2016, Volume 7, Number 2

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ISSN 2077–9879

Eurasian

Mathematical Journal

Founded in 2010 by

the L.N. Gumilyov Eurasian National University in cooperation with

the M.V. Lomonosov Moscow State University the Peoples’ Friendship University of Russia

the University of Padua

Supported by the ISAAC

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by the Kazakhstan Mathematical Society

Published by

the L.N. Gumilyov Eurasian National University

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EURASIAN MATHEMATICAL JOURNAL

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Editors–in–Chief

V.I. Burenkov, M. Otelbaev, V.A. Sadovnichy

Editors

Sh.A. Alimov (Uzbekistan), H. Begehr (Germany), T. Bekjan (China), O.V. Besov (Rus- sia), N.A. Bokayev (Kazakhstan), A.A. Borubaev (Kyrgyzstan), G. Bourdaud (France), A. Caetano (Portugal), M. Carro (Spain), A.D.R. Choudary (Pakistan), V.N. Chubarikov (Russia), A.S. Dzumadildaev (Kazakhstan), V.M. Filippov (Russia), H. Ghazaryan (Arme- nia), M.L. Goldman (Russia), V. Goldshtein (Israel), V. Guliyev (Azerbaijan), D.D. Haroske (Germany), A. Hasanoglu (Turkey), M. Huxley (Great Britain), M. Imanaliev (Kyrgyzstan), P. Jain (India), T.Sh. Kalmenov (Kazakhstan), B.E. Kangyzhin (Kazakhstan), K.K. Ken- zhibaev (Kazakhstan), S.N. Kharin (Kazakhstan), E. Kissin (Great Britain), V. Koki- lashvili (Georgia), V.I. Korzyuk (Belarus), A. Kufner (Czech Republic), L.K. Kussainova (Kazakhstan), P.D. Lamberti (Italy), M. Lanza de Cristoforis (Italy), V.G. Maz’ya (Swe- den), E.D. Nursultanov (Kazakhstan), R. Oinarov (Kazakhstan), K.N. Ospanov (Kaza- khstan), I.N. Parasidis (Greece), J. Peˇcari´c (Croatia), S.A. Plaksa (Ukraine), L.-E. Pers- son (Sweden), E.L. Presman (Russia), M.A. Ragusa (Italy), M.D. Ramazanov (Russia), M. Reissig (Germany), M. Ruzhansky (Great Britain), S. Sagitov (Sweden), T.O. Sha- poshnikova (Sweden), A.A. Shkalikov (Russia), V.A. Skvortsov (Poland), G. Sinnamon (Canada), E.S. Smailov (Kazakhstan), V.D. Stepanov (Russia), Ya.T. Sultanaev (Russia), I.A. Taimanov (Russia), T.V. Tararykova (Great Britain), J.A. Tussupov (Kazakhstan), U.U. Umirbaev (Kazakhstan), Z.D. Usmanov (Tajikistan), N. Vasilevski (Mexico), Dachun Yang (China), B.T. Zhumagulov (Kazakhstan)

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c

The Eurasian National University

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TYNYSBEK SHARIPOVICH KAL’MENOV

(to the 70th birthday)

On May 5, 2016 was the 70th birthday of Tynysbek Sharipovich Kal’menov, member of the Editorial Board of the Eurasian Math- ematical Journal, general director of the Institute of Mathematics and Mathematical Modeling of the Ministry of Education and Sci- ence of the Republic of Kazakhstan, laureate of the Lenin Komsomol Prize of the Kazakh SSR (1978), doctor of physical and mathematical sciences (1983), professor (1986), honoured worker of science and technology of the Republic of Kazakhstan (1996), academician of the National Academy of Sciences (2003), laureate of the State Prize in the field of science and technology (2013).

T.Sh. Kal’menov was born in the South-Kazakhstan region of the Kazakh SSR. He graduated from the Novosibirsk State University (1969) and completed his postgraduate studies there in 1972.

He obtained seminal scientific results in the theory of partial differential equations and in the spectral theory of differential operators.

