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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 265092, 27 pages
http://dx.doi.org/10.1155/2012/265092
Research Article

Weighted πœ•-Integral Representations of 𝐢𝟏-Functions in 𝐢𝑛

Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia

Received 28 July 2011; Accepted 13 August 2011

Academic Editor: RichardΒ Rochberg

Copyright Β© 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.

Abstract

For 𝐢1-functions 𝑓, given in the complex space 𝐢𝑛, integral representations of the form 𝑓=𝑃(𝑓)βˆ’π‘‡(πœ•π‘“) are obtained. Here, 𝑃 is the orthogonal projector of the space 𝐿2{𝐢𝑛;π‘’βˆ’πœŽ|𝑧|𝜌|𝑧|π›Ύπ‘‘π‘š(𝑧)} onto its subspace of entire functions and the integral operator 𝑇 appears by means of explicitly constructed kernel Ξ¦ which is investigated in detail.

1. Introduction

Let 𝑛β‰₯1, 1<𝑝<+∞, and 0<𝜌,𝜎<∞, 𝛾>βˆ’2𝑛. Denote by πΏπ‘πœŒ,𝜎,𝛾(𝐢𝑛) the space of all measurable complex-valued functions 𝑓(𝑧), π‘§βˆˆπΆπ‘›, satisfying the conditionξ€œπΆπ‘›||𝑓(𝑧)||π‘π‘’βˆ’πœŽ|𝑧|𝜌|𝑧|π›Ύπ‘‘π‘š(𝑧)<+∞.(1.1) Let π»π‘πœŒ,𝜎,𝛾(𝐢𝑛) be the corresponding subspace of entire functions. It was established in [1, 2] (when 𝑛=1) and [3] (when 𝑛>1) that arbitrary function π‘“βˆˆπ»π‘πœŒ,𝜎,𝛾(𝐢𝑛) has an integral representation of the form𝑓(𝑧)=πœŒπœŽπœ‡2πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)π‘’βˆ’πœŽ|𝑀|𝜌|𝑀|𝛾⋅𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©;πœ‡ξ€Έπ‘‘π‘š(𝑀),π‘§βˆˆπΆπ‘›,(1.2) where πœ‡=(𝛾+2𝑛)/𝜌 and𝐸(𝑛)𝜌/2(πœ‚;πœ‡)=βˆžξ“π‘˜=0Ξ“(π‘˜+𝑛)Ξ“(π‘˜+1)β‹…πœ‚π‘˜Ξ“(πœ‡+2π‘˜/𝜌),πœ‚βˆˆπΆ,(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 𝐿2𝜌,𝜎,𝛾(𝐢𝑛) onto its subspace 𝐻2𝜌,𝜎,𝛾(𝐢𝑛). Certainly, the condition (1.1) and corresponding properties of 𝐸(𝑛)𝜌/2(πœ‚;πœ‡) ensure an absolute convergence of the integral in (1.2).

Note that for 𝑝=2,𝜌=2, 𝛾=0, π»π‘πœŒ,𝜎,𝛾(𝐢𝑛) coincides with the well-known Fock space of entire functions and (1.2) takes the form𝑓(𝑧)=πœŽπ‘›πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)β‹…π‘’βˆ’πœŽ|𝑀|2β‹…π‘’πœŽβŸ¨π‘§,π‘€βŸ©π‘‘π‘š(𝑀),π‘§βˆˆπΆπ‘›.(1.4)

In [4] general weighted integral representations were obtained for differential forms. In particular, for functions π‘“βˆˆπΆ1(𝐢𝑛) (satisfying certain growth conditions), the following generalization of the formula (1.4) was established:𝑓(𝑧)=πœŽπ‘›πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)β‹…π‘’πœŽβŸ¨π‘§,π‘€βŸ©β‹…π‘’βˆ’πœŽ|𝑀|2π‘‘π‘š(𝑀)βˆ’Ξ“(𝑛)πœ‹π‘›ξ€œπΆπ‘›ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2π‘›β‹…π‘’πœŽβŸ¨π‘§,π‘€βŸ©β‹…π‘›βˆ’1ξ“πœˆ=0𝜎𝜈𝜈!|π‘€βˆ’π‘§|2πœˆβ‹…π‘’βˆ’πœŽ|𝑀|2π‘‘π‘š(𝑀),π‘§βˆˆπΆπ‘›,(1.5) whereπœ•π‘“πœ•π‘€(𝑀)=ξ‚΅πœ•π‘“πœ•π‘€1(𝑀),πœ•π‘“πœ•π‘€2(𝑀),…,πœ•π‘“πœ•π‘€π‘›(𝑀)ξ‚Ά(1.6)

and, consequently, ξƒ‘πœ•π‘“πœ•π‘€(𝑀),π‘€βˆ’π‘§ξƒ’=π‘›ξ“π‘˜=1πœ•π‘“πœ•π‘€π‘˜(𝑀)β‹…ξ€·π‘€π‘˜βˆ’π‘§π‘˜ξ€Έ.(1.7)

In [5] a canonical operator is constructed for πœ•-solution in a space of differential forms square integrable with the weight π‘’βˆ’|𝑀|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 πœ• (and πœ•πœ•) integral representations are well investigated: for unit ball of 𝐢𝑛 see [6–8]; for general strictly pseudoconvex domains see [9–13]; for Cartan matrix domain (β€œmatrix disc”) see [14]. The whole space 𝐢𝑛 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] β€œpolycylindric” weight function of the type βˆπ‘›π‘˜=1π‘’βˆ’πœŽπ‘˜|π‘€π‘˜|πœŒπ‘˜|π‘€π‘˜|π›Ύπ‘˜, 𝑀=(𝑀1,𝑀2,…,𝑀𝑛)βˆˆπΆπ‘›, was considered. In the present paper, the corresponding weighted πœ•-integral representations are obtained for the case of radial weight function of the type π‘’βˆ’πœŽ|𝑀|𝜌|𝑀|𝛾, 𝑀=(𝑀1,𝑀2,…,𝑀𝑛)βˆˆπΆπ‘›, |𝑀|=ξ”βˆ‘π‘›π‘˜=1|π‘€π‘˜|2 (see the condition (1.1)). More precisely, for functions π‘“βˆˆπΆ1(𝐢𝑛) (satisfying certain growth conditions), the integral representation of the form𝑓(𝑧)=πœŒπœŽπœ‡2πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)π‘’βˆ’πœŽ|𝑀|𝜌|𝑀|𝛾⋅𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©;πœ‡ξ€Έπ‘‘π‘š(𝑀)βˆ’Ξ“(𝑛)πœ‹π‘›ξ€œπΆπ‘›ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2𝑛⋅Φ(𝑧;𝑀)π‘‘π‘š(𝑀),π‘§βˆˆπΆπ‘›,(1.8) is established. Moreover, the kernel Ξ¦ 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 |𝑀|𝛾exp{βˆ’πœŽ|𝑀|𝜌}, the case of weight function of the type exp{βˆ’πœ‘(𝑀)} was considered in the assumption that πœ‘ is a convex function of class 𝐢2. In this case, a formula of type (1.8) was obtained. But in that formula the operator of orthogonal projection of the space 𝐿2(𝐢𝑛;exp{βˆ’πœ‘(𝑀)}π‘‘π‘š(𝑀)) onto its subspace of entire functions does not appear, except of the special case πœ‘(𝑀)=𝜎|𝑀|2 when we again obtain (1.5).

