# Weighted∂¯-Integral  Representations of C1-Functions in Cn

Authors: Arman H. Karapetyan
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

ForC1-functions f, given in the complex space Cn, integral representations of the form f=P(f)−T(∂¯f)
are obtained. Here, P is the orthogonal projector of the space L2{Cn;e−σ|z|ρ|z|γdm(z)} onto its subspace of entire functions and the integral operator T appears by means of explicitly constructed kernel Φ which is investigated in detail.


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## Body

## 1. Introduction

Letn≥1, 1<p<+∞, and 0<ρ,σ<∞, γ>-2n. Denote by Lρ,σ,γp(Cn) the space of all measurable complex-valued functions f(z), z∈Cn, satisfying the condition(1.1)∫Cn|f(z)|pe-σ|z|ρ|z|γdm(z)<+∞.
Let Hρ,σ,γp(Cn) be the corresponding subspace of entire functions. It was established in [1, 2] (when n=1) and [3] (when n>1) that arbitrary function f∈Hρ,σ,γp(Cn) has an integral representation of the form(1.2)f(z)=ρσμ2πn∫Cnf(w)e-σ|w|ρ|w|γ⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)dm(w),z∈Cn,
where μ=(γ+2n)/ρ and(1.3)Eρ/2(n)(η;μ)=∑k=0∞Γ(k+n)Γ(k+1)⋅ηkΓ(μ+2k/ρ),η∈C,
is the Mittag-Leffler type function. Moreover, the integral operator generated by the right-hand side of the formula (1.2) is an orthogonal projection of the space Lρ,σ,γ2(Cn) onto its subspace Hρ,σ,γ2(Cn). Certainly, the condition (1.1) and corresponding properties of Eρ/2(n)(η;μ) ensure an absolute convergence of the integral in (1.2).Note that forp=2,ρ=2, γ=0, Hρ,σ,γp(Cn) coincides with the well-known Fock space of entire functions and (1.2) takes the form(1.4)f(z)=σnπn∫Cnf(w)⋅e-σ|w|2⋅eσ〈z,w〉dm(w),z∈Cn.In [4] general weighted integral representations were obtained for differential forms. In particular, for functions f∈C1(Cn) (satisfying certain growth conditions), the following generalization of the formula (1.4) was established:(1.5)f(z)=σnπn∫Cnf(w)⋅eσ〈z,w〉⋅e-σ|w|2dm(w)-Γ(n)πn∫Cn〈(∂f/∂w̅)(w),w-z〉|w-z|2n⋅eσ〈z,w〉⋅∑ν=0n-1σνν!|w-z|2ν⋅e-σ|w|2dm(w),z∈Cn,
where(1.6)∂f∂w¯(w)=(∂f∂w1¯(w),∂f∂w2¯(w),…,∂f∂wn¯(w))and, consequently,(1.7)〈∂f∂w¯(w),w-z〉=∑k=1n∂f∂wk¯(w)⋅(wk¯-zk¯).In [5] a canonical operator is constructed for ∂¯-solution in a space of differential forms square integrable with the weight e-|w|2.The following natural question arises: as good as (1.4) is generalized for the case of smooth (not necessarily holomorphic) functions by the representation (1.5), is it possible to generalize the representation (1.2) in the similar way? Of course, such generalization should contain (1.5) as a particular case. Let us note (before we discuss this question) that in the case of bounded domains ∂¯ (and ∂∂¯) integral representations are well investigated: for unit ball of Cn see [6–8]; for general strictly pseudoconvex domains see [9–13]; for Cartan matrix domain (“matrix disc”) see [14]. The whole space Cn essentially differs from strictly pseudoconvex and bounded symmetric domains (the last ones have rich group of automorphisms!), so the above-mentioned generalization requires other methods. For n=1 it was done in [15]. For the case n>1, two essentially different generalizations are possible. In [16] “polycylindric” weight function of the type ∏k=1ne-σk|wk|ρk|wk|γk, w=(w1,w2,…,wn)∈Cn, was considered. In the present paper, the corresponding weighted ∂¯-integral representations are obtained for the case of radial weight function of the type e-σ|w|ρ|w|γ, w=(w1,w2,…,wn)∈Cn, |w|=∑k=1n|wk|2 (see the condition (1.1)). More precisely, for functions f∈C1(Cn) (satisfying certain growth conditions), the integral representation of the form(1.8)f(z)=ρσμ2πn∫Cnf(w)e-σ|w|ρ|w|γ⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)dm(w)-Γ(n)πn∫Cn〈(∂f/∂w̅)(w),w-z〉|w-z|2n⋅Φ(z;w)dm(w),z∈Cn,
is established. Moreover, the kernel Φ is written in an explicit form. Also, we prove certain important differential and integral properties of this kernel. As it will be seen below, in the case of radial weight function, a new approach is requested and significant analytical difficulties are arised.Remark 1.1.
In [4], instead of |w|γexp{-σ|w|ρ}, the case of weight function of the type exp{-φ(w)} was considered in the assumption that φ is a convex function of class C2. In this case, a formula of type (1.8) was obtained. But in that formula the operator of orthogonal projection of the space L2(Cn;exp{-φ(w)}dm(w)) onto its subspace of entire functions does not appear, except of the special case φ(w)=σ|w|2 when we again obtain (1.5).


