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ISSN 2079-3316 PROGRAM SYSTEMS: THEORY AND APPLICATIONS no.4(35), 2017, pp. 149-162
V. Markasheva, A. Mashtakov
Existence of Global Fundamental Solution to a Class of Fokker—Planck Equations
Abstract. In this paper, we investigate global solvability of the Fokker-Planck equations of a special type. Such equations arise in models of the primary visual cortex of the human brain and describe a process of anisotropic blurring of the image of the visual field on the retina of the eye. By modifying the Folland lifting technique for linear hypoelliptic differential operators satisfying the Hormander condition, we propose a method to saturate the system of vector fields in the equation to a basis of the tangent space at every point. We present the conditions that guarantee existence of a global fundamental solution to the considered equations.
Key words and phrases: nilpotent and stratified Lie groups, lifting of operator, saturation, fundamental solution.
1. Introduction
In this paper, we study a class of Fokker-Planck equations given by
(1)
^ = (Ex2-YI«
i=1
^ «(x, t),
where x = (x1,..., xn) G Rn, t G R and the vector fields
n q n
(2) xi = (x) QXT., i=1' 2,...,m, Y = 53a(m
J = 1 0 3 = 1
x) —
dxn
satisfy the following conditions:
(1 ) Xi and Y are C° vector fields on R", i.e. their coefficients are smooth functions aij (x) G C°
-TOO /Tn>n\
The work of A. Mashtakov is supported by the Russian Science Foundation under grant 17-11-01387.
© V. Markasheva!1, A. Mashtakov!2, 2017 © Moscow Institute of Physics and Technology!1, 2017 © Ailamazyan Program Systems Institute of RAS(2, 2017 © Program systems: Theory and Applications, 2017
DOI: 10.25209/2079-3316-2017-8-4-149-162
(2) X1,..., Xm are homogeneous of degree 1 and Y is homogeneous of degree 2 w.r.t. 6\(x) = (ACT1 x1,Aa2 x2,...,XŒn xn), 1 < a1 ... < an, i.e. for any smooth on 1" test function f
Xi(f o Sx(x)) = X(Xif (x)) o JA(x), F(/ o ôx(x)) = X2(Yf (x)) o Sx(x);
(3) X1,..., Xm,Y are linearly independent almost everywhere w.r.t. standard Lebesgue measure on 1"
rank(X1,..., Xm, Y) = m +1;
(4) X1,..., Xm, Y satisfy Hormander hypoellipticity condition
rankLie(X1,..., Xm, Y) = n.
The question of solvability of equation (1) is equivalent to the question of existence of a global fundamental solution to the corresponding partial differential operator of the second order
ft m
(3) I - E *? + K
i=1
The paper has the following structure. It starts from motivation that comes from modelling of the primary visual cortex of the human brain, where equations (1) describe a process of anisotropic blurring (diffusion) of an image of the visual field on the retina of the eye. Then, in Section 3, we prepare a necessary mathematical background. Afterwards, in Section 4, we present the main result, the conditions that guarantee existence of a global fundamental solution of (3), followed by its proof in Section 5. Finally, we summarize the work in Conclusion.
2. Motivation
Our motivation to study Fokker-Planck equations (1) comes from modelling of the primary visual cortex V1 of the human brain, see e.g. [1], where such equations describe anisotropic diffusion of the image transmitted from the retina of the eye to the visual cortex V1. Such a diffusion underlies a mechanism of contour completion. According to the Petitot-Citti-Sarti model [2,3], the primary visual cortex lifts the image from the retina 12 to the extended space of positions and directions 12 xS1 = SE(2) (the group of Euclidean motions of the plane [4]):
T : I = ((X1,X2) G 12 ^ [0,1]) ^ {(x1,x2,xs) G SE(2) ^ [0,1]) = Î,
where x3 G S1 is the direction angle. Thus, the original image of I(x) on SE(2) has the form î(x) = T(I)(x).
Denote by X¿ the basis left invariant vector fields on SE(2):
0 0 . 0 0 0
Xi = --, X2 = cos X3—--hsinx^--, A3 = - Sinx^---+COS X3——.
0x3 OXi 0x2 OXi 0x2
The Fokker-Planck equation that simulates the contour completion mechanism has the following form, see [5],
(4) ^T^ = (BX2 -*2>(x, t),
where B > 0 is the diffusion coefficient.
