JFSAJournal of Function Spaces and Applications0972-68022090-8997Hindawi Publishing Corporation26509210.1155/2012/265092265092Research ArticleWeighted ¯-Integral Representations of C1-Functions in C nKarapetyanArman H.RochbergRichard1Institute of MathematicsNational Academy of Sciences of ArmeniaYerevanArmeniasci.am2012822012201228072011130820112012Copyright © 2012 Arman H. Karapetyan.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For C1-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;eσ|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.

1. Introduction

Let n1, 1<p<+, and 0<ρ,σ<, γ>-2n. Denote by Lρ,σ,γp(Cn) the space of all measurable complex-valued functions f(z), zCn, satisfying the conditionCn|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  (when n>1) that arbitrary function fHρ,σ,γp(Cn) has an integral representation of the formf(z)=ρσμ2πnCnf(w)e-σ|w|ρ|w|γEρ/2(n)(σ2/ρz,w;μ)dm(w),zCn, where μ=(γ+2n)/ρ andEρ/2(n)(η;μ)=k=0Γ(k+n)Γ(k+1)ηkΓ(μ+2k/ρ),ηC, 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 for p=2,  ρ=2, γ=0, Hρ,σ,γp(Cn) coincides with the well-known Fock space of entire functions and (1.2) takes the formf(z)=σnπnCnf(w)e-σ|w|2eσz,wdm(w),zCn.

In  general weighted integral representations were obtained for differential forms. In particular, for functions fC1(Cn) (satisfying certain growth conditions), the following generalization of the formula (1.4) was established:f(z)=σnπnCnf(w)eσz,we-σ|w|2dm(w)-Γ(n)πnCn(f/w̅)(w),w-z|w-z|2neσz,wν=0n-1σνν!|w-z|2νe-σ|w|2dm(w),zCn, wherefw¯(w)=(fw1¯(w),fw2¯(w),,fwn¯(w))

and, consequently, fw¯(w),w-z=k=1nfwk¯(w)(wk¯-zk¯).

In  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 ; for general strictly pseudoconvex domains see ; for Cartan matrix domain (“matrix disc”) see . 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 . For the case n>1, two essentially different generalizations are possible. In  “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 fC1(Cn) (satisfying certain growth conditions), the integral representation of the formf(z)=ρσμ2πnCnf(w)e-σ|w|ρ|w|γEρ/2(n)(σ2/ρz,w;μ)dm(w)-Γ(n)πnCn(f/w̅)(w),w-z|w-z|2nΦ(z;w)dm(w),zCn, 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 , 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 Kernel <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M61"><mml:mrow><mml:mi mathvariant="normal">Φ</mml:mi></mml:mrow></mml:math></inline-formula>

In what follows, it is supposed that n1 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 fC1(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,wCn, to have the following (or similar) properties:Φ(z;z)=1,zCn,Φ(z;)=0,  zCn.

Denote by KMB(z;w) (for all zCn, wCn{z}) the well-known Martinelli-Bochner kernel, which has the following useful properties (see, for instance, [17, Chapter 16]):¯wKMB(z;w)0,wCn{z};¯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 fC1(Ω), where ΩCn{z} is an open set;limɛ0|w-z|=ɛu(w)KMB(z;w)=u(z) for arbitrary function u continuous in a neighborhood of z;Snf(ζ)KMB(0;ζ)=Γ(n)2πnSnf(ζ)dσ(ζ), where Sn={ζCn:|ζ|=1} is the unit sphere in Cn,fC(Sn), and σ is the surface measure on Sn.

Let us fix an arbitrary zCn and consider the following differential form:φ(w)=f(w)Φ(z;w)KMB(z;w),wCn{z}.

Then apply the Stokes formula to this form and to the domain {wCn:0<ɛ<|w-z|<R<+}:|w-z|=Rφ-|w-z|=ɛφ=ɛ<|w-z|<Rdφ. The integral |w-z|=Rφ0 (as R+) due to the property (2.2) (our argument is heuristic !!!). When ɛ0, then |w-z|=ɛφf(z) due to the properties (2.5) and (2.1). Thus after R+ and ɛ0, we have-f(z)=Cndφ,

or, in view of (2.3)-(2.4),f(z)=-Γ(n)πnCnf(w)(Φ/w̅)(z;w),w-z|w-z|2ndm(w)-Γ(n)πnCn(f/w̅)(w),w-zΦ(z;w)|w-z|2ndm(w).