## 2.  Heuristic Argument: Revealing of the KernelΦ

In what follows, it is supposed thatn≥1 and 0<ρ,σ<∞, γ>-2n, μ=(γ+2n)/ρ.We intend to reveal a formula of type (1.2) but this time for C1-functions. This means that the formula we search needs to have a second summand containing ∂¯-“part” of functions. In other words, for functions f∈C1(Cn) satisfying certain (indefinite yet) growth conditions at infinity, we search a formula of the type (1.8). Besides, it will be desirable for the kernel Φ(z;w),z,w∈Cn, to have the following (or similar) properties:(2.1)Φ(z;z)=1,∀z∈Cn,(2.2)Φ(z;∞)=0,∀z∈Cn.Denote byKMB(z;w) (for all z∈Cn, w∈Cn∖{z}) the well-known Martinelli-Bochner kernel, which has the following useful properties (see, for instance, [17, Chapter 16]):(2.3)∂¯wKMB(z;w)≡0,w∈Cn∖{z};(2.4)∂¯w{f(w)⋅KMB(z;w)}=∂¯f(w)∧KMB(z;w)≡Γ(n)πn⋅〈(∂f/∂w̅)(w),w-z〉|w-z|2n×dm(w)
for arbitrary function f∈C1(Ω), where Ω⊂Cn∖{z} is an open set;(2.5)lim⁡ɛ↓0∫|w-z|=ɛu(w)⋅KMB(z;w)=u(z)
for arbitrary function u continuous in a neighborhood of z;(2.6)∫Snf(ζ)⋅KMB(0;ζ)=Γ(n)2πn⋅∫Snf(ζ)dσ(ζ),
where Sn={ζ∈Cn:|ζ|=1} is the unit sphere in Cn,f∈C(Sn), and σ is the surface measure on Sn.Let us fix an arbitraryz∈Cn and consider the following differential form:(2.7)φ(w)=f(w)⋅Φ(z;w)⋅KMB(z;w),w∈Cn∖{z}.Then apply the Stokes formula to this form and to the domain{w∈Cn:0<ɛ<|w-z|<R<+∞}:(2.8)∫|w-z|=Rφ-∫|w-z|=ɛφ=∫ɛ<|w-z|<Rdφ.
The integral ∫|w-z|=Rφ→0 (as R→+∞) due to the property (2.2) (our argument is heuristic !!!). When ɛ→0, then ∫|w-z|=ɛφ→f(z) due to the properties (2.5) and (2.1). Thus after R→+∞ and ɛ→0, we have(2.9)-f(z)=∫Cndφ,or, in view of (2.3)-(2.4),(2.10)f(z)=-Γ(n)πn∫Cnf(w)⋅〈(∂Φ/∂w̅)(z;w),w-z〉|w-z|2ndm(w)-Γ(n)πn∫Cn〈(∂f/∂w̅)(w),w-z〉⋅Φ(z;w)|w-z|2ndm(w).Comparing (1.8) and (2.10), we arrive at the equality(2.11)〈∂Φ∂w¯(z;w),w-z〉=-ρσμ2Γ(n)Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ|w-z|2n,∀w∈Cn∖{z}.Let us fix arbitraryz,w∈Cn, z≠w and consider the function(2.12)φ(λ)=Φ(z;z+λ(w-z)),λ∈C.Then we have(2.13)∂φ∂λ¯=∑k=1n∂Φ∂wk¯(z;z+λ(w-z))⋅(wk¯-zk¯)=〈∂Φ∂w¯(z;z+λ(w-z)),w-z〉=-ρσμλ¯⋅2Γ(n)×Eρ/2(n)(σ2/ρ〈z,z+λ(w-z)〉;μ)e-σ|z+λ(w-z)|ρ×|z+λ(w-z)|γ|λ|2n|w-z|2n.The condition (2.2) implies φ(∞)=0; hence due to Cauchy-Green-Pompeiju formula,(2.14)φ(λ0)=-1π∫C∂φ/∂λ¯λ-λ0dm(λ),∀λ0∈C.Sinceφ(1)=Φ(z;w), we finally have
(2.15)Φ(z;w)=ρσμ2πΓ(n)|w-z|2n×∫C|λ|2nλ¯(λ-1)Eρ/2(n)(σ2/ρ〈z,z+λ(w-z)〉;μ)e-σ|z+λ(w-z)|ρ×|z+λ(w-z)|γdm(λ),∀z∈Cn,∀w∈Cn∖{z}.
This formula can be also written in the following (may be, more convenient) form:(2.16)Φ(z;z+w)=ρσμ2πΓ(n)|w|2n×∫C|λ|2nλ¯(λ-1)Eρ/2(n)(σ2/ρ〈z,z+λw〉;μ)e-σ|z+λw|ρ|z+λw|γdm(λ),∀z∈Cn,∀w∈Cn∖{0}.
Now it is natural to investigate the properties of the kernel introduced. But first we need some auxiliary results.