In the paper [6], the authors show that such a diffusion process can be modelled by the following Fokker-Planck equation of type (1) in R2:
(5) = (B^i2 -y2>(x,t),
where (x,y) G R2, t G R and the vector fields Y\ = y-^ — x, Y2 = . 3. Preliminaries
Let P be a linear partial differential operator of an arbitrary order with smooth on R" real-valued coefficients. We say that a function
r : {(x; y) G R" x R" : x = y} ^ R,
is a (global) fundamental solution for P if it satisfies the following assumptions:
(1) for every fixed x G R" the function r(x; •) is locally integrable on R" and
[ r(x; y)P'</>(y)dy = -<£(x) for every ^ G C~(R"), JMn
where P' denotes the usual formal adjoint of P (this condition can be rewritten as Prx = -Dirx in V(R™));
(2) r(x, y) > 0 whenever x = y;
(3) r(x; y) G Li,ioc(Rp x Rp) for every fixed y G R" the function r(-;y) is locally integrable on R";
(4) for every fixed x G R" the function y ^ r(x; y) vanishes as y ^ to;
(5) for every fixed x G R" the function y ^ r(x; y) tends to to as y ^ x.
Let P be a smooth linear partial differential operator on R". We say that a linear partial differential operator P, defined on a higher-dimensional space R" x Rp, is a lifting of P if the following conditions are fulfilled:
(1) P has smooth coefficients, possibly depending on x G 1" and £ G 1,
(2) for every fixed f G CTO(1"), one has
(6) P(f o n)(x, £) = Pf (x), for every(x, £) G 1" x 1 = 1N,
where N = n + p and ~k(x, £) = x is the canonical projection. It is obvious that (6) holds if and only if
P = P + R with R = ^ (x,OD°D?,
¡3=0
for a finite number of coefficients G CTO(1N), possibly identically vanishing on 1N. The use of the term 'lifting' here is more specific than commonly accepted in differential geometry.
Let P be a smooth linear partial differential equation on 1", and P = P + R be a lifting of P on 1N. We say that P is saturable lifting of P if the following conditions hold:
(1) Every summand of the formal adjoint R' to a given operator P collect as least one derivative along some £, i.e., R' has a form
R' = E(x,£)DZDl,
fi=0
for a finite number of possibly vanishing smooth coefficients r'a,p.
(2) There exists a sequence {Qj(£)}|=i : 1 ^ [0,1] of smooth function with compact supports such that
y Q j = 1 where Q j = {£ G 1 : 0j (£) = 1} and for Vj Q j C Qj+1.
je n
Moreover, for every compact set K C 1" and for any coefficient function r'a,p (x,£) of R' there are exist constants (K) such that
№
\r'a,p(x,€) Q^0j(£)\ < Ca,p(K) for every x G K,i G 1p,j G N.
In the paper of Bonfiglioli-Biagi [7] one can find some sufficient conditions for a lifting operator to be saturable. In particular, for any smooth second order operator on 12 the associated operator P = dt — P is a saturable lifting of P.
THEOREM 1. (Bonfiglioli-Biagi) Let P be a smooth linear partial differential equation on 1" and let P be a saturable lifting of P on 1N. Assume that there exists a fundamental solution L to P on the whole 1N which satisfies the following conditions
(1) for every fixed x, y G 1" such that x = y
r ^ f (x,0; y, r) G Li(1p);
(2) for every fixed x G 1" and for any compact K C 1"
(y, r) ^ f (x, 0; y, ri) G Li(K x 1P). Then the function T : {(x; y) G 1" x 1" : x = y} ^ 1, defined by
r(x; y) = / f (x, 0; y, rMr
jRp
is a global fundamental solution of P.
4. Main result
As a basic example, let us consider Grushin vector fields
Xi = A, x2 = x^,
ox ay
which are smooth on a plane 12 3 (x, y). The vector fields X1,X2 homogeneous of degree 1 w.r.t. <5^(x) = (Ax, A2y). Conditions (1)-(4) hold. 12 is a Lie group homogeneous w.r.t. <5^(x). Vector fields X1,X2 satisfy the Hormander rank condition (4), hence Hormander operator X2 + X| as well as Kolmogorov operator X1 + X2 are both hypoelliptic but there is no Lie group structure on 12 making these operators left-invariant on it.
