Holomorphic Functions

LetCdenote the set of complex numbers Throughout the book we fix a positive integernand let

The notation \( \mathbb{C}^n \) represents the Euclidean space of complex dimension \( n \), where operations such as addition, scalar multiplication, and conjugation are performed componentwise For two elements \( z = (z_1, \ldots, z_n) \) and \( w = (w_1, \ldots, w_n) \) in \( \mathbb{C}^n \), the product \( z \cdot w \) is defined as \( z \cdot w = z_1 \overline{w_1} + \ldots + z_n \overline{w_n} \), with \( \overline{w_k} \) representing the complex conjugate of \( w_k \).

The spaceC n becomes ann-dimensional Hilbert space when endowed with the inner product above The standard basis forC n consists of the following vectors: e 1 = (1,0,0,ã ã ã,0), e 2 = (0,1,0,ã ã ã,0), ã ã ã, e n = (0,0,ã ã ã,0,1).

Via this basis we will identify the linear transformations ofC n withn×nmatrices whose entries are complex numbers Another vector inC n that we often use is the zero vector,

The reader should have no problem accepting this slightly confusing notation. The open unit ball inC n is the set

The boundary ofB n will be denoted byS n and is called the unit sphere inC n Thus

Occasionally, we will also need the closed unit ball

Holomorphic functions in several complex variables have a more nuanced definition compared to the single-variable case, with multiple natural definitions that are ultimately equivalent While we will utilize these classical definitions, we will not delve into proving their equivalence, as comprehensive proofs can be found in textbooks such as [61] and [89].

Holomorphic functions in \( B^n \) are defined through complex partial derivatives A function \( f: B^n \to \mathbb{C} \) is considered holomorphic in \( B^n \) if, for every point \( z \in B^n \) and for each \( k \in \{1, 2, \ldots, n\} \), the limit \( \lim_{\lambda \to 0} \frac{f(z + \lambda e_k) - f(z)}{\lambda} \) exists and is finite, where \( \lambda \in \mathbb{C} \) When a function is holomorphic in \( B^n \), we denote it accordingly.

∂z k (z) to denote the above limit and call it the partial derivative off with respect toz k Equivalently, a functionf :B n →Cis holomorphic if f(z) m a m z m , z∈B n

Here the summation is over all multi-indexes m= (m 1 ,ã ã ã, m n ), where eachm k is a nonnegative integer and z m =z 1 m 1 ã ã ãz n m n

The Automorphism Group

The series above is called the Taylor series offat the origin; it converges absolutely and uniformly on each of the sets rB n ={z∈C n :|z| ≤r}, 0< r 0 \) and \( r + dr \) is expressed as \( dV = dS \cdot S(1) \cdot (V(r + dr) - V(r)) \).

Here, forr >0,V(r)is the actual volume of the ball

|z 1 | 2 +ã ã ã+|z n | 2 < r 2 , andS(r)is the actual surface area of the sphere

From the change of variablesz k =rw k ,1≤k≤n, we obtain

Omitting powers ofdrwith exponents greater than1, we get dV =V(1)

S(1) 2nr 2 n −1 dr dS, or dv= 2nr 2 n −1 dr dσ.

Lemma 1.8 will be referred to as integration in polar coordinates The next lemma deals with integration onS n of functions of fewer variables.

Lemma 1.9 Supposef is a function onS n that depends only onz 1 ,ã ã ã, z k , where

1≤k < n Thenf can be regarded as defined onB k and

(1− |w| 2 ) n − k −1 f(w)dv k (w), whereB k is the open unit ball inC k anddv k is the normalized volume measure on

Proof For the purpose of this proof letP k denote the orthogonal projection fromC n ontoC k Then

To establish the desired result, we can limit our focus to the case where the function \( f \) is continuous in \( C_k \) and has support within the region defined by \( r_0 B_k \), where \( r_0 \) is a constant in the interval \( (0,1) \) By fixing such a function \( f \), we can then examine the relevant integrals.

We integrate in polar coordinates to get

We then differentiate this to obtain

On the other hand, an application of Fubini’s theorem shows that

(r 2 − |w| 2 ) n − k f(w)dv k (w), wherer > r 0 andcis a certain constant depending on the normalization ofdvand dv k Differentiation then gives

Comparing this with the formula forI (1)in the previous paragraph, we obtain

1.3 Lebesgue Spaces 11 wherec is a constant independent of f Thus the lemma is proved except for the multiplicative constantc

To determine the value ofc , simply takef = 1and compute the integral

Two special situations are worth mentioning First, ifk=n−1, then

B k f dv k , (1.12) because the binomial coefficient in Lemma 1.9 becomes1in this case On the other hand, ifk = 1,n > 1, and f is a function of one complex variable, then for any η∈S n we have

This is because, by unitary invariance, we may assume that η = e 1 , and hence ζ, η=ζ 1

We will also need to use the following formulas for integration on the unit sphere, the first of which is called integration by slices, and the second generalizes Lemma 1.9.

Proof It is obvious that

S n f(e iθ ζ)dσ(ζ) for all0≤θ≤2π Integrate with respect toθ∈[0,2π]and apply Fubini’s theorem.

We then obtain (1.14), the formula of integration by slices.

If we writeζ= (ζ , ζ ), whereζ ∈C k andζ ∈C n − k , then

By the unitary invariance ofσ, we have

1− |ζ | 2 η)dσ(ζ), whereηis any fixed point onS n − k Integrating overη∈S n − k and applying Fubini’s theorem, we obtain

The inner integral above defines a function that only depends on the firstkvariables. Therefore, we can apply Lemma 1.9 to get

1− |z| 2 η)dσ n − k (η), which completes the proof of the lemma

One special case of (1.15) is especially useful, namely, ifk=n−1, we have

In the proof of Lemma 1.10 we used the obvious fact that bothvandσare invari- ant under unitary transformations More specifically, ifUis a unitary transformation ofC n , then

These equations are also referred to as the rotation invariance ofvandσ, respectively.

We will also need a class of weighted volume measures onB n Observe that ifα is a real parameter, then integration in polar coordinates shows that the integral

(1− |z| 2 ) α dv(z) is finite if and only ifα >−1 Whenα >−1, we define a finite measuredv α onB n by dv α (z) =c α (1− |z| 2 ) α dv(z), (1.18) wherec α is a normalizing constant so thatv α (B n ) = 1 Using polar coordinates, we easily calculate that

Whenα≤ −1, we simply write dv α (z) = (1− |z| 2 ) α dv(z).

All the measuresdv α ,−∞< α −1 Whenm=l, we have the following formulas.

Lemma 1.11 Supposem= (m 1 ,ã ã ã, m n )is a multi-index of nonnegative integers andα >−1 Then

In this article, we establish a connection between \( C^n \) and \( R^{2n} \) by utilizing the real and imaginary components of complex numbers, while denoting the standard Lebesgue measure on \( C^n \) as \( dV \) Furthermore, we note that if the Euclidean volume of the unit ball \( B^n \) is represented by \( c_n \), then it follows that \( c_n \, dv = dV \).