## 3.  Auxiliary Results

First of all let us put for brevity (see (1.3))(3.1)E(η)≡Eρ/2(n)(η;μ),η∈C.
It is an entire function of order ρ/2 and of type 1. The same is true for its derivative E′(η), η∈C. Consequently, we have(3.2)|E(η)|+|E′(η)|≤const(ρ;n;μ)⋅e2|η|ρ/2,η∈C.Let us introduce a convenient notation(3.3)φ(x)≡e-σxρ/2⋅xγ/2,x∈(0;+∞).
If γ≥0, we can suppose that x∈[0;+∞) in (3.3). Then obviously the function φ∈C[0;+∞) and(3.4)φ(x)≤const(ρ;σ;γ)⋅e(-σ/2)xρ/2,x∈[0;+∞).
Note that under additional assumptions γ≥2, ρ>0 or γ=0, ρ≥2, the function φ∈C1[0;+∞) and, moreover,(3.5)φ′(x)=e-σxρ/2⋅(γ2xγ/2-1-σρ2xρ/2-1+γ/2),x∈[0;+∞),|φ′(x)|≤const(ρ;σ;γ)⋅e(-σ/2)xρ/2,x∈[0;+∞).
In view of (3.1) and (3.3), the formula (2.16) can be written as follows:(3.6)Φ(z;z+w)=ρσμ2πΓ(n)|w|2n⋅∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λw〉)φ(|z+λw|2)dm(λ),∀z∈Cn,∀w∈Cn∖{0}.Further, assume thatz∈Cn, w∈Cn∖{0}, λ∈C, then evidently(3.7)|z+λw|2=|w|2⋅|λ+ã|2+δ̃,
where(3.8)ã=〈z,w〉|w|2,δ̃=|z|2|w|2-|〈z,w〉|2|w|2≥0.Note thatδ̃=0⇔z and w lies on the same complex “straight line” (i.e., complex plane) of Cn passing through the origin.Lemma 3.1.
Assume that0<R<+∞ and 0<m<M<+∞, then
(3.9)|E(σ2/ρ〈z,z+λw〉)|⋅e-σ|z+λw|ρ|z+λw|γ≡|E(σ2/ρ〈z,z+λw〉)|⋅φ(|z+λw|2)≤{c⋅e-b|λ|ρ,λ∈C,(γ≥0),c1⋅e-b1|λ|ρ⋅(|λ+ã|2+δ̃|w|2)γ/2≤c1⋅e-b1|λ|ρ⋅|λ+ã|γ,λ∈C∖{-ã},(-2n<γ<0),
uniformly in z and w with |z|≤R, m≤|w|≤M, where c,c1,b,b1 are positive constants and depend, in general, on n,ρ,σ,γ,R,m,M.Proof.
According to (3.2),
(3.10)|E(σ2/ρ〈z,z+λw〉)|≤const⋅e2σ|z|ρ/2(|z|+|λ∥w|)ρ/2≤const⋅e2σ|z|ρ/22ρ/2(|z|ρ/2+|λ|ρ/2|w|ρ/2)=const⋅e2(2+ρ)/2σ|z|ρ⋅e2(2+ρ)/2σ|z|ρ/2|w|ρ/2|λ|ρ/2≤const⋅e2(2+ρ)/2σRρ⋅e2(2+ρ)/2σRρ/2Mρ/2|λ|ρ/2≡const⋅ek|λ|ρ/2,λ∈C,
where k is a positive number.
Further, ifγ≥0, then in view of (3.4) and (3.7)
(3.11)φ(|z+λw|2)≤const(ρ,σ,γ)⋅e(-σ/2)|z+λw|ρ=const⋅e(-σ/2)|w|ρ(|λ+ã|2+δ̃/|w|2)ρ/2≤const⋅e(-σ/2)mρ|λ+ã|ρ.
Due to the conditions on z and w, we have |ã|≤R/m. Let us choose T>0 such that R/mT≤1/2, then for |λ|≥T(3.12)|λ+ã|≥|λ|-|ã|=|λ|(1-|ã||λ|)≥|λ|(1-RmT)≥|λ|2.
Hence
(3.13)e(-σ/2)mρ|λ+ã|ρ≤const⋅e(-σ/2)mρ(|λ|ρ/2ρ)(λ∈C).
Combining (3.11) and (3.13), we obtain
(3.14)φ(|z+λw|2)≤const⋅e-d|λ|ρ,λ∈C,
where d is a positive number.
Combination of (3.10) and (3.14) easily implies (3.9) for the case γ≥0. If -2n<γ<0, then similarly to (3.11)–(3.14) we have
(3.15)e-σ|z+λw|ρ≤const⋅e-d1|λ|ρ,λ∈C,
where d1 is a positive number. Also, in view of (3.7),
(3.16)|z+λw|γ=(|w|2⋅|λ+ã|2+δ̃)γ/2=|w|γ(|λ+ã|2+δ̃|w|2)γ/2≤mγ(|λ+ã|2+δ̃|w|2)γ/2.
Combination of (3.10), (3.15), and (3.16) establishes (3.9) for the case -2n<γ<0. The proof is complete.Taking into account (3.2), (3.4), (3.5) and repeating “word by word” the argument of Lemma 3.1, we obtain the following lemma.Lemma 3.2.
Assume that0<R<+∞ and 0<m<M<+∞. Then there exist positive constants c,b (depending, in general, on n,ρ,σ,γ,R,m,M) such that uniformly in z and w with |z|≤R, m≤|w|≤M.(a)
Ifγ≥0, then(3.17)|E(σ2/ρ〈z,z+λw〉)||E′(σ2/ρ〈z,z+λw〉)|×|φ(|z+λw|2)|≤c⋅e-b|λ|ρ,λ∈C.(b)
Ifγ≥2, ρ>0 or γ=0, ρ≥2, then(3.18)|E(σ2/ρ〈z,z+λw〉)||E′(σ2/ρ〈z,z+λw〉)|×|φ′(|z+λw|2)|≤c⋅e-b|λ|ρ,λ∈C.Lemma 3.3.
Letβ(η), η∈C, be an arbitrary function of class C1(C) such that it together with its first-order partial derivatives decreases (modulo) at infinity. For instance, the decreasing of type O(1/|η|1+ɛ), |η|→+∞, (for arbitrary small ɛ>0) is quite sufficient for us. Then the function
(3.19)α(η)≡-1π∫Cβ(ξ)ξ-ηdm(ξ),η∈C,
is of class C1(C) and (∂α/∂η¯)(η)≡β(η), η∈C.This assertion is of standard type so we omit the proof. Similar results one can find in [18, page 10, Theorem 1.1.3] for bounded open sets or in [19, page 300, Lemma] for arbitrary simply connected domains, but for C∞-functions.Lemma 3.4.
Assume thatψ(w), w∈Cn∖{0}, is continuously differentiable (i.e., of class C1) and g(w), w∈Cn∖{0}, is continuous. Then the following two relations are equivalent:
(3.20)〈∂ψ∂w¯(w),w〉≡g(w),w∈Cn∖{0},(3.21)∂∂η¯ψ(η⋅w)≡g(η⋅w)η¯,η∈C∖{0}(∀w∈Cn∖{0}).Proof.
Let us fix an arbitraryw∈Cn∖{0}; then
(3.22)∂∂η¯ψ(η⋅w)=∑k=1n∂ψ∂wk¯(η⋅w)⋅wk¯=〈∂ψ∂w¯(η⋅w),w〉=〈(∂ψ/∂w¯)(η⋅w),η⋅w〉η¯,η∈C∖{0}.
This immediately gives the implication (3.20)⇒(3.21). On the contrary, if (3.21) is valid, then
(3.23)〈(∂ψ/∂w¯)(η⋅w),η⋅w〉η¯=g(η⋅w)η¯,η∈C∖{0}.
Substitution of η=1 into the last relation gives (3.20). Thus, the assertion is proved.