In general case in such situation we need to use a special modification of Folland-Bonfiglioli-Biagi technique built in this paper. In the considered example we can use the Folland-Bonfiglioli-Biagi saturation-lifting technique without any modifications, which leads to a new set of generating vector fields (let us call them Kolmogorov vector fields)
d d d X = -T-, X2 = + x—. ax at ay
Now, 13 is a saturated Lie group with a group law •
(x, y, i) • (x',y',t') = (x + x', y + y' + t'x,i + t').
Let us construct Hormander and Kolmogorov operators H and K on these vector fields
d2 id d \2 " + X = d3 + ddt +xdT>)
K = X? + x = £
Notice that the well-known Kolmogorov operator K on R3 coincides with the Fokker-Planck operator on Grushin vector fields 0 — -is — x on
1 at ax2 ay
R2 x R which we would like to solve, cf. (3).
It is easy to check that K is invariant w.r.t. the left translations on R3 and commutes with the following dilations:
(7) 0« (x) = (Xx,X3y,X2t ).
Kolmogorov vector fields are homogeneous for these dilations family. Therefore, Lie group R3 is homogeneous w.r.t. (7). But this time X1 is 1-homogeneous whereas X2 is 2-homogeneous w.r.t. (7). We can see that H is also invariant w.r.t. left translations on R3 while H commutes with another family of dilations (x):
(8) (x) = (Xx,X2y,X2t ).
The homogeneity for hypoellipic operators guarantees that these operators have global fundamental solutions. Thus, by lifting technique we have proved the existence of a global solution to the Fokker-Planck equation on Grushin vector fields.
Let us formulate the main result of this paper
THEOREM 2. For any set of vector fields that satisfy conditions (1)-(4) there exists a global fundamental solution to Fokker-Planck differential operator F = i - £1=1 X2 + Y.
5. Proof of Theorem 5.1. Lifting construction
Let x = (x1,... ,xn) G R". Let us consider the family (by X > 0) of non-isotropic diagonal maps on Rn
S\(x) = (Xff1x1, Xff2x2,..., Xa~xn), 1 < CT1 < ^2 < ... < °n, and a set of vector fields Xi and Y such that 4 main conditions hold: (1 ) Coefficients are smooth functions aij(x) G CTO(Rn). So, X1,..., Xm and Y are Cvector fields in Rn;
(2) Xi,..., Xm are homogeneous of degree 1 and Y is homogeneous of degree 2 w.r.t. (5A(x), i.e. for an arbitrary vector filed X homogeneous in degree I and for any smooth on 1" test function f
X(/ o <5A(x))=A'(X/(x)) o <5A(x);
(3) Xj, Y are linearly independent (as linear differential operators);
(4) Xj, Y satisfy Hormander hypoellipticity condition:
rank Lie{Xi,..., Xm, Y} = n.
Condition (4) means that at any point of 1" one can find n linearly independent differential operators among Xi,...,Xm,Y and their non-zero commutators. We recall that a typical example of vector fields that satisfy conditions (1) — (4) is given by Grushin vector fields X\ = , X2 = x-J^, which are smooth in 12 and homogeneous of degree 1 w.r.t. (5A(x) = (Ax, A2y).
The innovative modification is that now we can extend x G 1" to (x, t) G 1n+1. The dilations can be extended as well
(9) <5+(x, i) = ( <5A(x),A2t).
Using the commutativity operation, let us construct Lie algebra a = Lie{X1,..., Xm, Y}. We denote
(10) dim a := N > n.
This algebra must be extended from a to a+ with one more vector filed = T, linearly independent from X1, X2,..., Xm, Y. We have changed the dilations for T to be homogeneous with degree 2. This field is independent from others in sense that [T, Xj] = [T, Y] = 0 and, thus, it gives an additional dimension dim a+ = N + 1.
From homogeneity of the extended set of generators X1,..., Xm, Y, T (property (2)) one can conclude that a+ is nilpotent of step r and stratified
a+ — ^ iPi — a1 ^ ... ^ ar .
Let us denote by A the set of vector fields which is a basis of algebra a+ =Lie{X1,X2,..,Xm,Y,T}.
As far as X1, X2,..., Xm, Y, T are linearly independent in 1n+1 (condition (3)) to fulfil condition (10) (to find N + 1 linearly independent differential operators and construct the basis a+) we can choose first X1, X2,..., Xm,Y,T, and then some additional operators among commutators of X1,..., Xm, Y or their linear combinations. Let us denote them Xm+2,..., Xw. We will call this set the additional part of the basis A^.
Let us notice that each vector field among A± belongs to some layer a+, 1 < k < r, consequently, it is ^(x)-homogeneous of degree k while X1 ,X2,... ,Xm having 1-homogeneity belong to a+ and Y,T G a+ for the same reason.
Thus, Campbell-Hausdorff formula for arbitrary X,Y G a+ is finite X o Y = X + Y + 2[X, Y]... + const[X, [, [...[...,...]]].
The last commutator has degree r (and we have r > n). Each element from a+ can be written in a unique way as a linear combination of vector fields from A.
Thus, we can rewrite Campbell-Hausdorff formula as a linear composition of the basis A and coefficients of linear combinations of vector fields X and Y
n
X oY = X + Y + £ piXi.
i =1
Here p-i are in our case finite polynomials of degree < k (in general, infinite polynomials) of coefficients of decompositions of X and Y on the basis vector fields. So a+ can be considered as a homogeneous Lie group.
Observe that since among vector fields X1,.., XN +1 there are n +1 linearly independent then to this set belong our generators X1,.., Xm, Y, T. Let us denote them
B = (Xi1, ...,Xin+1 ) and take B(0) as a basis for Rn+1.
As a consequence, B must be homogeneous with degrees a1,...,an, (<rm+2 = 2). We rearrange them saving the same notation to
1 <01 <02 < ... < On+1.
Now, let us reorder A^ as well and continue the rearrangement to get the matrix of coefficients (Xi1,..., Xin+1,..., XiN+1 ) with homogeneity degrees
01,02,...,0n,0n+1, sn+2, . . . , SN +1.
It was shown in the paper of Bofiglioli-Biagi [7] that by a smooth change of variables in a+ one can construct a basis J1,..., JN +1 such that matrix (Xi1,..., Xin+1,..., XiN+1 ) will be transformed to (Zi1,..., Zin+1, ..., ZiN+1 ) which at point 0 has the following form
0
0 I
(? 0 ) ■
where I is a unit matrix (which corresponds to Rn) and coincides with
( Xi1 + Ri1 ,...,Xin+1 + Rin+1 ,...,XiN+1 + RiN+1 ^ wtere each Ri satisfies
the following conditions:
(1) Pj is a vector field of a+ and it consists only from ^-derivatives with coefficients possibly depending from (x, £);
(2) Zj = Xj + Pj holds the same homogeneity as Xj.
Now, following the Folland technique [8], there is the possibility to construct 1-to-1 smooth map k
a+ - 1«+1 ^ 1»+1,
Exp( sX) = Exp( s 1J1) o ... o Exp(% +1 Jw+1), k(X ) = Exp(S X )|s=1 .
Here Exp(sX) is a smooth integral curve (we can call it a flow) which starts from the origin at time = 0 and moves always in the direction X with unit speed. This curve is a unique solution of the system of smooth ordinary differential equations 7 = X(7(s)). Thus, for any smooth function
a a
(11) -K~f (x1 (s),x2(s),. .. ,x„(s)) = —f (7(s)) = X/(x1,x2, ... ,x„). a a
In the paper of Folland [8] and in the consequent article of Bonfilgioli and Biagi [7] there was proved that if B(0) is a basis for 1n+1 then Jacobi matrix of the projection k at point 0 coincides with B(0) and one can find the neighborhood in which k is surjective. Moreover, it is polynomial map which preserves the dilations, k(x, i, £) = (x, i) and, the most important, d-(Jj)a = (Xj)w(a), V a G a+ — 1w+1, Vi. Thus, Zj are liftings of Xj, Vi = 1, ..,m, Zm+1 is a lift of Y. It easy to see also that Zm+2 = T is left without changes.
Let us consider the operator F = — ^j=1Zi2 + Zm+1. By the construction it is a saturable lifting of F. From Folland results one can conclude that F is homogeneous of degree 2 w.r.t. £+(x, i).
5.2. Solvability which depends from homogeneity of an operator
Proof of solvability for homogeneous hypoelliptic operators is based on two classical theorems. The first states the local solvability for hypoelliptic operators. It belongs to Fr. Treves
THEOREM 3. (Treves) (see [9] Theorem 52.2) If D is a hypoelliptic differential operator on an open domain Q C 1", then every point in Q has an open neighborhood in which formal adjoint operator D' has a fundamental kernel. If D' is also hypoelliptic, then every point of Q has neighborhood in which D has a two-sided fundamental kernel, which is very regular (belongs to Frechet space).