Comparing (1.8) and (2.10), we arrive at the equalityΦw¯(z;w),w-z=-ρσμ2Γ(n)Eρ/2(n)(σ2/ρ  z,w;μ)e-σ|w|ρ|w|γ|w-z|2n,wCn{z}.

Let us fix arbitrary z,wCn, zw and consider the functionφ(λ)=Φ(z;z+λ(w-z)),λC.

Then we haveφλ¯=k=1nΦwk¯(z;z+λ(w-z))(wk¯-zk¯)=Φw¯(z;z+λ(w-z)),w-z=-ρσμλ¯2Γ(n)×Eρ/2(n)(σ2/ρz,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,φ(λ0)=-1πCφ/λ¯λ-λ0dm(λ),λ0C.

Since φ(1)=Φ(z;w), we finally have Φ(z;w)=ρσμ2πΓ(n)|w-z|2n×C|λ|2nλ¯(λ-1)Eρ/2(n)(σ2/ρz,z+λ(w-z);μ)e-σ|z+λ(w-z)|ρ×|z+λ(w-z)|γdm(λ),zCn,wCn{z}. This formula can be also written in the following (may be, more convenient) form:Φ(z;z+w)=ρσμ2πΓ(n)|w|2n×C|λ|2nλ¯(λ-1)Eρ/2(n)(σ2/ρz,z+λw;μ)e-σ|z+λw|ρ|z+λw|γdm(λ),zCn,wCn{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))E(η)Eρ/2(n)(η;μ),ηC. It is an entire function of order ρ/2 and of type 1. The same is true for its derivative E(η), ηC. Consequently, we have|E(η)|+|E(η)|const(ρ;n;μ)e2|η|ρ/2,ηC.

Let us introduce a convenient notationφ(x)e-σxρ/2xγ/2,x(0;+). If γ0, we can suppose that x[0;+) in (3.3). Then obviously the function φC[0;+) andφ(x)const(ρ;σ;γ)e(-σ/2)xρ/2,x[0;+). Note that under additional assumptions γ2, ρ>0 or γ=0, ρ2, the function φC1[0;+) and, moreover,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:Φ(z;z+w)=ρσμ2πΓ(n)|w|2nC|λ|2nλ¯(λ-1)E(σ2/ρz,z+λw)φ(|z+λw|2)dm(λ),zCn,wCn{0}.

Further, assume that zCn, wCn{0}, λC, then evidently|z+λw|2=|w|2|λ+ã|2+δ̃, whereã=z,w|w|2,  δ̃=|z|2|w|2-|z,w|2|w|20.

Note that δ̃=0z and w lies on the same complex “straight line” (i.e., complex plane) of Cn passing through the origin.

Lemma 3.1.

Assume that 0<R<+ and 0<m<M<+, then |E(σ2/ρz,z+λw)|e-σ|z+λw|ρ|z+λw|γ|E(σ2/ρz,z+λw)|φ(|z+λw|2){ce-b|λ|ρ,  λC,(γ0),c1e-b1|λ|ρ(|λ+ã|2+δ̃|w|2)γ/2c1e-b1|λ|ρ|λ+ã|γ,λC{-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), |E(σ2/ρz,z+λw)|conste2σ|z|ρ/2(|z|  +|λw|)ρ/2conste2σ|z|ρ/22ρ/2(|z|ρ/2+|λ|ρ/2|w|ρ/2)=conste2(2+ρ)/2σ|z|ρe2(2+ρ)/2σ|z|ρ/2|w|ρ/2|λ|ρ/2conste2(2+ρ)/2σRρe2(2+ρ)/2σRρ/2Mρ/2|λ|ρ/2constek|λ|ρ/2,λC, where k is a positive number.

Further, if γ0, then in view of (3.4) and (3.7) φ(|z+λw|2)const(ρ,σ,γ)e(-σ/2)|z+λw|ρ=conste(-σ/2)|w|ρ(|λ+ã|2+δ̃/|w|2)ρ/2conste(-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/mT1/2, then for |λ|T|λ+ã||λ|-|ã|=|λ|(1-|ã||λ|)|λ|(1-RmT)|λ|2. Hence e(-σ/2)mρ|λ+ã|ρconste(-σ/2)mρ(|λ|ρ/2ρ)  (λC). Combining (3.11) and (3.13), we obtain φ(|z+λw|2)conste-d|λ|ρ,λC, where d is a positive number.