For the Lavrentiev-Bitsadze equation T.Sh. Kal’menov proved the criterion of strong solvability of the Tricomi problem in the L_p-spaces. He described all well-posed boundary value problems for the wave equation and equations of mixed type within the framework of the general theory of boundary value problems.

He solved the problem of existence of an eigenvalue of the Tricomi problem for the Lavrentiev-Bitsadze equation and the general Gellerstedt equation on the basis of the new extremum principle formulated by him.

T.Sh. Kal’menov proved the completeness of root vectors of main types of Bitsadze- Samarskii problems for a general elliptic operator. Green’s function of the Dirichlet problem for the polyharmonic equation was constructed. He established that the spectrum of general differential operators, generated by regular boundary conditions, is either an empty or an infinite set. The boundary conditions characterizing the volume Newton potential were found. A new criterion of well-posedness of the mixed Cauchy problem for the Poisson equation was found.

On the whole, the results obtained by T.Sh. Kal’menov have laid the groundwork for new perspective scientific directions in the theory of boundary value problems for hyperbolic equations, equations of the mixed type, as well as in the spectral theory.

More than 50 candidate of sciences and 9 doctor of sciences dissertations have been defended under his supervision. He has published more than 120 scientific papers. The list of his basic publications can be viewed on the web-page

https://scholar.google.com/citations?user=Zay4f xkAAAAJ&hl=ru&authuser = 1 The Editorial Board of the Eurasian Mathematical Journal congratulates Tynysbek Sharipovich Kal’menov on the occasion of his 70th birthday and wishes him good health and new creative achievements!

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EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879

Volume 7, Number 2 (2016), 19 – 37

INTEGRAL REPRESENTATION OF FUNCTIONS AND EMBEDDING THEOREMS FOR MULTIANISOTROPIC SPACES

IN THE THREE-DIMENSIONAL CASE G.A. Karapetyan

Communicated by V.I. Burenkov

Key words: integral representation, embedding theorem, multianisotropic spaces, completely regular polyhedron.

AMS Mathematics Subject Classification: 46E35.

Abstract. In this paper we obtain a special integral representation of functions with a set of multi-indices and use it to prove embedding theorems for multianisotropic spaces in three-dimensional case.

1 Introduction

The embedding theorems for multianisotropic spaces in two-dimensional case can be found in [4]. In this paper we prove embedding theorems for the Sobolev multianisotropic spaces in three-dimensional space when the completely regular polyhedron has one anisotropicity vertex. The obtained results generalize embedding theorems for isotropic and anisotropic spaces in [1-3, 5-9] (for an overview of the history of the problem and related results see [2]), and in the case of anisotropic spaces coincide with known theorems, although a method of integral representation of functions is applied, which is based on the usage of special multianisotropic kernels.

2 Multianisotropic kernels and their properties

LetR³ be the three-dimensional space, Z³+ the set of all three-dimensional multi-indices, i.e.

α= (α₁, α₂, α₃)∈ Z³+, if α₁, α₂, α₃ are non-negative integers. Forξ, η ∈R³, α ∈Z³+, t >0 let |α| = α₁+α₂ +α₃, ξ^α = ξ₁^α¹ξ₂^α²ξ₃^α³, t^η = (t^η¹, t^η², t^η³), D_k = ¹_i_∂x^∂

k, k = 1,2,3, and let D^α =D^α₁¹D₂^α²D₃^α³ be the weak derivative of order α.

For a set of multi-indices let N be the smallest convex polyhedron containing it. N is said to be completely regular if

a) it has vertices at the origin and further vertices on each coordinate axis

b) all components of the outer normals of all two-dimensional non-coordinate faces are positive. For a completely regular polyhedron N in R³ let N²_i (i = 1, ..., M) be the two- dimensional non-coordinate faces. Let µⁱ(i= 1, . . ., M) be the outer normal of the face N²_i,

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20 G.A. Karapetyan

such that the equation of that face is(α;µⁱ) = 1; (i= 1, ..., M). We study the case when the vertices ofNare multi-index: α¹ = (l₁,0,0), α² = (0, l₂,0), α³ = (0,0, l₃), α⁴ = (α₁, α₂, α₃).