C n |z m | 2 e −| z | 2 dV(z) by two different methods First, Fubini’s theorem gives

Then, integration in polar coordinates gives

Comparing the two answers, we obtain

Another integration in polar coordinates gives

Identity (1.23) then follows from (1.22) and the fact that

This completes the proof of the lemma

As a by-product of the above proof we obtained the actual volume of B n as π n /n! Therefore, the volume of the ballrB n is

V(r) =π n n! r 2 n ; see the proof of Lemma 1.8 If we useS(r)to denote the surface measure of the sphererS n , then

In particular, the surface area of the unit sphereS n is(2π n )/(n−1)!.

As another consequence of Lemma 1.11 we obtain the following asymptotic es- timates for certain important integrals on the ball and the sphere.

Theorem 1.12 Supposecis real andt >−1 Then the integrals

|1− z, w| n +1+ t + c , z∈B n , have the following asymptotic properties.

(1) Ifc -1 \) This leads to a natural change of variables that simplifies the integral.

This along with (1.5) produces the desired result.

Several Notions of Differentiation

Two special weights are of particular interest to us The first one isα= 0 In this case, we have

The other weight isα=−(n+ 1) In this case we denote the resulting measure by dτ(z) = dv(z)

(1− |z| 2 ) n +1 , (1.25) and call it the invariant measure onB n The usage of the term “invariant measure” is justified by the following formula:

In addition to the separable Lebesgue spaces L^p(B^n, dv_α) and L^p(S^n, dσ), we will also explore the spaces L^∞(B^n) and L^∞(S^n) The spaces of all continuous functions on B^n and S^n are denoted as C(B^n) and C(S^n), respectively.

C0(B n )consists of all functions inC(B n )that vanish on the unit sphere.

In this section we discuss several different notions of differentiation onB n The most basic one is of course the standard partial differentiation, namely,∂f /∂z k

The radial derivative is a key concept in differentiation on the unit ball, derived from the standard partial derivatives of a holomorphic function For a holomorphic function \( f \) defined in the unit ball \( B^n \), this derivative plays a crucial role in understanding the behavior of the function.

A simple verification shows that if f(z) ∞ k =0 f k (z) is the homogeneous expansion off, then

This is called the radial derivative offbecause

Rf(z) = lim r →0 f(z+rz)−f(z) r (1.29) wheneverf is holomorphic Hereris a real parameter, so thatz+rz is a radial variation of the pointz.

For every holomorphic functionf inB n , it is easy to see that f(z)−f(0) 1

Rf(tz) t dt (1.30) for allz ∈ B n This formula will come in handy when we need to recover a holo- morphic function from its radial derivative.

With the help of homogeneous expansions we can introduce a class of fractional radial derivatives on the spaceH(B n ) Thus for each real parametertwe define an operator

The operatorR t is clearly invertible onH(B n )/C, with its inverse given by

If we equip the spaceH(B n )with the topology of uniform convergence on compact sets, then the operatorsR t andR t are continuous onH(B n ).

More generally, for any two real parametersαandtwith the property that neither n+αnorn+α+tis a negative integer, we define an invertible operator

R α,t :H(B n )→H(B n ) as follows If f(z) ∞ k =0 f k (z) is the homogeneous expansion off, then

R α,t f(z) ∞ k =0 Γ(n+ 1 +α)Γ(n+ 1 +k+α+t) Γ(n+ 1 +α+t)Γ(n+ 1 +k+α)f k (z) (1.33) The inverse ofR α,t , denoted byR α,t , is given by

R α,t f(z) ∞ k =0 Γ(n+ 1 +α+t)Γ(n+ 1 +k+α) Γ(n+ 1 +α)Γ(n+ 1 +k+α+t)f k (z) (1.34)The following result gives an alternative description of these operators.

Proposition 1.14 Suppose neithern+αnorn+α+tis a negative integer Then the operatorR α,t is the unique continuous linear operator onH(B n )satisfying

(1− z, w) n +1+ α + t (1.35) for allw∈B n Similarly, the operatorR α,t is the unique continuous linear operator onH(B n )satisfying

(1− z, w) n +1+ α + t ∞ k =0 Γ(n+ 1 +k+α+t) k! Γ(n+ 1 +α+t) z, w k are actually homogeneous expansions It is then obvious that the operatorsR α,t and

R α,t have the desired mapping properties on kernel functions.

On the other hand, ifR α,t andR α,t have the stated mapping properties on kernel functions, then applying the differential operators

∂w m (0) to these kernel equations shows thatR α,t andR α,t have the desired effect on mono- mials and hence on general holomorphic functions

Proposition 1.15 states that for a positive integer N and a real number α, where n + α is not a negative integer, the operator R α,N acts on H(B n ) as a linear partial differential operator of order N with polynomial coefficients.

Proof Fixw∈B n , replace the numerator of

(1− z, w+z, w) N , and apply the binomial formula Then

For eachkwe apply the multi-nomial formula (1.1) to write z, w k | m |= k k! m!z m w m

It is then clear that there exist constantsc mk such that

Combining this with Proposition 1.14, we obtain

(1− z, w) n +1+ α for any fixedw∈B n Differentiating with respect towthen leads to

∂y k 2 be the ordinary Laplacian onC n Here

, provided we use the identificationz k =x k +iy k for1≤k≤n The complex-type derivatives are more convenient to use than the correponding real ones.

Supposef is a twice differentiable function inB n We define

The invariant Laplacian operator, denoted as (∆f)(z) = ∆(f◦ϕ z)(0) for z ∈ B n, is defined using the involutive automorphism ϕ z, which swaps the points 0 and z This operator exhibits a unique property in relation to the automorphism group, highlighting its significance in mathematical analysis.

Proposition 1.16 Supposef is twice differentiable inB n Then

Proof Fixz∈B n andϕ∈Aut(B n ) Leta=ϕ(z) Then the automorphism

U=ϕ a ◦ϕ◦ϕ z fixes the origin and hence is a unitary by Lemma 1.1 It is easy to see that

∆(g◦U)(0) = ∆(g)(0) for any twice differentiable functiong It follows that

The invariant Laplacian can be described in terms of ordinary partial derivatives as follows.

Proposition 1.17 Iffis twice differentiable inB n , then

∂z i ∂z¯ j (z) for allz∈B n , whereδ i,j is the Kronecker delta.

∂z k (0). The definition ofϕ z shows that it admits the expansion ϕ z (w) =z−s z w+ s z

1 +s z w, zz+ã ã ã, wheres z 1− |z| 2 and the omitted terms havew-degree2or higher It follows that ∂ϕ i

The Bergman Metric

(1−z w) n +1 is called the Bergman kernel ofB n and will be discussed in more detail in the next chapter For now let us use it to define a Hermitian metric onB n

We begin with then×ncomplex matrix

We will call this the Bergman matrix ofB n We also introduce an auxiliary matrix

In the context of linear transformations on C^n, we can represent these transformations using n x n matrices through the standard basis of C^n This identification allows us to relate the adjoint of a linear transformation to the conjugate transpose of its corresponding matrix Notably, when evaluating this relationship at the point z = 0, the properties of the transformation become more straightforward to analyze.

A(z) =|z| 2 P z , (1.38) whereP z is the orthogonal projection fromC n onto the one-dimensional subspace [z]generated byz.

Proposition 1.18 Forz∈B n the matricesA(z)andB(z)have the following prop- erties:

It follows that forn ≥2andz = 0the matrixB(z)has two eigenvalues, namely,

(1− |z| 2 ) −2 with eigenspace[z], and(1− |z| 2 ) −2 with eigenspaceC n [z].