## 4.   The Main Properties of the KernelΦ

Proposition 4.1.
Ifγ>-1, then for fixed z∈Cn, w∈Cn∖{0} and for arbitrary η∈C∖{0}(4.1)Φ(z;z+η⋅w)=-1π⋅∫Cβ(ξ)ξ-ηdm(ξ),
where
(4.2)β(ξ)=-ρσμ2Γ(n)|w|2n⋅|ξ|2nξ¯E(σ2/ρ〈z,z+ξw〉)e-σ|z+ξw|ρ|z+ξw|γ≡-ρσμ2Γ(n)|w|2n⋅|ξ|2nξ¯E(σ2/ρ〈z,z+ξw〉)φ(|z+ξw|2),ξ∈C∖{-ã}.
Moreover, assume that 0<R<+∞ and 0<m<M<+∞. Then there exist positive constants c,q (depending, in general, on n,ρ,σ,γ,R,m,M) such that uniformly in z and w with |z|≤R, m≤|w|≤M.(a)
Ifγ≥0, then β is a continuous function in C and(4.3)|β(ξ)|≤c⋅e-q|ξ|ρ,ξ∈C.(b)
Ifγ≥2, ρ>0 or γ=0, ρ≥2, then β is a C1-function in C and, in addition to (4.3),4.3’|∂β(ξ)∂ξ||∂β(ξ)∂ξ¯|≤c⋅e-q|ξ|ρ,ξ∈C.Proof.
Indeed, according to (2.16)
(4.4)Φ(z;z+η⋅w)=ρσμ2πΓ(n)|w|2n|η|2n×∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+ληw〉)e-σ|z+ληw|ρ|z+ληw|γdm(λ)==λη→ξ=ρσμ2πΓ(n)|w|2n⋅∫C|ξ|2nξ¯(ξ-η)E(σ2/ρ〈z,z+ξw〉)e-σ|z+ξw|ρ|z+ξw|γdm(ξ)≡-1π⋅∫Cβ(ξ)ξ-ηdm(ξ).
As to (4.3)-(4.3’), these inequalities immediately follow from Lemma 3.2 and the following relations:
(4.5)∂∂ξE(σ2/ρ〈z,z+ξw〉)=E′(σ2/ρ〈z,z+ξw〉)⋅0≡0,∂∂ξ¯E(σ2/ρ〈z,z+ξw〉)=E′(σ2/ρ〈z,z+ξw〉)⋅σ2/ρ〈z,w〉,∂∂ξφ(|z+ξw|2)=φ′(|z+ξw|2)⋅(〈w,z〉+ξ¯|w|2),∂∂ξ¯φ(|z+ξw|2)=φ′(|z+ξw|2)⋅(〈z,w〉+ξ|w|2).Proposition 4.2.
Ifγ≥0, then the kernel Φ(z;z+w) is continuous in z∈Cn, w∈Cn∖{0}.Proof.
Let us writeΦ(z;z+w) as follows:
(4.6)Φ(z;z+w)=ρσμ2πΓ(n)⋅|w|2n⋅∫CH(z;w;λ)dm(λ),
where
(4.7)H(z;w;λ)=|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λw〉)e-σ|z+λw|ρ|z+λw|γ≡|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λw〉)⋅φ(|z+λw|2),z∈Cn,w∈Cn∖{0},λ∈C∖{1}.
For arbitrary fixed positive numbersR,m,M(m<M), it suffices to construct a function h(λ)∈L1(C∖{1}), such that
(4.8)|H(z;w;λ)|≤h(λ),λ∈C∖{1},
uniformly in z and w with |z|≤R, m≤|w|≤M. According to Lemma 3.1 (the case γ≥0), the function
(4.9)c⋅|λ|2n-1|λ-1|e-b|λ|ρ
is suitable for a function h we seek.Proposition 4.3.
Ifγ≥0 and z∈Cn is arbitrary, then
(4.10)lim⁡ɛ↓0∫|w-z|=ɛf(w)⋅Φ(z;w)⋅KMB(z;w)≡lim⁡ɛ↓0∫|w|=ɛf(z+w)⋅Φ(z;z+w)⋅KMB(z;z+w)=f(z)
for arbitrary function f continuous in a neighborhood of z.Proof.
For sufficiently smallɛ>0, put
(4.11)Iɛ(z)=∫|w|=ɛf(z+w)⋅Φ(z;z+w)⋅KMB(z;z+w).
Taking into account the explicit formula (2.16) for Φ(z;z+w) and the well-known explicit form of the Martinelli-Bochner kernel KMB(z;z+w), we obtain
(4.12)Iɛ(z)=ρσμ2πΓ(n)⋅(-1)n(n-1)/2Γ(n)(2πi)n⋅∫|w|=ɛf(z+w)×|w|2n⋅{∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λw〉)⋅φ(|z+λw|2)dm(λ)}×1|w|2n⋅∑j=1n(-1)j-1wj¯dw¯[j]∧dw=====w=ɛ⋅ζ(ζ∈Sn)=ρσμ2πΓ(n)⋅(-1)n(n-1)/2Γ(n)(2πi)n⋅∫Snf(z+ɛ⋅ζ)×{∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λɛ⋅ζ〉)⋅φ(|z+λɛ⋅ζ|2)dm(λ)}×ɛ2n⋅∑j=1n(-1)j-1ζj¯dζ¯[j]∧dζ≡ρσμ2πΓ(n)⋅ɛ2n⋅∫Snf(z+ɛ⋅ζ)×{∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λɛ⋅ζ〉)⋅φ(|z+λɛ⋅ζ|2)dm(λ)}⋅KMB(0;ζ).
In view of (2.6), we obtain
(4.13)Iɛ(z)=ρσμɛ2n4πn+1⋅∫Snf(z+ɛ⋅ζ)×{∫C|λ|2nλ¯(λ-1)E(σ2/ρ〈z,z+λɛ⋅ζ〉)⋅φ(|z+λɛ⋅ζ|2)dm(λ)}dσ(ζ).
After the change of variable λ→λ/ɛ in the inner integral in (4.13), we have
(4.14)Iɛ(z)=ρσμ4πn+1⋅∫Snf(z+ɛ⋅ζ)×{∫C|λ|2nλ¯(λ-ε)E(σ2/ρ〈z,z+λ⋅ζ〉)⋅φ(|z+λ⋅ζ|2)dm(λ)}dσ(ζ)===λ→λ+ɛ=ρσμ4πn+1⋅∫Snf(z+ɛ⋅ζ)×{∫C(λ+ɛ)n(λ¯+ɛ)n-1λE(σ2/ρ〈z,z+(λ+ɛ)ζ〉)⋅φ(|z+(λ+ɛ)ζ|2)dm(λ)}dσ(ζ)≡ρσμ4πn+1⋅∫Snf(z+ɛ⋅ζ)⋅Iɛ(z;ζ)dσ(ζ),
where
(4.15)Iɛ(z;ζ)=∫CPɛ(z;ζ;λ)dm(λ),ζ∈Sn,
where
(4.16)Pɛ(z;ζ;λ)=(λ+ɛ)n(λ¯+ɛ)n-1λE(σ2/ρ〈z,z+(λ+ɛ)ζ〉)⋅φ(|z+(λ+ɛ)ζ|2),ζ∈Sn,λ∈C∖{0}.
Obviously, without loss of generality, we can suppose that 0<ɛ≤1.
According to Lemma3.1 (the case γ≥0) or, equivalently, to Lemma 3.2(a), there exist positive constants c,b (depending on n,ρ,σ,γ) such that
(4.17)|Pɛ(z;ζ;λ)|≤(|λ|+1)2n-1|λ|⋅c⋅e-b|λ+ɛ|ρ≤h(λ)≡{c⋅(|λ|+1)2n-1|λ|,0<|λ|≤1c⋅(|λ|+1)2n-1|λ|⋅e-b(|λ|-1)ρ,|λ|>1∈L1(C∖{0})
uniformly in ζ∈Sn, 0<ɛ≤1. Hence, due to the Lebesgue dominated convergence theorem, we can conclude that(i)
the functionsIɛ(z;ζ) are continuous in ζ∈Sn;(ii)
|Iɛ(z;ζ)|≤M<+∞, ζ∈Sn (uniformly in ɛ);(iii)
For for allζ∈Sn:limɛ↓0Iɛ(z;ζ)=I(z;ζ), where(4.18)I(z;ζ)=∫C|λ|2n-2⋅E(σ2/ρ〈z,z+λζ〉)⋅φ(|z+λζ|2)dm(λ).
Note that the function I(z;ζ) is also continuous in ζ∈Sn and |I(z;ζ)|≤M<+∞, ζ∈Sn.
Now remember (see (4.14)) that
(4.19)Iɛ(z)=ρσμ4πn+1⋅∫Snf(z+ɛ⋅ζ)⋅Iɛ(z;ζ)dσ(ζ).
