Weighted 𝜕-Integral Representations of 𝐶^𝟏-Functions in 𝐶^𝑛

Abstract

For 𝐶¹-functions 𝑓, given in the complex space 𝐶^𝑛, integral representations of the form 𝑓=𝑃(𝑓)−𝑇(𝜕𝑓) are obtained. Here, 𝑃 is the orthogonal projector of the space 𝐿²{𝐶^𝑛;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.

1. Introduction

Let 𝑛≥1, 1<𝑝<+∞, and 0<𝜌,𝜎<∞, 𝛾>−2𝑛. Denote by 𝐿^𝑝_{𝜌,𝜎,𝛾}(𝐶^𝑛) the space of all measurable complex-valued functions 𝑓(𝑧), 𝑧∈𝐶^𝑛, satisfying the condition_𝐶𝑛||𝑓(𝑧)||^𝑝𝑒^{−𝜎|𝑧|𝜌}|𝑧|^𝛾𝑑𝑚(𝑧)<+∞.(1.1) Let 𝐻^𝑝_{𝜌,𝜎,𝛾}(𝐶^𝑛) 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∈𝐻^𝑝_{𝜌,𝜎,𝛾}(𝐶^𝑛) has an integral representation of the form𝑓(𝑧)=𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝑓(𝑤)𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑑𝑚(𝑤),𝑧∈𝐶^𝑛,(1.2) where 𝜇=(𝛾+2𝑛)/𝜌 and𝐸^(𝑛)_𝜌/2(𝜂;𝜇)=^∞_𝑘=0Γ(𝑘+𝑛)Γ(𝑘+1)⋅𝜂^𝑘Γ(𝜇+2𝑘/𝜌),𝜂∈𝐶,(1.3) 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 𝐿²_{𝜌,𝜎,𝛾}(𝐶^𝑛) onto its subspace 𝐻²_{𝜌,𝜎,𝛾}(𝐶^𝑛). Certainly, the condition (1.1) and corresponding properties of 𝐸^(𝑛)_𝜌/2(𝜂;𝜇) ensure an absolute convergence of the integral in (1.2).

Note that for 𝑝=2,𝜌=2, 𝛾=0, 𝐻^𝑝_{𝜌,𝜎,𝛾}(𝐶^𝑛) coincides with the well-known Fock space of entire functions and (1.2) takes the form𝑓(𝑧)=𝜎^𝑛𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝑒^{−𝜎|𝑤|2}⋅𝑒^{𝜎⟨𝑧,𝑤⟩}𝑑𝑚(𝑤),𝑧∈𝐶^𝑛.(1.4)

In [4] general weighted integral representations were obtained for differential forms. In particular, for functions 𝑓∈𝐶¹(𝐶^𝑛) (satisfying certain growth conditions), the following generalization of the formula (1.4) was established:𝑓(𝑧)=𝜎^𝑛𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤−𝑧|^2𝜈⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤),𝑧∈𝐶^𝑛,(1.5) where𝜕𝑓𝜕𝑤(𝑤)=𝜕𝑓𝜕𝑤₁(𝑤),𝜕𝑓𝜕𝑤₂(𝑤),…,𝜕𝑓𝜕𝑤_𝑛(𝑤)(1.6)

and, consequently, 𝜕𝑓𝜕𝑤(𝑤),𝑤−𝑧=^𝑛_𝑘=1𝜕𝑓𝜕𝑤_𝑘(𝑤)⋅𝑤_𝑘−𝑧_𝑘.(1.7)

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 𝐶^𝑛 see [6–8]; for general strictly pseudoconvex domains see [9–13]; for Cartan matrix domain (“matrix disc”) see [14]. The whole space 𝐶^𝑛 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 ∏^𝑛_𝑘=1𝑒^{−𝜎𝑘|𝑤𝑘|𝜌𝑘}|𝑤_𝑘|^𝛾𝑘, 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)∈𝐶^𝑛, 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,…,𝑤_𝑛)∈𝐶^𝑛, |𝑤|=∑^𝑛_𝑘=1|𝑤_𝑘|² (see the condition (1.1)). More precisely, for functions 𝑓∈𝐶¹(𝐶^𝑛) (satisfying certain growth conditions), the integral representation of the form𝑓(𝑧)=𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝑓(𝑤)𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤),𝑧∈𝐶^𝑛,(1.8) 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(𝐶^𝑛;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 Φ

In what follows, it is supposed that 𝑛≥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(𝐶^𝑛) 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∈𝐶^𝑛, to have the following (or similar) properties:Φ(𝑧;𝑧)=1,∀𝑧∈𝐶^𝑛,(2.1)Φ(𝑧;∞)=0,∀𝑧∈𝐶^𝑛.(2.2)

Denote by 𝐾_MB(𝑧;𝑤) (for all 𝑧∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{𝑧}) the well-known Martinelli-Bochner kernel, which has the following useful properties (see, for instance, [17, Chapter 16]):𝜕_𝑤𝐾_MB(𝑧;𝑤)≡0,𝑤∈𝐶^𝑛⧵{𝑧};(2.3)𝜕_𝑤𝑓(𝑤)⋅𝐾_MB(𝑧;𝑤)=𝜕𝑓(𝑤)∧𝐾_MB(𝑧;𝑤)≡Γ(𝑛)𝜋^𝑛⋅𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛×𝑑𝑚(𝑤)(2.4) for arbitrary function 𝑓∈𝐶¹(Ω), where Ω⊂𝐶^𝑛⧵{𝑧} is an open set;lim_𝜀↓0_{|𝑤−𝑧|=𝜀}𝑢(𝑤)⋅𝐾_MB(𝑧;𝑤)=𝑢(𝑧)(2.5) for arbitrary function 𝑢 continuous in a neighborhood of 𝑧;_𝑆𝑛𝑓(𝜁)⋅𝐾_MB(0;𝜁)=Γ(𝑛)2𝜋^𝑛⋅_𝑆𝑛𝑓(𝜁)𝑑𝜎(𝜁),(2.6) where 𝑆_𝑛={𝜁∈𝐶^𝑛∶|𝜁|=1} is the unit sphere in 𝐶^𝑛,𝑓∈𝐶(𝑆_𝑛), and 𝜎 is the surface measure on 𝑆_𝑛.

Let us fix an arbitrary 𝑧∈𝐶^𝑛 and consider the following differential form:𝜑(𝑤)=𝑓(𝑤)⋅Φ(𝑧;𝑤)⋅𝐾_MB(𝑧;𝑤),𝑤∈𝐶^𝑛⧵{𝑧}.(2.7)

Then apply the Stokes formula to this form and to the domain {𝑤∈𝐶^𝑛∶0<𝜀<|𝑤−𝑧|<𝑅<+∞}:_{|𝑤−𝑧|=𝑅}𝜑−_{|𝑤−𝑧|=𝜀}𝜑=_{𝜀<|𝑤−𝑧|<𝑅}𝑑𝜑.(2.8) The integral ∫_{|𝑤−𝑧|=𝑅}𝜑→0 (as 𝑅→+∞) due to the property (2.2) (our argument is heuristic !!!). When 𝜀→0, then ∫_{|𝑤−𝑧|=𝜀}𝜑→𝑓(𝑧) due to the properties (2.5) and (2.1). Thus after 𝑅→+∞ and 𝜀→0, we have−𝑓(𝑧)=_𝐶𝑛𝑑𝜑,(2.9)

or, in view of (2.3)-(2.4),𝑓(𝑧)=−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝜕Φ/𝜕𝑤(𝑧;𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧⋅Φ(𝑧;𝑤)|𝑤−𝑧|^2𝑛𝑑𝑚(𝑤).(2.10)

Comparing (1.8) and (2.10), we arrive at the equality𝜕Φ𝜕𝑤(𝑧;𝑤),𝑤−𝑧=−𝜌𝜎^𝜇2Γ(𝑛)𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾|𝑤−𝑧|^2𝑛,∀𝑤∈𝐶^𝑛⧵{𝑧}.(2.11)

Let us fix arbitrary 𝑧,𝑤∈𝐶^𝑛, 𝑧≠𝑤 and consider the function𝜑(𝜆)=Φ(𝑧;𝑧+𝜆(𝑤−𝑧)),𝜆∈𝐶.(2.12)