Combination of (3.10) and (3.14) easily implies (3.9) for the case γ0. If -2n<γ<0, then similarly to (3.11)–(3.14) we have e-σ|z+λw|ρconste-d1|λ|ρ,  λC, where d1 is a positive number. Also, in view of (3.7), |z+λw|γ=(|w|2|λ+ã|2+δ̃)γ/2=|w|γ(|λ+ã|2+δ̃|w|2)γ/2mγ(|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 that 0<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.

If γ0, then

|E(σ2/ρz,z+λw)||E(σ2/ρz,z+λw)|×|φ(|z+λw|2)|ce-b|λ|ρ,λC.

If γ2, ρ>0 or γ=0, ρ2, then

|E(σ2/ρz,z+λw)||E(σ2/ρz,z+λw)|×|φ(|z+λw|2)|ce-b|λ|ρ,λC.

Lemma 3.3.

Let β(η), ηC, 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 α(η)-1πCβ(ξ)ξ-ηdm(ξ),  ηC, is of class C1(C) and (α/η¯)(η)β(η), ηC.

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 that ψ(w), wCn{0}, is continuously differentiable (i.e., of class C1) and g(w), wCn{0}, is continuous. Then the following two relations are equivalent: ψw¯(w),wg(w),wCn{0},  η¯ψ(ηw)g(ηw)η¯,ηC{0}(wCn{0}).

Proof.

Let us fix an arbitrary wCn{0}; then η¯ψ(ηw)=k=1nψwk¯(ηw)wk¯=ψw¯(ηw),w=(ψ/w¯)(ηw),ηwη¯,ηC{0}. This immediately gives the implication (3.20)(3.21). On the contrary, if (3.21) is valid, then (ψ/w¯)(ηw),ηwη¯=g(ηw)η¯,ηC{0}. Substitution of η=1 into the last relation gives (3.20). Thus, the assertion is proved.

4. The Main Properties of the Kernel <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M201"><mml:mrow><mml:mi mathvariant="normal">Φ</mml:mi></mml:mrow></mml:math></inline-formula>Proposition 4.1.

If γ>-1, then for fixed zCn, wCn{0} and for arbitrary ηC{0}Φ(z;z+ηw)=-1πCβ(ξ)ξ-ηdm(ξ), where β(ξ)=-ρσμ2Γ(n)|w|2n|ξ|2nξ¯E(σ2/ρz,z+ξw  )e-σ|z+ξw|ρ|z+ξw|γ-ρσμ2Γ(n)|w|2n|ξ|2nξ¯E(σ2/ρz,z+ξw)φ(|z+ξw|2),ξC{-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.

If γ0, then β is a continuous function in C and

|β(ξ)|ce-q|ξ|ρ,ξC.

If γ2, ρ>0 or γ=0, ρ2, then β is a C1-function in C and, in addition to (4.3),

|β(ξ)ξ||β(ξ)ξ¯|  ce-q|ξ|ρ,ξC.

Proof.

Indeed, according to (2.16) Φ(z;z+ηw)=ρσμ2πΓ(n)|w|2n|η|2n×C|λ|2nλ¯(λ-1)E(σ2/ρz,z+ληw)e-σ|z+ληw|ρ|z+ληw|γdm(λ)==ληξ=ρσμ2πΓ(n)|w|2nC|ξ|2nξ¯(ξ-η)E(σ2/ρz,z+ξw)e-σ|z+ξw|ρ|z+ξw|γdm(ξ)-1πCβ(ξ)ξ-ηdm(ξ). As to (4.3)-(4.3), these inequalities immediately follow from Lemma 3.2 and the following relations: ξE(σ2/ρz,z+ξw)=E(σ2/ρz,z+ξw)00,ξ¯E(σ2/ρz,z+ξw)=E(σ2/ρz,z+ξw)σ2/ρz,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.

If γ0, then the kernel Φ(z;z+w) is continuous in zCn, wCn{0}.

Proof.

Let us write Φ(z;z+w) as follows: Φ(z;z+w)=ρσμ2πΓ(n)|w|2nCH(z;w;λ)dm(λ), where H(z;w;λ)=|λ|2nλ¯(λ-1)E(σ2/ρz,z+λw)e-σ|z+λw|ρ|z+λw|γ|λ|2nλ¯(λ-1)E(σ2/ρz,z+λw)φ(|z+λw|2),zCn,wCn{0},λC{1}.