Proof Since logK(z, z) =−(n+ 1) log(1− |z| 2 ), we have

1.5 The Bergman Metric 23 forj = 1,ã ã ã, n It follows that

A direct computation using rows and columns shows that

The orthogonal projection \( P_z \) represents the projection of \( \mathbb{C}^n \) onto the one-dimensional subspace spanned by \( [z] \), while \( Q_z \) denotes the orthogonal projection from \( \mathbb{C}^n \) onto \( \mathbb{C}^n[z] \) In this context, linear transformations of \( \mathbb{C}^n \) are identified with \( n \times n \) matrices using the standard basis of \( \mathbb{C}^n \).

FromP z +Q z =Iwe then deduce that

The real Jacobian determinant of the mapping ϕ(z) corresponds to the squared modulus of its complex Jacobian determinant Given that the matrix ϕ(z) is self-adjoint, the relationship B(z) = (ϕ(z))^2 establishes that det(B(z)) equals the squared magnitude of det(ϕ(z)), which is equal to the real Jacobian J_R ϕ(z).

This along with the formula forJ R ϕ z (z)in (1.11) shows thatdet(B(z)) =K(z, z).

The calculations indicate that the Bergman matrix B(z) is both positive and invertible, a fact that is widely recognized and holds true in general Additionally, we derive a representation for the square root of the Bergman matrix.

Proposition 1.19 The Bergman matrix is invariant under automorphisms, that is,

Proof Without loss of generality we may assume thatϕ=ϕ a for somea∈B n In this case, it follows from (1.5) and (1.11) that the Bergman kernel satisfies

K(z, z) =|J C ϕ(z)| 2 K(ϕ(z), ϕ(z)) for allz∈B n andϕ∈Aut(B n ) Thus logK(z, z) = log|J C ϕ(z)| 2 + logK(ϕ(z), ϕ(z)).

By locally writing log|J C ϕ(z)| 2 = logJ C ϕ(z) + logJ C ϕ(z), we see that

Then the chain rule gives

Another application of the chain rule produces

∂z i for alli, j= 1,ã ã ã, n This proves the desired result

1.5 The Bergman Metric 25 For a smooth curveγ: [0,1]→B n we define l(γ) 1

This definition clearly generalizes to the case of a piecewise smooth curve.

The Bergman metric, denoted as β, is defined for any two points z and w in Bn as the infimum of the lengths of all piecewise smooth curves γ connecting z to w within Bn This definition ensures that β is a valid metric, as it adheres to the positivity property inherent in B(z).

Proposition 1.20 The Bergman metric is invariant under automorphisms, that is, β(ϕ(z), ϕ(w)) =β(z, w) for allz, w∈B n andϕ∈Aut(B n ).

Proof This follows easily from Proposition 1.19 and the definition of the Bergman metric

Proposition 1.21 Ifzandware points inB n , then β(z, w) = 1

1− |ϕ z (w)|, whereϕ z is the involutive automorphism ofB n that interchanges0andz.

Proof By invariance, we only need to prove the result forw= 0.

Fix a point \( z \) in the unit ball \( B^n \) and consider a piecewise smooth curve \( \gamma: [0,1] \to B^n \) connecting the points 0 and \( z \) By partitioning the interval \( [0,1] \) into a finite number of subintervals, we can assume that \( \gamma \) is smooth and regular, meaning that \( \gamma(t) \) does not vanish for all \( t \) in the interval.

[0,1] In this case, the functionα(t) =|γ(t)|is smooth on[0,1].

2α(t)α (t) = 2Reγ (t), γ(t) = 2ReP γ ( t ) γ (t), γ(t), whereP γ ( t ) is the orthogonal projection fromC n onto the one-dimensional subspace spanned byγ(t) It follows that

On the other hand, according to part (e) of Proposition 1.18,

It is easy to check that equality holds ifγ(t) =tz,t∈[0,1] This shows β(0, z) = 1

1− |z|, and completes the proof of the proposition

Corollary 1.22 ForzandwinB n let ρ(z, w) =|ϕ z (w)|. Thenρis a metric onB n Moreover,ρis invariant under automorphisms, that is, ρ(ϕ(z), ϕ(w)) =ρ(z, w) for allz, w∈B n andϕ∈Aut(B n ).

Proof A calculation shows that ρ(z, w) = tanhβ(z, w) (1.40) for all z, w ∈ B n The invariance ofρ, which can be checked directly, is thus a consequence of the invariance ofβ.

To demonstrate that ρ is a valid distance function, it is essential to verify its compliance with the positivity and symmetry conditions outlined in the definition of a distance The triangle inequality for ρ can be established by analyzing the function f(x) = tanh(x+h) - tanh(h) - tanh(x) for x in the interval [0, ∞), where h is a positive constant This leads to the expression f(x) = sech²(x+h) - sech²(x) for x ≥ 0, which is crucial for proving the triangle inequality.

Since the function sinc(x) is decreasing on the interval (0, ∞), we can conclude that f(x) is less than 0 for all x > 0, indicating that f(x) is strictly decreasing on (0, ∞) Additionally, given that f(0) = 0, it follows that tanh(x + h) is less than or equal to tanh(x) + tanh(h) for all x and h in the range [0, ∞) Consequently, the triangle inequality for ρ emerges from the monotonicity of tanh(x), the triangle inequality for β, and the established inequality.

The metricρwill be called the pseudo-hyperbolic metric onB n It is clear thatρ is bounded, whileβis not.

Forz∈B n andr >0we letD(z, r)denote the Bergman metric ball atz Thus

We now calculate the volume of the Bergman metric balls.

Lemma 1.23 For anyz∈B n andr >0we have v(D(z, r)) = R 2 n (1− |z| 2 ) n +1

(1−R 2 |z| 2 ) n +1 , (1.41) whereR = tanh(r) In particular, for anyr >0, there exist constantsc r >0and

Proof By Proposition 1.21, each D(0, r) is actually a Euclidean ball of radius

R = tanh(r)centered at the origin Since the Bergman metric is invariant under automorphisms ofB n , we have

Changing variables several times, we obtain v(D(z, r))

This proves (1.41) The estimates in (1.42) clearly follow from (1.41)

Recall that for any realαwe have dv α (z) =c α (1− |z| 2 ) α dv(z), wherec α >0is a constant For more generalαwe have the following asymptotic estimate ofv α (D(z, r)).

Lemma 1.24 For any realαand positiverthere exist constantsC >0andc >0 such that c(1− |z| 2 ) n +1+ α ≤v α (D(z, r))≤C(1− |z| 2 ) n +1+ α for allz∈B n

Proof LetR= tanh(r)again and make a change of variables according to Propo- sition 1.13 We obtain v α (D(z, r)) =c α

It is clear that there exist positive constantscandCsuch that c≤ c α (1− |w| 2 ) α

The Invariant Green’s Formula

In this section we discuss Green’s formula for the invariant Laplacian and the asso- ciated Green’s function.

Theorem 1.25 SupposeΩis an open subset ofB n ,Ω⊂B n , whose boundary∂Ωis sufficiently smooth Ifuandvare twice differentiable inΩand continuously differ- entiable onΩ, then

In the context of the Bergman metric, the volume element on the domain Ω is denoted by dτ, while σ represents the surface area element on the boundary ∂Ω, also defined by the Bergman metric Additionally, the notation ∂/∂n refers to the outward normal derivative along the boundary ∂Ω with respect to the Bergman metric.