Denote by µⁱ (i= 1,2,3) the outer normal of N²_i, which is the face N²₁ passing through all the vertices with the exception of αⁱ (i=1,2,3), i.e. µ¹ is the outer normal of the face passing through the vertices{α²;α³;α⁴}, and so on.

Forθ > 0and positive integer k denote

P (θ, ξ) = (θξ₁^l¹)^2k+ (θξ₂^l²)^2k+ (θξ₃^l³)^2k+ (θξ₁^α¹ξ₂^α²ξ₃^α³)^2k, (2.1)

G₀(ξ;θ) = e^−P^(θ,ξ), (2.2)

G_1,j(ξ, θ) = 2k(θξ^α^j)^2k−1e⁻^P^(θ,ξ) , j = 1,2,3,4. (2.3) It is obvious that for any value of θ >0 G₀, G_1,j∈S,(j = 1,2,3,4), where S =S(Rⁿ)is the Schwartz space of rapidly decreasing functions.

Let Gˆ₀(t, θ), Gˆ_1,j(t, θ) (j = 1,2,3,4) be the corresponding Fourier transform of G₀, G_1,j (j = 1,2,3,4).

Note that Gˆ₀ ,Gˆ_1,j ∈ S,(j = 1,2,3,4), because the Fourier transform is an automor- phism on S.

Suppose that the inequality α₁ < α₂ < α₃ < l₃ holds forα=α⁴ = (α₁, α₂, α₃). Consider the intersection of planesN²₁T

N²₂, i.e. the line passing through the pointsα³ = (0; 0;l₃)and α⁴ = (α₁, α₂, α₃). The parametric equation of that line isx =α₁t; y=α₂t; z =l₃−t(l₃−α₃).

Let β = (β₁, β₂,0) be the point of intersection of that line with the XOY plane. Note that β₁ = _l^α¹^l³

3−α3, β₂ = _l^α²^l³

3−α3, β ∈ N²₁T

N²₂ (i.e. (β;µ¹) = 1; (β;µ²) = 1), also β₁ < β₂ which follows from α₁ < α₂.

Consider the pointγ = (γ₁,0,0), such thatγ ∈N²₁. Let N be an even number, such that all components of multi-indices N α, N β, N γ are even. (A) Such N exists, because all components ofα, β, γ are rational. Let us prove the following lemma (an analogue of Lemma 2.1 in [4]).

Lemma 2.1. Let α⁴ = (α₁, α₂, α₃)∈Z³+, α₁ < α₂ < α₃ and θ ∈(0; 1). Then for any multi- index m = (m1, m2, m3) and for any even N satisfying (A) there exist constants Ci, (i = 0,1,2), such that

|D^mGˆ_1,j(t, θ)| ≤ θ⁻^i=1,2,3^max⁽|^µⁱ|^+(m,µⁱ⁾⁾

C0(lnθ)²+C1|lnθ|+C2

1 +θ^−N(t^{N α}₁ ¹t^{N α}₂ ²t^{N α}₃ ³ +t^{N β}₁ ¹t^{N β}₂ ² +t^{N γ}₁ ¹) . (2.4) An analogous inequality is true for Gˆ0(t, θ).

Proof. As in the proof of Lemma 2.1 in [4], for any multi-indexm= (m₁, m₂, m₃)we estimate the following integral:

I = Z ∞

0

Z ∞ 0

ξ^m₁ ¹ξ₂^m²ξ₃^m³e^−P^(θ,ξ)dξ₁dξ₂dξ₃. Consider _m^αⁱ

i+1 (i= 1,2,3). Leti₀ be an index, such that max

i=1,2,3 αi

mi+1 = _m^αⁱ⁰

i0+1 . There are 3 possible cases:

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Integral representation of functions and embedding theorems for multianisotropic spaces 21 1. The maximum is obtained only at one index i₀ , i.e. _m^αⁱ

i+1<_m^αⁱ⁰

i0+1 i6=i₀, i= 1,2,3.