Then we have𝜕𝜑𝜕𝜆=^𝑛_𝑘=1𝜕Φ𝜕𝑤_𝑘(𝑧;𝑧+𝜆(𝑤−𝑧))⋅𝑤_𝑘−𝑧_𝑘=𝜕Φ𝜕𝑤(𝑧;𝑧+𝜆(𝑤−𝑧)),𝑤−𝑧=−𝜌𝜎^𝜇𝜆⋅2Γ(𝑛)×𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑧+𝜆(𝑤−𝑧)⟩;𝜇𝑒^{−𝜎|𝑧+𝜆(𝑤−𝑧)|𝜌}×||𝑧+𝜆(𝑤−𝑧)||^𝛾||𝜆||^2𝑛|𝑤−𝑧|^2𝑛.(2.13)

The condition (2.2) implies 𝜑(∞)=0; hence due to Cauchy-Green-Pompeiju formula,𝜑𝜆₀=−1𝜋_𝐶𝜕𝜑/𝜕𝜆𝜆−𝜆₀𝑑𝑚(𝜆),∀𝜆₀∈𝐶.(2.14)

Since 𝜑(1)=Φ(𝑧;𝑤), we finally have Φ(𝑧;𝑤)=𝜌𝜎^𝜇2𝜋Γ(𝑛)|𝑤−𝑧|^2𝑛×_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑧+𝜆(𝑤−𝑧)⟩;𝜇𝑒^{−𝜎|𝑧+𝜆(𝑤−𝑧)|𝜌}×||𝑧+𝜆(𝑤−𝑧)||^𝛾𝑑𝑚(𝜆),∀𝑧∈𝐶^𝑛,∀𝑤∈𝐶^𝑛⧵{𝑧}.(2.15) This formula can be also written in the following (may be, more convenient) form:Φ(𝑧;𝑧+𝑤)=𝜌𝜎^𝜇2𝜋Γ(𝑛)|𝑤|^2𝑛×_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩;𝜇𝑒^{−𝜎|𝑧+𝜆𝑤|𝜌}||𝑧+𝜆𝑤||^𝛾𝑑𝑚(𝜆),∀𝑧∈𝐶^𝑛,∀𝑤∈𝐶^𝑛⧵{0}.(2.16) 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)≡𝐸^(𝑛)_𝜌/2(𝜂;𝜇),𝜂∈𝐶.(3.1) 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||𝐸(𝜂)||+||𝐸^(𝜂)||≤const(𝜌;𝑛;𝜇)⋅𝑒^{2|𝜂|𝜌/2},𝜂∈𝐶.(3.2)

Let us introduce a convenient notation𝜑(𝑥)≡𝑒^{−𝜎𝑥𝜌/2}⋅𝑥^𝛾/2,𝑥∈(0;+∞).(3.3) If 𝛾≥0, we can suppose that 𝑥∈[0;+∞) in (3.3). Then obviously the function 𝜑∈𝐶[0;+∞) and𝜑(𝑥)≤const(𝜌;𝜎;𝛾)⋅𝑒^{(−𝜎/2)𝑥𝜌/2},𝑥∈[0;+∞).(3.4) Note that under additional assumptions 𝛾≥2, 𝜌>0 or 𝛾=0, 𝜌≥2, the function 𝜑∈𝐶¹[0;+∞) and, moreover,𝜑^(𝑥)=𝑒^{−𝜎𝑥𝜌/2}⋅𝛾2𝑥^𝛾/2−1−𝜎𝜌2𝑥^{𝜌/2−1+𝛾/2},𝑥∈[0;+∞),||𝜑^(𝑥)||≤const(𝜌;𝜎;𝛾)⋅𝑒^{(−𝜎/2)𝑥𝜌/2},𝑥∈[0;+∞).(3.5) In view of (3.1) and (3.3), the formula (2.16) can be written as follows:Φ(𝑧;𝑧+𝑤)=𝜌𝜎^𝜇2𝜋Γ(𝑛)|𝑤|^2𝑛⋅_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩𝜑||𝑧+𝜆𝑤||²𝑑𝑚(𝜆),∀𝑧∈𝐶^𝑛,∀𝑤∈𝐶^𝑛⧵{0}.(3.6)

Further, assume that 𝑧∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{0}, 𝜆∈𝐶, then evidently||𝑧+𝜆𝑤||²=|𝑤|²⋅||𝜆+𝑎||̃²+𝛿̃‌,(3.7) where𝑎̃=⟨𝑧,𝑤⟩|𝑤|²,𝛿̃‌=|𝑧|²|𝑤|²−||⟨𝑧,𝑤⟩||²|𝑤|²≥0.(3.8)

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

Lemma 3.1. Assume that 0<𝑅<+∞ and 0<𝑚<𝑀<+∞, then ||𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||⋅𝑒^{−𝜎|𝑧+𝜆𝑤|𝜌}||𝑧+𝜆𝑤||^𝛾≡||𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||⋅𝜑||𝑧+𝜆𝑤||²≤⎧⎪⎨⎪⎩𝑐⋅𝑒^{−𝑏|𝜆|𝜌},𝜆∈𝐶,(𝛾≥0),𝑐₁⋅𝑒^{−𝑏1|𝜆|𝜌}⋅||𝜆+𝑎||̃²+𝛿̃‌|𝑤|²^𝛾/2≤𝑐₁⋅𝑒^{−𝑏1|𝜆|𝜌}⋅||𝜆+𝑎||̃^𝛾,𝜆∈𝐶⧵{−𝑎}̃,(−2𝑛<𝛾<0),(3.9) 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𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||≤const⋅𝑒^{2𝜎|𝑧|𝜌/2(|𝑧|+|𝜆‖𝑤|)𝜌/2}≤const⋅𝑒^{2𝜎|𝑧|𝜌/22𝜌/2(|𝑧|𝜌/2+|𝜆|𝜌/2|𝑤|𝜌/2)}=const⋅𝑒^{2(2+𝜌)/2𝜎|𝑧|𝜌}⋅𝑒^{2(2+𝜌)/2𝜎|𝑧|𝜌/2|𝑤|𝜌/2|𝜆|𝜌/2}≤const⋅𝑒^{2(2+𝜌)/2𝜎𝑅𝜌}⋅𝑒^{2(2+𝜌)/2𝜎𝑅𝜌/2𝑀𝜌/2|𝜆|𝜌/2}≡const⋅𝑒^{𝑘|𝜆|𝜌/2},𝜆∈𝐶,(3.10) where 𝑘 is a positive number.
Further, if 𝛾≥0, then in view of (3.4) and (3.7) 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.(3.11) Due to the conditions on 𝑧 and 𝑤, we have |𝑎|̃≤𝑅/𝑚. Let us choose 𝑇>0 such that 𝑅/𝑚𝑇≤1/2, then for |𝜆|≥𝑇||𝜆+𝑎||̃≥||𝜆||−||𝑎||̃=||𝜆||⎛⎜⎝1−||𝑎||̃||𝜆||⎞⎟⎠≥||𝜆||1−𝑅𝑚𝑇≥||𝜆||2.(3.12) Hence 𝑒^{(−𝜎/2)𝑚𝜌|𝜆+}𝑎̃_|^𝜌≤const⋅𝑒^{(−𝜎/2)𝑚𝜌(|𝜆|𝜌/2𝜌)}(𝜆∈𝐶).(3.13) Combining (3.11) and (3.13), we obtain 𝜑||𝑧+𝜆𝑤||²≤const⋅𝑒^{−𝑑|𝜆|𝜌},𝜆∈𝐶,(3.14) 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 𝑒^{−𝜎|𝑧+𝜆𝑤|𝜌}≤const⋅𝑒^{−𝑑1|𝜆|𝜌},𝜆∈𝐶,(3.15) where 𝑑₁ is a positive number. Also, in view of (3.7), ||𝑧+𝜆𝑤||^𝛾=|𝑤|²⋅||𝜆+𝑎||̃²+𝛿̃‌^𝛾/2=|𝑤|^𝛾||𝜆+𝑎||̃²+𝛿̃‌|𝑤|²^𝛾/2≤𝑚^𝛾||𝜆+𝑎||̃²+𝛿̃‌|𝑤|²^𝛾/2.(3.16) 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.(a)If 𝛾≥0, then||𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||||𝐸^𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||×||𝜑||𝑧+𝜆𝑤||²||≤𝑐⋅𝑒^{−𝑏|𝜆|𝜌},𝜆∈𝐶.(3.17)(b) If 𝛾≥2, 𝜌>0 or 𝛾=0, 𝜌≥2, then||𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||||𝐸^𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩||×||𝜑^||𝑧+𝜆𝑤||²||≤𝑐⋅𝑒^{−𝑏|𝜆|𝜌},𝜆∈𝐶.(3.18)