We will not prove this theorem in the book, because its proof is not complex analytic See [41] or [102].

The volume element ofΩin the Bergman metric is simply the restriction toΩof the M¨obius invariant measure, that is, dτ(z) = dv(z)

We will only apply Green’s formula in the case when Ωis a shell inB n On the surface|z|=r,0< r 1), and since

2n|z| 2 n −1 , it is easy to show that the second integral in (1.48) tends to−f(0)asr→0.

Subharmonic Functions

This section presents key results regarding subharmonic functions within the unit ball, essential for later discussions in the book We consider \( B_n \) as the open unit ball in the real Euclidean space \( \mathbb{R}^{2n} \) The following three properties of harmonic functions in multiple real variables will be utilized without proof.

(a) Every harmonic function has the mean value property.

(b) The maximum principle holds for harmonic functions.

(c) SupposeBis any ball with boundary sphereS Ifuis a continuous function on

S, thenucan be continuously extended to a harmonic function inB.

A functionf :B n →[−∞,∞)is said to be upper semi-continuous if lim sup z → z 0 f(z)≤f(z 0 ) for everyz 0 ∈B n An upper semi-continuous functionf :B n →[−∞,∞)is said to be subharmonic if f(a)≤

Theorem 1.28 Supposef : B n → [−∞,∞)is upper semi-continuous Then the following conditions are equivalent.

(b) For every pointainB n there exists some positive number 0the weighted Bergman spaceA p α consists of holomorphic functionsfinL p (B n , dv α ), that is,

It is clear thatA p α is a linear subspace ofL p (B n , dv α ).

When the weightα = 0, we simply writeA p forA p α These are the standard (unweighted) Bergman spaces.

We will use the notation f p,α

(2.4) forf ∈L p (B n , dv α ) Note that when1≤p 0,α > −1,0 < r < 1, andm = (m 1 ,ã ã ã, m n )is a multi-index of nonnegative integers Then there exists a positive constantCsuch that

≤Cf p,α for allf ∈A p α and allz∈B n with|z| ≤r.

Proof Fix someδ ∈ (r,1)and apply Theorem 2.2 in the special case α= 0 We obtain f(δz)

Making a change of variables and replacingδzbyz, we get f(z) =δ 2

We can then differentiate under the integral sign and find a positive constantCsuch that

≤Csup{|f(w)|:|w| ≤δ} for all|z| ≤ r This reduces the proof of the lemma to the case|m| = 0 When

|m|= 0, the desired estimate clearly follows from Theorem 2.1

Corollary 2.5 For eachp > 0andα > −1 the weighted Bergman spaceA p α is closed inL p (B n , dv α ).

In the context of a sequence {f_n} within the space A_p^α, if the limit of f_n - f_{p,α} approaches zero as n approaches infinity for some function f in L_p(B_n, dv_α), it follows that a subsequence of {f_n(z)} will converge to f(z) for almost every z in B_n Additionally, since {f_n} forms a Cauchy sequence in A_p^α, Lemma 2.4 indicates that the sequence {f_n(z)} is uniformly Cauchy on the set {z ∈ B_n : |z| < r} for any 0 < r < 1, leading to convergence to a holomorphic function within that domain This result holds for any arbitrary r, reinforcing the uniform convergence of the subsequence.

{f n (z)} converges to a holomorphic functiong(z) on B n By the uniqueness of pointwise limits, we havef(z) =g(z)for almost allz ∈B n This shows thatf is holomorphic inB n and hence belongs toA p α

Bergman Type Projections

It follows that the weighted Bergman spaceA p α , with topology inherited from

L p (B n , dv α ), is a Banach space when1 ≤p −1 Then the set of polynomials is dense inA p α

Proof Writing the ball as

In the context of the unit ball B n in complex analysis, we establish that for any function f belonging to the space A p α, the limit as r approaches 1 from the left of the difference between f r and f p,α is zero Here, f r is defined as f(rz) for a fixed r in the interval (0,1) Additionally, it can be shown that the function f r can be uniformly approximated by polynomials, allowing for the conclusion that each f r can also be approximated in the norm topology of A p α by polynomials.

Since each point evaluation inB n is a bounded linear functional on the Hilbert space

A 2 α , whereα >−1, the classical Riesz representation theory in functional analysis shows that for eachw∈B n there exists a unique functionK w α inA 2 α such that f(w) =f, K w α α

This will be called the reproducing formula forf inA 2 α The function

K α (z, w) =K w α (z), z, w∈B n , is called the reproducing kernel ofA 2 α Whenα= 0, the reproducing kernel

K(z, w) =K 0 (z, w) is also called the Bergman kernel.

Theorem 2.7 For eachα >−1the reproducing kernel ofA 2 α is given by

Proof This follows from Theorem 2.2 and the uniqueness of the Riesz representa- tion for a bounded linear functional on a Hilbert space

Since the functionK α (z, w)is bounded inzwheneverwis fixed, we can con- sider integrals of the form

B n f(z)K α (w, z)dv s (z), whereα >−1,s ∈R,w∈B n , andf ∈L 1 (B n , dv s ) In particular, we will make use of the following integral operator

Lemma 2.8 Supposeα > −1 Then the restriction of P α toL 2 (B n , dv α )is the orthogonal projection fromL 2 (B n , dv α )ontoA 2 α

Proof LetP be the orthogonal projection from L 2 (B n , dv α ) onto A 2 α For f ∈

L 2 (B n , dv α )andz∈B n the reproducing property ofK α and the self-adjointness of

SinceK z α ∈A 2 α , we haveP K z α =K z α , and so

This shows thatP is the restriction ofP α toL 2 (B n , dv α )

The lemma indicates that the operator \( P_\alpha \) maps the space \( L^2(B_n, dv_\alpha) \) boundedly onto the Bergman space \( A^2_\alpha \) Additionally, we seek to understand the action of the operator \( P_\alpha \) on other spaces, including \( L^p(B_n, dv_t) \) A valuable method for addressing these inquiries is Schur’s test.

Theorem 2.9 Suppose(X, à)is a measure space,1< p −pa.

If1 < p < ∞and1/p+ 1/q = 1, the boundedness ofT onL p (B n , dv t )is equivalent to the boundedness of the adjoint ofT onL q (B n , dv t ) It is easy to see that

Combining this with the conclusion of the previous paragraph, we conclude that t+ 1>−q(b−t), which is equivalent to t+ 1< p(b+ 1).

Similarly, the boundedness ofT onL 1 (B n , dv t )implies the boundedness ofT ∗ onL ∞ (B n ) ApplyingT ∗ to the constant function1then yieldsb−t≥ 0 To see that equality cannot occur here, consider the functions f z (w) = (1− z, w) n +1+ a + b

Eachf z is a unit vector inL ∞ (B n ) Ifb=t, then

By part (2) of Theorem 1.12, the above integral tends to+∞as|z| →1 − Thus the boundedness ofT onL 1 (B n , dv t )implies thatb−t > 0, ort+ 1 < b+ 1 This completes the proof that (a) implies (c).