2. The maximum is obtained at two indices (e.g. i₀ = 2 Рё i₀ = 3), i.e. _m^α¹

1+1 <

α3

m3+1 ,_m^α²

2+1 = _m^α³

3+1

3. All the ratios are equal, i.e. _m^α¹

1+1 = _m^α²

2+1 = _m^α³

3+1.

Consider the first case. Let i0 = 3. Note that it suffices to estimate the integral only on the closed upper half-space. By substituting ξ=θ^−µ³η in I, we have

I =θ⁻(|^µ³|⁺(^m,µ³))Z ∞ 0

Z ∞ 0

η₁^m¹η^m₂ ²η₃^m³e^−η^2kl¹ ¹e^−η^2kl² ²e^−η^2kα¹ ¹^η²^2kα²^η^2kα³ ³dη₁dη₂dη₃

=θ⁻(|^µ³|⁺(^m,µ³))Z +∞

0

e⁻

η1

α1 α3η2

α2 α3η3

2kα3

(η₁

α1 α3η₂

α2

α3η₃)^m³d η₁

α1 α3η₂

α2 α3η₃

· Z ∞

0

η^m¹⁻

α1 α3m3−^α_α¹

3

1 e^−η^2kl¹ ¹dη₁ Z ∞

0

η^m²⁻

α2 α3m3−^α_α²

3

2 e^−η^2kl² ²dη₂

=Cθ⁻(|^µ³|⁺(^m,µ³)) =Cθ⁻^i=1,2,3^max (|^µⁱ|⁺(^m,µⁱ))

. (2.5) The last relation follows from the convergence of the three integrals and from inequalities m₁−^α_α¹

3m₃−^α_α¹

3 >−1, m₂−^α_α²

3m₃−^α_α²

3 >−1, which are in turn inferred from ^α_α¹

3 < ^m_m¹⁺¹

3+1 ,^α_α²

3 <

m2+1 m3+1.

Consider the second case. We have ^α_α¹

3 < ^m_m¹⁺¹

3+1 ,^α_α²

3 = ^m_m²⁺¹

3+1 and thus ^α_α¹

2 < ^m_m¹⁺¹

2+1

The polyhedron N is completely regular, thus ^α_l¹

1 + ^α_l²

2 + ^α_l³

3 > 1 and µ³₃ < µ²₃. We can split I into 8 integrals:

I = Z θ^−µ³¹

0

dξ1

Z θ^−µ³² 0

dξ2

Z θ^−µ³³ 0

. . . dξ3+ Z θ^−µ³¹

0

dξ1

Z θ^−µ³² 0

dξ2

Z ∞ θ^−µ³³

. . . dξ3

+ Z θ^−µ³¹

0

dξ₁ Z ∞

θ^−µ³²

dξ₂ Z θ^−µ³³

0

. . . dξ₃+ Z ∞

θ^−µ³¹

dξ₁ Z θ^−µ³²

0

. . . dξ₃

+ Z θ^−µ³¹

0

dξ₁ Z ∞

θ^−µ³²

dξ₂ Z ∞

θ^−µ³³

. . . dξ₃+ Z ∞

θ^−µ³¹

dξ₁ Z ∞

θ^−µ³²

0

. . . dξ₃

+ Z ∞

θ^−µ³¹

dξ₁ Z θ^−µ³²

0

dξ₂ Z ∞

θ^−µ³³

. . . dξ₃ + Z ∞

θ^−µ³¹

dξ₁ Z ∞

θ^−µ³²

dξ₂ Z ∞

θ^−µ³³

. . . dξ₃ =I₁+· · ·+I₈. We will estimate each integral separately. The substitution ξ =θ^−µ³η in I₁ yields I₁ ≤ Cθ⁻(|^µ³|⁺(^m,µ³)).