Lemma 3.3. Let 𝛽(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+𝜀), |𝜂|→+∞, (for arbitrary small 𝜀>0) is quite sufficient for us. Then the function 𝛼(𝜂)≡−1𝜋_𝐶𝛽(𝜉)𝜉−𝜂𝑑𝑚(𝜉),𝜂∈𝐶,(3.19) 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 that 𝜓(𝑤), 𝑤∈𝐶^𝑛⧵{0}, is continuously differentiable (i.e., of class 𝐶¹) and 𝑔(𝑤), 𝑤∈𝐶^𝑛⧵{0}, is continuous. Then the following two relations are equivalent: 𝜕𝜓𝜕𝑤(𝑤),𝑤≡𝑔(𝑤),𝑤∈𝐶^𝑛⧵{0},(3.20)𝜕𝜕𝜂𝜓(𝜂⋅𝑤)≡𝑔(𝜂⋅𝑤)𝜂,𝜂∈𝐶⧵{0}(∀𝑤∈𝐶^𝑛⧵{0}).(3.21)

Proof. Let us fix an arbitrary 𝑤∈𝐶^𝑛⧵{0}; then 𝜕𝜕𝜂𝜓(𝜂⋅𝑤)=^𝑛_𝑘=1𝜕𝜓𝜕𝑤_𝑘(𝜂⋅𝑤)⋅𝑤_𝑘=𝜕𝜓𝜕𝑤(𝜂⋅𝑤),𝑤=𝜕𝜓/𝜕𝑤(𝜂⋅𝑤),𝜂⋅𝑤𝜂,𝜂∈𝐶⧵{0}.(3.22) This immediately gives the implication (3.20)⇒(3.21). On the contrary, if (3.21) is valid, then 𝜕𝜓/𝜕𝑤(𝜂⋅𝑤),𝜂⋅𝑤𝜂=𝑔(𝜂⋅𝑤)𝜂,𝜂∈𝐶⧵{0}.(3.23) Substitution of 𝜂=1 into the last relation gives (3.20). Thus, the assertion is proved.

4. The Main Properties of the Kernel Φ

Proposition 4.1. If 𝛾>−1, then for fixed 𝑧∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{0} and for arbitrary 𝜂∈𝐶⧵{0}Φ(𝑧;𝑧+𝜂⋅𝑤)=−1𝜋⋅_𝐶𝛽(𝜉)𝜉−𝜂𝑑𝑚(𝜉),(4.1) where 𝛽(𝜉)=−𝜌𝜎^𝜇2Γ(𝑛)|𝑤|^2𝑛⋅||𝜉||^2𝑛𝜉𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩𝑒^{−𝜎|𝑧+𝜉𝑤|𝜌}||𝑧+𝜉𝑤||^𝛾≡−𝜌𝜎^𝜇2Γ(𝑛)|𝑤|^2𝑛⋅||𝜉||^2𝑛𝜉𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩𝜑||𝑧+𝜉𝑤||²,𝜉∈𝐶⧵{−𝑎}̃.(4.2) 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) If 𝛾≥0, then 𝛽 is a continuous function in 𝐶 and||𝛽(𝜉)||≤𝑐⋅𝑒^{−𝑞|𝜉|𝜌},𝜉∈𝐶.(4.3)(b) If 𝛾≥2, 𝜌>0 or 𝛾=0, 𝜌≥2, then 𝛽 is a 𝐶¹-function in 𝐶 and, in addition to (4.3),||||𝜕𝛽(𝜉)𝜕𝜉||||||||𝜕𝛽(𝜉)𝜕𝜉||||≤𝑐⋅𝑒^{−𝑞|𝜉|𝜌},𝜉∈𝐶.(4.3′)

Proof. Indeed, according to (2.16) Φ(𝑧;𝑧+𝜂⋅𝑤)=𝜌𝜎^𝜇2𝜋Γ(𝑛)|𝑤|^2𝑛||𝜂||^2𝑛×_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜂𝑤⟩𝑒^{−𝜎|𝑧+𝜆𝜂𝑤|𝜌}||𝑧+𝜆𝜂𝑤||^𝛾𝑑𝑚(𝜆)==^{𝜆𝜂→𝜉}=𝜌𝜎^𝜇2𝜋Γ(𝑛)|𝑤|^2𝑛⋅_𝐶||𝜉||^2𝑛𝜉(𝜉−𝜂)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩𝑒^{−𝜎|𝑧+𝜉𝑤|𝜌}||𝑧+𝜉𝑤||^𝛾𝑑𝑚(𝜉)≡−1𝜋⋅_𝐶𝛽(𝜉)𝜉−𝜂𝑑𝑚(𝜉).(4.4) As to (4.3)-(4.3^’), these inequalities immediately follow from Lemma 3.2 and the following relations: 𝜕𝜕𝜉𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩=𝐸^𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩⋅0≡0,𝜕𝜕𝜉𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩=𝐸^𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩⋅𝜎^2/𝜌⟨𝑧,𝑤⟩,𝜕𝜕𝜉𝜑||𝑧+𝜉𝑤||²=𝜑^||𝑧+𝜉𝑤||²⋅⟨𝑤,𝑧⟩+𝜉|𝑤|²,𝜕𝜕𝜉𝜑||𝑧+𝜉𝑤||²=𝜑^||𝑧+𝜉𝑤||²⋅⟨𝑧,𝑤⟩+𝜉|𝑤|².(4.5)

Proposition 4.2. If 𝛾≥0, then the kernel Φ(𝑧;𝑧+𝑤) is continuous in 𝑧∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{0}.

Proof. Let us write Φ(𝑧;𝑧+𝑤) as follows: Φ(𝑧;𝑧+𝑤)=𝜌𝜎^𝜇2𝜋Γ(𝑛)⋅|𝑤|^2𝑛⋅_𝐶𝐻(𝑧;𝑤;𝜆)𝑑𝑚(𝜆),(4.6) where 𝐻(𝑧;𝑤;𝜆)=||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩𝑒^{−𝜎|𝑧+𝜆𝑤|𝜌}||𝑧+𝜆𝑤||^𝛾≡||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩⋅𝜑||𝑧+𝜆𝑤||²,𝑧∈𝐶^𝑛,𝑤∈𝐶^𝑛⧵{0},𝜆∈𝐶⧵{1}.(4.7)
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),𝜆∈𝐶⧵{1},(4.8) uniformly in 𝑧 and 𝑤 with |𝑧|≤𝑅, 𝑚≤|𝑤|≤𝑀. According to Lemma 3.1 (the case 𝛾≥0), the function 𝑐⋅||𝜆||^2𝑛−1||𝜆−1||𝑒^{−𝑏|𝜆|𝜌}(4.9) is suitable for a function ℎ we seek.

Proposition 4.3. If 𝛾≥0 and 𝑧∈𝐶^𝑛 is arbitrary, then lim_𝜀↓0_{|𝑤−𝑧|=𝜀}𝑓(𝑤)⋅Φ(𝑧;𝑤)⋅𝐾_MB(𝑧;𝑤)≡lim_𝜀↓0_|𝑤|=𝜀𝑓(𝑧+𝑤)⋅Φ(𝑧;𝑧+𝑤)⋅𝐾_MB(𝑧;𝑧+𝑤)=𝑓(𝑧)(4.10) for arbitrary function 𝑓 continuous in a neighborhood of 𝑧.