2. Heuristic Argument: Revealing of the Kernel Ξ¦

In what follows, it is supposed that 𝑛β‰₯1 and 0<𝜌,𝜎<∞, 𝛾>βˆ’2𝑛, πœ‡=(𝛾+2𝑛)/𝜌.

We intend to reveal a formula of type (1.2) but this time for 𝐢1-functions. This means that the formula we search needs to have a second summand containing πœ•-β€œpart” of functions. In other words, for functions π‘“βˆˆπΆ1(𝐢𝑛) 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 Ξ¦(𝑧;𝑀),𝑧,π‘€βˆˆπΆπ‘›, to have the following (or similar) properties:Ξ¦(𝑧;𝑧)=1,βˆ€π‘§βˆˆπΆπ‘›,(2.1)Ξ¦(𝑧;∞)=0,βˆ€π‘§βˆˆπΆπ‘›.(2.2)

Denote by 𝐾MB(𝑧;𝑀) (for all π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{𝑧}) the well-known Martinelli-Bochner kernel, which has the following useful properties (see, for instance, [17, Chapter 16]):πœ•π‘€πΎMB(𝑧;𝑀)≑0,π‘€βˆˆπΆπ‘›β§΅{𝑧};(2.3)πœ•π‘€ξ€½π‘“(𝑀)⋅𝐾MB(𝑧;𝑀)ξ€Ύ=πœ•π‘“(𝑀)∧𝐾MB(𝑧;𝑀)≑Γ(𝑛)πœ‹π‘›β‹…ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2π‘›Γ—π‘‘π‘š(𝑀)(2.4) for arbitrary function π‘“βˆˆπΆ1(Ξ©), where Ξ©βŠ‚πΆπ‘›β§΅{𝑧} is an open set;limπœ€β†“0ξ€œ|π‘€βˆ’π‘§|=πœ€π‘’(𝑀)⋅𝐾MB(𝑧;𝑀)=𝑒(𝑧)(2.5) for arbitrary function 𝑒 continuous in a neighborhood of 𝑧;ξ€œπ‘†π‘›π‘“(𝜁)⋅𝐾MB(0;𝜁)=Ξ“(𝑛)2πœ‹π‘›β‹…ξ€œπ‘†π‘›π‘“(𝜁)π‘‘πœŽ(𝜁),(2.6) where 𝑆𝑛={πœβˆˆπΆπ‘›βˆΆ|𝜁|=1} is the unit sphere in 𝐢𝑛,π‘“βˆˆπΆ(𝑆𝑛), and 𝜎 is the surface measure on 𝑆𝑛.

Let us fix an arbitrary π‘§βˆˆπΆπ‘› and consider the following differential form:πœ‘(𝑀)=𝑓(𝑀)β‹…Ξ¦(𝑧;𝑀)⋅𝐾MB(𝑧;𝑀),π‘€βˆˆπΆπ‘›β§΅{𝑧}.(2.7)

Then apply the Stokes formula to this form and to the domain {π‘€βˆˆπΆπ‘›βˆΆ0<πœ€<|π‘€βˆ’π‘§|<𝑅<+∞}:ξ€œ|π‘€βˆ’π‘§|=π‘…πœ‘βˆ’ξ€œ|π‘€βˆ’π‘§|=πœ€πœ‘=ξ€œπœ€<|π‘€βˆ’π‘§|<π‘…π‘‘πœ‘.(2.8) The integral ∫|π‘€βˆ’π‘§|=π‘…πœ‘β†’0 (as 𝑅→+∞) due to the property (2.2) (our argument is heuristic !!!). When πœ€β†’0, then ∫|π‘€βˆ’π‘§|=πœ€πœ‘β†’π‘“(𝑧) due to the properties (2.5) and (2.1). Thus after 𝑅→+∞ and πœ€β†’0, we haveβˆ’π‘“(𝑧)=ξ€œπΆπ‘›π‘‘πœ‘,(2.9)

or, in view of (2.3)-(2.4),𝑓(𝑧)=βˆ’Ξ“(𝑛)πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)β‹…ξ«ξ€·πœ•Ξ¦/πœ•π‘€ξ€Έ(𝑧;𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2π‘›π‘‘π‘š(𝑀)βˆ’Ξ“(𝑛)πœ‹π‘›ξ€œπΆπ‘›ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬β‹…Ξ¦(𝑧;𝑀)|π‘€βˆ’π‘§|2π‘›π‘‘π‘š(𝑀).(2.10)

Comparing (1.8) and (2.10), we arrive at the equalityξƒ‘πœ•Ξ¦πœ•π‘€(𝑧;𝑀),π‘€βˆ’π‘§ξƒ’=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©;πœ‡ξ€Έπ‘’βˆ’πœŽ|𝑀|𝜌|𝑀|𝛾|π‘€βˆ’π‘§|2𝑛,βˆ€π‘€βˆˆπΆπ‘›β§΅{𝑧}.(2.11)

Let us fix arbitrary 𝑧,π‘€βˆˆπΆπ‘›, 𝑧≠𝑀 and consider the functionπœ‘(πœ†)=Ξ¦(𝑧;𝑧+πœ†(π‘€βˆ’π‘§)),πœ†βˆˆπΆ.(2.12)

Then we haveπœ•πœ‘πœ•πœ†=π‘›ξ“π‘˜=1πœ•Ξ¦πœ•π‘€π‘˜(𝑧;𝑧+πœ†(π‘€βˆ’π‘§))β‹…ξ€·π‘€π‘˜βˆ’π‘§π‘˜ξ€Έ=ξƒ‘πœ•Ξ¦πœ•π‘€(𝑧;𝑧+πœ†(π‘€βˆ’π‘§)),π‘€βˆ’π‘§ξƒ’=βˆ’πœŒπœŽπœ‡πœ†β‹…2Ξ“(𝑛)×𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†(π‘€βˆ’π‘§)⟩;πœ‡ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ†(π‘€βˆ’π‘§)|πœŒΓ—||𝑧+πœ†(π‘€βˆ’π‘§)||𝛾||πœ†||2𝑛|π‘€βˆ’π‘§|2𝑛.(2.13)

The condition (2.2) implies πœ‘(∞)=0; hence due to Cauchy-Green-Pompeiju formula,πœ‘ξ€·πœ†0ξ€Έ=βˆ’1πœ‹ξ€œπΆπœ•πœ‘/πœ•πœ†πœ†βˆ’πœ†0π‘‘π‘š(πœ†),βˆ€πœ†0∈𝐢.(2.14)

Since πœ‘(1)=Ξ¦(𝑧;𝑀), we finally have Ξ¦(𝑧;𝑀)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)|π‘€βˆ’π‘§|2π‘›Γ—ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†(π‘€βˆ’π‘§)⟩;πœ‡ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ†(π‘€βˆ’π‘§)|πœŒΓ—||𝑧+πœ†(π‘€βˆ’π‘§)||π›Ύπ‘‘π‘š(πœ†),βˆ€π‘§βˆˆπΆπ‘›,βˆ€π‘€βˆˆπΆπ‘›β§΅{𝑧}.(2.15) This formula can be also written in the following (may be, more convenient) form:Ξ¦(𝑧;𝑧+𝑀)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)|𝑀|2π‘›Γ—ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©;πœ‡ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ†π‘€|𝜌||𝑧+πœ†π‘€||π›Ύπ‘‘π‘š(πœ†),βˆ€π‘§βˆˆπΆπ‘›,βˆ€π‘€βˆˆπΆπ‘›β§΅{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(πœ‚)≑𝐸(𝑛)𝜌/2(πœ‚;πœ‡),πœ‚βˆˆπΆ.(3.1) It is an entire function of order 𝜌/2 and of type 1. The same is true for its derivative πΈξ…ž(πœ‚), πœ‚βˆˆπΆ. Consequently, we have||𝐸(πœ‚)||+||πΈξ…ž(πœ‚)||≀const(𝜌;𝑛;πœ‡)⋅𝑒2|πœ‚|𝜌/2,πœ‚βˆˆπΆ.(3.2)