By substituting ξ =θ^−µ²η inI₂, we get

I₂ ≤Cθ⁻(|^µ²|⁺(^m,µ²))Z ∞ 0

dη₁ Z ∞

0

dη₂ Z ∞

θ^−µ³³

η₁^m¹η^m₂ ²η₃^m³e^−η¹^2kl¹e^−η³^2kl³e^−η^2kα¹ ¹^η^2kα² ²^η³^2kα³ dη₃

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22 G.A. Karapetyan

=Cθ⁻(|^µ²|⁺(^m,µ²))Z ∞ 0

η^m¹⁻

α1 α2m2−^α¹

α2

1 e^−η^2kl¹ ¹dη₁ Z ∞

θ^−µ³³^+µ²³

e^−η^2kl³ ³ η3

dη₃

· Z ∞

0

η

α1 α2

1 η₂η

α3 α2

3

m2

e

− η

α1 α2 1 η2η

α3 α2 3

!2kα2

d(η

α1 α2

1 η₂η

α3 α2

3 )≤θ⁻(|^µ²|⁺(^m,µ²)) (C₁|lnθ|+C₂). The first integral converges, because ^α_α¹

3 < ^m_m¹⁺¹

3+1. By substituting t=η

α1 α2

1 η₂η

α3 α2

3 , it follows that the third integral is convergent. The second integral can be estimated via(C1|lnθ|+C2).

By substituting ξ =θ^−µ³η inI₃, we get

I₃ =θ⁻(|^µ³|⁺(^m,µ³))Z 1 0

dη₁ Z ∞

0

dη₂ Z 1

0

(η^m₁ ¹η₂^m²η^m₃ ³e^−η^2kl¹ ¹e^−η^2kl² ²e^−η¹^2kα¹^η^2kα² ²^η^2kα³ ³)dη₃

≤Cθ⁻(|^µ³|⁺(^m,µ³)).

We substitute ξ =θ^−µ³η inI₄ and estimate it similarly to I₃. By substituting ξ =θ^−µ³η inI₅, we get

I₅ ≤Cθ⁻(|^µ³|⁺(^m,µ³))Z 1 0

η^m¹⁻

α1 α3m2−^α_α¹

3

1 e^−η^2kl¹ ¹dη₁ Z ∞

1

e^−η²^2kl² η₂ dη₂

Z ∞ 0

t^m³e^−t^2kα³dt

≤Cθ⁻(|^µ³|⁺(^m,µ³)). By substituting ξ =θ^−µ²η inI₆, we get

I₆ ≤Cθ⁻(|^µ²|⁺(^m,µ²))Z ∞ 0

η^m¹⁻

α1 α2m2−^α¹

α2

1 e^−η^2kl¹ ¹dη₁ Z ∞

1

e^−η³^2kl³ η₃ dη₃

Z ∞ 0

t^m²e^−t^2kα²dt

≤θ⁻(|^µ²|⁺(^m,µ²)) (C₁|lnθ|+C2). By substituting ξ =θ^−µ³η inI₇, we get

I₇ ≤Cθ⁻(|^µ³|⁺(^m,µ³))Z 1 0

η^m¹⁻

α1 α3m2−^α¹

α3

1 e^−η^2kl¹ ¹dη₁ Z ∞

1

e^−η²^2kl² η₂ dη₂

Z ∞ 0

t^m³e^−t^2kα³dt

≤Cθ⁻(|^µ³|⁺(^m,µ³)). Finally, by substituting ξ =θ^−µ³η inI₈, we get

I8 ≤Cθ⁻(|^µ³|⁺(^m,µ³)). As a result, if ^α_α¹

3 < ^m_m¹⁺¹

3+1 ,^α_α²

3 = ^m_m²⁺¹

3+1 then for some constants C₁, C₂ the following inequality holds:

I ≤θ⁻^i=1,2,3^max (^|µⁱ^|+(m,µⁱ⁾)

(C₁|lnθ|+C₂). (2.6) Consider the third case, i.e. ^α_α¹

3 = ^m_m¹⁺¹

3+1 ,^α_α²

3 = ^m_m¹⁺¹

3+1, thus ^α_α²

3 = ^m_m²⁺¹

3+1.