For arbitrary fixed positive numbers R,m,M(m<M), it suffices to construct a function h(λ)L1(C{1}), such that |H(z;w;λ)|h(λ),λC{1}, uniformly in z and w with |z|R, m|w|M. According to Lemma 3.1 (the case γ0), the function c|λ|2n-1|λ-1|e-b|λ|ρ is suitable for a function h we seek.

Proposition 4.3.

If γ0 and zCn is arbitrary, then limɛ0|w-z|=ɛf(w)Φ(z;w)KMB(z;w)limɛ0|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 small ɛ>0, put 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 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/ρz,z+λw)φ(|z+λw|2)dm(λ)}×1|w|2nj=1n(-1)j-1wj¯dw¯[j]dw=====w=ɛζ(ζSn)  =ρσμ2πΓ(n)(-1)n(n-1)/2Γ(n)(2πi)nSnf(z+ɛζ)×{C|λ|2nλ¯(λ-1)E(σ2/ρz,z+λɛζ)φ(|z+λɛζ|2)dm(λ)}×ɛ2nj=1n(-1)j-1ζj¯dζ¯[j]dζρσμ2πΓ(n)ɛ2nSnf(z+ɛζ)×{C|λ|2nλ¯(λ-1)E(σ2/ρz,z+λɛζ)φ(|z+λɛζ|2)dm(λ)}KMB(0;ζ). In view of (2.6), we obtain Iɛ(z)=ρσμɛ2n4πn+1Snf(z+ɛζ)×{C|λ|2nλ¯(λ-1)E(σ2/ρz,z+λɛζ)φ(|z+λɛζ|2)dm(λ)}dσ(ζ). After the change of variable λλ/ɛ in the inner integral in (4.13), we have Iɛ(z)=ρσμ4πn+1Snf(z+ɛζ)×{C|λ|2nλ¯(λ-ε)E(σ2/ρz,z+λζ)φ(|z+λζ|2)dm(λ)}dσ(ζ)===λλ+ɛ=ρσμ4πn+1Snf(z+ɛζ)×{C(λ+ɛ)n(λ¯+ɛ)n-1λE(σ2/ρz,z+(λ+ɛ)ζ)φ(|z+(λ+ɛ)ζ|2)dm(λ)}dσ(ζ)ρσμ4πn+1Snf(z+ɛζ)Iɛ(z;ζ)dσ(ζ), where Iɛ(z;ζ)=CPɛ(z;ζ;λ)dm(λ),  ζSn, where Pɛ(z;ζ;λ)=(λ+ɛ)n(λ¯+ɛ)n-1λE(σ2/ρz,z+(λ+ɛ)ζ)φ(|z+(λ+ɛ)ζ|2),ζSn,λC{0}. Obviously, without loss of generality, we can suppose that 0<ɛ1.

According to Lemma 3.1 (the case γ0) or, equivalently, to Lemma 3.2(a), there exist positive constants c,b (depending on n,ρ,σ,γ) such that |Pɛ(z;ζ;λ)|(|λ|+1)2n-1|λ|ce-b|λ+ɛ|ρh(λ){c(|λ|+1)2n-1|λ|,0<|λ|1c(|λ|+1)2n-1|λ|e-b(|λ|-1)ρ,|λ|>1L1(C{0}) uniformly in ζSn, 0<ɛ1. Hence, due to the Lebesgue dominated convergence theorem, we can conclude that

the functions Iɛ(z;ζ) are continuous in ζSn;

|Iɛ(z;ζ)|M<+, ζSn (uniformly in ɛ);

For for all ζSn:limɛ0Iɛ(z;ζ)=I(z;ζ), where

I(z;ζ)=C|λ|2n-2E(σ2/ρz,z+λζ)φ(|z+λζ|2)dm(λ). Note that the function I(z;ζ) is also continuous in ζSn and |I(z;ζ)|M<+, ζSn.