It remains to prove that (c) implies (b) The casep= 1is a direct consequence of Fubini’s theorem and part (3) of Theorem 1.12.

If1< p t.

In particular, we see that there exist a lot of bounded projections from the space

L 1 (B n , dv α )ontoA 1 α This is in sharp contrast with the classical theory of Hardy spaces.

(A p α ) ∗ =A q β (with equivalent norms) under the integral pairing f, g γ

(1− |z| 2 ) α/p f(z)(1− |z| 2 ) β/q g(z)dv(z), f ∈A p α , it follows from H¨older’s inequality thatFis a bounded linear functional onA p α with

In the context of functional analysis, if \( F \) is a bounded linear functional on the space \( A_{p, \alpha} \), then by the Hahn-Banach extension theorem, it can be extended to a bounded linear functional on \( L^p(B_n, dv_\alpha) \) without increasing its norm Furthermore, utilizing the duality properties of \( L^p \) spaces, there exists a function \( h \in L^q(B_n, dv_\alpha) \) that corresponds to \( F \).

It is easy to check that the conditionα > −1is equivalent toq(γ+ 1) > β+ 1, and the conditionβ >−1is equivalent top(γ+ 1)> α+ 1 So, by Theorem 2.11,

P γ is a bounded projection fromL p (B n , dv α )ontoA p α , andP γ is also a bounded projection fromL q (B n , dv β )ontoA q β Letg=P γ (H) Theng∈A q β and

F(f) =f, H γ =P γ (f), H γ =f, P γ (H) γ =f, g γ for allf ∈A p α The proof is now complete

A special case of the preceding theorem is whenα=β In this case, we clearly haveγ=αas well.

The dual ofA p α for0 < p ≤ 1 will be identified in the next chapter after we introduce the Bloch space.

Other Characterizations

In this section we characterize the weighted Bergman spaces in terms of various derivatives of a function First recall that

∂z k (z) is the radial derivative off atz.

2.3 Other Characterizations 49 For a holomorphic functionf inB n we write

(2.10) and call|∇f(z)|the (holomorphic) gradient off atz Similarly, we define

∇f(z) =∇(f ◦ϕ z )(0), (2.11) whereϕ z is the biholomorphic mapping ofB n that interchanges0 andz, and call

Lemma 2.13 Iff is holomorphic inB n , then

Proof For any holomorphic functionfinB n we have

4|∇f(z)| 2 = 4|∇(f ◦ϕ z )(0)| 2 =∆( |f◦ϕ z | 2 )(0) =∆( |f| 2 )(z) (2.12) The desired result now follows from Proposition 1.17

Since the invariant Laplacian is invariant under the action of the automorphism group, the relation (2.12) in the preceding proof shows that |∇f|is also M¨obius invariant, namely,

|∇(f◦ϕ)(z)|=|(∇f)◦ϕ(z)| (2.13) for allf and allϕ∈Aut(B n ).

Lemma 2.14 Iff is holomorphic inB n , then

Proof By the Cauchy-Schwarz inequality forC n ,

This proves the first inequality The second inequality follows from Lemma 2.13 and the fact that|Rf(z)| ≤ |z||∇f(z)|

The following lemma is critical for many problems concerning the spacesA p α when0< p −1,p > 0, andf is holomorphic inB n Then the following conditions are equivalent:

Proof Lemma 2.14 shows that (b) implies (c), and (c) implies (d).

To prove (a) implies (b), we fixβ > αand observe that there exists a constant

|g(w)| p dv β (w) for all holomorphicginB n ; see Lemma 2.4 Letg=f ◦ϕ z , wherez∈B n andϕ z is the biholomorphic mapping ofB n that interchanges0andz, and make an obvious change of variables according to Proposition 1.13 We obtain

An application of Fubini’s theorem and part (3) of Theorem 1.12 then gives

|f(z)| p dv α (z) for some constantC 2 > 0and allf holomorphic inB n Actually, replacingf by f −f(0), we have

To prove (d) implies (a), we assume thatf is a holomorphic function inB n such that the function(1− |z| 2 )Rf(z)is inL p (B n , dv α ) Letβ be a sufficiently large positive constant Then

(1− z, w) n +1+ β , z∈B n , by Theorem 2.2 SinceRf(0) = 0, we have

Rf(w)L(z, w)dv β (w), where the kernel

|1− z, w| n + β for allzandwinB n ; see Exercise 2.24 So

If1≤p −1 By Lemma 2.15, we have

(1− z, w) n + β p dv β (w), whereC 7 >0is a constant independent off A use of Fubini’s theorem and part (3) of Theorem 1.12 then gives

This completes the proof of the theorem

Note that the proof of the above theorem actually produces equivalent norms on

A p α in terms of the radial derivative, the gradient, and the invariant gradient off.

Theorem 2.17 Supposeα >−1,p >0,N is a positive integer, andf is holomor- phic inB n Thenf ∈A p α if and only if the functions

Proof The caseN = 1follows from the equivalence of (a) and (c) in Theorem 2.16.

We prove the caseN = 2here; the general case can then be proved using the same idea and induction.

So we assumef ∈A p α By the equivalence of (a) and (c) in Theorem 2.16, each function∂f /∂z i is inA p α + p , where1≤i≤n This in turn implies that each function

∂z i ∂z j (z) is inL p (B n , dv α + p ), or equivalently, each function

The arguments in the previous paragraph can be reversed So the desired result is proved forN = 2

It should be clear that the integral

|f(z)| p dv α (z) is comparable to the quantity

∂z m (z) p dv α (z) wheneverf is holomorphic inB n

Lemma 2.18 Suppose neithern+snorn+s+tis a negative integer Ifβ =s+N for some positive integerN, then there exists a one-variable polynomialhof degree

(1− z, w) n +1+ β + t There also exists a polynomialP(z, w)such that

(1− z, w) λ ∞ k =0 Γ(k+λ) k! Γ(λ) z, w k (2.14) for anyλ= 0,−1,−2,ã ã ã It follows from the definition ofR s,t that forβ =s+ 1 we can find a constantCsuch that

∞ k =0 k+n+ 1 +s k! Γ(n+ 1 +k+s+t)z, w k , which easily breaks into the sum of

Adding these up proves the desired result forR s,t whenβ=s+ 1.

In general, ifβ=s+N, then there exists a constantC N such that

∞ k =0 p(k)Γ(n+ 1 +k+s+t) k! z, w k , wherep(k)is a polynomial ofkof degreeN We can writep(k)as a linear combi- nation of

And the proof forR s,t proceeds exactly the same as in the caseN= 1.

To prove the result forR s,t , we use Proposition 1.14 to write

We then use the commutativity ofR s,t andR s + t,N to obtain

Use Proposition 1.14 again We have

The desired result forR s,t now follows from Proposition 1.15

The lemma allows us to compare the behaviors of \( R_{s,t} \) and \( R_{\beta,t} \) In a general sense, \( R_{s,t} f \) and \( R_{\beta,t} f \) are comparable for any holomorphic function \( f \) To illustrate this, we can express \( h(z, w) \) as a series expansion with \( N \) terms, starting from \( k = 0 \).

