:0) Φ A ( z ) (cid:1) θ χk (cid:0) Φ A ( w ) (cid:1)i F − b χ ( w )= X χ ∈ b G A F − b χ ( z ) B η χ Ω ∗ (cid:0) Φ A ( z ) , Φ A ( w ) (cid:1) F − b χ ( w ) . This is the relation (1.22). The relation (1.24) follows from (1.22) by setting z = w , com-pleting the proof. (cid:3) Proof of Proposition 1.6.
Proof.
Since the inclusion A (Ω , ω ) ⊆ A (Ω ∗ , ω ) holds trivially for any domain Ω and anyweight ω , we will focus on the reverse inclusion. In other words, given any admissible weight ω of monomial type on Ω, and any f ∈ A (Ω ∗ , ω ), our goal is to show that f admits aholomorphic extension to Ω ∩ H .Suppose that u = ( u , · · · , u n ) ∈ Ω ∩ H . Let m denote the number of indices 1 ≤ j ≤ n such that u j = 0. Without loss of generality suppose that u = u = · · · = u m = 0. Write u = ( u ′ , u ′′ ), where u ′ = ( u , · · · , u m ) = ′ ∈ C m and u ′′ = ( u m +1 , · · · , u n ) ∈ C n − m . Choose ǫ > D n ( u ; ǫ ) = (cid:8) z = ( z , · · · , z n ) : | z j − u j | < ǫ, for 1 ≤ j ≤ n (cid:9) is contained in Ω. In particular, choosing ǫ < min {| u j | / m < j ≤ n } ensures that D n − m ( u ′′ ; ǫ ′′ ) avoids the coordinate hyperplanes in C n − m .Let A m ( ′ , ǫ ′ ) denote the m -fold Cartesian product of the punctured disk { z ∈ C : 0 < | z | < ǫ } . The choice of ǫ shows that U n = A m ( ′ ; ǫ ′ ) × D n − m ( u ′′ ; ǫ ′′ ) ⊆ Ω ∗ . Since f ∈ A (Ω ∗ , ω ), it restricts to a function in A ( U n , ω ). Let us recall that ω ( z ) = (cid:12)(cid:12) F µµµ ( z ) (cid:12)(cid:12) ϑ ( z ) is of monomial type on Ω. In view of the definition (1.26), this implies that ϑ is bounded above and below by positive constants on D n ( ; ǫ ). Hence A ( U n , ω ) = A ( U n ; | F µµµ | ), and the A ( U n , ω ) norm of f is bounded above and below by constant multiplesof Z U n (cid:12)(cid:12) f ( z ) (cid:12)(cid:12) × (cid:12)(cid:12) F µµµ ( z ) (cid:12)(cid:12) dV ( z ) . Write µµµ = ( µ ′ µ ′ µ ′ , µ ′′ µ ′′ µ ′′ ) ∈ R m × R n − m . Since U n is the Cartesian product of a Reinhardt domainwith a polydisk, and the weight | F µµµ | is also of product type, the weighted Bergman space A ( U n ; | F µµµ | ) is well-understood. A complete orthogonal basis for A ( U n ; | F µµµ | ) is given bythe set of monomial-type functions (cid:8) F k ( z ) F ℓℓℓ ( z ′′ − u ′′ ) : k ∈ K [ µ ′ µ ′ µ ′ ] , ℓℓℓ ∈ ( N ∪ { } ) n − m (cid:9) , where K [ µ ′ µ ′ µ ′ ] := (cid:8) k ∈ Z m : F k ∈ L (cid:0) A m ( ′ ; ǫ ′ ) , | F µµµ ′ | (cid:1)(cid:9) = { k ∈ Z m : 2 k + 2 µ ′ µ ′ µ ′ + ′ has strictly positive entries } . Since ω is admissible, we know from the definition (1.27) that µ j < / ≤ j ≤ m .Therefore any k = ( k , · · · , k m ) ∈ K [ µ ′ µ ′ µ ′ ] must obey k j > − µ j − / > −
1, and hence musthave non-negative integer entries. Thus any f ∈ A ( U n ; ω ) is of the form(5.6) f ( z ) = ′ X k F k ( z ′ ) f k ( z ′′ ) , where z = ( z ′ , z ′′ ) ∈ U n , where the sum P ′ ranges over multi-indices k ∈ K [ µµµ ′ ], and the functions f k are analytic on D n − m ( u ′′ ; ǫ ′′ ). The series converges both in L ( U n ; ω ) and also absolutely and uniformlyover compact subsets of U n . Since the series (5.6) only admits non-negative integer powersof z ′ , an application of the iterated Cauchy integral formula shows that such a function f extends holomorphically to D n ( u , ǫ ), and hence to u . (cid:3) Proof of Theorem 1.7.