Proof. For sufficiently small 𝜀>0, put 𝐼_𝜀(𝑧)=_|𝑤|=𝜀𝑓(𝑧+𝑤)⋅Φ(𝑧;𝑧+𝑤)⋅𝐾_MB(𝑧;𝑧+𝑤).(4.11) Taking into account the explicit formula (2.16) for Φ(𝑧;𝑧+𝑤) and the well-known explicit form of the Martinelli-Bochner kernel 𝐾_MB(𝑧;𝑧+𝑤), we obtain 𝐼_𝜀(𝑧)=𝜌𝜎^𝜇2𝜋Γ(𝑛)⋅(−1)^{𝑛(𝑛−1)/2}Γ(𝑛)(2𝜋𝑖)^𝑛⋅_|𝑤|=𝜀𝑓(𝑧+𝑤)×|𝑤|^2𝑛⋅⎧⎨⎩_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝑤⟩⋅𝜑||𝑧+𝜆𝑤||²𝑑𝑚(𝜆)⎫⎬⎭×1|𝑤|^2𝑛⋅^𝑛_𝑗=1(−1)^𝑗−1𝑤_𝑗𝑑𝑤_[𝑗]∧𝑑𝑤=====^{𝑤=𝜀⋅𝜁𝜁∈𝑆𝑛}=𝜌𝜎^𝜇2𝜋Γ(𝑛)⋅(−1)^{𝑛(𝑛−1)/2}Γ(𝑛)(2𝜋𝑖)^𝑛⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)×⎧⎨⎩_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜀⋅𝜁⟩⋅𝜑||𝑧+𝜆𝜀⋅𝜁||²𝑑𝑚(𝜆)⎫⎬⎭×𝜀^2𝑛⋅^𝑛_𝑗=1(−1)^𝑗−1𝜁_𝑗𝑑𝜁_[𝑗]∧𝑑𝜁≡𝜌𝜎^𝜇2𝜋Γ(𝑛)⋅𝜀^2𝑛⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)×⎧⎨⎩_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜀⋅𝜁⟩⋅𝜑||𝑧+𝜆𝜀⋅𝜁||²𝑑𝑚(𝜆)⎫⎬⎭⋅𝐾_MB(0;𝜁).(4.12) In view of (2.6), we obtain 𝐼_𝜀(𝑧)=𝜌𝜎^𝜇𝜀^2𝑛4𝜋^𝑛+1⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)×⎧⎨⎩_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜀⋅𝜁⟩⋅𝜑||𝑧+𝜆𝜀⋅𝜁||²𝑑𝑚(𝜆)⎫⎬⎭𝑑𝜎(𝜁).(4.13) After the change of variable 𝜆→𝜆/𝜀 in the inner integral in (4.13), we have 𝐼_𝜀(𝑧)=𝜌𝜎^𝜇4𝜋^𝑛+1⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)×⎧⎨⎩_𝐶||𝜆||^2𝑛𝜆(𝜆−𝜀)𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆⋅𝜁⟩⋅𝜑||𝑧+𝜆⋅𝜁||²𝑑𝑚(𝜆)⎫⎬⎭𝑑𝜎(𝜁)===^{𝜆→𝜆+𝜀}=𝜌𝜎^𝜇4𝜋^𝑛+1⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)×⎧⎪⎨⎪⎩_𝐶(𝜆+𝜀)^𝑛𝜆+𝜀^𝑛−1𝜆𝐸𝜎^2/𝜌⟨𝑧,𝑧+(𝜆+𝜀)𝜁⟩⋅𝜑||𝑧+(𝜆+𝜀)𝜁||²𝑑𝑚(𝜆)⎫⎪⎬⎪⎭𝑑𝜎(𝜁)≡𝜌𝜎^𝜇4𝜋^𝑛+1⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)⋅𝐼_𝜀(𝑧;𝜁)𝑑𝜎(𝜁),(4.14) where 𝐼_𝜀(𝑧;𝜁)=_𝐶𝒫_𝜀(𝑧;𝜁;𝜆)𝑑𝑚(𝜆),𝜁∈𝑆_𝑛,(4.15) where 𝒫_𝜀(𝑧;𝜁;𝜆)=(𝜆+𝜀)^𝑛𝜆+𝜀^𝑛−1𝜆𝐸𝜎^2/𝜌⟨𝑧,𝑧+(𝜆+𝜀)𝜁⟩⋅𝜑||𝑧+(𝜆+𝜀)𝜁||²,𝜁∈𝑆_𝑛,𝜆∈𝐶⧵{0}.(4.16) Obviously, without loss of generality, we can suppose that 0<𝜀≤1.
According to Lemma 3.1 (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 ||𝒫_𝜀(𝑧;𝜁;𝜆)||≤||𝜆||+1^2𝑛−1||𝜆||⋅𝑐⋅𝑒^{−𝑏|𝜆+𝜀|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})(4.17) uniformly in 𝜁∈𝑆_𝑛, 0<𝜀≤1. Hence, due to the Lebesgue dominated convergence theorem, we can conclude that(i)the functions 𝐼_𝜀(𝑧;𝜁) are continuous in 𝜁∈𝑆_𝑛;(ii)|𝐼_𝜀(𝑧;𝜁)|≤𝑀<+∞, 𝜁∈𝑆_𝑛 (uniformly in 𝜀);(iii)For for all 𝜁∈𝑆_𝑛∶lim_𝜀↓0𝐼_𝜀(𝑧;𝜁)=𝐼(𝑧;𝜁), where𝐼(𝑧;𝜁)=_𝐶||𝜆||^2𝑛−2⋅𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜁⟩⋅𝜑||𝑧+𝜆𝜁||²𝑑𝑚(𝜆).(4.18) Note that the function 𝐼(𝑧;𝜁) is also continuous in 𝜁∈𝑆_𝑛 and |𝐼(𝑧;𝜁)|≤𝑀<+∞, 𝜁∈𝑆_𝑛.
Now remember (see (4.14)) that 𝐼_𝜀(𝑧)=𝜌𝜎^𝜇4𝜋^𝑛+1⋅_𝑆𝑛𝑓(𝑧+𝜀⋅𝜁)⋅𝐼_𝜀(𝑧;𝜁)𝑑𝜎(𝜁).(4.19) Therefore, the application of the Lebesgue dominated convergence theorem once again gives lim_𝜀↓0𝐼_𝜀(𝑧)=𝑓(𝑧)⋅𝜌𝜎^𝜇4𝜋^𝑛+1_𝑆𝑛𝐼(𝑧;𝜁)𝑑𝜎(𝜁)=𝑓(𝑧)⋅𝜌𝜎^𝜇4𝜋^𝑛+1_𝑆𝑛𝑑𝜎(𝜁)_𝐶||𝜆||^2𝑛−2⋅𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜆𝜁⟩⋅𝜑||𝑧+𝜆𝜁||²𝑑𝑚(𝜆)=𝑓(𝑧)⋅𝜌𝜎^𝜇4𝜋^𝑛+1_𝑆𝑛𝑑𝜎(𝜁)^+∞₀^2𝜋₀𝑟^2𝑛−1⋅𝐸𝜎^2/𝜌𝑧,𝑧+𝑟𝑒^𝑖𝜗⋅𝜁⋅𝜑||𝑧+𝑟𝑒^𝑖𝜗⋅𝜁||²𝑑𝑟𝑑𝜗===^{𝑟⋅𝜁=𝑤}=𝑓(𝑧)⋅𝜌𝜎^𝜇4𝜋^𝑛+1^2𝜋₀𝑑𝜗_𝐶𝑛𝐸𝜎^2/𝜌𝑧,𝑧+𝑒^𝑖𝜗⋅𝑤⋅𝜑||𝑧+𝑒^𝑖𝜗⋅𝑤||²𝑑𝑚(𝑤)=====^{𝑧+𝑒𝑖𝜗⋅𝑤→𝑤}=𝑓(𝑧)⋅𝜌𝜎^𝜇4𝜋^𝑛+1^2𝜋₀𝑑𝜗_𝐶𝑛𝐸𝜎^2/𝜌⟨𝑧,𝑤⟩⋅𝜑|𝑤|²𝑑𝑚(𝑤)=𝑓(𝑧)⋅𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝐸𝜎^2/𝜌⟨𝑧,𝑤⟩⋅𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾𝑑𝑚(𝑤)=𝑓(𝑧)⋅1=𝑓(𝑧)(4.20) 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 𝛾≥2, 𝜌>0 or 𝛾=0, 𝜌≥2, the kernel Φ(𝑧;𝑧+𝑤) is a function of class 𝐶¹ in 𝑧∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{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∈𝐶^𝑛, 𝑤∈𝐶^𝑛⧵{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 ||||𝜕𝜕𝑤_𝑘𝐻(𝑧;𝑤;𝜆)||||,||||𝜕𝜕𝑤_𝑘𝐻(𝑧;𝑤;𝜆)||||≤ℎ(𝜆),𝜆∈𝐶⧵{1},||||𝜕𝜕𝑧_𝑘𝐻(𝑧;𝑤;𝜆)||||,||||𝜕𝜕𝑧_𝑘𝐻(𝑧;𝑤;𝜆)||||≤ℎ(𝜆),𝜆∈𝐶⧵{1},(4.21) 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|𝜌}𝑑𝑚(𝜆)(𝑏>0,𝜏=−1;0;1),(4.22) which is evident.