(1− z, w) n +1+ β + t − k , which, according to Proposition 1.14, is the same as

Differentiating with respect tow, we obtain

It is evident that C₀ = 0 For each integer k within the range of 1 to N, the function R(β + t - k, k) R(β, t) f serves as the k-th integral of R(β, t) f, exhibiting greater regularity than R(β, t) f itself This indicates that the behaviors of R(s, t) f and R(β, t) f frequently align, as illustrated in Exercise 2.22.

Theorem 2.19 Supposeα >−1,p >0, andt >0 If neithern+snorn+s+t is a negative integer, then there exist positive constantscandCsuch that c

|f(z)| p dv α (z) for all holomorphic functionsfinB n

Proof First assume that f ∈ A p α Ifβ = s+N, whereN is a sufficiently large positive integer, we have the integral representation f(z)

Apply the operatorR s,t inside the integral and use Lemma 2.18 We find a constant

Ifp≥1andNis large enough so that α+ 1< p(β+ 1), then it follows from Theorem 2.10 that

|f(z)| p dv α (z) for some constantC 2 >0(independent off).

Here we assume thatNis large enough so thatα > α By (2.15) and Lemma 2.15, there exists a constantC 3 >0such that

An application of Fubini’s theorem in combination of Theorem 1.12 shows that

|f(w)| p dv α (w), whereC 4 is a positive constant independent off.

Next assume that the function(1− |z| 2 ) t R s,t f(z)belongs toL p (B n , dv α ) By the remark following Lemma 2.18 (also see Exercise 2.22), the function g(z) =c β + t c β

(1− |z| 2 ) t R β,t f(z) also belongs to L p (B n , dv α ) Using Corollary 2.3, Fubini’s theorem, and Theo- rem 2.2, we check thatf = P β g If1 ≤ p < ∞, then Theorem 2.11 shows that f is inA p α When0< p −1 We can then apply Lemma 2.15 to obtain

|1− w, z| ( n +1+ β ) p dv α (w) for allz∈B n Observe that

(n+ 1 +β)p=n+ 1 +α −pt=n+ 1 +α+ (α −pt−α), and that we may assume thatN is so large that α −pt−α >0.

Using Fubini’s theorem and Theorem 1.12, we deduce that

(1− |w| 2 ) pt |R s,t f(w)| p dv α (w), whereC 7 is a positive constant This completes the proof of the theorem

Whenp = 2, both integrals in the preceding theorem can be evaluated using the Taylor expansions off andR s,t f The desired result then follows easily fromStirling’s formula for the gamma function.

Carleson Type Measures

In this section we are interested in measuresàon the unit ball with the property that

L p (B n , à)contains the Bergman spaceA p α Such measures will be termed Carleson type measures It turns out that the requirements onàare independent ofp.

Recall that forr >0andz∈B n the set

D(z, r) ={w∈B n :β(z, w)< r} (2.16) is a Bergman metric ball atz.

Lemma 2.20 For eachr >0there exists a positive constantC r such that

|1− a, z|≤C r (2.18) for allaandzinB n withβ(a, z)< r Moreover, ifris bounded above, then we may chooseC r to be independent ofr.

Proof Given any two points aandz inB n withβ(a, z) < r, we can writez ϕ a (w)for somew∈B n withβ(0, w)< r It follows from Lemma 1.2 that

SinceD(0, r)is actually a Euclidean ball centered at the origin with Euclidean radius less than1, we can easily find a positive constantCsuch that

1− |z| 2 ≤C for allaandzwithβ(a, z)< r, and (2.17) is proved.

Sincez = ϕ a (w)if and only ifw =ϕ a (z), we can also apply Lemma 1.2 to obtain

By the previous paragraph,1−|w| 2 is bounded by positive constants from both above and below, and1− |a| 2 is comparable to1− |z| 2 , the estimates in (2.18) are now obvious

Corollary 2.21 Suppose−∞< α < ∞,r 1 >0,r 2 >0, andr 3 >0 Then there exists a constantC >0such that

Proof This follows from Lemmas 1.24 and 2.20

Various analytical techniques utilize lemmas, which are methods for breaking down the underlying domain into manageable components In this context, we introduce a valuable decomposition of the open unit ball \( B^n \) into Bergman metric balls.

Lemma 2.22 Given any positive numberRand natural numberM, there exists a natural numberN such that every Bergman metric ball of radiusr, wherer ≤R, can be covered byNBergman metric balls of radiusr/M.

To establish a covering of a Bergman metric ball D(a, r) with radius r (where 0 < r ≤ R), we define δ as r/M We consider a finite collection of balls {D(a_k, δ/2)} that covers D(a, r), with each center a_k located within D(a, r) We start with a_1 = a_1 and identify a_2 as the first element in the sequence {a_2, a_3, } such that the Bergman distance β(a_2, a_1) is at least δ/2, if possible Next, we select a_3 as the first term following a_2 that maintains a Bergman distance of at least δ/2 from both a_1 and a_2 This iterative process continues until no further terms can be added, resulting in a covering {D(a_k, δ)} of D(a, r) where the distance between any two distinct centers a_i and a_j satisfies β(a_i, a_j) ≥ δ/2.

Since the sets{D(a k , r/(4M))}are disjoint and contained inD(a, r+r/(4M)), we have k v

By Corollary 2.21, there exists a constantC >0, independent ofrbut dependent on

4M for eachk It follows thatk≤C, so the natural numberN = [C] + 1has the desired properties

Theorem 2.23 There exists a positive integerNsuch that for any0< r≤1we can find a sequence{a k }inB n with the following properties:

(2) The setsD(a k , r/4)are mutually disjoint.

(3) Each pointz∈B n belongs to at mostNof the setsD(a k ,4r).

Proof Fix anyr∈(0,1] It is easy to find a sequence{a k }such that

D(a k , r) and thatβ(a i , a j )≥r/2for alli=j; see the first part of the proof of Lemma 2.22.

Property (2) then follows from the triangle inequality.

According to Lemma 2.22, there exists a positive integer N, independent of r, such that any Bergman metric ball with a radius of 4r can be covered by N Bergman metric balls, each with a radius of r/4 This establishes the necessity of property (3) Specifically, if z is an element of the defined space, this condition must be satisfied.

2.4 Carleson Type Measures 59 thena k i ∈ D(z,4r)for1 ≤ i ≤N + 1 LetD(z i , r/4),1 ≤ i ≤N, be a cover ofD(z,4r) Then at least one ofD(z i , r/4) must contain two ofa k j ,1 ≤ j ≤

According to the triangle inequality, the two points in question must have a Bergman distance of less than r/2, which contradicts the earlier assumption regarding the sequence {a_k} This contradiction concludes the proof of the theorem.

The radius \( r/4 \) is arbitrary, as the result can be established for any fixed radius exceeding \( r/4 \) We define \( r \) as the separation constant for the sequence \( \{a_k\} \), and we refer to \( \{a_k\} \) as an \( r \)-lattice in the Bergman metric.

Lemma 2.24 Supposer > 0,p > 0, and α > −1 Then there exists a constant

D ( z,r ) |f(w)| p dv α (w) for allf ∈H(B n )and allz∈B n

Proof Recall from Proposition 1.21 thatD(0, r)is a Euclidean ball centered at the origin with Euclidean radiusR= tanh(r) So the subharmonicity of|f| p and Corol- lary 1.29 show that

D (0 ,r ) |f(w)| p dv α (w) for all holomorphicf inB n Replacef byf ◦ϕ z and make a change of variables according to Proposition 1.13 Then

The desired result then follows from Lemma 2.20

Note that the above result can be restated as follows:

D ( z,r ) |f(w)| p dv α (w), z∈B n , (2.19) wherefis holomorphic andCis a constant independent offandz; see Lemma 1.24.

