Homogeneously polyanalytic kernels on the unit ball and the Siegel domain
Christian Rene Leal-Pacheco, Egor A. Maximenko, Gerardo Ramos-Vazquez
aa r X i v : . [ m a t h . C V ] F e b Homogeneously polyanalytic kernelson the unit ball and the Siegel domain
Christian Rene Leal-Pacheco, Egor A. Maximenko,Gerardo Ramos-VazquezFebruary 3, 2021
Abstract
We prove that the homogeneously polyanalytic functions of total order m , de-fined by the system of equations D ( k ,...,k n ) f = 0 with k + · · · + k n = m , canbe written as polynomials of total degree < m in variables z , . . . , z n , with someanalytic coefficients. We establish a weighted mean value property for such func-tions, using a reproducing property of Jacobi polynomials. After that, we give ageneral recipe to transform a reproducing kernel by a weighted change of variables.Applying these tools, we compute the reproducing kernel of the Bergman space ofhomogeneously polyanalytic functions on the unit ball in C n and on the Siegel do-main. For the one-dimensional case, analogous results were obtained by Koshelev(1977), Pessoa (2014), Hachadi and Youssfi (2019).Mathematical Subject Classification (2020): 32A25, 32K99, 30H20, 46E22, 47B32,47B37, 33C45.Keywords: polyanalytic function of several variables, reproducing kernel, meanvalue property, Bergman space, Jacobi polynomial, M¨obius transform, pseudohy-perbolic distance. Contents Introduction
Bergman [7] comprehensively studied spaces of square-integrable analytic functions onone-dimensional domains, considering them as reproducing kernel Hilbert spaces (RKHS).For some of multidimensional generalizations, see [11,28,29]. Polyanalytic functions, havebeen attracted attention of many mathematicians since the beginning of the 20th century.See some of their properties, applications, and history in [1, 2, 5, 12, 14].Koshelev [16] proved that every integrable m -analytic function f on D fulfills an analogof the mean value property: f (0) = 1 π Z D f ( z ) P ( | z | ) d µ ( z ) , where P is a certain polynomial of degree m − A m ( D ) is a RKHS and gave anexplicit formula for the reproducing kernel (RK) at the arbitrary point z of the disk, usingthe M¨obius transormation ϕ z that interchanges z with the origin. Due to the formatof the journal, his explanation was extremely short: “although the class of polyanalyticfunctions is not invariant relative to fractional-linear transformations, this device is stillusefull thanks to the presence of K n ( z, z ) under the integral sign”. Pessoa [20] identified P with a certain shifted Jacobi polynomial and explained very clearly, how to translatethe reproducing property from the origin to an arbitrary point z of the disk. Namely, hefound a correcting factor that restores the polyanalyticity and converts the compositionoperator f f ◦ ϕ z into a unitary operator in A m ( D ). He also computed [19] the RK ofthe space A m ( H ) of m -analytic functions on the upper halfplane H in C . Hachadi andYoussfi [13] studied polyanalytic functions on the disk and on the entire complex plane,provided with radial measures. In particular, they computed the RK of A m ( D , µ α ), whered µ α ( z ) = π (1 − | z | ) α d µ ( z ).In this paper, we extend some of these results to the unit ball B n in C n and to theSiegel domain H n := { ξ ∈ C n : Im( ξ n ) > | ξ | + · · · + | ξ n | } .Let N = { , , . . . } , N = { , , , . . . } , n ∈ N . We employ the usual notation for themulti-indices and the notation | · | for the norm in C n , see [25, Section 1.1]. Given anopen set Ω in C n , a multi-index k = ( k , . . . , k n ) in N n and a function of the class C | k | (Ω),we denote by D k f the Wirtinger derivative of f of the order k (such derivatives werepreviously used by Poincar´e, Pompeiu, and Kolossov). In a more classical notation, D k f ( z ) := ∂ | k | ∂ k z · · · ∂ k n z n f ( z ) ( z ∈ Ω) . Let A (Ω) be the class of all analytic functions on Ω. It is defined by the system ofequations D (1 , ,..., f = 0 , D (0 , ,..., f = 0 , . . . , D (0 , ,..., f = 0 . (1.1)2iven an open subset Ω of C n and a multi-index k = ( k , . . . , k n ) in N n , k -analyticfunctions on Ω are defined [5, Section 6.4] as functions that can be represented in theform f ( z ) = k − ,...,k n − X j ,...,j n =0 g j ( z ) z j , (1.2)where all functions g j are analytic. We denote by A k (Ω) the class of all functions of theform (1.2). For simply connected domains Ω, such functions can also be characterized assmooth solutions of the system of differential equations D ( k , ,..., f = 0 , D (0 ,k ,..., f = 0 , . . . , D (0 , ,...,k n ) f = 0 . (1.3)Instead of considering polyanalytic functions of a given multi-order k , we prefer to workwith the following classes of “homogeneously polyanalytic” functions. Definition 1.1.
Let Ω be an open set in C n and m ∈ N . We say that f : Ω → C is homogeneously polyanalytic of total order m or just m -analytic , if f belongs to the class C m (Ω) and D k f = 0 for every k in N n with | k | = m . We denote by A m (Ω) the set of allsuch functions.The multi-indices k with | k | = m can be associated with m -multisubsets of the set { , . . . , n } , and the number of such multi-indices is (cid:0) n + m − m (cid:1) . For example, the class A (Ω) = A (Ω) is defined by n differential equations (1.1). Definition 1.2.