Proposition 4.6. If 𝛾≥2, 𝜌>0 or 𝛾=0, 𝜌≥2, then for arbitrary fixed 𝑧∈𝐶^𝑛 we have 𝜕Φ𝜕𝑤(𝑧;𝑧+𝑤),𝑤=−𝜌𝜎^𝜇2Γ(𝑛)|𝑤|^2𝑛𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝑤⟩𝑒^{−𝜎|𝑧+𝑤|𝜌}|𝑧+𝑤|^𝛾,∀𝑤∈C^𝑛⧵{0},(4.23) or, equivalently, 𝜕Φ𝜕𝑤(𝑧;𝑤),𝑤−𝑧=−𝜌𝜎^𝜇2Γ(𝑛)|𝑤−𝑧|^2𝑛𝐸𝜎^2/𝜌⟨𝑧,𝑤⟩𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾,∀𝑤∈𝐶^𝑛⧵{𝑧}.(4.23^)

Proof. We intend to use Lemma 3.4. To this end, let us put for 𝑤∈𝐶^𝑛⧵{0}𝜓(𝑤)=Φ(𝑧;𝑧+𝑤),𝑔(𝑤)=−𝜌𝜎^𝜇2Γ(𝑛)|𝑤|^2𝑛𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝑤⟩𝑒^{−𝜎|𝑧+𝑤|𝜌}|𝑧+𝑤|^𝛾.(4.24) It suffices to establish (3.21) for the introduced functions 𝜓 and 𝑔. Fix 𝑤∈𝐶^𝑛⧵{0}, then in view of Proposition 4.1 we have 𝜓(𝜂⋅𝑤)=−1𝜋⋅_𝐶𝛽(𝜉)𝜉−𝜂𝑑𝑚(𝜉),∀𝜂∈𝐶⧵{0},(4.25) 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 𝜕𝜕𝜂𝜓(𝜂⋅𝑤)≡𝛽(𝜂)=−𝜌𝜎^𝜇2Γ(𝑛)|𝑤|^2𝑛⋅||𝜂||^2𝑛𝜂𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜂𝑤⟩𝑒^{−𝜎|𝑧+𝜂𝑤|𝜌}||𝑧+𝜂𝑤||^𝛾≡𝑔(𝜂⋅𝑤)𝜂,∀𝜂∈𝐶⧵{0}.(4.26) 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 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. If 𝛾≥0, then(a)for arbitrary 𝑧∈𝐶^𝑛 and for arbitrary 𝑤∈𝐶^𝑛⧵{0}, we have lim_𝜂→0Φ(𝑧;𝑧+𝜂⋅𝑤)=𝜌𝜎^𝜇⋅|𝑤|^2𝑛2𝜋⋅Γ(𝑛)×_𝐶||𝜉||^2𝑛−2⋅𝐸𝜎^2/𝜌⟨𝑧,𝑧+𝜉𝑤⟩⋅𝑒^{−𝜎|𝑧+𝜉𝑤|𝜌}||𝑧+𝜉𝑤||^𝛾𝑑𝑚(𝜉);(4.27)(b) if 𝑧≠0 and [𝑧] is the complex plane generated by the vector 𝑧, then lim_{𝑤→0,𝑤⟂[𝑧]}Φ(𝑧;𝑧+𝑤)=𝜌𝜎^𝜇2Γ(𝑛)⋅𝐸𝜎^2/𝜌|𝑧|²⋅^+∞₀𝑥^𝑛−1⋅𝑒^{−𝜎{|𝑧|2+𝑥}𝜌/2}|𝑧|²+𝑥^𝛾/2𝑑𝑥,(4.28) and for arbitrary 𝑅>0 and for (arbitrary small) 𝜀>0, there exist a positive constant 𝑐=𝑐(𝑛,𝜌,𝜎,𝛾,𝜀,𝑅) such that ||Φ(𝑧;𝑧+𝑤)||≤𝑐⋅𝑒^{−(𝜎−𝜀)|𝑤|𝜌},∀𝑤⟂[𝑧](𝑤≠0),(4.29) uniformly in 𝑧 with |𝑧|<𝑅;(c) if 𝑧≠0, then lim_{𝑤→0,𝑤∈[𝑧]}Φ(𝑧;𝑧+𝑤)=^𝑛−1_𝑚=0(−1)^{𝑛−1−𝑚}⋅𝐶^𝑚_𝑛−1⋅|𝑧|²𝜎^2/𝜌^{𝑛−1−𝑚}⋅ℒ^𝛾,𝜌_𝑛,𝑚,(4.30) where the coefficients ℒ^𝛾,𝜌_𝑛,𝑚(𝑚=1,2,…,𝑛−1) can be written in an explicit form;(d) 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}(4.31) uniformly in 𝑤∈𝐶^𝑛⧵{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, lim_𝑤→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∈𝐶^𝑛⧵{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:(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 put𝜒_𝑅(𝑤)≡𝜒|𝑤|²−𝑅²⎧⎪⎨⎪⎩=1,0≤|𝑤|≤𝑅,∈[0;1],𝑅≤|𝑤|≤√𝑅²+1,=0,|𝑤|≥√𝑅²+1.(5.1) Note that 𝜒_𝑅∈𝐶¹(𝐶^𝑛) and𝜕𝜒_𝑅𝜕𝑤(𝑤)≡𝜕𝜒_𝑅𝜕𝑤₁(𝑤),𝜕𝜒_𝑅𝜕𝑤₂(𝑤),…,𝜕𝜒_𝑅𝜕𝑤_𝑛(𝑤)=⎧⎪⎨⎪⎩(0,0,…,0),0≤|𝑤|≤𝑅,𝜒^|𝑤|²−𝑅²⋅𝑤₁,𝑤₂,…,𝑤_𝑛,𝑅≤|𝑤|≤√𝑅²+1,(0,0,…,0),|𝑤|≥√𝑅²+1.(5.2)

Moreover,||||𝜕𝜒_𝑅𝜕𝑤(𝑤)||||≡⎷^𝑛_𝑘=1||||𝜕𝜒_𝑅𝜕𝑤_𝑘(𝑤)||||²⎧⎪⎨⎪⎩=0,0≤|𝑤|≤𝑅,≤𝑀⋅|𝑤|,𝑅≤|𝑤|≤√𝑅²+1,=0,|𝑤|≥√𝑅²+1.(5.3)