Therefore, the application of the Lebesgue dominated convergence theorem once again gives
(4.20)lim⁡ɛ↓0Iɛ(z)=f(z)⋅ρσμ4πn+1∫SnI(z;ζ)dσ(ζ)=f(z)⋅ρσμ4πn+1∫Sndσ(ζ)∫C|λ|2n-2⋅E(σ2/ρ〈z,z+λζ〉)⋅φ(|z+λζ|2)dm(λ)=f(z)⋅ρσμ4πn+1∫Sndσ(ζ)∫0+∞∫02πr2n-1⋅E(σ2/ρ〈z,z+reiϑ⋅ζ〉)⋅φ(|z+reiϑ⋅ζ|2)drdϑ===r⋅ζ=w=f(z)⋅ρσμ4πn+1∫02πdϑ∫CnE(σ2/ρ〈z,z+eiϑ⋅w〉)⋅φ(|z+eiϑ⋅w|2)dm(w)=====z+eiϑ⋅w→w=f(z)⋅ρσμ4πn+1∫02πdϑ∫CnE(σ2/ρ〈z,w〉)⋅φ(|w|2)dm(w)=f(z)⋅ρσμ2πn∫CnE(σ2/ρ〈z,w〉)⋅e-σ|w|ρ|w|γdm(w)=f(z)⋅1=f(z)
in view of (1.2). Thus (4.10) is established.Remark 4.4.
Assume for a moment thatΦ(z;z+w) can be defined at w=0 such that Φ(z;z)=1 and, moreover, that after this Φ(z;z+w) becomes continuous at w=0 (unfortunately, this is not, in general, true, as it will be mentioned below). Then the relation (4.10) is a simple consequence of (2.5). Hence (4.10) can be considered as a substitute for the natural (and “very desired”) property Φ(z;z)=1.Proposition 4.5.
Ifγ≥2, ρ>0 or γ=0, ρ≥2, the kernel Φ(z;z+w) is a function of class C1 in z∈Cn, w∈Cn∖{0}.Proof.
In view of (4.6)-(4.7), we have to show that ∫CH(z;w;λ)dm(λ) is of class C1 in z∈Cn, w∈Cn∖{0}. In other words, for arbitrary fixed positive numbers R,m,M(m<M), it suffices to construct a function h(λ)∈L1(C∖{1}), such that
(4.21)|∂∂wkH(z;w;λ)|,|∂∂wk¯H(z;w;λ)|≤h(λ),λ∈C∖{1},|∂∂zkH(z;w;λ)|,|∂∂zk¯H(z;w;λ)|≤h(λ),λ∈C∖{1},
uniformly in z and w with |z|≤R, m≤|w|≤M, and k=1,…,n.
Explicitly computing the corresponding partial derivatives and taking note of Lemma3.2, we find that we are reduced to the question of the finiteness of integrals of the type
(4.22)∫C|λ|2n+τ|λ-1|e-b|λ|ρdm(λ)(b>0,τ=-1;0;1),
which is evident.Proposition 4.6.
Ifγ≥2, ρ>0 or γ=0, ρ≥2, then for arbitrary fixed z∈Cn we have
(4.23)〈∂Φ∂w¯(z;z+w),w〉=-ρσμ2Γ(n)|w|2nE(σ2/ρ〈z,z+w〉)e-σ|z+w|ρ|z+w|γ,∀w∈Cn∖{0},
or, equivalently,
(4.23′)〈∂Φ∂w¯(z;w),w-z〉=-ρσμ2Γ(n)|w-z|2nE(σ2/ρ〈z,w〉)e-σ|w|ρ|w|γ,∀w∈Cn∖{z}.Proof.
We intend to use Lemma3.4. To this end, let us put for w∈Cn∖{0}(4.24)ψ(w)=Φ(z;z+w),g(w)=-ρσμ2Γ(n)|w|2nE(σ2/ρ〈z,z+w〉)e-σ|z+w|ρ|z+w|γ.
It suffices to establish (3.21) for the introduced functions ψ and g. Fix w∈Cn∖{0}, then in view of Proposition 4.1 we have
(4.25)ψ(η⋅w)=-1π⋅∫Cβ(ξ)ξ-ηdm(ξ),∀η∈C∖{0},
where β(ξ)∈C1(C) is defined by (4.2) and satisfies (4.3)-(4.3’). Consequently, β satisfies all the conditions of Lemma 3.3. Hence
(4.26)∂∂η¯ψ(η⋅w)≡β(η)=-ρσμ2Γ(n)|w|2n⋅|η|2nη¯E(σ2/ρ〈z,z+ηw〉)e-σ|z+ηw|ρ|z+ηw|γ≡g(η⋅w)η¯,∀η∈C∖{0}.
By this (3.21) has been established for the pair of introduced functions ψ and g. The proof is complete.The next assertion describes the behaviour of the kernelΦ(z;z+w) when w→0 or w→∞. Since these properties will not be used in what follows, we omit the proof, which, by the way, is not easy.Proposition 4.7.
Ifγ≥0, then(a)
for arbitraryz∈Cn and for arbitrary w∈Cn∖{0}, we have
(4.27)lim⁡η→0Φ(z;z+η⋅w)=ρσμ⋅|w|2n2π⋅Γ(n)×∫C|ξ|2n-2⋅E(σ2/ρ〈z,z+ξw〉)⋅e-σ|z+ξw|ρ|z+ξw|γdm(ξ);(b)
ifz≠0 and [z] is the complex plane generated by the vector z, then
(4.28)lim⁡w→0,w⊥[z]Φ(z;z+w)=ρσμ2Γ(n)⋅E(σ2/ρ|z|2)⋅∫0+∞xn-1⋅e-σ{|z|2+x}ρ/2{|z|2+x}γ/2dx,
and for arbitrary R>0 and for (arbitrary small) ɛ>0, there exist a positive constant c=c(n,ρ,σ,γ,ɛ,R) such that
(4.29)|Φ(z;z+w)|≤c⋅e-(σ-ɛ)|w|ρ,∀w⊥[z](w≠0),
uniformly in z with |z|<R;(c)
ifz≠0, then
(4.30)lim⁡w→0,w∈[z]Φ(z;z+w)=∑m=0n-1(-1)n-1-m⋅Cn-1m⋅(|z|2σ2/ρ)n-1-m⋅Ln,mγ,ρ,
where the coefficients ℒn,mγ,ρ(m=1,2,…,n-1) can be written in an explicit form;(d)
for arbitraryR>0, there exist positive constants c,δ1 (depending, in general, on n,ρ,σ,γ,R) such that
(4.31)|Φ(z;z+w)|≤c⋅(|w|+e-δ1|w|ρ)
uniformly in w∈Cn∖{0} and z with |z|<R. In particular, the kernel Φ(z;z+w) remains bounded (uniformly in z with |z|<R) as w→0.Remark 4.8.
As it follows from (4.28) and (4.30), in general, limw→0Φ(z;z+w) (when w tends to zero arbitrarily) cannot be properly defined (i.e., this limit does not exist). In view of this fact, it seems surprising the existence of the limit in (4.27). In fact, it only means that, nevertheless, the restrictions of the kernel Φ(z;z+w) on complex planes [w] (generated by arbitrary w∈Cn∖{0}) have limit values at the origin.