Let Ω be an open set in C n and m ∈ N . We denote by e A m (Ω) the set ofall functions f : Ω → C that can be written in the form f ( z ) = X | j | Let Ω be an open set in C n , m ∈ N , W : Ω → (0 , + ∞ ) be a continuousfunction, and d ν = W d µ . We denote by A m (Ω , ν ) the set of all functions f ∈ A m (Ω)that are square-integrable with respect to ν . We consider this space with the innerproduct inherited from L (Ω , ν ). Furthermore, we denote by A m ) (Ω , ν ) the orthogonalcomplement of A m − (Ω , ν ) in A m (Ω , ν ). Here A (Ω , ν ) := { } .3ection 3 contains a weighted mean-value property for integrable functions belongingto A m (Ω). As a consequence of this property, A m (Ω , ν ) is a RKHS. In Section 4 we showhow the RK transforms under a weighted change of variables. In Section 5 we use theprevious tools to compute the RK of A m ( B n , µ α ), where B n is the unit ball in C n and µ α is the standard radial measure on B n , see (3.13). Finally, in Section 6 we compute theRK of A m ( H n , ν α ), where H n is the standard Siegel domain in C n and ν α is a weightedLebesgue measure (see (6.1) and (6.2)).There are many recent investigations on Toeplitz operators, acting in polyanalyticBergman spaces over one-dimensional domains [8, 15, 17, 18, 24, 27]. We hope that thispaper can serve as a basis for some multidimensional generalizations, see Remarks 5.10,5.11, and 6.17.Finalizing this introduction, we mention several multidimensional results about poly-analytic spaces and kernels in other settings. Askour, Intissar, and Mouayn [4] computedthe RK of the space of polyanalytic functions on C n , square-integrable with respect tothe Gaussian weight (i.e., the polyanalytic Bargmann–Segal–Fock space). If k ∈ N n and(Ω , ν ) is a direct product of one-dimensional domains with some weights (for example,Ω = C n or Ω = D n ), then the RK of A k (Ω , ν ) can be obtained as the tensor product ofthe corresponding reproducing kernels on one-dimensional domains [13]. Ram´ırez Ortegaand S´anchez Nungaray [23] defined some polyanalytic-type spaces on the Siegel domain H n by other systems of differential equations, involving non-constant coefficients. Let Ω be an open set in C n and m ∈ N . In this section we show that A m (Ω) = e A m (Ω)and mention some other properties of A m (Ω). Lemma 2.1. Let f ∈ A m (Ω) . Then the following function is analytic: g ( z ) := X k ∈ N n | k | Let p ∈ { , . . . , n } and e p be the p -th canonical vector in N n , i.e., e p := ( δ p,s ) ns =1 ,where δ is the Kronecker’s delta. We have to show that D e p g = 0. By the product rule,( D e p g )( z ) = S ( z ) + S ( z ) + S ( z ) + S ( z ) , where S ( z ) = X | k | 4e have that S ( z ) = 0, because f ∈ A m (Ω) and | k + e p | = m in the sum defining S .Also S ( z ) = 0, because D e p z k = 0 when k p = 0. Finally, with the change of variable j = k − e p , we rewrite S ( z ) as S ( z ) = − X | j | Let f ∈ e A p (Ω) and g ∈ e A q (Ω) . Then f g ∈ e A p + q − (Ω) .Proof. This lemma follows from the elementary observation that if j ∈ N n and k ∈ N n ,with | j | < p and | k | < q , then z j z k = z j + k and | j + k | = | j | + | k | < p + q − Theorem 2.3. Let Ω be an open set in C n and m ∈ N . Then A m (Ω) = e A m (Ω) .Proof. It is well known that A (Ω) = e A (Ω). Let m > 1. It is obvious that e A m (Ω) ⊆A m (Ω). We show, by induction on m , that A m (Ω) ⊆ e A m (Ω). Suppose A p (Ω) ⊆ e A p (Ω)for every p < m and let f ∈ A m (Ω). Define g as in Lemma 2.1, then observe that f ( z ) = − X < | k | Let f ∈ A m (Ω) and a ∈ Ω . Then there exists a family of functions ( h k ) | k | First, we write f as (1.4). Then, expanding z j = ( z − a + a ) j into multi-powers of z − a and regrouping the summands, we obtain a sum of the form (2.1). Corollary 2.5. Let f ∈ A m (Ω) , a ∈ Ω , and r > such that a + r B n ⊆ Ω . Then thereexists a family ( β ) j,k ∈ N n , | k | Let Ω be a connected open set in C n , Ω be an open subset of Ω , and f ∈ A m (Ω) such that f ( z ) = 0 for every z in Ω . Then f ( z ) = 0 for every z in Ω .Proof. For k in N n , the uniqueness property of k -analytic functions is proven in [5, Sec-tion 6.4]. The uniqueness property for m -analytic functions is a corollary of this fact,since A m (Ω) ⊆ A ( m,...,m ) (Ω).To finish this section, we will show that the class A m with m in N is closed underlinear changes of variables, while the classes A k with k ∈ N n are generally not. Proposition 2.7. Let M be an invertible n × n complex matrix and f in A m (Ω) . Define g : M Ω → C by g ( z ) := f ( M − z ) . Then g ∈ A m ( M Ω) .Proof. Theorem 2.3 allows us to work with e A m instead of A m . Let f be like in Defini-tion 1.2. Then g ( z ) = X | j | Proposition 