Proof.
Part (a): If A (Ω ∗ , ω ) = A (Ω , ω ), then B Ω ∗ ( · , · ; ω ) ≡ B Ω ( · , · ; ω ). Thus (1.28)and (1.29) follow respectively from (1.23) and (1.24). The second statement in part (a) is aconsequence of Proposition 1.6. Part (b):
Suppose now that ω is a weight of monomial type on Ω that is not necessarilyadmissible, say ω ( w ) = (cid:12)(cid:12) F ννν ( w ) (cid:12)(cid:12) ϑ ( w ) for some ννν ∈ R n , inf (cid:8) w : ϑ ( w ) : w ∈ Ω (cid:9) > . Given any χ ∈ b G A , let b ∗ ∈ Z n be a vector such that ϕ ([[ b ∗ ]] = χ . It follows from Lemma3.5(b) that a vector b ∈ Z n has the same property if and only if b − b ∗ = m · A for m ∈ Z n .We now compute the values of c , as given by Theorem 1.3 (d), corresponding to these twochoices of vectors: c ( b ) − c ( b ∗ ) = ( b ∗ − b ) A − = − m Choosing m ∈ Z n to have sufficiently large positive entries relative to c ( b ∗ ) and ννν , we canensure that every entry of the vector c ( b ) + ννν is < , so that the weight function η b givenby η b ( w ) = det( A ) − | F c ( w ) | ω ( w ) = det( A ) − | F c + ννν ( w ) | ϑ ( w )is admissible of monomial type on Ω . Set b χ = b . Invoking Proposition 1.6 yields A (Ω , η b χ ) = A (Ω ∗ , η b χ ). Therefore the two spaces share the same Bergman kernel. Sub-stituting B Ω instead of B Ω ∗ into (1.23) and (1.24) leads to the desired claim. Part (c):
This follows by combining parts (a) and (b), completing the proof. (cid:3) Examples and applications
Here we apply the conclusions of this paper to a few specific domains and obtain Bergmankernel identities and/or estimates for these.
Example 1: Complex ellipsoids.
Let Ω = n ( z , z ) ∈ C : | z | p + | z | q < (cid:9) , ω ( z , z ) ≡ , B = n ( ζ , ζ ) ∈ C : | ζ | + | ζ | < o , ω ( ζ , ζ ) ≡ . The domain Ω is an example of a complex ellipsoid and B is the unit ball. If Φ A ( z ) = ( z p , z q )then Φ A : Ω → B is a proper holomorphic mapping. In this case A = (cid:18) p q (cid:19) and A − = (cid:18) p − q − (cid:19) . Then G A ∼ = Z p ⊕ Z q = (cid:8) ( m , m ) ∈ Z : 0 ≤ m ≤ p − , ≤ m ≤ q − (cid:9) . The action of G A on C is given by ξ ( m , m ) ⊗ ( z , z ) = (cid:16) e πi m p z , e πi m q z (cid:17) . If b = ( b , b ) ∈ Z , the action of the corresponding character χ b on [ m ] ∈ G A is given by χ b ([ m ]) = exp (cid:2) πi h m , b · A − i (cid:3) = exp h πi (cid:16) m b p + m b q (cid:17)i . For χ ≡ χ b , the operator Π χ defined in (1.15) becomes in this caseΠ χ [ f ]( z , z ) = 1 pq X [ m ] ∈ G A exp h πi (cid:16) m b p + m b q (cid:17)i f (cid:16) e πi m p z , e πi m q z (cid:17) . The property (1.16) shows that the function( z , z ) ∈ Ω z b z b pq X [ m ] ∈ G A exp h πi (cid:16) m b p + m b q (cid:17)i f (cid:16) e πi m p z , e πi m q z (cid:17) is invariant under the action of G A . Finally, according to Theorem 1.3(d), we compute c = ( − b ) · A − − = (cid:18) − b − pp , − b − qq (cid:19) = ( c , c ) , and so η b ( ζ , ζ ) = ( pq ) − | ζ | − b − p ) p − | ζ | − b − q ) q − . Thus according to Theorem 1.7, B Ω ( z , w ) = p − X b =0 q − X b =0 ( z w ) − pb ( z w ) − qb B B ∗ (cid:16) ( z p , z q ) , ( w p , w q ); η b (cid:17) . Thus the Bergman kernel for the complex ellipsoid in C can be written as a sum of weightedBergman kernels in the punctured unit ball B ∗ of C . For other formulas see for example[23]. Example 2: Variants of the Hartogs triangle.