Let us introduce a convenient notationπœ‘(π‘₯)β‰‘π‘’βˆ’πœŽπ‘₯𝜌/2β‹…π‘₯𝛾/2,π‘₯∈(0;+∞).(3.3) If 𝛾β‰₯0, we can suppose that π‘₯∈[0;+∞) in (3.3). Then obviously the function πœ‘βˆˆπΆ[0;+∞) andπœ‘(π‘₯)≀const(𝜌;𝜎;𝛾)⋅𝑒(βˆ’πœŽ/2)π‘₯𝜌/2,π‘₯∈[0;+∞).(3.4) Note that under additional assumptions 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2, the function πœ‘βˆˆπΆ1[0;+∞) and, moreover,πœ‘ξ…ž(π‘₯)=π‘’βˆ’πœŽπ‘₯𝜌/2⋅𝛾2π‘₯𝛾/2βˆ’1βˆ’πœŽπœŒ2π‘₯𝜌/2βˆ’1+𝛾/2,π‘₯∈[0;+∞),||πœ‘ξ…ž(π‘₯)||≀const(𝜌;𝜎;𝛾)⋅𝑒(βˆ’πœŽ/2)π‘₯𝜌/2,π‘₯∈[0;+∞).(3.5) In view of (3.1) and (3.3), the formula (2.16) can be written as follows:Ξ¦(𝑧;𝑧+𝑀)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)|𝑀|2π‘›β‹…ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έπœ‘ξ€·||𝑧+πœ†π‘€||2ξ€Έπ‘‘π‘š(πœ†),βˆ€π‘§βˆˆπΆπ‘›,βˆ€π‘€βˆˆπΆπ‘›β§΅{0}.(3.6)

Further, assume that π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{0}, πœ†βˆˆπΆ, then evidently||𝑧+πœ†π‘€||2=|𝑀|2β‹…||πœ†+Μƒπ‘Ž||2+Μƒβ€Œπ›Ώ,(3.7) whereΜƒπ‘Ž=βŸ¨π‘§,π‘€βŸ©|𝑀|2,Μƒβ€Œπ›Ώ=|𝑧|2|𝑀|2βˆ’||βŸ¨π‘§,π‘€βŸ©||2|𝑀|2β‰₯0.(3.8)

Note that Μƒβ€Œπ›Ώ=0⇔𝑧 and 𝑀 lies on the same complex β€œstraight line” (i.e., complex plane) of 𝐢𝑛 passing through the origin.

Lemma 3.1. Assume that 0<𝑅<+∞ and 0<π‘š<𝑀<+∞, then ||πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||β‹…π‘’βˆ’πœŽ|𝑧+πœ†π‘€|𝜌||𝑧+πœ†π‘€||𝛾≑||πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||β‹…πœ‘ξ€·||𝑧+πœ†π‘€||2ξ€Έβ‰€βŽ§βŽͺ⎨βŽͺβŽ©π‘β‹…π‘’βˆ’π‘|πœ†|𝜌,πœ†βˆˆπΆ,(𝛾β‰₯0),𝑐1β‹…π‘’βˆ’π‘1|πœ†|πœŒβ‹…ξ‚΅||πœ†+Μƒπ‘Ž||2+Μƒβ€Œπ›Ώ|𝑀|2𝛾/2≀𝑐1β‹…π‘’βˆ’π‘1|πœ†|πœŒβ‹…||πœ†+Μƒπ‘Ž||𝛾,πœ†βˆˆπΆβ§΅{βˆ’Μƒπ‘Ž},(βˆ’2𝑛<𝛾<0),(3.9) uniformly in 𝑧 and 𝑀 with |𝑧|≀𝑅, π‘šβ‰€|𝑀|≀𝑀, where 𝑐,𝑐1,𝑏,𝑏1 are positive constants and depend, in general, on 𝑛,𝜌,𝜎,𝛾,𝑅,π‘š,𝑀.

Proof. According to (3.2), ||πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||≀const⋅𝑒2𝜎|𝑧|𝜌/2(|𝑧|+|πœ†β€–π‘€|)𝜌/2≀const⋅𝑒2𝜎|𝑧|𝜌/22𝜌/2(|𝑧|𝜌/2+|πœ†|𝜌/2|𝑀|𝜌/2)=const⋅𝑒2(2+𝜌)/2𝜎|𝑧|πœŒβ‹…π‘’2(2+𝜌)/2𝜎|𝑧|𝜌/2|𝑀|𝜌/2|πœ†|𝜌/2≀const⋅𝑒2(2+𝜌)/2πœŽπ‘…πœŒβ‹…π‘’2(2+𝜌)/2πœŽπ‘…πœŒ/2π‘€πœŒ/2|πœ†|𝜌/2≑constβ‹…π‘’π‘˜|πœ†|𝜌/2,πœ†βˆˆπΆ,(3.10) where π‘˜ is a positive number.
Further, if 𝛾β‰₯0, then in view of (3.4) and (3.7) πœ‘ξ€·||𝑧+πœ†π‘€||2≀const(𝜌,𝜎,𝛾)⋅𝑒(βˆ’πœŽ/2)|𝑧+πœ†π‘€|𝜌=const⋅𝑒(βˆ’πœŽ/2)|𝑀|𝜌(|πœ†+Μƒπ‘Ž|2+Μƒβ€Œπ›Ώ/|𝑀|2)𝜌/2≀const⋅𝑒(βˆ’πœŽ/2)π‘šπœŒ|πœ†+Μƒπ‘Ž|𝜌.(3.11) Due to the conditions on 𝑧 and 𝑀, we have |Μƒπ‘Ž|≀𝑅/π‘š. Let us choose 𝑇>0 such that 𝑅/π‘šπ‘‡β‰€1/2, then for |πœ†|β‰₯𝑇||πœ†+Μƒπ‘Ž||β‰₯||πœ†||βˆ’||Μƒπ‘Ž||=||πœ†||βŽ›βŽœβŽ1βˆ’||Μƒπ‘Ž||||πœ†||⎞⎟⎠β‰₯||πœ†||ξ‚€1βˆ’π‘…π‘šπ‘‡ξ‚β‰₯||πœ†||2.(3.12) Hence 𝑒(βˆ’πœŽ/2)π‘šπœŒ|πœ†+Μƒπ‘Ž|πœŒβ‰€const⋅𝑒(βˆ’πœŽ/2)π‘šπœŒ(|πœ†|𝜌/2𝜌)(πœ†βˆˆπΆ).(3.13) Combining (3.11) and (3.13), we obtain πœ‘ξ€·||𝑧+πœ†π‘€||2≀constβ‹…π‘’βˆ’π‘‘|πœ†|𝜌,πœ†βˆˆπΆ,(3.14) where 𝑑 is a positive number.
Combination of (3.10) and (3.14) easily implies (3.9) for the case 𝛾β‰₯0. If βˆ’2𝑛<𝛾<0, then similarly to (3.11)–(3.14) we have π‘’βˆ’πœŽ|𝑧+πœ†π‘€|πœŒβ‰€constβ‹…π‘’βˆ’π‘‘1|πœ†|𝜌,πœ†βˆˆπΆ,(3.15) where 𝑑1 is a positive number. Also, in view of (3.7), ||𝑧+πœ†π‘€||𝛾=ξ€·|𝑀|2β‹…||πœ†+Μƒπ‘Ž||2+Μƒβ€Œπ›Ώξ€Έπ›Ύ/2=|𝑀|𝛾||πœ†+Μƒπ‘Ž||2+Μƒβ€Œπ›Ώ|𝑀|2𝛾/2β‰€π‘šπ›Ύξ‚΅||πœ†+Μƒπ‘Ž||2+Μƒβ€Œπ›Ώ|𝑀|2𝛾/2.(3.16) Combination of (3.10), (3.15), and (3.16) establishes (3.9) for the case βˆ’2𝑛<𝛾<0. The proof is complete.