Theorem 2.25 Supposep > 0,r >0,α >−1, andàis a positive Borel measure onB n Then the following conditions are equivalent:

(a) There exists a constantC >0such that

|f(z)| p dv α (z) for all holomorphicf inB n

(b) There exists a constantC >0such that

(c) There exists a constantC >0such that à(D(a, r))≤C(1− |a| 2 ) n +1+ α for alla∈B n

(d) There exists a constantC >0such that à(D(a k , r))≤C(1− |a k | 2 ) n +1+ α for allk≥1, where{a k }is the sequence in Theorem 2.23.

Proof It is easy to see that (a) implies (b) In fact, setting f(z) (1− |a| 2 ) n + α +1

|1− z, a| 2( n +1+ α ) dà(z)≤C for alla∈B n This along with Lemma 2.20 shows that (c) must be true.

It remains to prove that (d) implies (a) So we assume that there exists a constant

C 1 >0such that à(D(a k , r))≤C 1 (1− |a k | 2 ) n +1+ α for allk≥1 Iff is holomorphic inB n , then

By Lemmas 2.24 and 2.20, there exists a constantC 2 >0such that sup{|f(z)| p :z∈D(a k , r)} ≤ C 2

D ( a k , 2 r ) |f(w)| p dv α (w) for allk≥1 It follows that

D ( a k , 2 r ) |f(w)| p dv α (w) for all holomorphicfinB n Since every point inB n belongs to at mostNof the sets

|f(w)| p dv α (w) for allf holomorphic inB n This completes the proof of the theorem

Since(1− |a| 2 ) n +1+ α is comparable tov α (D(a, r)), conditions (c) and (d) in Theorem 2.25 are equivalent to à(D(a, r))≤Cv α (D(a, r)), a∈B n , and à(D(a k , r))≤Cv α (D(a k , r)), k≥1, respectively.

We say that a sequence{f k }inA p α converges to0ultra-weakly if{f k p,α }is bounded andf k (z)→0for everyz∈B n

Theorem 2.26 Supposep > 0,r >0,α >−1, andàis a positive Borel measure onB n Then the following conditions are equivalent:

(a) Whenever{f k }converges ultra-weakly to0inA p α , we have lim k →∞

(c) The measureàhas the property that

(d) For the sequence{a k }from Theorem 2.23 we have lim k →∞ à(D(a k , r))

Proof The proof is similar to that of Theorem 2.25 We leave the details to the interested reader

Once again, conditions (c) and (d) in Theorem 2.26 can be reformulated as

Atomic Decomposition

In this section, we demonstrate that every function within the Bergman space \(A_p^\alpha\) can be expressed as a series of well-defined functions known as atoms These atoms, which are formulated using kernel functions, serve as a foundational basis for the space \(A_p^\alpha\).

Lemma 2.27 For anyR >0and any realbthere exists a constantC >0such that

Proof Ifuandvsatisfyβ(u, v)≤R, we can writev=ϕ u (w)with|w| ≤r, where r= tanhR∈(0,1) Letz =ϕ u (z) Then by Lemma 1.3,

Also, because|u, w| < rand|z , w| < r, there exists a constantC 1 >0, de- pending onrandb, such that

On the relatively compact set |w| < r, the Bergman metric is equivalent to the

Euclidean metric Thus there exists a constantC 2 >0such that

|w| ≤C 2 β(0, w) =C 2 β(0, ϕ u (v)) =C 2 β(u, v) wheneverv=ϕ u (w)withβ(u, v)< R This completes the proof of the lemma

Letb = 1 in the preceding lemma and apply the triangle inequality, then let b =−1and apply the triangle inequality We see that for anyR >0there exists a constantC >0such that

|1− z, v| ≤C (2.20) for allz ∈ B n and alluandv withβ(u, v)≤R This generalizes the estimates in

In the remainder of this section we fix a sequence{a k } chosen according to Theorem 2.23 Letrbe the separation constant for{a k }.

Lemma 2.28 For eachk ≥ 1there exists a Borel setD k satisfying the following conditions:

Then eachE k containsD(a k , r/4)and is contained inD(a k , r) Also,{E k }covers

B n In fact, ifz ∈B n , thenz ∈D(a k , r)for somek Ifz ∈ D(a j , r/4)for some j=k, thenz∈E j ; otherwise,z∈E k

Then{D k }is clearly a disjoint cover ofB n In fact, ifz∈B n , thenz∈E k for some k Ifk= 1, thenz∈D 1 Ifk >1, then either we havez∈D i for some1≤i < k, or we havez∈D k

It is clear that D 1 = E 1 contains D(a 1 , r/4) To see that each D k +1 contains

D(a k +1 , r/4), we fixk ≥ 1 and fixz ∈ D(a k +1 , r/4) ⊂ E k +1 Thenz ∈ E i for any1≤i≤k, which implies thatz∈D i for any1≤i≤k This shows that z∈E k +1 − k i =1

D i =D k +1 , and the proof of the lemma is complete

We fix a real parameterb > nand letβ=b−(n+ 1) Define an operatorT as follows.

|1− z, w| b f(w)dv(w), (2.21) wheref ∈L 1 (B n , dv β ) We emphasize that the operatorTdepends on the parameter b.

To enhance the analysis presented in Lemma 2.28, we will further partition the sets {D_k} Specifically, we will start by partitioning the set D_1 and then utilize automorphisms to extend this partition to the other sets D_k For this purpose, we define η as a positive radius that is significantly smaller than the separation constant r, ensuring that the ratio η/r remains small We will also establish a finite sequence {z_1, , z_J} to aid in this process.

D(0, r), depending onη, such that{D(z j , η)}coverD(0, r)and that{D(z j , η/4)} are disjoint We then enlarge each setD(z j , η/4)∩D(0, r)to a Borel setE j in such a way thatE j ⊂D(z j , η)and that

E j is a disjoint union; see the proof of Lemma 2.28 for how to achieve this.

Fork≥1and1≤j≤Jwe definea kj =ϕ a k (z j )and

It is clear thata kj ∈D(a k , r)for allk≥1and1≤j≤J Since

D kj is a disjoint union for everyk, we obtain a disjoint decomposition

We define an operatorSonH(B n )as follows.

We emphasize that the operatorSdepends on both the parameterband the partition

{D kj }(and hence also on the separation constantsrandη).

The following lemma is the key to atomic decompositions for Bergman spaces,the Bloch space, and BMOA.

Lemma 2.29 For anyp >0andα >−1there exists a constantC >0, independent of the separation constantsrandη, such that

1 p for allr≤1,z∈B n , andf∈H(B n ), where σ=η+ tanh(η)

Proof Without loss of generality we may assume that f ∈ A 1 β Then by Theo- rem 2.2, f(z)

Since{D kj }is a partition ofB n , we can write f(z)−Sf(z) ∞ k =1

We first estimateI(z) For any1≤k 0 \), which remains independent of \( r \) and \( \eta \).