## 5.  The Main Integral Representation

Now we are ready to formulate and prove the main result: an integral representation of the type (1.8). To this end, we have to repeat the heuristic argument of Section 2, but this time it should be well reasoned.In what follows, we need a functionχ(t), t∈(-∞;+∞), satisfying the following conditions:(i)
χ∈C1(R);(ii)
0≤χ(t)≤1, t∈(-∞;+∞);(iii)
χ(t)≡1, t∈(-∞;0];(iv)
χ(t)≡0, t∈[1;+∞);(v)
χ↓[0;1];(vi)
|χ′(t)|≤M<+∞, t∈[0;1] and (obviously) χ′(t)≡0 otherwise.The existence of such functions is evident. Then put(5.1)χR(w)≡χ(|w|2-R2){=1,0≤|w|≤R,∈[0;1],R≤|w|≤R2+1,=0,|w|≥R2+1.
Note that χR∈C1(Cn) and(5.2)∂χR∂w¯(w)≡(∂χR∂w1¯(w),∂χR∂w2¯(w),…,∂χR∂wn¯(w))={(0,0,…,0),0≤|w|≤R,χ′(|w|2-R2)⋅(w1,w2,…,wn),R≤|w|≤R2+1,(0,0,…,0),|w|≥R2+1.Moreover,(5.3)|∂χR∂w¯(w)|≡∑k=1n|∂χR∂wk¯(w)|2{=0,0≤|w|≤R,≤M⋅|w|,R≤|w|≤R2+1,=0,|w|≥R2+1.Theorem 5.1.
Assume thatn≥1, 1<p<+∞, σ>0 and μ=(γ+2n)/ρ, where either γ≥2, ρ>0 or γ=0, ρ≥2. If the kernel Φ(z;w),for all z∈Cn, for all w∈Cn∖{z}, is defined by the formula (2.15), then the integral representation of the form
(5.4)f(z)=ρσμ2πn∫Cnf(w)⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)⋅e-σ|w|ρ|w|γdm(w)-Γ(n)πn∫Cn〈(∂f/∂w¯)(w),w-z〉|w-z|2nΦ(z;w)dm(w),z∈Cn,
is valid for each function f∈C1(Cn) satisfying the following conditions:(a)
f∈Lρ,σ,γp(Cn);(b)
for any fixedz∈Cn,
(5.5)|f(w)|⋅Φ(z;w)|w-z|2n-2∈L1(Cn;dm(w))⟺|f(w)|⋅Φ(z;w)|w|2n-2∈L1(Cn;dm(w));(c)
for any fixedz∈Cn(5.6)|(∂f/∂w¯)(w)|⋅Φ(z;w)|w-z|2n-1∈L1(Cn;dm(w))⟺|(∂f/∂w¯)(w)|⋅Φ(z;w)|w|2n-1∈L1(Cn;dm(w)).Proof.
Let us fix an arbitraryz∈Cn, and for ∀R>0 consider the following differential form:(5.7)ψ(z;w)=f(w)⋅χR(w-z)⋅Φ(z;w)⋅KMB(z;w),w∈Cn∖{z}.
Then choose an ɛ∈(0;R) and apply the Stokes formula to this form and to the domain {w∈Cn:0<ɛ<|w-z|<R2+1}:
(5.8)∫|w-z|=R2+1ψ-∫|w-z|=ɛψ=∫ɛ<|w-z|<R2+1dψ.
In view of (5.1) and (2.4), the last relation can be written as follows:
(5.9)-∫|w-z|=ɛf(w)⋅Φ(z;w)⋅KMB(z;w)=Γ(n)πn∫ɛ<|w-z|<R2+1〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅χR(w-z)⋅Φ(z;w)dm(w)+Γ(n)πn∫ɛ<|w-z|<R2+1〈(∂χR/∂w¯)(w-z),w-z〉|w-z|2n⋅f(w)⋅Φ(z;w)dm(w)+Γ(n)πn∫ɛ<|w-z|<R2+1〈(∂Φ/∂w¯)(z;w),w-z〉|w-z|2n⋅f(w)⋅χR(w-z)dm(w).
Moreover, (5.2) and (4.23′) imply
(5.10)-∫|w-z|=ɛf(w)⋅Φ(z;w)⋅KMB(z;w)=Γ(n)πn∫ɛ<|w-z|<R2+1〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅χR(w-z)⋅Φ(z;w)dm(w)+Γ(n)πn∫R<|w-z|<R2+1χ′(|w-z|2-R2)|w-z|2n-2⋅f(w)⋅Φ(z;w)dm(w)-ρσμ2πn∫ɛ<|w-z|<R2+1Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ⋅f(w)⋅χR(w-z)dm(w).
When ɛ→0, then due to (4.10) we obtain
(5.11)f(z)=ρσμ2πn∫|w-z|<R2+1Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ⋅f(w)⋅χR(w-z)dm(w)-Γ(n)πn∫|w-z|<R2+1〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅χR(w-z)⋅Φ(z;w)dm(w)-Γ(n)πn∫R<|w-z|<R2+1χ′(|w-z|2-R2)|w-z|2n-2⋅f(w)⋅Φ(z;w)dm(w)≡A1(R)-A2(R)-A3(R).
Further,
(5.12)A1(R)=ρσμ2πn∫|w-z|<REρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ⋅f(w)dm(w)+ρσμ2πn∫R<|w-z|<R2+1Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ⋅f(w)⋅χR(w-z)dm(w)≡A1(1)(R)+A1(2)(R).
Since (via the condition (a))
(5.13)f(w)⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γ∈L1(Cn;dm(w)),
the summand A1(1)(R) tends to the same integral but taken now over the whole space Cn and A1(2)(R) tends to zero (as R→+∞).
In other words,(5.14)lim⁡R→+∞A1(R)=ρσμ2πn∫Cnf(w)⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)e-σ|w|ρ|w|γdm(w).
Similarly,
(5.15)A2(R)=Γ(n)πn∫|w-z|<R〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅Φ(z;w)dm(w)+Γ(n)πn∫R<|w-z|<R2+1〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅χR(w-z)⋅Φ(z;w)dm(w)≡A2(1)(R)+A2(2)(R).
In view of the condition (c) of the theorem
(5.16)〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅Φ(z;w)∈L1(Cn;dm(w)),
so A2(1)(R) tends to the same integral but taken now over the whole space Cn and A2(2)(R) tends to zero (as R→+∞). In other words,
(5.17)lim⁡R→+∞A2(R)=Γ(n)πn∫Cn〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅Φ(z;w)dm(w).
Finally,
(5.18)|A3(R)|≤Γ(n)πn⋅M⋅∫R<|w-z|<R2+1|f(w)|⋅Φ(z;w)|w-z|2n-2dm(w)≤Γ(n)πn⋅M⋅∫R<|w-z|<+∞|f(w)|⋅Φ(z;w)|w-z|2n-2dm(w).
Hence, due to the condition (b) of the theorem,
(5.19)lim⁡R→+∞A3(R)=0.
Combining (5.11)–(5.19), we ultimately obtain
(5.20)f(z)=ρσμ2πn∫Cnf(w)⋅Eρ/2(n)(σ2/ρ〈z,w〉;μ)⋅e-σ|w|ρ|w|γdm(w)-Γ(n)πn∫Cn〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅Φ(z;w)dm(w),
which coincides with (5.4). Thus the theorem is proved.Remark 5.2.
The idea of introducing an auxiliary functionχ is borrowed from [16].