Taking into account (3.2), (3.4), (3.5) and repeating β€œword by word” the argument of Lemma 3.1, we obtain the following lemma.

Lemma 3.2. Assume that 0<𝑅<+∞ and 0<π‘š<𝑀<+∞. Then there exist positive constants 𝑐,𝑏 (depending, in general, on 𝑛,𝜌,𝜎,𝛾,𝑅,π‘š,𝑀) such that uniformly in 𝑧 and 𝑀 with |𝑧|≀𝑅, π‘šβ‰€|𝑀|≀𝑀.(a)If 𝛾β‰₯0, then||πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||||πΈξ…žξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||Γ—||πœ‘ξ€·||𝑧+πœ†π‘€||2ξ€Έ||β‰€π‘β‹…π‘’βˆ’π‘|πœ†|𝜌,πœ†βˆˆπΆ.(3.17)(b) If 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2, then||πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||||πΈξ…žξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έ||Γ—||πœ‘ξ…žξ€·||𝑧+πœ†π‘€||2ξ€Έ||β‰€π‘β‹…π‘’βˆ’π‘|πœ†|𝜌,πœ†βˆˆπΆ.(3.18)

Lemma 3.3. Let 𝛽(πœ‚), πœ‚βˆˆπΆ, be an arbitrary function of class 𝐢1(𝐢) such that it together with its first-order partial derivatives decreases (modulo) at infinity. For instance, the decreasing of type 𝑂(1/|πœ‚|1+πœ€), |πœ‚|β†’+∞, (for arbitrary small πœ€>0) is quite sufficient for us. Then the function 𝛼(πœ‚)β‰‘βˆ’1πœ‹ξ€œπΆπ›½(πœ‰)πœ‰βˆ’πœ‚π‘‘π‘š(πœ‰),πœ‚βˆˆπΆ,(3.19) is of class 𝐢1(𝐢) and (πœ•π›Ό/πœ•πœ‚)(πœ‚)≑𝛽(πœ‚), πœ‚βˆˆπΆ.

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 𝐢∞-functions.

Lemma 3.4. Assume that πœ“(𝑀), π‘€βˆˆπΆπ‘›β§΅{0}, is continuously differentiable (i.e., of class 𝐢1) and 𝑔(𝑀), π‘€βˆˆπΆπ‘›β§΅{0}, is continuous. Then the following two relations are equivalent: ξƒ‘πœ•πœ“πœ•π‘€(𝑀),𝑀≑𝑔(𝑀),π‘€βˆˆπΆπ‘›β§΅{0},(3.20)πœ•πœ•πœ‚πœ“(πœ‚β‹…π‘€)≑𝑔(πœ‚β‹…π‘€)πœ‚,πœ‚βˆˆπΆβ§΅{0}(βˆ€π‘€βˆˆπΆπ‘›β§΅{0}).(3.21)

Proof. Let us fix an arbitrary π‘€βˆˆπΆπ‘›β§΅{0}; then πœ•πœ•πœ‚πœ“(πœ‚β‹…π‘€)=π‘›ξ“π‘˜=1πœ•πœ“πœ•π‘€π‘˜(πœ‚β‹…π‘€)β‹…π‘€π‘˜=ξƒ‘πœ•πœ“πœ•π‘€(πœ‚β‹…π‘€),𝑀=ξ«ξ€·πœ•πœ“/πœ•π‘€ξ€Έ(πœ‚β‹…π‘€),πœ‚β‹…π‘€ξ¬πœ‚,πœ‚βˆˆπΆβ§΅{0}.(3.22) This immediately gives the implication (3.20)β‡’(3.21). On the contrary, if (3.21) is valid, then ξ«ξ€·πœ•πœ“/πœ•π‘€ξ€Έ(πœ‚β‹…π‘€),πœ‚β‹…π‘€ξ¬πœ‚=𝑔(πœ‚β‹…π‘€)πœ‚,πœ‚βˆˆπΆβ§΅{0}.(3.23) Substitution of πœ‚=1 into the last relation gives (3.20). Thus, the assertion is proved.

4. The Main Properties of the Kernel Ξ¦

Proposition 4.1. If 𝛾>βˆ’1, then for fixed π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{0} and for arbitrary πœ‚βˆˆπΆβ§΅{0}Ξ¦(𝑧;𝑧+πœ‚β‹…π‘€)=βˆ’1πœ‹β‹…ξ€œπΆπ›½(πœ‰)πœ‰βˆ’πœ‚π‘‘π‘š(πœ‰),(4.1) where 𝛽(πœ‰)=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|𝑀|2𝑛⋅||πœ‰||2π‘›πœ‰πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ‰π‘€|𝜌||𝑧+πœ‰π‘€||π›Ύβ‰‘βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|𝑀|2𝑛⋅||πœ‰||2π‘›πœ‰πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έπœ‘ξ€·||𝑧+πœ‰π‘€||2ξ€Έ,πœ‰βˆˆπΆβ§΅{βˆ’Μƒπ‘Ž}.(4.2) Moreover, assume that 0<𝑅<+∞ and 0<π‘š<𝑀<+∞. Then there exist positive constants 𝑐,π‘ž (depending, in general, on 𝑛,𝜌,𝜎,𝛾,𝑅,π‘š,𝑀) such that uniformly in 𝑧 and 𝑀 with |𝑧|≀𝑅, π‘šβ‰€|𝑀|≀𝑀.(a) If 𝛾β‰₯0, then 𝛽 is a continuous function in 𝐢 and||𝛽(πœ‰)||β‰€π‘β‹…π‘’βˆ’π‘ž|πœ‰|𝜌,πœ‰βˆˆπΆ.(4.3)(b) If 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2, then 𝛽 is a 𝐢1-function in 𝐢 and, in addition to (4.3),||||πœ•π›½(πœ‰)πœ•πœ‰||||||||πœ•π›½(πœ‰)πœ•πœ‰||||β‰€π‘β‹…π‘’βˆ’π‘ž|πœ‰|𝜌,πœ‰βˆˆπΆ.(4.3β€²)