2.8. Let n ≥ , Ω be an open subset of C n , k ∈ N n , k = (1 , , . . . , . Thenthere exists a function f in A k (Ω) and an invertible matrix M in C n × n such that thefunction g : M Ω → C , defined by g ( z ) := f ( M − z ) , does not belong to A k ( M Ω) .Proof. To simplify the notation, we suppose that k > 1. The general case is analogous.Define M in such a manner that M − z = ( z + z , z − z , z , . . . , z n ) . Consider f : Ω → C , f ( z ) := z k − z k − . Then g ( z ) = ( z + z ) k − ( z − z ) k − . In the expansion of the last polynomial, one of the terms is z k + k − . Since k + k − >k − 1, we obtain g / ∈ A k ( M Ω), though f ∈ A k (Ω).6 Weighted mean value property In this section we prove that the value of a m -analytic function at the center of the unitball B n can be expressed as the integral of this function over the ball, with a certainreal radial weight (Theorem 3.3). Similar results in the one-dimensional case were provedin [13, 16, 20]. Jacobi polynomials and their reproducing property Some integrals over the unit ball, written in the spherical coordinates, reduce to integralsover the unit interval (0 , 1) with weights of power type at the boundary points 0 and 1.Thereby Jacobi polynomials naturally appear. They can be defined by Rodrigues formula: P ( ξ,η ) m ( x ) := ( − m m m ! (1 − x ) − ξ (1 + x ) − η d m d x m (cid:16) (1 − x ) m + ξ (1 + x ) m + η (cid:17) . (3.1)Here are well-known explicit formulas for P m : P ( ξ,η ) m ( x ) = m X s =0 (cid:18) ξ + η + m + ss (cid:19)(cid:18) ξ + mm − s (cid:19) (cid:18) x − (cid:19) s (3.2)= m X s =0 ( − s (cid:18) ξ + η + m + ss (cid:19)(cid:18) η + mm − s (cid:19) (cid:18) x + 12 (cid:19) s . (3.3)If ξ, η > − 1, then ( P ( ξ,η ) m ) ∞ m =0 is an orthogonal family on the interval ( − , 1) with respectto the weight (1 − x ) ξ (1 + x ) η . Using (3.1) and integrating by parts yields the followingintegral formula: Z − P ( α,β +1) m ( x ) (1 − x ) α (1 + x ) β d x = 2 α + β +1 ( − m B( α + m + 1 , β + 1) . (3.4) Definition 3.1. Let m ∈ N and α, β > − 1. We denote by R ( α,β ) m the following polynomial: R ( α,β ) m ( t ) := ( − m B( α + 1 , β + 1)B( α + m + 1 , β + 1) P ( α,β +1) m (2 t − . (3.5)Equivalently, by the symmetry relation for Jacobi polynomials, R ( α,β ) m ( t ) = B( α + 1 , β + 1)B( α + m + 1 , β + 1) P ( β +1 ,α ) m (1 − t ) . (3.6)Combining (3.5) with (3.3) or (3.6) with (3.2), we get more explicit formulas for R ( α,β ) m : R ( α,β ) m ( t ) = Γ( α + 1) Γ( β + m + 2)Γ( α + β + 2) Γ( α + m + 1) m X s =0 ( − s Γ( α + β + m + s + 2) s ! ( m − s )! Γ( β + s + 2) t s (3.7)= Γ( α + 1) Γ( β + m + 2)Γ( α + β + 2) ( α + m ) m ! m X s =0 ( − s (cid:0) ms (cid:1) B( α + m, β + s + 2) t s . (3.8)7he next simple result was proven in [6] using the orthogonality of the Jacobi polynomialsand formula (3.4). Previously, Hachadi and Youssfi [13, formula (5.7)] gave another prooffor the case β = 0. Proposition 3.2. Let m ∈ N and α, β > − . Then for every univariate polynomial h with complex coefficients and deg( h ) ≤ m , α + 1 , β + 1) Z h ( t ) R ( α,β ) m ( t ) (1 − t ) α t β d t = h (0) . (3.9)The polynomials of degree ≤ m , considered as square-integrable functions on theinterval (0 , 1) with the normalized weight α +1 ,β +1) (1 − t ) α t β , form a RKHS. Formula (3.9)means that R ( α,β ) m is the RK of this space at the point 0.As a particular case of (3.9), for every k in N with k ≤ m ,1B( α + 1 , β + 1) Z R ( α,β ) m ( t )(1 − t ) α t β + k d t = δ k, . (3.10) Weighted mean value property of homogeneously polyanalyticfunctions We denote by µ the Lebesgue measure on C n , by S n the unit sphere in C n , and by µ S n the (non-normalized) area measure on S n . It is well known [25, Section 1.4] that µ ( B n ) = π n n ! , µ S n ( S n ) = 2 π n ( n − , and Z S n ζ j ζ k d µ S n ( ζ ) = 2 π n j !( n − | j | )! · δ j,k ( j, k ∈ N n ) . (3.11)Given an integrable function f on B n , its integral over B n can be written as Z B n f d µ = Z r n − (cid:18)Z S n f ( rζ ) d µ S n ( ζ ) (cid:19) d r. (3.12)For α > − 1, we denote by µ α the Lebesgue measure on B n with the standard radialweight: d µ α ( z ) = c α (1 − | z | ) α d µ ( z ) . (3.13)The normalizing constant c α is chosen so that µ α ( B n ) = 1: c α := Γ( n + α + 1) π n Γ( α + 1) . (3.14)8 heorem 3.3. Let f ∈ A m ( B n ) such that f ∈ L ( B n , µ α ) . Then f (0) = Z B n f ( z ) R ( α,n − m − ( | z | ) d µ α ( z ) . (3.15) Proof. We represent f in the form (2.2) with a = 0, then make the change of variables z = rζ with 0 ≤ r < ζ ∈ S n : f ( z ) = X j ∈ N n X k ∈ N n | k | 1. Passing to this limit andusing (3.10), we finally obtain I = Γ( n + α + 1)Γ( α + 1) X k ∈ N n | k | Let Ω be an open subset of C n , f ∈ A m (Ω) , a ∈ Ω , and r > such that a + r B n ⊆ Ω . Suppose that f ∈ L ( a + r B n , µ ) . Then f ( a ) = n ! π n r n Z a + r B n f ( z ) R (0 ,n − m − (cid:18) | z − a | r (cid:19) d µ ( z ) . (3.17)9 ergman spaces of homogeneously polyanalytic functions In the rest of this section, we suppose that Ω, m , W , ν are like in Definition 1.3. Us-ing (3.17), it is easy to prove the upcoming Lemma 3.5 and Proposition 3.6. See similarproofs for the one-dimensional case in [6, Lemma 4.3, Proposition 4.4]. Lemma 3.5. Let K be a compact subset of Ω . Then there exists a number C m,W,K > such that for every f in A m (Ω , ν ) and every z in K , | f ( z ) | ≤ C m,W,K k f k A m (Ω ,ν ) . (3.18) Proposition 3.6. A m (Ω , ν ) is a RKHS. As a corollary, the spaces A m ) (Ω , ν ) are also RKHS. Proposition 3.7. In the conditions of Definition 1.3, suppose additionally that Ω isbounded and ν is finite. Then L (Ω , ν ) = ∞ M m =1 A m ) (Ω , ν ) . (3.19) Proof. This is a simple consequence of three facts: 1) the continuous functions with com-pact supports form a dense subset of L (Ω , ν ); 2) by the Stone–Weierstrass theorem, everycontinuous function on the closure of Ω can be uniformly approximated by polynomialsin z , . . . , z n , z , . . . , z n ; and 3) the norm of L (Ω , ν ) can be estimated from above by aconstant multiple of the maximum-norm.In the one-dimensional case, the “true- m -analytic” spaces A m ) were studied by Ra-mazanov [22] and Vasilevski [26, 27]. According to [26] (see also another proof in [18]),the decomposition (3.19) holds for the poly-Fock space A m ( C , e −| z | d µ ). On the otherhand, if Ω is the upper halfplane H with the Lebesgue measure, then L ( H ) decomposesinto the orthogonal sum of the spaces A m ) ( H ) and their conjugates [27, Theorem 3.3.5],and (3.19) fails. It is natural to ask if Proposition 3.7 remains true if ν (Ω) < + ∞ , withoutassuming Ω to be bounded. In this section we show how to transform a RK using a weighted change of variables.First, we deal with abstract positive kernels [3], then we consider reproducing kernels inHilbert spaces.Let X be a non-empty set. We denote by C X the complex vector space of all functions X → C with pointwise operations. A family ( K x ) x ∈ X with values in C X is called a positivekernel on X if for every m in N , every x , . . . , x m in X and every α , . . . , α m in C , m X r,s =1 α r α s K x r ( x s ) ≥ . roposition 4.1. Let X, Y be non-empty sets, ψ : Y → X and J : Y → C be somefunctions, and ( K x ) x ∈ X be a positive kernel on X . Then the family ( L u ) u ∈ Y , defined by L u ( v ) := J ( u ) J ( v ) K ψ ( u ) ( ψ ( v )) , is a positive kernel on Y .Proof. Let m ∈ N , u , . . . , u m ∈ Y , α , . . . , α m ∈ C . For every s in { , . . . , m } put x s := ψ ( u s ) and β s := J ( u s ) α s . Then m X r,s =1 α r α s L u r ( u s ) = m X r,s =1 β r β s K x r ( x s ) ≥ . Let X be a non-empty set. We say that H is a Hilbert space of functions on X if H is a vector subspace of C X , provided with an inner product and complete with respect tothe corresponding norm. Furthermore, if x ∈ X , K ∈ H and h f, Ki = f ( x ) for every f in H , then we say that K is a reproducing kernel of H at the point x . In case of existence,this function is unique. Proposition 4.2. Let X, Y be non-empty sets, ψ : Y → X and J : Y → C be somefunctions, H be a Hilbert space of functions over X , H be a Hilbert space of functionsover Y , and ( U f )( z ) := J ( z ) f ( ψ ( z )) be a well-defined unitary operator mapping H onto H . Suppose that u ∈ Y and K be thereproducing kernel of H at the point ψ ( u ) . Then the function L : Y → C , defined by thefollowing rule, is the reproducing kernel of H at the point u : L ( v ) := J ( u ) J ( v ) K ( ψ ( v )) . Proof. Let g ∈ H and f = U − g . Then g ( u ) = J ( u ) f ( ψ ( u )) = J ( u ) h f, Ki H = h g, J ( u ) U Ki H . Defining L by L ( v ) = J ( u )( U K )( v ) = J ( u ) J ( v ) K ( ψ ( v )), we get the RK of H at u . Proposition 4.3. Let X, Y be non-empty sets, ψ : Y → X and J : Y → C be somefunctions, H be a Hilbert space of functions over X with reproducing kernel ( K x ) x ∈ X , H be a Hilbert space of functions over Y , and ( U f )( z ) := J ( z ) f ( ψ ( z )) be a well-defined unitary operator mapping H onto H . Then H is a RKHS, and itsreproducing kernel ( L u ) u ∈ Y is given by L u ( v ) = J ( u ) J ( v ) K ψ ( u ) ( ψ ( v )) . roof. Apply Proposition 4.2 at every point u of Y .As a simple application of the this scheme, let us express the Berezin transform in H via the Berezin transform in H . Given a Hilbert space H , we denote by B ( H ) theC*-algebra of all bounded linear operators acting in H . Given a set X , we denote by