Now remember (see (4.14)) that Iɛ(z)=ρσμ4πn+1Snf(z+ɛζ)Iɛ(z;ζ)dσ(ζ). Therefore, the application of the Lebesgue dominated convergence theorem once again gives limɛ0Iɛ(z)=f(z)ρσμ4πn+1SnI(z;ζ)dσ(ζ)=f(z)ρσμ4πn+1Sndσ(ζ)C|λ|2n-2E(σ2/ρz,z+λζ)φ(|z+λζ|2)dm(λ)=f(z)ρσμ4πn+1Sndσ(ζ)0+02πr2n-1E(σ2/ρz,z+reiϑζ)φ(|z+reiϑζ|2)drdϑ===rζ=w=f(z)ρσμ4πn+102πdϑCnE(σ2/ρz,z+eiϑw)φ(|z+eiϑw|2)dm(w)=====z+eiϑww=f(z)ρσμ4πn+102πdϑCnE(σ2/ρz,w)φ(|w|2)dm(w)=f(z)ρσμ2πnCnE(σ2/ρz,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 that Φ(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.

If γ2, ρ>0 or γ=0, ρ2, the kernel Φ(z;z+w) is a function of class C1 in zCn, wCn{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 zCn, wCn{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 |wkH(z;w;λ)|,|wk¯H(z;w;λ)|h(λ),λC{1},|zkH(z;w;λ)|,|zk¯H(z;w;λ)|h(λ),λC{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 Lemma 3.2, we find that we are reduced to the question of the finiteness of integrals of the type C|λ|2n+τ|λ-1|e-b|λ|ρdm(λ)(b>0,τ=-1;0;1), which is evident.

Proposition 4.6.

If γ2, ρ>0 or γ=0, ρ2, then for arbitrary fixed zCn we have Φw¯(z;z+w),w  =-ρσμ2Γ(n)|w|2nE(σ2/ρz,z+w)e-σ|z+w|ρ|z+w|γ,wCn{0}, or, equivalently, Φw¯(z;w),w-z=-ρσμ2Γ(n)|w-z|2nE(σ2/ρz,w)e-σ|w|ρ|w|γ,wCn{z}.

Proof.

We intend to use Lemma 3.4. To this end, let us put for wCn{0}ψ(w)=Φ(z;z+w),g(w)=-ρσμ2Γ(n)|w|2nE(σ2/ρz,z+w)e-σ|z+w|ρ|z+w|γ. It suffices to establish (3.21) for the introduced functions ψ and g. Fix wCn{0}, then in view of Proposition 4.1 we have ψ(ηw)=-1πCβ(ξ)ξ-ηdm(ξ),  ηC{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 η¯ψ(ηw)β(η)=-ρσμ2Γ(n)|w|2n|η|2nη¯E(σ2/ρz,z+ηw)e-σ|z+ηw|ρ|z+ηw|γg(ηw)η¯,ηC{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 kernel Φ(z;z+w) when w0 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.

If γ0, then

for arbitrary zCn and for arbitrary wCn{0}, we have limη0Φ(z;z+ηw)=ρσμ|w|2n2πΓ(n)×C|ξ|2n-2E(σ2/ρz,z+ξw)e-σ|z+ξw|ρ|z+ξw|γdm(ξ);

if z0 and [z] is the complex plane generated by the vector z, then limw0,w[z]Φ(z;z+w)=ρσμ2Γ(n)E(σ2/ρ|z|2)  0+xn-1e-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 |Φ(z;z+w)|ce-(σ-ɛ)|w|ρ,  w[z](w0), uniformly in z with |z|<R;

if z0, then limw0,w[z]Φ(z;z+w)=m=0n-1(-1)n-1-mCn-1m(|z|2σ2/ρ)n-1-mLn,mγ,ρ, where the coefficients n,mγ,ρ(m=1,2,,n-1) can be written in an explicit form;

for arbitrary R>0, there exist positive constants c,δ1 (depending, in general, on n,ρ,σ,γ,R) such that |Φ(z;z+w)|c(|w|+e-δ1|w|ρ) uniformly in wCn{0} and z with |z|<R. In particular, the kernel Φ(z;z+w) remains bounded (uniformly in z with |z|<R) as w0.

Remark 4.8.

As it follows from (4.28) and (4.30), in general, limw0Φ(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 wCn{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 function χ(t), t(-;+), satisfying the following conditions:

χC1(R);

0χ(t)1, t(-;+);

χ(t)1, t(-;  0];

χ(t)0, t[1;+);

χ[0;1];

|χ(t)|M<+, t[0;1] and (obviously) χ(t)0 otherwise.

The existence of such functions is evident. Then putχR(w)χ(|w|2-R2){=1,0|w|R,[0;1],R|w|R2+1,=0,|w|R2+1. Note that χRC1(Cn) andχRw¯(w)(χRw1¯(w),χRw2¯(w),,χRwn¯(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,|χRw¯(w)|  k=1n|χRwk¯(w)|2{=0,0|w|R,M|w|,R|w|R2+1,=0,|w|R2+1.