Theorem 5.1. Assume that 𝑛≥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∈𝐶^𝑛, for all 𝑤∈𝐶^𝑛⧵{𝑧}, is defined by the formula (2.15), then the integral representation of the form 𝑓(𝑧)=𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇⋅𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛Φ(𝑧;𝑤)𝑑𝑚(𝑤),𝑧∈𝐶^𝑛,(5.4) is valid for each function 𝑓∈𝐶¹(𝐶^𝑛) satisfying the following conditions:(a)𝑓∈𝐿^𝑝_{𝜌,𝜎,𝛾}(𝐶^𝑛);(b)for any fixed𝑧∈𝐶^𝑛, ||𝑓(𝑤)||⋅Φ(𝑧;𝑤)|𝑤−𝑧|^2𝑛−2∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤))⟺||𝑓(𝑤)||⋅Φ(𝑧;𝑤)|𝑤|^2𝑛−2∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤));(5.5)(c) for any fixed 𝑧∈𝐶^𝑛||𝜕𝑓/𝜕𝑤(𝑤)||⋅Φ(𝑧;𝑤)|𝑤−𝑧|^2𝑛−1∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤))⟺||𝜕𝑓/𝜕𝑤(𝑤)||⋅Φ(𝑧;𝑤)|𝑤|^2𝑛−1∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤)).(5.6)

Proof. Let us fix an arbitrary 𝑧∈𝐶^𝑛, and for ∀𝑅>0 consider the following differential form:𝜓(𝑧;𝑤)=𝑓(𝑤)⋅𝜒_𝑅(𝑤−𝑧)⋅Φ(𝑧;𝑤)⋅𝐾_MB(𝑧;𝑤),𝑤∈𝐶^𝑛⧵{𝑧}.(5.7) Then choose an 𝜀∈(0;𝑅) and apply the Stokes formula to this form and to the domain {𝑤∈𝐶^𝑛∶0<𝜀<|𝑤−𝑧|<√𝑅²+1}: _{|𝑤−𝑧|=√𝑅2+1}𝜓−_{|𝑤−𝑧|=𝜀}𝜓=_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝑑𝜓.(5.8) In view of (5.1) and (2.4), the last relation can be written as follows: −_{|𝑤−𝑧|=𝜀}𝑓(𝑤)⋅Φ(𝑧;𝑤)⋅𝐾_MB(𝑧;𝑤)=Γ(𝑛)𝜋^𝑛_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝜒_𝑅(𝑤−𝑧)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)+Γ(𝑛)𝜋^𝑛_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝜕𝜒_𝑅/𝜕𝑤(𝑤−𝑧),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝑓(𝑤)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)+Γ(𝑛)𝜋^𝑛_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝜕Φ/𝜕𝑤(𝑧;𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝑓(𝑤)⋅𝜒_𝑅(𝑤−𝑧)𝑑𝑚(𝑤).(5.9) Moreover, (5.2) and (4.23^) imply −_{|𝑤−𝑧|=𝜀}𝑓(𝑤)⋅Φ(𝑧;𝑤)⋅𝐾_MB(𝑧;𝑤)=Γ(𝑛)𝜋^𝑛_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝜒_𝑅(𝑤−𝑧)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)+Γ(𝑛)𝜋^𝑛_{𝑅<|𝑤−𝑧|<√𝑅2+1}𝜒^|𝑤−𝑧|²−𝑅²|𝑤−𝑧|^2𝑛−2⋅𝑓(𝑤)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)−𝜌𝜎^𝜇2𝜋^𝑛_{𝜀<|𝑤−𝑧|<√𝑅2+1}𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝑓(𝑤)⋅𝜒_𝑅(𝑤−𝑧)𝑑𝑚(𝑤).(5.10) When 𝜀→0, then due to (4.10) we obtain 𝑓(𝑧)=𝜌𝜎^𝜇2𝜋^𝑛_{|𝑤−𝑧|<√𝑅2+1}𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝑓(𝑤)⋅𝜒_𝑅(𝑤−𝑧)𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_{|𝑤−𝑧|<√𝑅2+1}𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝜒_𝑅(𝑤−𝑧)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_{𝑅<|𝑤−𝑧|<√𝑅2+1}𝜒^|𝑤−𝑧|²−𝑅²|𝑤−𝑧|^2𝑛−2⋅𝑓(𝑤)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)≡𝐴₁(𝑅)−𝐴₂(𝑅)−𝐴₃(𝑅).(5.11) Further, 𝐴₁(𝑅)=𝜌𝜎^𝜇2𝜋^𝑛_{|𝑤−𝑧|<𝑅}𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝑓(𝑤)𝑑𝑚(𝑤)+𝜌𝜎^𝜇2𝜋^𝑛_{𝑅<|𝑤−𝑧|<√𝑅2+1}𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾⋅𝑓(𝑤)⋅𝜒_𝑅(𝑤−𝑧)𝑑𝑚(𝑤)≡𝐴⁽¹⁾₁(𝑅)+𝐴⁽²⁾₁(𝑅).(5.12) Since (via the condition (a)) 𝑓(𝑤)⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤)),(5.13) the summand 𝐴⁽¹⁾₁(𝑅) tends to the same integral but taken now over the whole space 𝐶^𝑛 and 𝐴⁽²⁾₁(𝑅) tends to zero (as 𝑅→+∞).
In other words, lim_𝑅→+∞𝐴₁(𝑅)=𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾𝑑𝑚(𝑤).(5.14) Similarly, 𝐴₂(𝑅)=Γ(𝑛)𝜋^𝑛_{|𝑤−𝑧|<𝑅}𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)+Γ(𝑛)𝜋^𝑛_{𝑅<|𝑤−𝑧|<√𝑅2+1}𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝜒_𝑅(𝑤−𝑧)⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤)≡𝐴⁽¹⁾₂(𝑅)+𝐴⁽²⁾₂(𝑅).(5.15) In view of the condition (c) of the theorem 𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅Φ(𝑧;𝑤)∈𝐿¹(𝐶^𝑛;𝑑𝑚(𝑤)),(5.16) so 𝐴⁽¹⁾₂(𝑅) tends to the same integral but taken now over the whole space 𝐶^𝑛 and 𝐴⁽²⁾₂(𝑅) tends to zero (as 𝑅→+∞). In other words, lim_𝑅→+∞𝐴₂(𝑅)=Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤).(5.17) Finally, ||𝐴₃(𝑅)||≤Γ(𝑛)𝜋^𝑛⋅𝑀⋅_{𝑅<|𝑤−𝑧|<√𝑅2+1}||𝑓(𝑤)||⋅Φ(𝑧;𝑤)|𝑤−𝑧|^2𝑛−2𝑑𝑚(𝑤)≤Γ(𝑛)𝜋^𝑛⋅𝑀⋅_{𝑅<|𝑤−𝑧|<+∞}||𝑓(𝑤)||⋅Φ(𝑧;𝑤)|𝑤−𝑧|^2𝑛−2𝑑𝑚(𝑤).(5.18) Hence, due to the condition (b) of the theorem, lim_𝑅→+∞𝐴₃(𝑅)=0.(5.19) Combining (5.11)–(5.19), we ultimately obtain 𝑓(𝑧)=𝜌𝜎^𝜇2𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝐸^(𝑛)_𝜌/2𝜎^2/𝜌⟨𝑧,𝑤⟩;𝜇⋅𝑒^{−𝜎|𝑤|𝜌}|𝑤|^𝛾𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅Φ(𝑧;𝑤)𝑑𝑚(𝑤),(5.20) which coincides with (5.4). Thus the theorem is proved.

Remark 5.2. The idea of introducing an auxiliary function 𝜒 is borrowed from [16].