Proof. Indeed, according to (2.16) Ξ¦(𝑧;𝑧+πœ‚β‹…π‘€)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)|𝑀|2𝑛||πœ‚||2π‘›Γ—ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœ‚π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ†πœ‚π‘€|𝜌||𝑧+πœ†πœ‚π‘€||π›Ύπ‘‘π‘š(πœ†)πœ†πœ‚β†’πœ‰===πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)|𝑀|2π‘›β‹…ξ€œπΆ||πœ‰||2π‘›πœ‰(πœ‰βˆ’πœ‚)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ‰π‘€|𝜌||𝑧+πœ‰π‘€||π›Ύπ‘‘π‘š(πœ‰)β‰‘βˆ’1πœ‹β‹…ξ€œπΆπ›½(πœ‰)πœ‰βˆ’πœ‚π‘‘π‘š(πœ‰).(4.4) As to (4.3)-(4.3’), these inequalities immediately follow from Lemma 3.2 and the following relations: πœ•πœ•πœ‰πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έ=πΈξ…žξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έβ‹…0≑0,πœ•πœ•πœ‰πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έ=πΈξ…žξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έβ‹…πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©,πœ•πœ•πœ‰πœ‘ξ€·||𝑧+πœ‰π‘€||2ξ€Έ=πœ‘ξ…žξ€·||𝑧+πœ‰π‘€||2ξ€Έβ‹…ξ‚€βŸ¨π‘€,π‘§βŸ©+πœ‰|𝑀|2,πœ•πœ•πœ‰πœ‘ξ€·||𝑧+πœ‰π‘€||2ξ€Έ=πœ‘ξ…žξ€·||𝑧+πœ‰π‘€||2ξ€Έβ‹…ξ€·βŸ¨π‘§,π‘€βŸ©+πœ‰|𝑀|2ξ€Έ.(4.5)

Proposition 4.2. If 𝛾β‰₯0, then the kernel Ξ¦(𝑧;𝑧+𝑀) is continuous in π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{0}.

Proof. Let us write Ξ¦(𝑧;𝑧+𝑀) as follows: Ξ¦(𝑧;𝑧+𝑀)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)β‹…|𝑀|2π‘›β‹…ξ€œπΆπ»(𝑧;𝑀;πœ†)π‘‘π‘š(πœ†),(4.6) where 𝐻(𝑧;𝑀;πœ†)=||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ†π‘€|𝜌||𝑧+πœ†π‘€||𝛾≑||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†π‘€||2ξ€Έ,π‘§βˆˆπΆπ‘›,π‘€βˆˆπΆπ‘›β§΅{0},πœ†βˆˆπΆβ§΅{1}.(4.7)
For arbitrary fixed positive numbers 𝑅,π‘š,𝑀(π‘š<𝑀), it suffices to construct a function β„Ž(πœ†)∈𝐿1(𝐢⧡{1}), such that ||𝐻(𝑧;𝑀;πœ†)||β‰€β„Ž(πœ†),πœ†βˆˆπΆβ§΅{1},(4.8) uniformly in 𝑧 and 𝑀 with |𝑧|≀𝑅, π‘šβ‰€|𝑀|≀𝑀. According to Lemma 3.1 (the case 𝛾β‰₯0), the function 𝑐⋅||πœ†||2π‘›βˆ’1||πœ†βˆ’1||π‘’βˆ’π‘|πœ†|𝜌(4.9) is suitable for a function β„Ž we seek.

Proposition 4.3. If 𝛾β‰₯0 and π‘§βˆˆπΆπ‘› is arbitrary, then limπœ€β†“0ξ€œ|π‘€βˆ’π‘§|=πœ€π‘“(𝑀)β‹…Ξ¦(𝑧;𝑀)⋅𝐾MB(𝑧;𝑀)≑limπœ€β†“0ξ€œ|𝑀|=πœ€π‘“(𝑧+𝑀)β‹…Ξ¦(𝑧;𝑧+𝑀)⋅𝐾MB(𝑧;𝑧+𝑀)=𝑓(𝑧)(4.10) for arbitrary function 𝑓 continuous in a neighborhood of 𝑧.