The second theorem belongs to F. Folland [10]
THEOREM 4. (Folland) Let D be a homogeneous of degree a differential operator on the homogeneous Lie group G (0 < a < Q, Q = £Oi is a homogeneous dimension of G) such that D and his adjoint D' are both hypoelliptic on G. Then there is a unique kernel Kq of type a which is a fundamental solution for D at point 0, i.e. satisfies in distributional meaning the equation DKq = Dirx. Here Dirx is a Dirac distribution.
To prove Theorem 4 Folland has used the so called "local-to-global" or blow up argument to construct from local solution the global one.
According to these theorems operator F has a global fundamental solution with properties (1)-(4) from definition of a fundamental solution, and following the Bofiglioli-Biagi theorem we can construct the fundamental solution to F. Thus, the proof is complete.
6. Conclusion
In this paper, we have stated global solvability of the Fokker-Planck equations of a special type. Our motivation comes from modelling of the primary visual cortex of the human brain, where equations (1) describe a process of anisotropic blurring (diffusion) of an image of the visual field on the retina of the eye. By modifying the Folland lifting technique for linear hypoelliptic differential operators satisfying the Hormander condition, we have obtained a method to saturate the system of vector fields in the equation to a basis of the tangent space at every point. Finally, in Theorem 2 we have presented the conditions that guarantee existence of a global fundamental solution to the considered equations.
Sections 1, 2 and 6 of the paper are written by A. Mashtakov, and Sections 3, 4 and 5 are written by V. Markasheva.
References
[1] A. Mashtakov, V. Yumaguzhin, V. Yumaguzhina. "On Solutions To Fokker-Planck Equations", Journal of Mathematical Sciences, 2017 (to appear). t 150
[2] G. Citti, A. Sarti. "A cortical based model of perceptual completion in the roto-translation space", Journal of Mathematical Imaging and Vision, 24:3 (2006), pp. 307-326. t 150
[3] J. Petitot. "The neurogeometry of pinwheels as a sub-Riemannian contact structure", Journal Physiology Paris, 97:2-3 (2003), pp. 265-309. t 150
[4] A. P. Mashtakov, A. A. Ardentov, Yu. L. Sachkov. "Parallel algorithm and software for image inpainting via sub-Riemannian minimizers on the group of rototranslations", Numerical Mathematics: Theory, Methods and Applications, 6:1 (2013), pp. 95-115. t 150
[5] G. Sanguinetti, G. Citti, A. Sarti. "A model of natural image edge co-occurrence in the rototranslation group", Journal of Vision, 10:14 (2010), pp. 37. t 151
[6] D. Barbieri, G. Citti, G. Sanguinetti, A. Sarti. "An uncertainty principle underlying the functional architecture of V1", Journal of Physiology Paris, 106:5-6 (2012), pp. 183-193. t 151
[7] S. Baigi, A. Bonfiglioli. The existence of a global fundamental solution for homogeneous Hormander operators via a global lifting method, 2016, arXiv: 1604.025 99.1 152>156>157
[8] G. B. Folland. "On the Rothschild-Stein lifting theorem", Comm. Partial Differential Equations, 2:2 (1977), pp. 161-207. t 157
[9] F. Treves. Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967. t 157
[10] G. B. Folland. "Subelliptic estimates and function spaces on nilpotent Lie groups", Ark. Mat., 13:1-2 (1975), pp. 161-207. t158
Submitted by Prof. Yuriy Sachkov
Sample citation of this publication:
V. Markasheva, A. Mashtakov. "Existence of Global Fundamental Solution to a Class of Fokker-Planck Equations", Program systems: Theory and applications, 2017, 8:4(35), pp. 149-162.