6. The Computable Form of the Kernel Φ

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,𝑤∈𝐶^𝑛(𝑧≠𝑤), 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|²≥0.(6.1) Note that a slight change of (3.7) yields||𝑧+𝜆(𝑤−𝑧)||²≡|𝑤−𝑧|²⋅||||𝜆+𝑎|𝑤−𝑧|||||²+𝛿.(6.2) Let 𝜌,𝜎>0, 𝛾≥0, and 𝜇=(𝛾+2𝑛)/𝜌. Consider the following functions:𝜓(𝑠)=𝑠−𝑎^𝑛−1𝐸^(𝑛)_𝜌/2𝜎^2/𝜌(𝑎⋅𝑠+𝛿);𝜇,𝑠∈𝐶,(6.3)𝜓^(𝜈)(𝑠)≡𝑑^𝜈𝜓(𝑠)𝑑𝑠^𝜈,𝜈=0,1,2,3,….(6.4) Besides, put (𝑥≥0):𝜙(𝑥)=𝑒^{−𝜎(𝑥+𝛿)𝜌/2}(𝑥+𝛿)^𝛾/2,(6.5)Φ_𝑘(𝑥)=1Γ(𝑘)^+∞_𝑥𝜙(𝑡)(𝑡−𝑥)^𝑘−1𝑑𝑡,𝑘=1,2,3,….(6.6) Note that Φ_𝑘 is the 𝑘-th primitive of the function 𝜙, that is, Φ_𝑘(𝑥)=^+∞_𝑥𝑑𝑥^+∞_𝑥𝑑𝑥⋯^+∞_{𝑥}_{𝑘𝑡𝑖𝑚𝑒𝑠}𝜙(𝑥)𝑑𝑥.(6.7) In what follows we also put Φ₀(𝑥)≡𝜙(𝑥), 𝑥≥0. Obviously, the following simple relations are valid:𝑑𝑑𝑥Φ_𝑘+1(𝑥)=−Φ_𝑘(𝑥)⟹𝑑^𝑘Φ_𝑘(𝑥)𝑑𝑥^𝑘=(−1)^𝑘𝜙(𝑥),𝑥≥0,𝑘=0,1,2,3,…,(6.8)Φ_𝑘(0)=1Γ(𝑘)^+∞₀𝜙(𝑡)𝑡^𝑘−1𝑑𝑡,𝑘=1,2,3.…,(6.9)_{𝑟<|𝜆|<𝑅}𝜆^𝑙𝜆^𝑚𝜙||𝜆||²𝑑𝑚(𝜆)=0,(6.10) 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 kernel Φ can be computed by the following formula: Φ(𝑧;𝑤)=𝜌𝜎^𝜇2Γ(𝑛)⋅⎧⎨⎩^𝑛−1_𝜈=0𝜓^(𝜈)(0)⋅Φ_𝜈+1(0)⋅^{𝑛−1−𝜈}_𝑘=0𝐶^𝑘+1+𝜈_𝑛(−𝑎)^{𝑛−1−𝜈−𝑘}𝑐^𝑘⎫⎬⎭−𝜌𝜎^𝜇2Γ(𝑛)⋅𝑐(𝑐−𝑎)^𝑛⋅¹₀𝜓𝑐⋅𝑡𝜙|𝑐|²⋅𝑡𝑑𝑡.(6.11)

Proof. In view of (2.15) and (6.2), we have Φ(𝑧;𝑤)=𝜌𝜎^𝜇2Γ(𝑛)⋅𝐼,(6.12) where 𝐼=|𝑤−𝑧|^2𝑛𝜋_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝐸^(𝑛)_𝜌/2𝜎^2/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−𝑐𝐸^(𝑛)_𝜌/2𝜎^2/𝜌𝑎⋅𝜆+𝛿;𝜇⋅𝑒^{−𝜎(|𝜆|2+𝛿)𝜌/2}||𝜆||²+𝛿^𝛾/2𝑑𝑚(𝜆).(6.13) In view of notations (6.3) and (6.5), we have 𝐼=1𝜋_𝐶(𝜆−𝑎)^𝑛𝜆−𝑐⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆).(6.14) In order to simplify the last integral, let us consider the following auxiliary integrals: 𝐴_𝜈=1𝜋_𝐶𝜆^𝜈⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆),𝜈=0,1,2,3,…,𝐵=1𝜋_𝐶𝜓𝜆𝜆−𝑐⋅𝜙||𝜆||²𝑑𝑚(𝜆).(6.15) Using the Maclaurin expansion 𝜓(𝜆)=∑^∞_𝑘=0(𝜓^(𝑘)(0)/Γ(𝑘+1))⋅𝜆^𝑘 and (6.10), we obtain 𝐴_𝜈=1𝜋⋅𝜓^(𝜈)(0)Γ(𝜈+1)⋅_𝐶||𝜆||^2𝜈⋅𝜙||𝜆||²𝑑𝑚(𝜆)=𝜓^(𝜈)(0)Γ(𝜈+1)⋅^∞₀𝑡^𝜈𝜙(𝑡)𝑑𝑡=𝜓^(𝜈)(0)⋅Φ_𝜈+1(0).(6.16)
Further, 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⁽⁻⁾.(6.17) If 𝑐=0, then, naturally, 𝐵⁽⁺⁾=0, and 𝐵⁽⁻⁾=1𝜋_|𝜆|>0𝜓𝜆𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆)=1𝜋_|𝜆|>0𝜆⋅𝜓𝜆⋅𝜙||𝜆||²||𝜆||²𝑑𝑚(𝜆)=0,(6.18) that is, 𝐵=0. If 𝑐≠0, then 1𝜆−𝑐=⎧⎪⎪⎨⎪⎪⎩−1𝑐⋅^∞_𝑘=0𝜆^𝑘𝑐^𝑘,||𝜆||<|𝑐|,1𝜆⋅^∞_𝑘=0𝑐^𝑘𝜆^𝑘,||𝜆||>|𝑐|.(6.19) Consequently, 𝐵⁽⁻⁾=1𝜋^∞_𝑘=0𝑐^𝑘⋅_{|𝜆|>|𝑐|}𝜓𝜆𝜆^𝑘+1⋅𝜙||𝜆||²𝑑𝑚(𝜆)=1𝜋^∞_𝑘=0𝑐^𝑘⋅_{|𝜆|>|𝑐|}𝜆^𝑘+1⋅𝜓𝜆⋅𝜙||𝜆||²||𝜆||^2𝑘+2𝑑𝑚(𝜆)=0.(6.20) Hence 𝐵=𝐵⁽⁺⁾=−1𝜋𝑐^∞_𝑘=0_{|𝜆|<|𝑐|}𝜆^𝑘𝑐^𝑘⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆)=−1𝜋𝑐^∞_𝑘=0_{|𝜆|<|𝑐|}𝜓^(𝑘)(0)Γ(𝑘+1)⋅𝑐^k⋅||𝜆||^2𝑘⋅𝜙||𝜆||²𝑑𝑚(𝜆)=−1𝑐^∞_𝑘=0^|𝑐|2₀𝜓^(𝑘)(0)Γ(𝑘+1)⋅𝑐^𝑘⋅𝑡^𝑘𝜙(𝑡)𝑑𝑚(𝑡)=−1𝑐^|𝑐|2₀^∞_𝑘=0𝜓^(𝑘)(0)Γ(𝑘+1)𝑡𝑐^𝑘𝜙(𝑡)𝑑𝑚(𝑡)=−1𝑐^|𝑐|2₀𝜓𝑡𝑐𝜙(𝑡)𝑑𝑚(𝑡)===^{𝑡→|𝑐|2𝑡}−𝑐⋅¹₀𝜓𝑐⋅𝑡𝜙|𝑐|²⋅𝑡𝑑𝑚(𝑡).(6.21) Using the binomial expansion of (𝜆−𝑎)^𝑛 and combining (6.14)–(6.21), we obtain 𝐼=^𝑛_𝑘=0𝐶^𝑘_𝑛(−𝑎)^𝑛−𝑘⋅1𝜋_𝐶𝜆^𝑘𝜆−𝑐⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆)=^𝑛_𝑘=1𝐶^𝑘_𝑛(−𝑎)^𝑛−𝑘⋅1𝜋_𝐶𝜆^𝑘−𝑐^𝑘+𝑐^𝑘𝜆−𝑐⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆)+(−𝑎)^𝑛⋅𝐵=^𝑛_𝑘=1𝐶^𝑘_𝑛(−𝑎)^𝑛−𝑘⋅1𝜋_𝐶(𝜆−𝑐)⋅∑^𝑘−1_𝜈=0𝑐^{𝑘−1−𝜈}𝜆^𝜈𝜆−𝑐⋅𝜓𝜆⋅𝜙||𝜆||²𝑑𝑚(𝜆)+^𝑛_𝑘=1𝐶^𝑘_𝑛(−𝑎)^𝑛−𝑘⋅𝑐^𝑘⋅𝐵+(−𝑎)^𝑛⋅𝐵=^𝑛_𝑘=1𝐶^𝑘_𝑛(−𝑎)^𝑛−𝑘⋅^𝑘−1_𝜈=0𝑐^{𝑘−1−𝜈}⋅𝐴_𝜈+(𝑐−𝑎)^𝑛⋅𝐵===^{𝑘→𝑘+1}=^𝑛−1_𝑘=0𝐶^𝑘+1_𝑛(−𝑎)^{𝑛−1−𝑘}⋅^𝑘_𝜈=0𝑐^𝑘−𝜈⋅𝜓^(𝜈)(0)⋅Φ_𝜈+1(0)−𝑐(𝑐−𝑎)^𝑛⋅¹₀𝜓𝑐⋅𝑡𝜙|𝑐|²⋅𝑡𝑑𝑚(𝑡)=^𝑛−1_𝜈=0𝜓^(𝜈)(0)⋅Φ_𝜈+1(0)⋅^{𝑛−1−𝜈}_𝑘=0𝐶^𝑘+1+𝜈_𝑛(−𝑎)^{𝑛−1−𝜈−𝑘}𝑐^𝑘−𝑐(𝑐−𝑎)^𝑛⋅¹₀𝜓𝑐⋅𝑡𝜙|𝑐|²⋅𝑡𝑑𝑚(𝑡).(6.22) 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 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 𝜌𝜎^𝜇/2Γ(𝑛) transforms into 𝜎^𝑛/Γ(𝑛). Next,𝐸^(𝑛)_𝜌/2(𝜂;𝜇)=^∞_𝑘=0Γ(𝑘+𝑛)Γ(𝑘+1)⋅𝜂^𝑘Γ(𝜇+2𝑘/𝜌)=^∞_𝑘=0𝜂^𝑘Γ(𝑘+1)≡𝑒^𝜂,𝜂∈𝐶.(7.1) As a consequence, the formula (2.15) takes the following form:Φ(𝑧;𝑤)=𝑒^{−𝜎|𝑤|2}⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤−𝑧|^2𝜈,∀𝑧∈𝐶^𝑛,∀𝑤∈𝐶^𝑛⧵{𝑧},(7.2) or, equivalently,Φ(𝑧;𝑧+𝑤)=𝑒^{−𝜎|𝑤|2}⋅𝑒^{−𝜎⟨𝑤,𝑧⟩}⋅^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤|^2𝜈,∀𝑧∈𝐶^𝑛,∀𝑤∈𝐶^𝑛⧵{0}.(7.3) Indeed, we haveΦ(𝑧;𝑤)=𝜎^𝑛𝜋Γ(𝑛)|𝑤−𝑧|^2𝑛⋅_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝑒^{𝜎⟨𝑧,𝑧+𝜆(𝑤−𝑧)⟩}⋅𝑒^{−𝜎|𝑧+𝜆(𝑤−𝑧)|2}𝑑𝑚(𝜆)=𝜎^𝑛𝜋Γ(𝑛)|𝑤−𝑧|^2𝑛⋅_𝐶||𝜆||^2𝑛𝜆(𝜆−1)𝑒^{−𝜎⟨𝜆(𝑤−𝑧),𝑧+𝜆(𝑤−𝑧)⟩}𝑑𝑚(𝜆)=𝜎^𝑛𝜋Γ(𝑛)|𝑤−𝑧|^2𝑛⋅_𝐶||𝜆||^2𝑛−2⋅𝑒^{−𝜎|𝑤−𝑧|2|𝜆|2}⋅𝜆⋅𝑒^{−𝜎𝜆⟨𝑤−𝑧,𝑧⟩}𝜆−1𝑑𝑚(𝜆)=2𝜎^𝑛Γ(𝑛)|𝑤−𝑧|^2𝑛⋅^+∞₀𝑟^2𝑛−1⋅𝑒^{−𝜎|𝑤−𝑧|2𝑟2}⋅12𝜋𝑖_|𝜁|=𝑟𝑒^{−𝜎𝜁⟨𝑤−𝑧,𝑧⟩}𝜁−1𝑑𝜁𝑑𝑟=2𝜎^𝑛Γ(𝑛)|𝑤−𝑧|^2𝑛⋅^+∞₀𝑟^2𝑛−1⋅𝑒^{−𝜎|𝑤−𝑧|2𝑟2}×0,0<𝑟<1𝑒^{−𝜎⟨𝑤−𝑧,𝑧⟩},1<𝑟<∞×𝑑𝑟=𝜎^𝑛Γ(𝑛)|𝑤−𝑧|^2𝑛⋅𝑒^{−𝜎⟨𝑤−𝑧,𝑧⟩}⋅^+∞₁𝑡^𝑛−1⋅𝑒^{−𝜎|𝑤−𝑧|2𝑡}𝑑𝑡=𝜎^𝑛Γ(𝑛)|𝑤−𝑧|^2𝑛⋅𝑒^{−𝜎⟨𝑤−𝑧,𝑧⟩}⋅𝑒^{−𝜎|𝑤−𝑧|2}⋅^+∞₀(𝑡+1)^𝑛−1⋅𝑒^{−𝜎|𝑤−𝑧|2𝑡}𝑑𝑡=𝜎^𝑛Γ(𝑛)|𝑤−𝑧|^2𝑛⋅𝑒^{−𝜎|𝑤|2}⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅^𝑛−1_𝜈=0𝐶^𝜈_𝑛−1⋅Γ(𝑛−𝜈)𝜎^𝑛−𝜈|𝑤−𝑧|^{2𝑛−2𝜈}=𝑒^{−𝜎|𝑤|2}⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤−𝑧|^2𝜈.(7.4)