Theorem 5.1.

Assume that n1, 1<p<+, σ>0 and μ=(γ+2n)/ρ, where either γ2, ρ>0 or γ=0, ρ2. If the kernel Φ(z;w),for all zCn, for all wCn{z}, is defined by the formula (2.15), then the integral representation of the form f(z)=ρσμ2πnCnf(w)Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γdm(w)-Γ(n)πnCn(f/w¯)(w),w-z|w-z|2nΦ(z;w)dm(w),zCn, is valid for each function fC1(Cn) satisfying the following conditions:

fLρ,σ,γp(Cn);

for any fixed  zCn, |f(w)|Φ(z;w)|w-z|2n-2L1(Cn;dm(w))|f(w)|Φ(z;w)|w|2n-2L1(Cn;dm(w));

for any fixed zCn

|(f/w¯)(w)|Φ(z;w)|w-z|2n-1L1(Cn;dm(w))|(f/w¯)(w)|Φ(z;w)|w|2n-1L1(Cn;dm(w)).

Proof.

Let us fix an arbitrary zCn, and for R>0 consider the following differential form:  ψ(z;w)=f(w)χR(w-z)Φ(z;w)KMB(z;w),wCn{z}. Then choose an ɛ(0;R) and apply the Stokes formula to this form and to the domain {wCn:0<ɛ<|w-z|<R2+1}: |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: -|w-z|=ɛf(w)Φ(z;w)KMB(z;w)=Γ(n)πnɛ<|w-z|<R2+1(f/w¯)(w),w-z|w-z|2nχR(w-z)Φ(z;w)dm(w)  +Γ(n)πnɛ<|w-z|<R2+1(χR/w¯)(w-z),w-z|w-z|2nf(w)Φ(z;w)dm(w)+Γ(n)πnɛ<|w-z|<R2+1(Φ/w¯)(z;w),w-z|w-z|2nf(w)χR(w-z)dm(w). Moreover, (5.2) and (4.23) imply -|w-z|=ɛf(w)Φ(z;w)KMB(z;w)=Γ(n)πnɛ<|w-z|<R2+1(f/w¯)(w),w-z  |w-z|2nχR(w-z)Φ(z;w)dm(w)+Γ(n)πnR<|w-z|<R2+1χ(|w-z|2-R2)|w-z|2n-2f(w)Φ(z;w)dm(w)-ρσμ2πnɛ<|w-z|<R2+1Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γf(w)χR(w-z)dm(w). When ɛ0, then due to (4.10) we obtain f(z)=ρσμ2πn|w-z|<R2+1Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γf(w)χR(w-z)dm(w)-Γ(n)πn|w-z|<R2+1(f/w¯)(w),w-z|w-z|2nχR(w-z)Φ(z;w)dm(w)-Γ(n)πnR<|w-z|<R2+1χ(|w-z|2-R2)|w-z|2n-2f(w)Φ(z;w)dm(w)A1(R)-A2(R)-A3(R). Further, A1(R)=ρσμ2πn|w-z|<REρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γf(w)dm(w)+ρσμ2πnR<|w-z|<R2+1Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γf(w)χR(w-z)dm(w)A1(1)(R)+A1(2)(R). Since (via the condition (a)) f(w)Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γL1(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, limR+A1(R)=ρσμ2πnCnf(w)Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γdm(w). Similarly, A2(R)=Γ(n)πn|w-z|<R(f/w¯)(w),w-z|w-z|2nΦ(z;w)dm(w)  +Γ(n)πnR<|w-z|<R2+1(f/w¯)(w),w-z|w-z|2nχR(w-z)Φ(z;w)dm(w)A2(1)(R)+A2(2)(R). In view of the condition (c) of the theorem (f/w¯)(w),w-z|w-z|2nΦ(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, limR+A2(R)=Γ(n)πnCn(f/w¯)(w),w-z|w-z|2nΦ(z;w)dm(w). Finally, |A3(R)|Γ(n)πnMR<|w-z|<R2+1|f(w)|Φ(z;w)|w-z|2n-2dm(w)Γ(n)πnMR<|w-z|<+|f(w)|Φ(z;w)|w-z|2n-2dm(w). Hence, due to the condition (b) of the theorem, limR+A3(R)=0. Combining (5.11)–(5.19), we ultimately obtain f(z)=ρσμ2πnCnf(w)Eρ/2(n)(σ2/ρz,w;μ)e-σ|w|ρ|w|γdm(w)-Γ(n)πnCn(f/w¯)(w),w-z|w-z|2nΦ(z;w)dm(w), which coincides with (5.4). Thus the theorem is proved.