In the rest of this proof, we letCdenote a positive constant (independent ofr,η,k, andj) whose exact value may change from one occurence to another.

Writer = tanh(r),η = tanh(η), andR=η /r Sinceηis much smaller than r, we may as well assume thatR≤ 1 2 By Lemma 2.4, there exists a constantC >0 such that

, |z| ≤R, wherehis any function inH(B n ) Considerh(z) =g(r z), where g(z) =f◦ϕ a kj (z), z∈B n

After a change of variables, we obtain r |∇g(r z)| ≤C

1 /p for allz∈D(0, η) For anyw∈E kj ⊂D(0, η), the identity g(w)−g(0) 1

So, going back toI kj , we have

2.5 Atomic Decomposition 67 Combining this with the earlier estimate on the complex gradient ofg, we obtain

By a change of variables again,

By Lemma 2.27, the quantities1− |a kj | 2 and|1− w, a kj |are both comparable to

1−|a k | 2 , wherew∈D(a kj , r) This along with the fact thatD(a kj , r)⊂D(a k ,2r) shows that

Since1− |a k | 2 is comparable to1− |w| 2 forw∈D(a k ,2r), we have

Combining this with the estimate in the previous paragraph, we obtain

J j =1 v(D(z j , η/4))≥CJ(η ) 2 n , where the last inequality follows from Lemma 1.23, we have

Combining this with the estimate in the previous paragraph, we obtain

For eachk≥1and1 ≤j ≤J it follows from Lemma 2.27 that|1− z, a kj | b is comparable to|1− z, a k | b Therefore,

|f(w)|dv β (w) fork≥1and1≤j≤J By Lemma 2.27 and (2.17) ,

For everyw∈D kj we use Lemma 2.24 to get

According to Lemma 2.27,|1− z, a kj | b is comparable to|1− z, a k | b It follows that

This completes the proof of the lemma

We can now prove the main result of this section.

Then there exists a sequence{a k }inB n such thatA p α consists exactly of functions of the form f(z) ∞ k =1 c k

(1− z, a k ) b , z∈B n , (2.24) where{c k }belongs to the sequence spacel p and the series converges in the norm topology ofA p α

In this proof, we examine the function \( f(z) \) as defined by equation (2.24), with the sequence \( \{a_k\} \) representing an \( r \)-lattice in the Bergman metric, as established by Theorem 2.23 Our objective is to demonstrate that the function \( f \) belongs to the space \( A_p^\alpha \) To achieve this, we express \( f_k(z) \) in the form \( (1 - |a_k|^2) (p b - n - 1 - \alpha) / p \).

The assumption onbimplies thatpb > n+ 1 +αfor allp > 0 Thus{f k } is a bounded sequence inA p α by Theorem 1.12 Recall that the norm inL p (B n , dv α )is denoted by p,α

Since{c k } ∈l p and{f k }is bounded inA p α , we see thatfis inA p α

Whenp > 1, we let{D k }denote the sets from Lemma 2.28 and consider the function

|c k |v α (D k ) −1 /p χ k (z), z∈B n , whereχ k is the characteristic function ofD k It is clear that

The assumption onbimplies that b > n+α+ 1 p , or, p(b−n)> α+ 1, when p > 1 By Theorem 2.10, the operator T defined in (2.21) is bounded on

SinceF is defined as a sum of nonnegative functions, we can apply T to the functionFand integrate term by term The result is

Also, (2.20) tells us that|1− z, w|is comparable to|1− z, a k |whenw∈D k It follows that there exists a constantδ >0such that

|1− z, a k | b for allz∈B n By the triangle inequality, we have

SinceF ∈L p (B n , dv α )andT is bounded onL p (B n , dv α )(see Theorem 2.10), we conclude thatf ∈A p α with

|c k | p for some positive constantCindependent off.

The above proof, after some obvious minor adjustments, still works if{a k }is replaced by{a kj } In fact, if f(z) ∞ k =1

(1− z, a kj ) b , then we can use the facts that1−|a kj | 2 ∼1−|a k | 2 and|1−z, a kj | ∼ |1−z, a k | to obtain a constantC >0such that

Therefore, the sequence{d k }is inl p and it follows from the earlier proof thatf ∈

We have successfully completed the initial proof of the theorem, demonstrating that every function defined by (2.24) is a member of \( A_{p}^{\alpha} \) This conclusion holds true when utilizing a sequence \( \{a_k\} \) as established by Theorem 2.23 or an associated sequence \( \{a_{kj}\} \) constructed prior to Lemma 2.29 It is important to note that our findings thus far do not rely on any assumptions regarding the separation constants \( r \) and \( \eta \).

Every function \( f \in A_{p}^{\alpha} \) can be represented as indicated in equation (2.24) We establish this by selecting an \( r \)-lattice \( \{ a_k \} \) within the Bergman metric and examining the (almost) \( \eta \)-lattice \( \{ a_{kj} \} \) along with the corresponding finer partition \( \{ D_{kj} \} \) of \( B^n \), as outlined prior to Lemma 2.29 According to Lemma 2.29 and the initial part of this proof, there exists a constant \( C_1 > 0 \) that supports this representation.

D ( a k , 2 r ) |f(z)| p dv α (z), whereσis the constant given in Lemma 2.29 Since each point ofB n belongs to at most ofN ofD(a k ,2r), we have

If η is sufficiently small such that C1Nσp < 1, the operator I - S on A p α has a norm of less than 1, indicating that the operator S is invertible on A p α Consequently, every function f in A p α can be expressed in the form f(z) = Σ c_kj (1 - |a_kj|^2)^(p - n - 1 - α)/p.

(1− z, a kj ) b , where c kj = v β (D kj )g(a kj )

(1− |a kj | 2 ) ( pb − n −1− α ) /p andg=S −1 f By Lemma 1.24, v β (D kj )≤v β (D k )∼(1− |a k | 2 ) n +1+ β = (1− |a k | 2 ) b

Since1− |a kj | 2 is comparable to1− |a k | 2 , we can find a constantC 2 >0, indepen- dent off, such that k,j

Applying Lemma 2.24 to eachg(a kj ), using the facts that1− |a kj | 2 is comparable to1− |a k | 2 and thatD(a kj , r) ⊂D(a k ,2r), we obtain another constantC 3 > 0 such that kj

Since every point ofB n belongs to at mostNof the setsD(a k ,2r), we have kj

This completes the proof of the theorem

The proof of Theorem 2.30 tells us that if{c k } ∈ l p and iff is given by the series representation (2.24), then

|c k | p for some positive constantCindependent off On the other hand, for anyf ∈A p α , the proof of Theorem 2.30 tells us that we can choose a sequence{c k }to represent f as in (2.24) which also satisfies k

|f(z)| p dv α (z), whereCis a positive constant independent off It follows that

We state two special cases of the preceding theorem The first case is whenp >1 andb=n+ 1 +α.

Corollary 2.31 For anyα >−1andp >1there exists a sequence{a k }inB n such thatA p α consists exactly of functions of the form f(z) k c k (1− |a k | 2 ) ( n +1+ α ) /q

The next case is forp= 1andb= 2(n+ 1 +α).

Corollary 2.32 For anyα >−1there exists a sequence{a k }inB n such thatA 1 α consists exactly of functions of the form f(z) k c k (1− |a k | 2 ) n +1+ α