It is interesting to note (see Remark 4.4) thatlim_𝑤→0Φ(𝑧;𝑧+𝑤)=Φ(𝑧;𝑧)=1.(7.5) Moreover (compare with Proposition 4.7), for arbitrary 𝑅>0 and for (arbitrary small) 𝜀>0, there exists a positive constant 𝑐=𝑐(𝑛,𝜎,𝜀,𝑅) such that||Φ(𝑧;𝑧+𝑤)||≤𝑐⋅𝑒^{−(𝜎−𝜀)|𝑤|2},∀𝑤∈𝐶^𝑛,(7.6) uniformly in 𝑧 with |𝑧|<𝑅. At last, the main Theorem 5.1 takes the following form.

Theorem 7.1. Assume that 𝑛≥1, 1<𝑝<+∞ and 𝜎>0, then the integral representation of the form 𝑓(𝑧)=𝜎^𝑛𝜋^𝑛_𝐶𝑛𝑓(𝑤)⋅𝑒^{𝜎⟨𝑧,𝑤⟩}⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤)−Γ(𝑛)𝜋^𝑛_𝐶𝑛𝜕𝑓/𝜕𝑤(𝑤),𝑤−𝑧|𝑤−𝑧|^2𝑛⋅𝑒^{𝜎⟨𝑧,𝑤⟩}×^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤−𝑧|^2𝜈⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤),𝑧∈𝐶^𝑛,(7.7) is valid for each function 𝑓∈𝐶¹(𝐶^𝑛) satisfying the following conditions:(1)∫_𝐶𝑛|𝑓(𝑤)|^𝑝⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤)<+∞;(2) for any fixed 𝑧∈𝐶^𝑛,_𝐶𝑛||𝜕𝑓/𝜕𝑤(𝑤)|||𝑤−𝑧|^2𝑛−1⋅||𝑒^{𝜎⟨𝑧,𝑤⟩}||⋅^𝑛−1_𝜈=0𝜎^𝜈𝜈!|𝑤−𝑧|^2𝜈⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤)<+∞,(7.8) or, equivalently, _𝐶𝑛||𝜕𝑓/𝜕𝑤(𝑤)|||𝑤−𝑧|⋅||𝑒^{𝜎⟨𝑧,𝑤⟩}||⋅𝑒^{−𝜎|𝑤|2}𝑑𝑚(𝑤)<+∞,(7.9) or, not equivalently (but more conveniently), _𝐶𝑛|||𝜕𝑓𝜕𝑤(𝑤)|||⋅𝑒^{−(𝜎−𝜀)|𝑤|2}𝑑𝑚(𝑤)<+∞,(7.10) where 𝜀 is arbitrary small.