Remark 5.2.

The idea of introducing an auxiliary function χ is borrowed from .

6. The Computable Form of the Kernel <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M429"><mml:mrow><mml:mi mathvariant="normal">Φ</mml:mi></mml:mrow></mml:math></inline-formula>

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 arbitrary z,wCn(zw), put (compare with (3.8))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|20. Note that a slight change of (3.7) yields|z+λ(w-z)|2|w-z|2|λ+a|w-z||2+δ. Let ρ,σ>0, γ0, and μ=(γ+2n)/ρ. Consider the following functions:ψ(s)=(s-a¯)n-1Eρ/2(n)(σ2/ρ(as+δ);μ),sC,ψ(ν)(s)dνψ(s)dsν,ν=0,1,2,3,. Besides, put (x0):ϕ(x)=e-σ(x+δ)ρ/2(x+δ)γ/2,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, Φk(x)=x+dxx+dxx+k  times  ϕ(x)dx. In what follows we also put Φ0(x)ϕ(x), x0. Obviously, the following simple relations are valid:ddx[Φk+1(x)]=-Φk(x)dkΦk(x)dxk=(-1)kϕ(x),x0,k=0,1,2,3,,Φk(0)=1Γ(k)0+ϕ(t)tk-1dt,k=1,2,3.,r<|λ|<Rλlλ¯mϕ(|λ|2)dm(λ)=0, where 0r<R and l,m=0,1,2,3,,lm. 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 kernel Φ can be computed by the following formula: 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)n01ψ(c¯t)ϕ(|c|2t)dt.

Proof.

In view of (2.15) and (6.2), we have Φ(z;w)=ρσμ2Γ(n)I, where I=|w-z|2nπC|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+aλ=1πC(λ-a)n(λ¯-a¯)n-1λ-cEρ/2(n)(σ2/ρ(aλ¯+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 I=1πC(λ-a)nλ-cψ(λ¯)ϕ(|λ|2)dm(λ). In order to simplify the last integral, let us consider the following auxiliary integrals: Aν=1πCλνψ(λ¯)ϕ(|λ|2)dm(λ),ν=0,1,2,3,,B=1πCψ(λ¯)λ-cϕ(|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 Aν=1πψ(ν)(0)Γ(ν+1)C|λ|2νϕ(|λ|2)dm(λ)=ψ(ν)(0)Γ(ν+1)0tνϕ(t)dt=ψ(ν)(0)Φν+1(0).

Further, B=1π|λ|<|c|ψ(λ¯)λ-cϕ(|λ|2)dm(λ)+1π|λ|>|c|ψ(λ¯)λ-cϕ(|λ|2)dm(λ)B(+)+B(-). If c=0, then, naturally, B(+)=0, and B(-)=1π|λ|>0ψ(λ¯)λϕ(|λ|2)dm(λ)=1π|λ|>0λ¯ψ(λ¯)ϕ(|λ|2)|λ|2dm(λ)=0, that is, B=0. If c0, then 1λ-c={-1ck=0λkck,|λ|<|c|,1λk=0ckλk,|λ|>|c|. Consequently, B(-)=1πk=0ck|λ|>|c|ψ(λ¯)λk+1ϕ(|λ|2)dm(λ)=1πk=0ck|λ|>|c|λ¯k+1ψ(λ¯)ϕ(|λ|2)|λ|2k+2dm(λ)=0. Hence B=B(+)=-1πck=0|λ|<|c|λkckψ(λ¯)ϕ(|λ|2)dm(λ)=-1πck=0|λ|<|c|ψ(k)(0)Γ(k+1)ck|λ|2kϕ(|λ|2)dm(λ)=-1ck=00|c|2ψ(k)(0)Γ(k+1)cktkϕ(t)dm(t)=-1c0|c|2k=0ψ(k)(0)Γ(k+1)(tc)kϕ(t)dm(t)=-1c0|c|2ψ(tc)ϕ(t)dm(t)===t|c|2t-c¯01ψ(c¯t)ϕ(|c|2t)dm(t). Using the binomial expansion of (λ-a)n and combining (6.14)–(6.21), we obtain I=k=0nCnk(-a)n-k1πCλkλ-cψ(λ¯)ϕ(|λ|2)dm(λ)=k=1nCnk(-a)n-k1πC(λk-ck)+ckλ-cψ(λ¯)ϕ(|λ|2)dm(λ)+(-a)nB=k=1nCnk(-a)n-k1πC(λ-c)ν=0k-1ck-1-νλνλ-cψ(λ¯)ϕ(|λ|2)dm(λ)+k=1nCnk(-a)n-kckB+(-a)nB=k=1nCnk(-a)n-kν=0k-1ck-1-νAν+(c-a)nB===kk+1=k=0n-1Cnk+1(-a)n-1  -kν=0kck-νψ(ν)(0)Φν+1(0)-c¯(c-a)n01ψ(c¯t)ϕ(|c|2t)dm(t)=ν=0n-1ψ(ν)(0)Φν+1(0)k=0n-1-νCnk+1+ν(-a)n-1-ν-kck-c¯(c-a)n01ψ(c¯t)ϕ(|c|2t)dm(t). Finally, note that (6.12) and (6.22) imply (6.11) and the proof is complete.