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Integral representation of functions and embedding theorems for multianisotropic spaces 23 Let

µ⁰_i = min

j=1,2,3µ^j_i.

Then µ⁰₁ = µ¹₁, µ⁰₂ = µ²₂, µ⁰₃ = µ³₃. As in the first case, we split I into 8 integrals by µ⁰₁, µ⁰₂ ,µ⁰₃. We consider each integral separately. By substituting ξ=θ^−µ³η in I₁, we get

I₁ ≤Cθ⁻(|^µ³|⁺(^m,µ³)). By substituting ξ =θ^−µ²η inI₂, we get

I₂ ≤Cθ⁻(|^µ²|⁺(^m,µ²))Z ∞ 0

η₁^m¹e^−η¹^2kl¹dη₁ Z 1

0

η₁^m²e^−η¹^2kα¹^η²^2kα²^η³^2kα³dη₂ Z ∞

0

η₃^m³e^−η³^2kl³dη₃

≤Cθ⁻(|^µ²|⁺(^m,µ²)).

By substitutingξ=θ^−µ³η inI₃ and I₄ and estimating them similarly toI₂, an analogous estimate can be obtained.

By substituting ξ =θ^−µ¹η inI₅, we get I₅ ≤Cθ⁻(|^µ¹|⁺(^m,µ¹))Z 1

0

dη₁ Z ∞

0

dη₂ Z ∞

0

e^−η²^2kl²e^−η³^2kl³e^−η¹^2kα¹^η²^2kα²^η³^2kα³η₁^m¹η₂^m²η₃^m³dη₃

≤Cθ⁻(|^µ¹|⁺(^m,µ¹)).

Similarly, we make the substitutions ξ =θ^−µ²η and ξ=θ^−µ³η in I₆ and I₇, respectively.

Let us estimate I₈. By substituting ξ =θ^−µ³η, we get I8 ≤Cθ⁻(|^µ³|⁺(^m,µ³))Z ∞

θ^−µ⁰¹^+µ³¹

e^−η¹^2kl¹ η₁ dη1

Z ∞ θ^−µ⁰²^+µ³²

e^−η²^2kl² η₂ dη2

Z ∞ 1

t^m³e^2kα³dt

≤ C₀(lnθ)²+C₁|lnθ|+C₂

θ⁻(|^µ³|⁺(^m,µ³)).

Finally, we get that in the third case there exist constants C₀, C₁, C₂, such that I ≤(C0(lnθ)²+C1|lnθ|+C2)θ^i=1,2,3^max |^µⁱ|^+(m,µⁱ⁾

. (2.7)

Using inequalities (2.5), (2.6) and (2.7), we proceed with the proof of Lemma 2.1. Let us estimate Gˆ_1,j(t, θ), j = 1,2,3,4. For any multi-index m= (m₁, m₂, m₃)

D^mGˆ_1,j(t, θ) = 1 (2π)³²

Z

R³

ξ₁^m¹ξ₂^m²ξ₃^m³e^−i(t,ξ)2k(θξ^α^j)^2k−1e^−P^(θ,ξ)dξ₁dξ₂dξ₃. (2.8)

Consider the vertexα = (α₁, α₂, α₃)ofN, and the multi-index(m₁+α^j₁, m₂+α^j₂, m₃+α₃^j).

Suppose that:

α_i

α_r 6= m_i +α^j_i + 1

mr+α^jr+ 1, i, r= 1,2,3. (2.9) Leti0 be the index, for which the following relation holds:

i=1,2,3max

α_i

m_i+α^j_i + 1 = α_i₀ m_i₀ +α^j_i₀ + 1.

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24 G.A. Karapetyan

Then let us substitute ξ=θ^−µⁱ⁰η in integral (2.8).