## 6. The Computable Form of the KernelΦ

Up to now, we base on the formulae (2.15)-(2.16) defining the kernel Φ, and this makes it possible to investigate the properties of the kernel (see Section 4). Now we intend to simplify (2.15) or, more precisely, to bring the formula to a more algorithmical form (in the sense of explicit computability). To this end, let us start with several notations.For arbitraryz,w∈Cn(z≠w), put (compare with (3.8))(6.1)a=〈z,w-z〉|w-z|,c=〈w,w-z〉|w-z|,δ=|z|2|w-z|2-|〈z,w-z〉|2|w-z|2≡|z|2|w|2-|〈z,w〉|2|w-z|2≥0.
Note that a slight change of (3.7) yields(6.2)|z+λ(w-z)|2≡|w-z|2⋅|λ+a|w-z||2+δ.
Let ρ,σ>0, γ≥0, and μ=(γ+2n)/ρ. Consider the following functions:(6.3)ψ(s)=(s-a¯)n-1Eρ/2(n)(σ2/ρ(a⋅s+δ);μ),s∈C,(6.4)ψ(ν)(s)≡dνψ(s)dsν,ν=0,1,2,3,….
Besides, put (x≥0):(6.5)ϕ(x)=e-σ(x+δ)ρ/2(x+δ)γ/2,(6.6)Φk(x)=1Γ(k)∫x+∞ϕ(t)(t-x)k-1dt,k=1,2,3,….
Note that Φk is the k-th primitive of the function ϕ, that is,  (6.7)Φk(x)=∫x+∞dx∫x+∞dx⋯∫x+∞︸ktimesϕ(x)dx.
In what follows we also put Φ0(x)≡ϕ(x), x≥0. Obviously, the following simple relations are valid:(6.8)ddx[Φk+1(x)]=-Φk(x)⟹dkΦk(x)dxk=(-1)kϕ(x),x≥0,k=0,1,2,3,…,(6.9)Φk(0)=1Γ(k)∫0+∞ϕ(t)tk-1dt,k=1,2,3.…,(6.10)∫r<|λ|<Rλlλ¯mϕ(|λ|2)dm(λ)=0,
where 0≤r<R≤∞ and l,m=0,1,2,3,…,l≠m. Certainly, here ϕ can be replaced by any other radial (i.e., depending only on |λ|) function if only the corresponding integrals exist.Theorem 6.1.
The kernelΦ can be computed by the following formula:
(6.11)Φ(z;w)=ρσμ2Γ(n)⋅{∑ν=0n-1ψ(ν)(0)⋅Φν+1(0)⋅∑k=0n-1-νCnk+1+ν(-a)n-1-ν-kck}-ρσμ2Γ(n)⋅c¯(c-a)n⋅∫01ψ(c¯⋅t)ϕ(|c|2⋅t)dt.Proof.
In view of (2.15) and (6.2), we have
(6.12)Φ(z;w)=ρσμ2Γ(n)⋅I,
where
(6.13)I=|w-z|2nπ∫C|λ|2nλ¯(λ-1)Eρ/2(n)(σ2/ρ(|z|2+|w-z|λ¯⋅a);μ)×e-σ(|w-z|2⋅|λ+(a/|w-z|)|2+δ)ρ/2⋅(|w-z|2⋅|λ+a|w-z||2+δ)γ/2dm(λ)=====|w-z|λ+a→λ=1π∫C(λ-a)n⋅(λ¯-a¯)n-1λ-cEρ/2(n)(σ2/ρ(a⋅λ¯+δ);μ)⋅e-σ(|λ|2+δ)ρ/2(|λ|2+δ)γ/2dm(λ).
In view of notations (6.3) and (6.5), we have
(6.14)I=1π∫C(λ-a)nλ-c⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ).
In order to simplify the last integral, let us consider the following auxiliary integrals:
(6.15)Aν=1π∫Cλν⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ),ν=0,1,2,3,…,B=1π∫Cψ(λ¯)λ-c⋅ϕ(|λ|2)dm(λ).
Using the Maclaurin expansion ψ(λ¯)=∑k=0∞(ψ(k)(0)/Γ(k+1))·λ¯k and (6.10), we obtain
(6.16)Aν=1π⋅ψ(ν)(0)Γ(ν+1)⋅∫C|λ|2ν⋅ϕ(|λ|2)dm(λ)=ψ(ν)(0)Γ(ν+1)⋅∫0∞tνϕ(t)dt=ψ(ν)(0)⋅Φν+1(0).
Further,(6.17)B=1π∫|λ|<|c|ψ(λ¯)λ-c⋅ϕ(|λ|2)dm(λ)+1π∫|λ|>|c|ψ(λ¯)λ-c⋅ϕ(|λ|2)dm(λ)≡B(+)+B(-).
If c=0, then, naturally, B(+)=0, and
(6.18)B(-)=1π∫|λ|>0ψ(λ¯)λ⋅ϕ(|λ|2)dm(λ)=1π∫|λ|>0λ¯⋅ψ(λ¯)⋅ϕ(|λ|2)|λ|2dm(λ)=0,
that is, B=0. If c≠0, then
(6.19)1λ-c={-1c⋅∑k=0∞λkck,|λ|<|c|,1λ⋅∑k=0∞ckλk,|λ|>|c|.
Consequently,
(6.20)B(-)=1π∑k=0∞ck⋅∫|λ|>|c|ψ(λ¯)λk+1⋅ϕ(|λ|2)dm(λ)=1π∑k=0∞ck⋅∫|λ|>|c|λ¯k+1⋅ψ(λ¯)⋅ϕ(|λ|2)|λ|2k+2dm(λ)=0.
Hence
(6.21)B=B(+)=-1πc∑k=0∞∫|λ|<|c|λkck⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ)=-1πc∑k=0∞∫|λ|<|c|ψ(k)(0)Γ(k+1)⋅ck⋅|λ|2k⋅ϕ(|λ|2)dm(λ)=-1c∑k=0∞∫0|c|2ψ(k)(0)Γ(k+1)⋅ck⋅tkϕ(t)dm(t)=-1c∫0|c|2∑k=0∞ψ(k)(0)Γ(k+1)(tc)kϕ(t)dm(t)=-1c∫0|c|2ψ(tc)ϕ(t)dm(t)===t→|c|2t-c¯⋅∫01ψ(c¯⋅t)ϕ(|c|2⋅t)dm(t).
Using the binomial expansion of (λ-a)n and combining (6.14)–(6.21), we obtain
(6.22)I=∑k=0nCnk(-a)n-k⋅1π∫Cλkλ-c⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ)=∑k=1nCnk(-a)n-k⋅1π∫C(λk-ck)+ckλ-c⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ)+(-a)n⋅B=∑k=1nCnk(-a)n-k⋅1π∫C(λ-c)⋅∑ν=0k-1ck-1-νλνλ-c⋅ψ(λ¯)⋅ϕ(|λ|2)dm(λ)+∑k=1nCnk(-a)n-k⋅ck⋅B+(-a)n⋅B=∑k=1nCnk(-a)n-k⋅∑ν=0k-1ck-1-ν⋅Aν+(c-a)n⋅B===k→k+1=∑k=0n-1Cnk+1(-a)n-1-k⋅∑ν=0kck-ν⋅ψ(ν)(0)⋅Φν+1(0)-c¯(c-a)n⋅∫01ψ(c¯⋅t)ϕ(|c|2⋅t)dm(t)=∑ν=0n-1ψ(ν)(0)⋅Φν+1(0)⋅∑k=0n-1-νCnk+1+ν(-a)n-1-ν-kck-c¯(c-a)n⋅∫01ψ(c¯⋅t)ϕ(|c|2⋅t)dm(t).
Finally, note that (6.12) and (6.22) imply (6.11) and the proof is complete.