B ( X ) the Banach space of all bounded functions on X , with the supremum norm. If H is a RKHS over X and its RK satisfies k K x k H = 0 for every x in X , then the Berezintransform Ber H : B ( H ) → B ( X ) is defined byBer H ( A )( x ) := h AK x , K x i H h K x , K x i H ( A ∈ B ( H ) , x ∈ X ) . Proposition 4.4. In the conditions of Proposition 4.3, suppose that k K x k H = 0 forevery x in X and J ( u ) = 0 for every u in Y . Then Ber H ( A )( u ) = Ber H ( U ∗ AU )( ψ ( u )) ( A ∈ B ( H ) , u ∈ Y ) . Proof. As we have seen in Proposition 4.2, L u ( v ) = J ( u )( U K ψ ( u ) )( v ). Therefore,Ber H ( A )( u ) = h AL u , L u i H k L u k = | J ( u ) | h AU K ψ ( u ) , U K ψ ( u ) i H | J ( u ) | k U K ψ ( u ) k = h U ∗ AU K ψ ( u ) , K ψ ( u ) i H k K ψ ( u ) k = Ber H ( U ∗ AU )( ψ ( u )) . Corollary 4.5. In the conditions of Proposition 4.4, suppose that Ber H is injective.Then Ber H is also injective. Moreover, if ψ is a bijection, than the injectivity of Ber H is equivalent to the injectivity of Ber H . In this section we consider the domain Ω = B n with the standard radial measure µ α , givenby (3.13). Using the weighted mean value property and appropriate unitary operators,we compute the RK of A m ( B n , µ α ). On the unit ball biholomorphisms For a fixed a in B n \ { } , we denote by ϕ a the function B n → B n , defined by ϕ a ( z ) := a − h z,a ih a,a i a − p − | a | (cid:16) z − h z,a ih a,a i a (cid:17) − h z, a i . (5.1)12or a = 0, ϕ a ( z ) := z . It is well known [25, Theorem 2.2.2] that for every a in B n , ϕ a is abiholomorphism of B n , ϕ a ( ϕ a ( z )) = z for every z in B n , ϕ a (0) = a , ϕ a ( a ) = 0, and1 − h ϕ a ( z ) , ϕ a ( w ) i = (1 − h a, a i )(1 − h z, w i )(1 − h z, a i )(1 − h a, w i ) . (5.2)Here are particular cases of (5.2), with w = z and w = 0, respectively:1 − | ϕ a ( z ) | = (1 − | a | )(1 − | z | ) | − h z, a i| , (5.3)1 − h ϕ a ( z ) , a i = 1 − | a | − h z, a i . (5.4)The real Jacobian of ϕ a is [29, Lemma 1.7]( J R ϕ a )( z ) = (cid:18) − | a | | − h z, a i| (cid:19) n +1 . (5.5)We denote by ρ B n ( z, w ) the expression | ϕ z ( w ) | , known as the pseudohyperbolic distance between z and w , see [29, Corollary 1.22] or [9]. Formula (5.3) provides a simple recipeto compute ρ B n ( z, w ). A factor to preserve the polyanalyticity Definition 5.1. Given a in B n , we define p m,a : B n → C by p m,a ( z ) := (cid:18) − h a, z i − h z, a i (cid:19) m − . In the one-dimensional case, the function p m,a was introduced and studied by Pes-soa [20]. As it is shown in the proof of Lemma 5.3, the main purpose of p m,a is toeliminate the denominators in the multi-powers of ϕ a ( z ). Lemma 5.2. For every a, z in B n , | p m,a ( z ) | = 1 , (5.6) p m,a ( ϕ a ( z )) p m,a ( z ) = 1 . (5.7) Proof. Formula (5.6) follows directly from the definition of p m,a . Identity (5.7) is easy toverify using (5.4). Lemma 5.3. Let a ∈ B n and f ∈ A m ( B n ) . Then ( f ◦ ϕ a ) · p m,a ∈ A m ( B n ) . roof. Let f be of the form (1.4). Denote by N a ( z ) the numerator of (5.1); it is apolynomial of degree 1 in z , . . . , z n . Then, f ( ϕ a ( z )) p m,α ( z ) = (cid:18) − h a, z i − h z, a i (cid:19) m − X | j | 1, with some analytic coefficients. A factor to preserve the norm Remark 5.4. In the upcoming formula for g α,a and in some other formulas of this paper,we work with (non necesarily integer) powers of complex numbers. Given t in C \ { } and β in C , we define t β as exp( β log( t )), where log( t ) = log R | t | + i arg( t ), log R | t | is the reallogarithm of | t | , and arg( t ) is the principal argument of t , belonging to ( − π, π ].Given a in B n , we denote by g α,a the following function B n → C : g α,a ( z ) := (1 − | a | ) n +1+ α (1 − h z, a i ) n +1+ α . (5.8)This function and their properties stated below appear in Vukoti´c [28]. See also [29,Proposition 1.13] or [10, formula (2.4)]. By (5.4), g α,a ( ϕ a ( z )) g α,a ( z ) = 1 . (5.9)By (5.9), (5.3), and (5.5), | g α,a ( ϕ a ( w )) | ( J R ϕ a )( w )(1 − | ϕ a ( w ) | ) α = (1 − | w | ) α . (5.10)Using (5.10) and the change of variables w = ϕ a ( z ), one easily shows that for every f in f ∈ L ( B n , µ α ), k ( f ◦ ϕ a ) · g α,a k L ( B n ,µ α ) = k f k L ( B n ,µ α ) . (5.11) A weighted shift operator preserving A m ( B n , µ α ) Definition 5.5. Given a in B n , we define U a : A m ( B n , µ α ) → A m ( B n , µ α ) by( U a f )( z ) := f ( ϕ a ( z )) p m,a ( z ) g α,a ( z ) . roposition 5.6. Let a ∈ B n . Then U a is a unitary operator in A m ( B n , µ α ) , and U a = I .Proof. Given f in A m ( B n , µ α ), Lemma 5.3 assures that U a f ∈ A m ( B n ). Formula (5.11),combined with (5.6), implies that U a is an isometry. Finally, (5.7) and (5.9) yield theinvolutive property U a = I . Computation of the RK on the unit ball Recall that R ( α,β ) m is defined by (3.5) and ρ B n ( z, w ) denotes | ϕ z ( w ) | . Theorem 5.7. Let n, m ∈ N and α > − . Then for every z in B n , the following function K z is the reproducing kernel of A m ( B n , µ α ) at the point z : K z ( w ) = (1 − h z, w i ) m − (1 − h w, z i ) n + m + α R ( α,n − m − ( ρ B n ( z, w ) ) . (5.12) Proof. For z = 0, the function defined by the right-hand side of (5.12) simplifies to K ( w ) = R ( α,n − m − ( | w | ) . Theorem 3.3 means that K is indeed the RK at the point 0. Now, for z in B n , we applyProposition 4.2 with H = H = A m ( B n ), ϕ z instead of ψ , and J z := p m,z g α,z . Since ϕ z ( z ) = 0, we obtain K z ( w ) = J z ( z ) J z ( w ) K ( ϕ z ( w )) . (5.13)It is easy to see that J z ( z ) = (1 − | z | ) − n +1+ α . So, after some simplifications, we arriveat (5.12): K z ( w ) = 1(1 − | z | ) n +1+ α (cid:18) − h z, w i − h w, z i (cid:19) m − (1 − | z | ) n +1+ α (1 − h w, z i ) n +1+ α R ( α,n − m − ( | ϕ z ( w ) | )= (1 − h z, w i ) m − (1 − h w, z i ) n + m + α R ( α,n − m − ( ρ B n ( z, w ) ) . Formula (5.12) is a natural generalization of previous results: [16, 19] for n = 1 and α = 0, [13] for n = 1 and α > − 1, and [29, Theorem 2.7] for m = 1. Corollary 5.8. Let n, m ∈ N and α > − . Then for every z in B n , k K z k A m ( B n ,µ α ) = K z ( z ) = (cid:18) n + m − n (cid:19) B( α + 1 , n )B( α + m, n ) 1(1 − | z | ) n + α +1 . (5.14)15 emark 5.9. We get other formulas, equivalent to (5.12), using (5.3) and (3.2): K z ( w ) = (1 − h z, w i ) m − (1 − h w, z i ) n + m + α ( − m − B( α + 1 , n )B( α + m, n ) P ( α,n ) m − (2 ρ B n ( z, w ) − 1) (5.15)= (1 − h z, w i ) m − (1 − h w, z i ) n + m + α ( − m − B( α + 1 , n )B( α + m, n ) P ( α,n ) m − (cid:18) − − | z | )(1 − | w | ) | − h w, z i| (cid:19) (5.16)= (1 − h z, w i ) m − (1 − h w, z i ) n + m + α ( − m − Γ( α + 1)Γ( α + n + 1) ( m − ×× m − X s =0 ( − s (cid:18) m − s (cid:19) Γ( α + m + n + s )Γ( α + s + 1) (cid:18) (1 − | z | )(1 − | w | ) | − h w, z i| (cid:19) s . (5.17) Remark 5.10. If M is a unitary n by n matrix, then the RK computed in Theorem 5.7is invariant under the simultaneous action of M in both arguments: K Mz ( M w ) = K z ( w ) ( z, w ∈ B n ) . Therefore, by [18, Proposition 4.1], the space A m ( B n , µ α ) is invariant under the action ofthe rotation operator ( R M f )( z ) := f ( M − z ) . This follows also directly from Proposition 2.7. Notice that the unitary matrices includepermutation matrices, diagonal matrices with unimodular complex entries, and real rota-tions in any two coordinates. Remark 5.11. Generalizing ideas of this section, it is possible to construct a unitaryweighted shift operator U ϕ acting in A m ( B n , µ α ), for every biholomorphism ϕ of B n .The next result was published by Engliˇs [10, Section 2] for RKHS of harmonic func-tions. We reformulate it for our situation and recall the idea of the proof. Proposition 5.12. Let H = A m ( B n , µ α ) , with n ≥ and m ≥ . Then Ber H is notinjective.Proof. The functions f ( z ) = z and g ( z ) = z are linearly independent elements of H .Therefore, the operator Sh = h h, f i H f − h h, g i H g is not zero, but the Berezin transformmaps it into the zero function. 16 Reproducing kernel on the Siegel domain Let n, m ∈ N and α > − 1. In this section we compute the RK of the space A m ( H n , ν α ),where H n is the standard Siegel domain (which can be considered as an unboundedrealization of the unit ball) and ν α is a usual weighted measure on H n : H n := { ξ = ( ξ ′ , ξ n ) ∈ C n − × C : Im( ξ n ) − | ξ ′ | > } , (6.1)d ν α ( ξ ) := c α ξ n ) − | ξ ′ | ) α d µ ( ξ ) . (6.2)For this purpose, we will construct a unitary operator V : A m ( B n , µ α ) → A m ( H n , ν α ),using some recipes from [21, Section 2] and an analog of the Pessoa factor which helps topreserve the polyanalyticity. Cayley transform Following [21, Section 2], we employ the biholomorphism ω : B n → H n defined by ω ( z ) := (cid:18) i z z n , . . . , i z n − z n , i 1 − z n z n (cid:19) . Its inverse ψ : H n → B n is given by ψ ( ξ ) := (cid:18) − ξ − i ξ n , . . . , − ξ n − − i ξ n , ξ n − i ξ n (cid:19) . By a direct computation,1 − h ψ ( ξ ) , ψ ( η ) i = 4 ξ n − η n − h ξ ′ , η ′ i (1 − iξ n )(1 + iη n ) . (6.3)In particular, 1 − | ψ ( ξ ) | = 4 Im( ξ n ) − | ξ ′ | | − iξ n | . (6.4)The complex Jacobian matrices of ψ and ω are triangular, and their determinants areeasy to compute:( J C ω )( z ) = − n (1 + z n ) n +1 , ( J C ψ )( ξ ) = − ( − n (1 − i ξ n ) n +1 . (6.5)Therefore, the real Jacobians of ω and ψ are( J R ω )( z ) = 4 | z n | n +1) , ( J R ψ )( ξ ) = 4 n | − i ξ n | n +1) . (6.6)17 seudohyperbolic distance on the Siegel domain Definition 6.1. Define a distance on H n by ρ H n ( ξ, η ) := ρ B n ( ψ ( ξ ) , ψ ( η )) . (6.7)The following proposition provides an efficient formula to compute ρ H n ( ξ, η ). Proposition 6.2. For every ξ, η in H n , − ρ H n ( ξ, η ) = (Im( ξ n ) − | ξ ′ | )(Im( η n ) − | η ′ | ) (cid:12)(cid:12) ξ n − η n − h ξ ′ , η ′ i (cid:12)(cid:12) . (6.8) Proof. Substitute ψ ( ξ ) and ψ ( η ) instead of z and w in (5.3):1 − ρ H n ( ξ, η ) = 1 − ρ B n ( ψ ( ξ ) , ψ ( η )) = (1 − | ψ ( ξ ) | )(1 − | ψ ( η ) | ) | − h ψ ( ξ ) , ψ ( η ) i| . Applying (6.3) and (6.4) we obtain (6.8). Remark 6.3. For n = 1, formulas (6.7) and (6.8) simplify to ρ H ( ξ, η ) = | ξ − η || ξ − η | , − ρ H ( ξ, η ) = 4 Im( ξ ) Im( η ) | ξ − η | . (6.9) A factor to preserve the norm when passing from H n to B n The material of this subsection is equivalent to some computations from [21, Section 2].Define h α : H n → C by h α ( ξ ) := (cid:18) − i ξ n (cid:19) n + α +1 . (6.10) Lemma 6.4. For every ξ in H n , | h α ( ξ ) | = 4(1 − | ψ ( ξ ) | ) α ( J R ψ )( ξ )(Im( ξ n ) − | ξ ′ | ) α . (6.11) For any z in B n , | h α ( ω ( z )) | (cid:18) − | z | | z n | (cid:19) α ( J R ω )( z ) = (1 − | z | ) α . (6.12) Proof. Formula (6.11) is obtained by (6.4) and (6.6). Then (6.12) follows from (6.11) andthe well-known formula for the Jacobian of the inverse function.18 emma 6.5. Let u ∈ L ( B n , µ α ) . Then k ( u ◦ ψ ) · h α k L ( H n ,ν α ) = k u k L ( B n ,µ α ) . (6.13) Proof. First, using (6.4), we observe that the change of variable z = ψ ( ζ ) transforms theweight function in the following way:(Im( ζ n ) − | ζ ′ | ) α = (cid:18) − | z | | z n | (cid:19) α . Apply this change of variables in the integral: k ( u ◦ ψ ) · h α k L ( H n ,ν α ) = c α Z H n | u ( ψ ( ζ )) | | h α ( ζ ) | (Im( ζ n ) − | ζ ′ | ) α d µ ( ζ )= c α Z B n | u ( z ) | | h α ( ω ( z )) | (cid:18) − | z | | z n | (cid:19) α ( J R ω )( z ) d µ ( z )= c α Z B n | u ( z ) | (1 − | z | ) α d µ ( z ) = k u k L ( B n ,µ α ) . A factor to preserve the polyanalyticity when passing from theunit ball to the Siegel domain Definition 6.6. Define q m : H n → C , q m ( ξ ) := (cid:18) ξ n − i ξ n (cid:19) m − . Lemma 6.7. Let f ∈ A m ( B n ) . Then ( f ◦ ψ ) · q m ∈ A m ( H n ) .Proof. This proof is similar to the proof of Lemma 5.3. The main idea is that the factor(1 + i ξ n ) m − , appearing in the numerator of q m ( ξ ), cancels the denominators of the ex-pressions ψ ( ξ ) j , where | j | < m . We represent f in the form (1.4), compose with ψ , andmultiply by q m : u ( ξ ) := ( f ◦ ψ )( ξ ) q m ( ξ ) = X | j |≤ m h j ( ψ ( ξ )) n − Y s =1 (2i ξ s ) j s (1 + i ξ n ) j s ! (1 − i ξ n ) j n (1 + i ξ n ) j n (1 + i ξ n ) m − (1 − i ξ n ) m − = X | j |≤ m h j ( ψ ( ξ ))(1 − i ξ n ) m − n − Y s =1 (2i ξ s ) j s ! (1 − i ξ n ) j n (1 + i ξ n ) m −| j |− . For each j , the corresponding summand is the product of an analytic function by apolynomial in ξ , . . . , ξ n of total degree m − emark 6.8. Another way to prove Lemma 6.7, computing D j u , seems to be morecomplicated. We will show it only for n = 2 and m = 2. In this case, u ( ξ ) = 1 + i ξ − i ξ f (cid:18) − ξ − i ξ , ξ − i ξ (cid:19) , By the well-known chain rule and product rule for Wirtinger derivatives,( D (1 , u )( ξ ) = 2i( D (1 , f )( ψ ( ξ ))1 − i ξ , ( D (0 , u )( ξ ) = i f ( ψ ( ξ ))1 − i ξ + 2 (cid:16)(cid:16) ξ D (1 , − i D (0 , (cid:17) f (cid:17) ( ψ ( ξ ))(1 − i ξ )(1 + i ξ ) , ( D (2 , u )( ξ ) = − D (2 , f )( ψ ( ξ ))(1 − i ξ )(1 + i ξ ) , ( D (1 , u )( ξ ) = 4 (cid:16)(cid:16) i ξ D (2 , + D (1 , (cid:17) f (cid:17) ( ψ ( ξ ))(1 − i ξ )(1 + i ξ ) , ( D (0 , u )( ξ ) = 4 (cid:16)(cid:16) ξ D (2 , − ξ D (1 , − D (0 , (cid:17) f (cid:17) ( ψ ( ξ ))(1 − i ξ )(1 + i ξ ) . Since f ∈ A ( B n ), we conclude that u ∈ A ( H n ). A weighted change of variables which unitarily maps A m ( B n , µ α ) onto A m ( H n , ν α ) Definition 6.9. Define V : A m ( B n , µ α ) → A m ( H n , ν α ) by V u := ( u ◦ ψ ) · h α · q m , i.e.,( V u )( ξ ) := u ( ψ ( ξ )) h α ( ξ ) q m ( ξ ) . Proposition 6.10. V is a well-defined unitary operator A m ( B n , µ α ) → A m ( H n , ν α ) .Proof. Lemma 6.7 assures that V u ∈ A m ( H n ) for every u in A m ( B n , µ α ). Lemma 6.5,combined with the identity | q m ( ξ ) | = 1, provides the isometric property of V . It is easyto verify that the adjoint operator V ∗ acts by( V ∗ f )( z ) = f ( ω ( z )) h α ( ω ( z )) q m ( ω ( z )) , (6.14)and that V ∗ is the inverse operator to V . 