According to (2.9), ^αⁱ

mi+α^j_i+1 < ^αⁱ⁰

mi0+α^j_i

0+1 i6=i₀. Considering 1−l₁µⁱ₁⁰ ≥0, 0< θ <1

|D^mGˆ_1,j(t, θ)| ≤Cθ^−(|µⁱ⁰^|+(m,µⁱ⁰⁾⁾. (2.10) In the second case, i.e. when there is one equality in (2.9), (2.6) is obtained for

|D^mGˆ_1,j(t, θ)|. In the third case, i.e. when there are only equalities in (2.9), (2.7) is obtained for |D^mGˆ_1,j(t, θ)|.

Let us estimatet^{N α}₁ ¹ t^{N α}₂ ² t^{N α}₃ ³D^mGˆ1,j(t, θ). Using the properties of the Fourier transform we have

θ^−Nt^{N α}₁ ¹ t^{N α}₂ ² t^{N α}₃ ³D^mGˆ1,j(t, θ)

= θ^−N (2π)³²

Z

R³

D^{N α}_ξ ¹

1 D_ξ^{N α}²

2 D_ξ^{N α}³

3 e^−i(t,ξ)ξ₁^m¹ξ^m₂ ²ξ₃^m³2k(θξ^α^j)^2k−1e^−P^(θ,ξ)dξ₁dξ₂dξ₃. Integration by parts makes it suffice to estimate the integral

θ^−N Z ∞

0

Z ∞ 0

D_ξ^{N α}¹

1 D^{N α}_ξ ²

2 D_ξ^{N α}³

3 (ξ₁^m¹ξ^m₂ ²ξ₃^m³(θξ^α^j)^2k−1e^−P^(θ,ξ))dξ₁dξ₂dξ₃.

As in [4], we apply the following formula for the derivative of Φ(ξ)e^−P^(θ,ξ) of order N α, whereΦ(ξ) = ξ^m₁ ¹ξ₂^m²ξ₃^m³

θξ^α^j^2k−1

θ^−N X

β+γ=N α

C_{|N α|}^|β|

Z ∞ 0

D^β_ξ(ξ₁^m¹ξ₂^m²ξ₃^m³(θξ^α^j)^2k−1·

· X

σ¹+···+σ^|γ|=γ

e^−P^(θ,ξ)

|γ|

Y

j=1

D^σ_ξ^jP(θ, ξ)dξ₁dξ₂dξ₃,

(2.11)

where the product is taken over all σ^j, satisfying |σ^j|>0.

Suppose that the substitution ξ = θ^−µⁱη is made in (2.11) for some i(i= 1,2,3), then, considering the equalities (αN, µⁱ) = N(i= 1,2,3), D_ξ^β =θ^(µⁱ^,β)D^β_η, we find that the "con- tribution" of θ to (2.11) is

θ(2k−1)(1−(α^j,µⁱ))

|γ|

Y

j=1

θ^(2k)(1−(^α^r^,µⁱ)⁾θ⁻⁽|^µⁱ|⁺(^m,µⁱ)⁾.

As (αN, µⁱ) = N,(α^r, µⁱ) ≤ 1 , r = 1,2,3,4 , i = 1,2,3, the exponent of θ is at least

−(|µⁱ|+ (m, µⁱ)).

Let ρ_i be an exponent of ξ_i in (2.11) (i = 1,2,3). Consider ρ = (ρ₁;ρ₂;ρ₃) and α = (α₁;α₂;α₃). For the ratios _ρ^αⁱ

i+1 (i = 1,2,3) cases 1-3 may arise. Then (2.5) is true for

|t^{N α}₁ ¹ t^{N α}₂ ² t^{N α}₃ ³D^mGˆ_1,j(t, θ)|in the first case, (2.6) in the second case, and (2.7) in the third case.