## 7. Important Special Case:ρ=2, γ=0

In this section, we analyze the special caseρ=2, γ=0, when the formulas become more simple and more explicit.First of all,μ=(γ+2n)/ρ=n. Hence, the coefficient ρσμ/2Γ(n) transforms into σn/Γ(n). Next,(7.1)Eρ/2(n)(η;μ)=∑k=0∞Γ(k+n)Γ(k+1)⋅ηkΓ(μ+2k/ρ)=∑k=0∞ηkΓ(k+1)≡eη,η∈C.
As a consequence, the formula (2.15) takes the following form:(7.2)Φ(z;w)=e-σ|w|2⋅eσ〈z,w〉⋅∑ν=0n-1σνν!|w-z|2ν,∀z∈Cn,∀w∈Cn∖{z},
or, equivalently,(7.3)Φ(z;z+w)=e-σ|w|2⋅e-σ〈w,z〉⋅∑ν=0n-1σνν!|w|2ν,∀z∈Cn,∀w∈Cn∖{0}.
Indeed, we have(7.4)Φ(z;w)=σnπΓ(n)|w-z|2n⋅∫C|λ|2nλ¯(λ-1)eσ〈z,z+λ(w-z)〉⋅e-σ|z+λ(w-z)|2dm(λ)=σnπΓ(n)|w-z|2n⋅∫C|λ|2nλ¯(λ-1)e-σ〈λ(w-z),z+λ(w-z)〉dm(λ)=σnπΓ(n)|w-z|2n⋅∫C|λ|2n-2⋅e-σ|w-z|2|λ|2⋅λ⋅e-σλ〈w-z,z〉λ-1dm(λ)=2σnΓ(n)|w-z|2n⋅∫0+∞r2n-1⋅e-σ|w-z|2r2⋅12πi∫|ζ|=re-σζ〈w-z,z〉ζ-1dζdr=2σnΓ(n)|w-z|2n⋅∫0+∞r2n-1⋅e-σ|w-z|2r2×{0,0<r<1e-σ〈w-z,z〉,1<r<∞×dr=σnΓ(n)|w-z|2n⋅e-σ〈w-z,z〉⋅∫1+∞tn-1⋅e-σ|w-z|2tdt=σnΓ(n)|w-z|2n⋅e-σ〈w-z,z〉⋅e-σ|w-z|2⋅∫0+∞(t+1)n-1⋅e-σ|w-z|2tdt=σnΓ(n)|w-z|2n⋅e-σ|w|2⋅eσ〈z,w〉⋅∑ν=0n-1Cn-1ν⋅Γ(n-ν)σn-ν|w-z|2n-2ν=e-σ|w|2⋅eσ〈z,w〉⋅∑ν=0n-1σνν!|w-z|2ν.It is interesting to note (see Remark4.4) that(7.5)lim⁡w→0Φ(z;z+w)=Φ(z;z)=1.
Moreover (compare with Proposition 4.7), for arbitrary R>0 and for (arbitrary small) ɛ>0, there exists a positive constant c=c(n,σ,ɛ,R) such that(7.6)|Φ(z;z+w)|≤c⋅e-(σ-ɛ)|w|2,∀w∈Cn,
uniformly in z with |z|<R. At last, the main Theorem 5.1 takes the following form.Theorem 7.1.
Assume thatn≥1, 1<p<+∞ and σ>0, then the integral representation of the form
(7.7)f(z)=σnπn∫Cnf(w)⋅eσ〈z,w〉⋅e-σ|w|2dm(w)-Γ(n)πn∫Cn〈(∂f/∂w¯)(w),w-z〉|w-z|2n⋅eσ〈z,w〉×∑ν=0n-1σνν!|w-z|2ν⋅e-σ|w|2dm(w),z∈Cn,
is valid for each function f∈C1(Cn) satisfying the following conditions:(1)
∫Cn|f(w)|p·e-σ|w|2dm(w)<+∞;(2)
for any fixedz∈Cn,(7.8)∫Cn|(∂f/∂w¯)(w)||w-z|2n-1⋅|eσ〈z,w〉|⋅∑ν=0n-1σνν!|w-z|2ν⋅e-σ|w|2dm(w)<+∞,
or, equivalently,
(7.9)∫Cn|(∂f/∂w¯)(w)||w-z|⋅|eσ〈z,w〉|⋅e-σ|w|2dm(w)<+∞,
or, not equivalently (but more conveniently),
(7.10)∫Cn|∂f∂w¯(w)|⋅e-(σ-ɛ)|w|2dm(w)<+∞,
where ɛ is arbitrary small.



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*Source: 265092-2012-02-08.xml*