Proof. For sufficiently small πœ€>0, put πΌπœ€(𝑧)=ξ€œ|𝑀|=πœ€π‘“(𝑧+𝑀)β‹…Ξ¦(𝑧;𝑧+𝑀)⋅𝐾MB(𝑧;𝑧+𝑀).(4.11) Taking into account the explicit formula (2.16) for Ξ¦(𝑧;𝑧+𝑀) and the well-known explicit form of the Martinelli-Bochner kernel 𝐾MB(𝑧;𝑧+𝑀), we obtain πΌπœ€(𝑧)=πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)β‹…(βˆ’1)𝑛(π‘›βˆ’1)/2Ξ“(𝑛)(2πœ‹π‘–)π‘›β‹…ξ€œ|𝑀|=πœ€π‘“(𝑧+𝑀)Γ—|𝑀|2π‘›β‹…βŽ§βŽ¨βŽ©ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†π‘€βŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†π‘€||2ξ€Έπ‘‘π‘š(πœ†)βŽ«βŽ¬βŽ­Γ—1|𝑀|2𝑛⋅𝑛𝑗=1(βˆ’1)π‘—βˆ’1𝑀𝑗𝑑𝑀[𝑗]βˆ§π‘‘π‘€π‘€=πœ€β‹…πœξ€·πœβˆˆπ‘†π‘›ξ€Έ======πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)β‹…(βˆ’1)𝑛(π‘›βˆ’1)/2Ξ“(𝑛)(2πœ‹π‘–)π‘›β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)Γ—βŽ§βŽ¨βŽ©ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœ€β‹…πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†πœ€β‹…πœ||2ξ€Έπ‘‘π‘š(πœ†)βŽ«βŽ¬βŽ­Γ—πœ€2𝑛⋅𝑛𝑗=1(βˆ’1)π‘—βˆ’1πœπ‘—π‘‘πœ[𝑗]βˆ§π‘‘πœβ‰‘πœŒπœŽπœ‡2πœ‹Ξ“(𝑛)β‹…πœ€2π‘›β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)Γ—βŽ§βŽ¨βŽ©ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœ€β‹…πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†πœ€β‹…πœ||2ξ€Έπ‘‘π‘š(πœ†)βŽ«βŽ¬βŽ­β‹…πΎMB(0;𝜁).(4.12) In view of (2.6), we obtain πΌπœ€(𝑧)=πœŒπœŽπœ‡πœ€2𝑛4πœ‹π‘›+1β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)Γ—βŽ§βŽ¨βŽ©ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’1)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœ€β‹…πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†πœ€β‹…πœ||2ξ€Έπ‘‘π‘š(πœ†)βŽ«βŽ¬βŽ­π‘‘πœŽ(𝜁).(4.13) After the change of variable πœ†β†’πœ†/πœ€ in the inner integral in (4.13), we have πΌπœ€(𝑧)=πœŒπœŽπœ‡4πœ‹π‘›+1β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)Γ—βŽ§βŽ¨βŽ©ξ€œπΆ||πœ†||2π‘›πœ†(πœ†βˆ’πœ€)πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†β‹…πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†β‹…πœ||2ξ€Έπ‘‘π‘š(πœ†)βŽ«βŽ¬βŽ­π‘‘πœŽ(𝜁)πœ†β†’πœ†+πœ€====πœŒπœŽπœ‡4πœ‹π‘›+1β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)Γ—βŽ§βŽͺ⎨βŽͺβŽ©ξ€œπΆ(πœ†+πœ€)π‘›ξ‚€πœ†+πœ€ξ‚π‘›βˆ’1πœ†πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+(πœ†+πœ€)πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+(πœ†+πœ€)𝜁||2ξ€Έπ‘‘π‘š(πœ†)⎫βŽͺ⎬βŽͺβŽ­π‘‘πœŽ(𝜁)β‰‘πœŒπœŽπœ‡4πœ‹π‘›+1β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)β‹…πΌπœ€(𝑧;𝜁)π‘‘πœŽ(𝜁),(4.14) where πΌπœ€(𝑧;𝜁)=ξ€œπΆπ’«πœ€(𝑧;𝜁;πœ†)π‘‘π‘š(πœ†),πœβˆˆπ‘†π‘›,(4.15) where π’«πœ€(𝑧;𝜁;πœ†)=(πœ†+πœ€)π‘›ξ‚€πœ†+πœ€ξ‚π‘›βˆ’1πœ†πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+(πœ†+πœ€)πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+(πœ†+πœ€)𝜁||2ξ€Έ,πœβˆˆπ‘†π‘›,πœ†βˆˆπΆβ§΅{0}.(4.16) Obviously, without loss of generality, we can suppose that 0<πœ€β‰€1.
According to Lemma 3.1 (the case 𝛾β‰₯0) or, equivalently, to Lemma 3.2(a), there exist positive constants 𝑐,𝑏 (depending on 𝑛,𝜌,𝜎,𝛾) such that ||π’«πœ€(𝑧;𝜁;πœ†)||≀||πœ†||+1ξ€Έ2π‘›βˆ’1||πœ†||β‹…π‘β‹…π‘’βˆ’π‘|πœ†+πœ€|πœŒβ‰€β„Ž(πœ†)β‰‘βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘β‹…ξ€·||πœ†||+1ξ€Έ2π‘›βˆ’1||πœ†||,0<||πœ†||≀1𝑐⋅||πœ†||+1ξ€Έ2π‘›βˆ’1||πœ†||β‹…π‘’βˆ’π‘(|πœ†|βˆ’1)𝜌,||πœ†||>1∈𝐿1(𝐢⧡{0})(4.17) uniformly in πœβˆˆπ‘†π‘›, 0<πœ€β‰€1. Hence, due to the Lebesgue dominated convergence theorem, we can conclude that(i)the functions πΌπœ€(𝑧;𝜁) are continuous in πœβˆˆπ‘†π‘›;(ii)|πΌπœ€(𝑧;𝜁)|≀𝑀<+∞, πœβˆˆπ‘†π‘› (uniformly in πœ€);(iii)For for all πœβˆˆπ‘†π‘›βˆΆlimπœ€β†“0πΌπœ€(𝑧;𝜁)=𝐼(𝑧;𝜁), where𝐼(𝑧;𝜁)=ξ€œπΆ||πœ†||2π‘›βˆ’2β‹…πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†πœ||2ξ€Έπ‘‘π‘š(πœ†).(4.18) Note that the function 𝐼(𝑧;𝜁) is also continuous in πœβˆˆπ‘†π‘› and |𝐼(𝑧;𝜁)|≀𝑀<+∞, πœβˆˆπ‘†π‘›.
Now remember (see (4.14)) that πΌπœ€(𝑧)=πœŒπœŽπœ‡4πœ‹π‘›+1β‹…ξ€œπ‘†π‘›π‘“(𝑧+πœ€β‹…πœ)β‹…πΌπœ€(𝑧;𝜁)π‘‘πœŽ(𝜁).(4.19) Therefore, the application of the Lebesgue dominated convergence theorem once again gives limπœ€β†“0πΌπœ€(𝑧)=𝑓(𝑧)β‹…πœŒπœŽπœ‡4πœ‹π‘›+1ξ€œπ‘†π‘›πΌ(𝑧;𝜁)π‘‘πœŽ(𝜁)=𝑓(𝑧)β‹…πœŒπœŽπœ‡4πœ‹π‘›+1ξ€œπ‘†π‘›π‘‘πœŽ(𝜁)ξ€œπΆ||πœ†||2π‘›βˆ’2β‹…πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ†πœβŸ©ξ€Έβ‹…πœ‘ξ€·||𝑧+πœ†πœ||2ξ€Έπ‘‘π‘š(πœ†)=𝑓(𝑧)β‹…πœŒπœŽπœ‡4πœ‹π‘›+1ξ€œπ‘†π‘›π‘‘πœŽ(𝜁)ξ€œ+∞0ξ€œ2πœ‹0π‘Ÿ2π‘›βˆ’1β‹…πΈξ€·πœŽ2/πœŒξ«π‘§,𝑧+π‘Ÿπ‘’π‘–πœ—β‹…πœξ¬ξ€Έβ‹…πœ‘ξ‚€||𝑧+π‘Ÿπ‘’π‘–πœ—β‹…πœ||2ξ‚π‘‘π‘Ÿπ‘‘πœ—π‘Ÿβ‹…πœ=𝑀====𝑓(𝑧)β‹…πœŒπœŽπœ‡4πœ‹π‘›+1ξ€œ2πœ‹0π‘‘πœ—ξ€œπΆπ‘›πΈξ€·πœŽ2/πœŒξ«π‘§,𝑧+π‘’π‘–πœ—β‹…π‘€ξ¬ξ€Έβ‹…πœ‘ξ‚€||𝑧+π‘’π‘–πœ—β‹…π‘€||2ξ‚π‘‘π‘š(𝑀)𝑧+π‘’π‘–πœ—β‹…π‘€β†’π‘€======𝑓(𝑧)β‹…πœŒπœŽπœ‡4πœ‹π‘›+1ξ€œ2πœ‹0π‘‘πœ—ξ€œπΆπ‘›πΈξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©ξ€Έβ‹…πœ‘ξ€·|𝑀|2ξ€Έπ‘‘π‘š(𝑀)=𝑓(𝑧)β‹…πœŒπœŽπœ‡2πœ‹π‘›ξ€œπΆπ‘›πΈξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©ξ€Έβ‹…π‘’βˆ’πœŽ|𝑀|𝜌|𝑀|π›Ύπ‘‘π‘š(𝑀)=𝑓(𝑧)β‹…1=𝑓(𝑧)(4.20) in view of (1.2). Thus (4.10) is established.

Remark 4.4. Assume for a moment that Ξ¦(𝑧;𝑧+𝑀) can be defined at 𝑀=0 such that Ξ¦(𝑧;𝑧)=1 and, moreover, that after this Ξ¦(𝑧;𝑧+𝑀) becomes continuous at 𝑀=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 β€œvery desired”) property Ξ¦(𝑧;𝑧)=1.

Proposition 4.5. If 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2, the kernel Ξ¦(𝑧;𝑧+𝑀) is a function of class 𝐢1 in π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{0}.

Proof. In view of (4.6)-(4.7), we have to show that ∫𝐢𝐻(𝑧;𝑀;πœ†)π‘‘π‘š(πœ†) is of class 𝐢1 in π‘§βˆˆπΆπ‘›, π‘€βˆˆπΆπ‘›β§΅{0}. In other words, for arbitrary fixed positive numbers 𝑅,π‘š,𝑀(π‘š<𝑀), it suffices to construct a function β„Ž(πœ†)∈𝐿1(𝐢⧡{1}), such that ||||πœ•πœ•π‘€π‘˜π»(𝑧;𝑀;πœ†)||||,||||πœ•πœ•π‘€π‘˜π»(𝑧;𝑀;πœ†)||||β‰€β„Ž(πœ†),πœ†βˆˆπΆβ§΅{1},||||πœ•πœ•π‘§π‘˜π»(𝑧;𝑀;πœ†)||||,||||πœ•πœ•π‘§π‘˜π»(𝑧;𝑀;πœ†)||||β‰€β„Ž(πœ†),πœ†βˆˆπΆβ§΅{1},(4.21) uniformly in 𝑧 and 𝑀 with |𝑧|≀𝑅, π‘šβ‰€|𝑀|≀𝑀, and π‘˜=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 ξ€œπΆ||πœ†||2𝑛+𝜏||πœ†βˆ’1||π‘’βˆ’π‘|πœ†|πœŒπ‘‘π‘š(πœ†)(𝑏>0,𝜏=βˆ’1;0;1),(4.22) which is evident.