7. Important Special Case: <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M474"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M475"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

In this section, we analyze the special case ρ=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,Eρ/2(n)(η;μ)=k=0Γ(k+n)Γ(k+1)ηkΓ(μ+2k/ρ)=k=0ηkΓ(k+1)eη,ηC. As a consequence, the formula (2.15) takes the following form:Φ(z;w)=e-σ|w|2eσz,wν=0n-1σνν!|w-z|2ν,zCn,wCn{z}, or, equivalently,Φ(z;z+w)=e-σ|w|2e-σw,zν=0n-1σνν!|w|2ν,zCn,wCn{0}. Indeed, we haveΦ(z;w)=σnπΓ(n)|w-z|2nC|λ|2nλ¯(λ-1)eσz,z+λ(w-z)e-σ|z+λ(w-z)|2dm(λ)=σnπΓ(n)|w-z|2nC|λ|2nλ¯(λ-1)e-σλ(w-z),z+λ(w-z)dm(λ)=σnπΓ(n)|w-z|2nC|λ|2n-2e-σ|w-z|2|λ|2λe-σλw-z,zλ-1dm(λ)=2σnΓ(n)|w-z|2n0+r2n-1e-σ|w-z|2r212πi|ζ|=re-σζw-z,zζ-1dζdr=2σnΓ(n)|w-z|2n0+r2n-1e-σ|w-z|2r2×{0,0<r<1e-σw-z,z,1<r<×dr=σnΓ(n)|w-z|2ne-σw-z,z1+tn-1e-σ|w-z|2tdt=σnΓ(n)|w-z|2ne-σw-z,ze-σ|w-z|20+(t+1)n-1e-σ|w-z|2tdt=σnΓ(n)|w-z|2ne-σ|w|2eσz,wν=0n-1Cn-1νΓ(n-ν)σn-ν|w-z|2n-2ν=e-σ|w|2eσz,wν=0n-1σνν!|w-z|2ν.

It is interesting to note (see Remark 4.4) thatlimw0Φ(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|Φ(z;z+w)|ce-(σ-ɛ)|w|2,wCn, uniformly in z with |z|<R. At last, the main Theorem 5.1 takes the following form.

Theorem 7.1.

Assume that n1, 1<p<+ and σ>0, then the integral representation of the form f(z)=σnπnCnf(w)eσz,we-σ|w|2dm(w)-Γ(n)πnCn(f/w¯)(w),w-z|w-z|2neσz,w×ν=0n-1σνν!|w-z|2νe-σ|w|2dm(w),zCn, is valid for each function fC1(Cn) satisfying the following conditions:

Cn|f(w)|p·e-σ|w|2dm(w)<+;

for any fixed zCn,

Cn|(f/w¯)(w)||w-z|2n-1|eσz,w|ν=0n-1σνν!|w-z|2νe-σ|w|2dm(w)<+, or, equivalently, Cn|(f/w¯)(w)||w-z||eσz,w|e-σ|w|2dm(w)<+, or, not equivalently (but more conveniently), Cn|fw¯(w)|e-(σ-ɛ)|w|2dm(w)<+, where ɛ is arbitrary small.

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