Let Ω = n ( z , z ) ∈ C : 0 < | z | p < | z | q < (cid:9) , ω ( z , z ) ≡ , ∆ = n ( ζ , ζ ) ∈ C : | ζ | < , | ζ | < o , ω ( ζ , ζ ) ≡ . Ω is a variant of the Hartog’s triangle (where p = q = 1) and ∆ is the bidisk. If Φ A ( z , z ) =( z p z − q , z q ) then Φ A : Ω → ∆ is a proper holomorphic map. This time the action of G A on C is given by ξ ( m , m ) ⊗ ( z , z ) = (cid:0) e πi m m p z , e πi m q z (cid:1) , and if b = ( b , b ) ∈ Z , theaction of the corresponding character on [ m ] ∈ G A is given by χ b ([ m ]) = exp h πi (cid:16) ( m + m ) b p + m b q (cid:17)i . As in Example 1, we can write the Bergman kernel for Ω in terms of weighted Bergmankernels on the bidisk.6.3. Example 3: Complex monomial balls.
Let P = (cid:8) p , . . . , p d (cid:9) ⊂ Z n be a spanning set of vectors in R n , each with non-negativeinteger entries. For a ∈ C n , let us define B P ( a , µ ) := n z ∈ C n : d X j =1 | F p j ( z ) − F p j ( a ) | < µ o . We refer to B P ( a , µ ) as a complex monomial ball with center a and radius µ >
0. The studyof such domains is part of a larger research program (see [16, 17, 18, 19, 20]). Using theresults of this paper we obtain sharp estimates for B B P ( a ,µ ) ( a , a ) ( i.e. diagonal estimates atthe center) which are uniform in the parameter a .We briefly sketch how this is done. Note that ifΩ = (cid:8) z ∈ C n : | F p j ( z ) − F p j ( a ) | < d − µ for 1 ≤ j ≤ d (cid:9) , Ω = (cid:8) z ∈ C n : | F p j ( z ) − F p j ( a ) | < µ for 1 ≤ j ≤ d (cid:9) , then Ω ⊂ B P ( a , µ ) ⊂ Ω . It follows from equation (1.4) that(6.1) B Ω ( a , a ) ≤ B B P ( a ,µ ) ( a , a ) ≤ B Ω ( a , a )and it suffices to obtain estimates at ( a , a ) for the comparable domains Ω , Ω . Suppose that a = ( a , . . . , a n ) / ∈ H so that each a j = 0, and let Ψ a ( z , . . . , z n ) = ( a z , . . . , a n z n ). ThenΨ − a (Ω ) = n z ∈ C n : | F p j ( z ) − | < d − µ | F p j ( a ) | − for 1 ≤ j ≤ d o , Ψ − a (Ω ) = n z ∈ C n : | F p j ( z ) − | < µ | F p j ( a ) | − for 1 ≤ j ≤ d o . Since Ψ a is a biholomorphic mapping, it follows from equation (6.1) that (cid:16) n Y j =1 a j (cid:17) − B Ψ − a (Ω ) (1 , ≤ B B P ( a ,µ ) ( a , a ) ≤ (cid:16) n Y j =1 a j (cid:17) − B Ψ − a (Ω ) (1 , . This suggests that we obtain estimates for the Bergman kernel for domains of the form(6.2) B P ( ~δ ) = n z ∈ C n : | F p j ( z ) − | < δ j for 1 ≤ j ≤ d o which are uniform in ~δ = ( δ , . . . , δ d ) ∈ (0 , ∞ ) d . In [18], we obtain a structure theorem fordomains B P ( ~δ ): after a monomial change of coordinates and depending on the componentsof ~δ the domain B P is comparable to a Cartesian product of disks with small radius and axisdeleted Reinhardt domains. More precisely, there are absolute positive constants c and C so that for every ~δ ∈ (0 , ∞ ) d , there exist the following: • A linearly independent subset { a i , · · · , a i n } ⊆ P with the corresponding n × n matrix A ∈ M n ( Z ). • A partition { , . . . , d } = J ∪ K with J = ( j , . . . , j m ), K = ( k , . . . , k d − m ), and J ∩ K = ∅ .Either J or K may be empty. • a set R [ K ] = { r j : j ∈ K } ⊆ Z n − m ,such that D m (cid:0) c ~δ ( J ) (cid:1) × W R (cid:0) c ~δ ( K ) (cid:1) ⊂ Φ A (cid:0) B P ( ~δ ) (cid:1) ⊂ D m (cid:0) C ~δ ( J ) (cid:1) × W R (cid:0) C ~δ ( K ) (cid:1) . Here ~δ ( J ) = ( δ j , · · · , δ j m ), ~δ ( K ) = ( δ k , . . . , δ k d − m ), and D m ( λ~δ ( J )) = (cid:8) ( w , · · · , w m ) ∈ C m : | w j − | < λδ j for all j ∈ J (cid:9) , and(6.3) W ∗R ( λ~δ ( I )) = (cid:8) ( w m +1 , .., w n ) ∈ C n − m : 0 < | F r j ( w m +1 , .., w n ) | < λδ j for all j ∈ I (cid:9) . (6.4)The domain W R is a Reinhardt domain which may or may not be axes-deleted. The papers[19] and [20] provide geometric estimates and a computationally effective algorithm for ob-taining sharp estimates for the Bergman kernel on the diagonal for Reinhardt domains ofthe form W R and W ∗R . Since the Bergman kernel of the polydisk D m is well understood, thestructure theorem and the results of this paper provide the desired estimates.We show how this procedure works in a simple case. Let n = 2 and P = (cid:8) (1 , , (0 , , (1 , (cid:9) so that(6.5) B P ( ~δ ) = n ( z , z ) ∈ C : | z − | < δ , | z − | < δ , | z z − | < δ o . The nature of B P ( ~δ ) depends on the sizes of δ , δ , δ . Consider the case in which δ and δ are large and δ is small. Explicitly suppose that(6.6) 32 < δ < ∞ , < δ < ∞ , < δ < . Then n | z | < δ , | z | < δ , | z z − | < δ o ⊂ B P ( ~δ ) ⊂ n | z | < δ , | z | < δ , | z z − | < δ o , so B P ( ~δ ) is comparable to B ′P ( ~δ ) = (cid:8) ( z , z ) ∈ C : | z | < δ , | z | < δ , | z z − | < δ (cid:9) . Let Φ A ( z , z ) = ( z z , z ), soΦ A ( B ′P ( ~δ )) = n ( w , w ) : | w w − | < δ , | w | < δ , | w − | < δ o . If ( z , z ) ∈ B ′P ( ~δ ) then < − δ < | z z | < δ < and so if ( w , w ) ∈ Φ A (cid:0) B P ( ~δ ) (cid:1) then < | w | < . Thus Φ A ( B ′P ) is comparable to (cid:8) ( w , w ) : | w − | < δ , δ − < | w | < δ (cid:9) = D × W . Here D is a disk in C of radius δ and W is an annulus with inner and outer radii δ − and δ respectively. It follows that under the size hypotheses in (6.6), we have c δ − log( δ δ ) < B B P ( ~δ ) ( , ) < C δ − log( δ δ )where the constants c, C are independent of ~δ . References [1] M. Artin,
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E-mail address : [email protected] Malabika PramanikDepartment of MathematicsUniversity of British Columbia1984 Mathematics RoadVancouver, BC V6T 1Z2 Canada
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