Proposition 4.6. If 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2, then for arbitrary fixed π‘§βˆˆπΆπ‘› we have ξƒ‘πœ•Ξ¦πœ•π‘€(𝑧;𝑧+𝑀),𝑀=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|𝑀|2π‘›πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+𝑀|𝜌|𝑧+𝑀|𝛾,βˆ€π‘€βˆˆC𝑛⧡{0},(4.23) or, equivalently, ξƒ‘πœ•Ξ¦πœ•π‘€(𝑧;𝑀),π‘€βˆ’π‘§ξƒ’=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|π‘€βˆ’π‘§|2π‘›πΈξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑀|𝜌|𝑀|𝛾,βˆ€π‘€βˆˆπΆπ‘›β§΅{𝑧}.(4.23ξ…ž)

Proof. We intend to use Lemma 3.4. To this end, let us put for π‘€βˆˆπΆπ‘›β§΅{0}πœ“(𝑀)=Ξ¦(𝑧;𝑧+𝑀),𝑔(𝑀)=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|𝑀|2π‘›πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+𝑀|𝜌|𝑧+𝑀|𝛾.(4.24) It suffices to establish (3.21) for the introduced functions πœ“ and 𝑔. Fix π‘€βˆˆπΆπ‘›β§΅{0}, then in view of Proposition 4.1 we have πœ“(πœ‚β‹…π‘€)=βˆ’1πœ‹β‹…ξ€œπΆπ›½(πœ‰)πœ‰βˆ’πœ‚π‘‘π‘š(πœ‰),βˆ€πœ‚βˆˆπΆβ§΅{0},(4.25) where 𝛽(πœ‰)∈𝐢1(𝐢) is defined by (4.2) and satisfies (4.3)-(4.3’). Consequently, 𝛽 satisfies all the conditions of Lemma 3.3. Hence πœ•πœ•πœ‚πœ“(πœ‚β‹…π‘€)≑𝛽(πœ‚)=βˆ’πœŒπœŽπœ‡2Ξ“(𝑛)|𝑀|2𝑛⋅||πœ‚||2π‘›πœ‚πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‚π‘€βŸ©ξ€Έπ‘’βˆ’πœŽ|𝑧+πœ‚π‘€|𝜌||𝑧+πœ‚π‘€||𝛾≑𝑔(πœ‚β‹…π‘€)πœ‚,βˆ€πœ‚βˆˆπΆβ§΅{0}.(4.26) By this (3.21) has been established for the pair of introduced functions πœ“ and 𝑔. The proof is complete.

The next assertion describes the behaviour of the kernel Ξ¦(𝑧;𝑧+𝑀) when 𝑀→0 or π‘€β†’βˆž. 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 𝛾β‰₯0, then(a)for arbitrary π‘§βˆˆπΆπ‘› and for arbitrary π‘€βˆˆπΆπ‘›β§΅{0}, we have limπœ‚β†’0Ξ¦(𝑧;𝑧+πœ‚β‹…π‘€)=πœŒπœŽπœ‡β‹…|𝑀|2𝑛2πœ‹β‹…Ξ“(𝑛)Γ—ξ€œπΆ||πœ‰||2π‘›βˆ’2β‹…πΈξ€·πœŽ2/πœŒβŸ¨π‘§,𝑧+πœ‰π‘€βŸ©ξ€Έβ‹…π‘’βˆ’πœŽ|𝑧+πœ‰π‘€|𝜌||𝑧+πœ‰π‘€||π›Ύπ‘‘π‘š(πœ‰);(4.27)(b) if 𝑧≠0 and [𝑧] is the complex plane generated by the vector 𝑧, then lim𝑀→0,π‘€βŸ‚[𝑧]Ξ¦(𝑧;𝑧+𝑀)=πœŒπœŽπœ‡2Ξ“(𝑛)β‹…πΈξ€·πœŽ2/𝜌|𝑧|2ξ€Έβ‹…ξ€œ+∞0π‘₯π‘›βˆ’1β‹…π‘’βˆ’πœŽ{|𝑧|2+π‘₯}𝜌/2ξ€½|𝑧|2+π‘₯𝛾/2𝑑π‘₯,(4.28) and for arbitrary 𝑅>0 and for (arbitrary small) πœ€>0, there exist a positive constant 𝑐=𝑐(𝑛,𝜌,𝜎,𝛾,πœ€,𝑅) such that ||Ξ¦(𝑧;𝑧+𝑀)||β‰€π‘β‹…π‘’βˆ’(πœŽβˆ’πœ€)|𝑀|𝜌,βˆ€π‘€βŸ‚[𝑧](𝑀≠0),(4.29) uniformly in 𝑧 with |𝑧|<𝑅;(c) if 𝑧≠0, then lim𝑀→0,π‘€βˆˆ[𝑧]Ξ¦(𝑧;𝑧+𝑀)=π‘›βˆ’1ξ“π‘š=0(βˆ’1)π‘›βˆ’1βˆ’π‘šβ‹…πΆπ‘šπ‘›βˆ’1β‹…ξ€·|𝑧|2𝜎2/πœŒξ€Έπ‘›βˆ’1βˆ’π‘šβ‹…β„’π›Ύ,πœŒπ‘›,π‘š,(4.30) where the coefficients ℒ𝛾,πœŒπ‘›,π‘š(π‘š=1,2,…,π‘›βˆ’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 ||Ξ¦(𝑧;𝑧+𝑀)||≀𝑐⋅|𝑀|+π‘’βˆ’π›Ώ1|𝑀|πœŒξ€Έ(4.31) uniformly in π‘€βˆˆπΆπ‘›β§΅{0} and 𝑧 with |𝑧|<𝑅. In particular, the kernel Ξ¦(𝑧;𝑧+𝑀) remains bounded (uniformly in 𝑧 with |𝑧|<𝑅) as 𝑀→0.

Remark 4.8. As it follows from (4.28) and (4.30), in general, lim𝑀→0Ξ¦(𝑧;𝑧+𝑀) (when 𝑀 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 Ξ¦(𝑧;𝑧+𝑀) on complex planes [𝑀] (generated by arbitrary π‘€βˆˆπΆπ‘›β§΅{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 πœ’(𝑑), π‘‘βˆˆ(βˆ’βˆž;+∞), satisfying the following conditions:(i)πœ’βˆˆπΆ1(𝑅); (ii)0β‰€πœ’(𝑑)≀1, π‘‘βˆˆ(βˆ’βˆž;+∞);(iii)πœ’(𝑑)≑1,β€‰π‘‘βˆˆ(βˆ’βˆž;0];(iv)πœ’(𝑑)≑0,β€‰π‘‘βˆˆ[1;+∞);(v)πœ’β†“[0;1]; (vi)|πœ’ξ…ž(𝑑)|≀𝑀<+∞, π‘‘βˆˆ[0;1] and (obviously) πœ’ξ…ž(𝑑)≑0 otherwise.