URL: http : //psta.psiras . ru/read/psta2017_4_149-162.pdf
About the authors:
H
Vera Markasheva
V. Markasheva currently is a Senior Tutor at the Department of High Mathematics in Moscow Institute of Physics and Technology, Dolgoprudny, Russian Federation. Received M.S. at Donetsk National University (Ukraine) in 2005, PhD. at the Institute of Applied Mathematics and Mechanics of NAS of Ukraine in 2012, was an Erasmus Mundus grantee at the University of Bologna in 2014-2015, had a postdoctoral position at the University of Bologna 2015-2016 under the supervision of Giovanna Citti (coordinator, together with A.Sarti, of the local interdepartmental group of "Neuromathematics and Visual Cognition"). Scientific interests are degenerate PDE on Lie groups and homogeneous spaces and applications in Neuromathematics and Visual Cognition.
e-mail: [email protected], [email protected]
Alexey Mashtakov
A. Mashtakov received M.S. in Applied Mathematics and Computer Science at University of Pereslavl in 2009. He received Ph.D. in Applied Mathematics and Computer Science in PSI RAS in 2013. He had a post-doctoral position at Department of Biomedical Engineering at the TU/e from 2014 till 2016. Currently, he is a senior researcher of Control Processes Research Centre at Program Systems Institute, Pereslavl-Zalessky (Russian Academy of Sciences). His research interests are geometric control theory, optimal control, control theory on Lie Groups, nonholonomic systems, sub-Riemannian geometry and their applications in image processing, robotics and modelling of
e-mail:
УДК 517.958
В. А. Маркашева, А. П. Маштаков. Существование глобального фундаментального Решения для Класса Уравнений Фоккера-Планка.
Аннотация. В статье исследован вопрос глобальной разрешимости уравнений Фоккера—Планка специального вида. Уравнения такого вида возникают в моделях первичной зрительной коры головного мозга человека и описывают процесс анизотропного размытия изображения, поступающего на сетчатку глаза. Модифицируя технику лифтинга Фолланда для линейных гипоэллиптических дифференциальных операторов, удовлетворяющих условию Хермандера, был предложен метод насыщения системы векторных полей в уравнении до базиса касательного пространства в каждой точке. Найдены условия, гарантирующие существование глобального фундаментального решения для уравнений рассматриваемого вида.
Ключевые слова и фразы: Уравнение Фоккера-Планка, группа Ли, фундаментальное решение, насыщение.
Пример ссылки на эту публикацию:
В. А. Маркашева, А. П. Маштаков. «Существование глобального фундаментального Решения для Класса Уравнений Фоккера—Планка», Программные системы: теория и приложения, 2017, 8:4(35), с. 149-162. (Англ). URL: http://psta.psiras.ru/read/psta2017_4_149-162.pdf
References
[1] A. Mashtakov, V. Yumaguzhin, V. Yumaguzhina. "On Solutions To Fokker-Planck Equations", Journal of Mathematical ¡Sciences, 2017 (to appear).
[2] G. Citti, A. Sarti. "A cortical based model of perceptual completion in the roto-translation space", Journal of Mathematical Imaging and Vision, 24:3 (2006), pp. 307-326.
[3] J. Petitot. "The neurogeometry of pinwheels as a sub-Riemannian contact structure", Journal Physiology Paris, 97:2-3 (2003), pp. 265-309.
[4] A. P. Mashtakov, A. A. Ardentov, Yu. L. Sachkov. "Parallel algorithm and software for image inpainting via sub-Riemannian minimizers on the group of
rototranslations", Numerical Mathematics: Theory, Methods and Applications, 6:1 (2013), pp. 95-115.
[5] G. Sanguinetti, G. Citti, A. Sarti. "A model of natural image edge co-occurrence in the rototranslation group", Journal of Vision, 10:14 (2010), pp. 37.
[6] D. Barbieri, G. Citti, G. Sanguinetti, A. Sarti. "An uncertainty principle underlying the functional architecture of V1", Journal of Physiology Paris, 106:5-6 (2012), pp. 183-193.
Исследование А.П. Маштакова выполнено за счет гранта Российского научного фонда (проект № 17-11-01387).
© В. А. МАРКАШЕВА! , А. П. МАШТАКОВ! , 2017
© Московский физико-тЕхничЕский институт! , 2017
© Институт программных систем имени А. К. Айламазяна РАН!, 2017
© Программные системы: теория и приложения, 2017
ЭС1: 10.25209/2079-3316-2017-8-4-149-162
[7] S. Baigi, A. Bonfiglioli. The existence of a global fundamental solution for homogeneous Hormander operators via a global lifting method, 2016, arXiv: 1604.02599.
[8] G.B. Folland. "On the Rothschild—Stein lifting theorem", Comm. Partial Differential Equations, 2:2 (1977), pp. 161-207.
[9] F. Treves. Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
[10] G. B. Folland. "Subelliptic estimates and function spaces on nilpotent Lie groups", Ark. Mat., 13:1-2 (1975), pp. 161-207.