20 omputation of the RK on the Siegel domain We define t β via the principal argument of t , see Remark 5.4. The formulas ( tu ) β = t β u β and ( t/u ) β are not always true. Let us recall some sufficient conditions for these formulasto be true. Lemma 6.11. Let t, u ∈ C \ { } and β ∈ C .1. If Re( t ) > and Re( u ) > , then ( tu ) β = t β u β .2. If Re( t ) > and Re( t/u ) > , then ( t/u ) β = t β /u β .Proof. 1. The assumptions on t and u imply that arg( tu ) = arg( t ) + arg( u ).2. Follows from part 1 applied to t and u/t . Lemma 6.12. Let ξ, η ∈ H n and β ≥ . Then (1 − h ψ ( ξ ) , ψ ( η ) i ) β = 4 β (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) β (1 − i ξ n ) β (1 + i η n ) β . Proof. Due to (6.3), 1 − h ψ ( ξ ) , ψ ( η ) i = t/ ( uv ), where t := 4 (cid:18) ξ n − η n − h ξ ′ , η ′ i (cid:19) , u := 1 − i ξ n , v := 1 + i η n . Since | ψ ( ξ ) | < | ψ ( η ) | < 1, we obtainRe( t/ ( uv )) = Re(1 − h ψ ( ξ ) , ψ ( η ) i ) > . Furthermore, Re( u ) = 1 + Im( ξ n ) > v ) = 1 + Im( η n ) > 0. So, by Lemma 6.11, (cid:18) tuv (cid:19) β = t β ( uv ) β = t β u β v β . Theorem 6.13. Let n, m ∈ N and α > − . Then for every ξ in H n , the followingfunction e K ξ is the reproducing kernel of A m ( H n , ν α ) at the point ξ : e K ξ ( η ) = (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) m − (cid:16) η n − ξ n − h η ′ , ξ ′ i (cid:17) n + m + α R ( α,n − m − ( ρ H n ( ξ, η ) ) . (6.15) Proof. Due to Proposition 6.10, we can apply Proposition 4.1 with H = A m ( B n , µ α ), H = A m ( H n , ν α ), and J ( ξ ) := h α ( ξ ) q m ( ξ ). So, for every ξ in H n , the next function is theRK of A m ( H n , ν α ) associated to the point ξ : e K ξ ( η ) = h α ( ξ ) q m ( ξ ) h α ( η ) q m ( η ) K ψ ( ξ ) ( ψ ( η )) . K : e K ξ ( η ) = h α ( ξ ) q m ( ξ ) h α ( η ) q m ( η ) (1 − h ψ ( ξ ) , ψ ( η ) i ) m − (1 − h ψ ( η ) , ψ ( ξ ) i ) n + m + α R ( α,n − m − ( ρ H n ( ξ, η ) ) . Then, substitute the definitions of h α , q m and use Lemma 6.12: e K ξ ( η ) = R ( α,n − m − ( ρ H n ( ξ, η ) ) 2 n + α +1 (1 + i η n ) m − (1 − i η n ) n + m + α n + α +1 (1 − i ξ n ) m − (1 + i ξ n ) n + m + α × m − (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) m − (1 − i ξ n ) m − (1 + i η n ) m − (1 + i ξ n ) n + m + α (1 − i η n ) n + m + α n + m + α (cid:16) η n − ξ n − h η ′ , ξ ′ i (cid:17) n + m + α . Simplifying this expression we obtain the right-hand side of (6.15). Corollary 6.14. Let n, m ∈ N and α > − . Then for every ξ in H n , k e K ξ k A m ( H n ,ν α ) = e K ξ ( ξ ) = (cid:18) n + m − n (cid:19) B( α + 1 , n )B( α + m, n ) 1(Im( ξ n ) − | ξ ′ | ) α + n +1 . (6.16) Remark 6.15. Analogously to the case of the unit ball, we get some formulas equivalentto (6.15), using (6.8) and (3.2): e K ξ ( η ) = (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) m − (cid:16) η n − ξ n − h η ′ , ξ ′ i (cid:17) n + m + α ( − m − B( α + 1 , n )B( α + m, n ) P ( α,n ) m − (2 ρ H n ( ξ, η ) − 1) (6.17)= (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) m − (cid:16) η n − ξ n − h η ′ , ξ ′ i (cid:17) n + m + α ( − m − B( α + 1 , n )B( α + m, n ) ×× P ( α,n ) m − − ξ n ) − | ξ ′ | )(Im( η n ) − | η ′ | ) (cid:12)(cid:12) ξ n − η n − h ξ ′ , η ′ i (cid:12)(cid:12) ! (6.18)= (cid:0) ξ n − η n − h ξ ′ , η ′ i (cid:1) m − (cid:16) η n − ξ n − h η ′ , ξ ′ i (cid:17) n + m + α ( − m − Γ( α + 1)Γ( α + n + 1) ( m − ×× m − X s =0 ( − s (cid:18) m − s (cid:19) Γ( α + m + n + s )Γ( α + s + 1) (Im( ξ n ) − | ξ ′ | )(Im( η n ) − | η ′ | ) (cid:12)(cid:12) ξ n − η n − h ξ ′ , η ′ i (cid:12)(cid:12) ! s . (6.19) Remark 6.16. In the case n = 1, i.e., for the upper halfplane H , formula (6.15) simplifiesto e K ξ ( η ) = (cid:0) ξ − η (cid:1) m − (cid:16) η − ξ (cid:17) m + α +1 R ( α, m − (cid:18) | ξ − η | | ξ − η | (cid:19) ( ξ, η ∈ H ) . (6.20)22n particular, for α = 0, this expression coincides with formula [19, Corollary 2.5] obtainedby another method. Remark 6.17. Generalizing ideas of this paper, it is possible to associate a unitaryoperator (namely, a certain weighted shift) in A m ( H n , ν α ) to every biholomorphism of theSiegel domain H n . 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Springer-Verlag, NewYork (2005).Christian Rene Leal-PachecoCentro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico NacionalDepartamento de Matem´aticasApartado Postal 07360Ciudad de M´exicoMexicoe-mail: [email protected] https://orcid.org/0000-0001-5738-4904 Egor A. MaximenkoInstituto Polit´ecnico NacionalEscuela Superior de F´ısica y Matem´aticas 25partado Postal 07730Ciudad de M´exicoMexicoe-mail: [email protected] https://orcid.org/0000-0002-1497-4338 Gerardo Ramos-VazquezCentro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico NacionalDepartamento de Matem´aticasApartado Postal 07360Ciudad de M´exicoMexicoe-mail: [email protected] https://orcid.org/0000-0001-9363-8043https://orcid.org/0000-0001-9363-8043