The existence of such functions is evident. Then putπœ’π‘…(𝑀)β‰‘πœ’ξ€·|𝑀|2βˆ’π‘…2ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩=1,0≀|𝑀|≀𝑅,∈[0;1],𝑅≀|𝑀|β‰€βˆšπ‘…2+1,=0,|𝑀|β‰₯βˆšπ‘…2+1.(5.1) Note that πœ’π‘…βˆˆπΆ1(𝐢𝑛) andπœ•πœ’π‘…πœ•π‘€(𝑀)β‰‘ξ‚΅πœ•πœ’π‘…πœ•π‘€1(𝑀),πœ•πœ’π‘…πœ•π‘€2(𝑀),…,πœ•πœ’π‘…πœ•π‘€π‘›(𝑀)ξ‚Ά=⎧βŽͺ⎨βŽͺ⎩(0,0,…,0),0≀|𝑀|≀𝑅,πœ’ξ…žξ€·|𝑀|2βˆ’π‘…2⋅𝑀1,𝑀2,…,𝑀𝑛,𝑅≀|𝑀|β‰€βˆšπ‘…2+1,(0,0,…,0),|𝑀|β‰₯βˆšπ‘…2+1.(5.2)

Moreover,||||πœ•πœ’π‘…πœ•π‘€(𝑀)||||β‰‘ξ„Άξ„΅ξ„΅βŽ·π‘›ξ“π‘˜=1||||πœ•πœ’π‘…πœ•π‘€π‘˜(𝑀)||||2⎧βŽͺ⎨βŽͺ⎩=0,0≀|𝑀|≀𝑅,≀𝑀⋅|𝑀|,𝑅≀|𝑀|β‰€βˆšπ‘…2+1,=0,|𝑀|β‰₯βˆšπ‘…2+1.(5.3)

Theorem 5.1. Assume that 𝑛β‰₯1, 1<𝑝<+∞, 𝜎>0 and πœ‡=(𝛾+2𝑛)/𝜌, where either 𝛾β‰₯2, 𝜌>0 or 𝛾=0, 𝜌β‰₯2. If the kernel Ξ¦(𝑧;𝑀),for all π‘§βˆˆπΆπ‘›, for all π‘€βˆˆπΆπ‘›β§΅{𝑧}, is defined by the formula (2.15), then the integral representation of the form 𝑓(𝑧)=πœŒπœŽπœ‡2πœ‹π‘›ξ€œπΆπ‘›π‘“(𝑀)⋅𝐸(𝑛)𝜌/2ξ€·πœŽ2/πœŒβŸ¨π‘§,π‘€βŸ©;πœ‡ξ€Έβ‹…π‘’βˆ’πœŽ|𝑀|𝜌|𝑀|π›Ύπ‘‘π‘š(𝑀)βˆ’Ξ“(𝑛)πœ‹π‘›ξ€œπΆπ‘›ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2𝑛Φ(𝑧;𝑀)π‘‘π‘š(𝑀),π‘§βˆˆπΆπ‘›,(5.4) is valid for each function π‘“βˆˆπΆ1(𝐢𝑛) satisfying the following conditions:(a)π‘“βˆˆπΏπ‘πœŒ,𝜎,𝛾(𝐢𝑛);(b)for any fixedπ‘§βˆˆπΆπ‘›, ||𝑓(𝑀)||β‹…Ξ¦(𝑧;𝑀)|π‘€βˆ’π‘§|2π‘›βˆ’2∈𝐿1(𝐢𝑛;π‘‘π‘š(𝑀))⟺||𝑓(𝑀)||β‹…Ξ¦(𝑧;𝑀)|𝑀|2π‘›βˆ’2∈𝐿1(𝐢𝑛;π‘‘π‘š(𝑀));(5.5)(c) for any fixed π‘§βˆˆπΆπ‘›||ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀)||β‹…Ξ¦(𝑧;𝑀)|π‘€βˆ’π‘§|2π‘›βˆ’1∈𝐿1(𝐢𝑛;π‘‘π‘š(𝑀))⟺||ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀)||β‹…Ξ¦(𝑧;𝑀)|𝑀|2π‘›βˆ’1∈𝐿1(𝐢𝑛;π‘‘π‘š(𝑀)).(5.6)

Proof. Let us fix an arbitrary π‘§βˆˆπΆπ‘›, and for βˆ€π‘…>0 consider the following differential form:πœ“(𝑧;𝑀)=𝑓(𝑀)β‹…πœ’π‘…(π‘€βˆ’π‘§)β‹…Ξ¦(𝑧;𝑀)⋅𝐾MB(𝑧;𝑀),π‘€βˆˆπΆπ‘›β§΅{𝑧}.(5.7) Then choose an πœ€βˆˆ(0;𝑅) and apply the Stokes formula to this form and to the domain {π‘€βˆˆπΆπ‘›βˆΆ0<πœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1}: ξ€œ|π‘€βˆ’π‘§|=βˆšπ‘…2+1πœ“βˆ’ξ€œ|π‘€βˆ’π‘§|=πœ€πœ“=ξ€œπœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1π‘‘πœ“.(5.8) In view of (5.1) and (2.4), the last relation can be written as follows: βˆ’ξ€œ|π‘€βˆ’π‘§|=πœ€π‘“(𝑀)β‹…Ξ¦(𝑧;𝑀)⋅𝐾MB(𝑧;𝑀)=Ξ“(𝑛)πœ‹π‘›ξ€œπœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2π‘›β‹…πœ’π‘…(π‘€βˆ’π‘§)β‹…Ξ¦(𝑧;𝑀)π‘‘π‘š(𝑀)+Ξ“(𝑛)πœ‹π‘›ξ€œπœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1ξ«ξ€·πœ•πœ’π‘…/πœ•π‘€ξ€Έ(π‘€βˆ’π‘§),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2𝑛⋅𝑓(𝑀)β‹…Ξ¦(𝑧;𝑀)π‘‘π‘š(𝑀)+Ξ“(𝑛)πœ‹π‘›ξ€œπœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1ξ«ξ€·πœ•Ξ¦/πœ•π‘€ξ€Έ(𝑧;𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2𝑛⋅𝑓(𝑀)β‹…πœ’π‘…(π‘€βˆ’π‘§)π‘‘π‘š(𝑀).(5.9) Moreover, (5.2) and (4.23ξ…ž) imply βˆ’ξ€œ|π‘€βˆ’π‘§|=πœ€π‘“(𝑀)β‹…Ξ¦(𝑧;𝑀)⋅𝐾MB(𝑧;𝑀)=Ξ“(𝑛)πœ‹π‘›ξ€œπœ€<|π‘€βˆ’π‘§|<βˆšπ‘…2+1ξ«ξ€·πœ•π‘“/πœ•π‘€ξ€Έ(𝑀),π‘€βˆ’π‘§ξ¬|π‘€βˆ’π‘§|2π‘›β‹…πœ’π‘…(π‘€βˆ’π‘§)β‹…Ξ¦(𝑧;𝑀)π‘‘π‘š(𝑀)+Ξ“(𝑛)πœ‹π‘›ξ€œπ‘…<|π‘€βˆ’π‘§|<βˆšπ‘…2+1πœ’ξ…žξ€·|π‘€βˆ’π‘§|2βˆ’π‘…2ξ€Έ|π‘€βˆ’π‘§|2π‘›βˆ’2⋅𝑓(𝑀)β‹…Ξ¦(𝑧;𝑀)