To estimate θ^−N|t^{N β}₁ ¹t^{N β}₂ ²t^{N β}₃ ³D^mGˆ1,j(t, θ)| a similar argument can be used. The exponent of θ will be at least − max

i=1,2,3(|µⁱ|+ (m, µⁱ)), because the multi-index (β₁, β₂,0) lies

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Integral representation of functions and embedding theorems for multianisotropic spaces 25 on the intersection of the plane passing through the points {α₂, α₃, α₄} and the plane passing through the points {α₁, α₃, α₄}. Thus, a substitution by µ¹ or µ² will yield (N β;µ¹) = (N β;µ²) =N, and a substitution by µ³ will yield (N β;µ³)>N.

The estimate for θ^−N

t^{N γ}₁ ¹ D^mGˆ_1,j(t, θ)

is done similarly, taking into account that γ = (γ₁,0,0) lies on the plane, passing through the points {α₂, α₃, α₄}, and (N γ;µ¹) = N, (N γ;µ²)> N, (N γ;µ³)> N.

In either case, one of (2.5), (2.6), (2.7) holds.

Remark 1. If (2.9) holds, then in (2.4) the constants C₀ and C₁ are 0. If there is at least one equality in (2.9), then C₀ = 0.

Lemma 2.2. Let 0< θ <1, α₁ < α₂ < α₃. Then there is a constant РЎ, such that Z

R³

dt1dt2dt3

1 +θ^−N

t^{N α}₁ ¹t^{N α}₂ ²t^{N α}₃ ³ +t^{N β}₁ ¹t^{N β}₂ ² +t^{N γ}₁ ¹ =Cθ|^µ¹|. (2.12) Proof. Recall, thatβ₁ = _l^α¹^l³

3−α3, β₂ = _l^α²^l³

3−α3, and thus ^β_β¹

2 = ^α_α¹

2, β₁ < β₂.

By substituting t =θ^µ¹η in (2.12) (note that it suffices to estimate the integral only on the closed upper half-space), we have

I = Z ∞

0

Z ∞ 0

dt₁dt₂dt₃ 1 +θ^−N

t^{N α}₁ ¹t^{N α}₂ ²t^{N α}₃ ³ +t^{N β}₁ ¹t^{N β}₂ ² +t^{N γ}₁ ¹

=θ^|µ¹^| Z ∞

0

Z ∞ 0

dη1dη2dη3

1 +

η

α1 α3

1 η

α2 α3

2 η₃ N α3

+

η

β1 β2

1 η₂ N β2

+η₁^{N γ}¹

!.

By applying the change of variables τ₁ =η₁, τ₂ =η

α1 α2

1 η₂, τ₃ =η

α1 α3

1 η

α2 α3

2 η₃, we get I =Cθ^|µ¹^|

Z ∞ 0

dτ₁dτ₂dτ₃ τ

α1 α2

1 τ

α2 α3

2

1 +τ₃^{N α}³ +τ₂^{N β}² +τ₁^{N γ}¹. (2.13) Since ^α_α¹

2 <1, ^α_α²

3 <1, the integral converges and I =Cθ^|µ¹^|.

Lemma 2.3. Let 0< θ <1, α1 < α2 =α3. Then for any multi-index

m= (m₁, m₂, m₃) and any N satisfying (A) there are constants C_i(i= 0,1,2), such that

D^mGˆ_1,j(t, θ)

≤θ⁻^i=1,2,3^max(^|µⁱ^|+(^m,µⁱ)) C₀(lnθ)²+C₁|lnθ|+C₂ 1 +θ^−N

t^{N α}₁ ¹t^{N α}₂ ²t^{N α}₃ ³ +t^{N β}₁ ¹t^{N β}₂ ² +t^{N γ}₁ ¹

· 1

1 +θ^−N

t^{N α}₁ ¹t^{N α}₂ ²t^{N α}₃ ³ +t^{N σ}₁ ¹t^{N σ}₃ ³ +t^{N γ}₁ ¹ ,

(2.14)

where γ = (γ₁, γ₂,0) is the point of intersection of the planes µ² and µ³ and XOY, σ = (σ1,0, σ3) is the point of intersection of the planes µ¹ and µ³ andXOZ, γ = (γ1,0,0) is the point of intersection of the µ³ plane with the x-axis.