Crystallographic groups for Ho¨ rmander fields Текст научной статьи по специальности «Математика»

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DOI: https://doi.Org/10.15688/mpcm.jvolsu.2017.3.4

UDC 517 LBC 22.161

CRYSTALLOGRAPHIC GROUPS FOR HORMANDER FIELDS

Boguslaw Zegarlinski

Professor, Faculty of Natural Sciences, Department of Mathematics,

Imperial College London

[email protected]

South Kensington Campus, London SW7 2AZ, UK

Abstract. This is a preview paper on Crystallographic Groups of Hormander Fields. We describe an emerging picture in analysis of extended groups. In particular, we introduce and provide examples of Crystallographic Groups associated to a Hormander system of fields as well as discuss some related analysis.

Key words: extended Lie groups, noncommutative Dunkl-type operators, Markov semigroups, entropy & heat kernel bounds.

1. Introduction

In [78] we have introduced an infinite Coxeter group on two generators associated to the Heisenberg group Hi and studied related analysis. In this paper we present some possible outlook how that theory could be extended to include other noncompact Lie groups.

2. Crystallographic Groups

In this paper we will consider an algebraic and analytic structures associated to a given set of fields Xj, j = 1,...,N, on a differentiable manifold G, given by a family of maps ak : G ^ G, k G N, satisfying the following condition

xj (f ° ak) = Ujk,u (Xif ) o at (1)

i,i

§ (possibly only on certain sub-space of functions) with some a.jk,a G R. In particular we pq introduce the following definition. ^ Definition 2.1. A map a : G ^ G satisfying

& X3(f o a) = -(X3f ) ° a, a ° a = id (2)

N

@ will be called a reflection.

3. Examples: Coxeter Groups of Systems of Fields

In this section we present explicit examples of maps indicated above for certain types of nilpotent Lie groups (as classified in section 4.1 in [15]).

3.1. B-Groups

Let G = R x Rra with composition • of elements w = (t, x),w' = (t', x') defined as follows

(t, x) • (t', x') = (t + t', x' + etBx)

Then for

a(w) = (-t,e-2tBx)

and

X = dt + (Bx) ■ V,

where V denotes the gradient with respect to x, we have

X(f o a) = -(Xf) o a

For components dj of V the reflections are simply given by maps Xj ^ (-1)5iJ' Xj, j = 1,..,n. the corresponding set (1) of maps includes also linear maps (with respect to x).

3.2. K-Groups

Let G = Rra° x Rrai x ... x R"-, with n0 > m > .. > nr, r e N. This is similar to B-groups except that t is replaced by a n0-vector t and the matrix in the definition of composition is lower triangular with the sub-diagonal part consisting of blocks ns x ns-i with rank equal to ns for each s = 1,..,r. In this framework we consider the following fields (homogeneous of order one with respect to a natural gradation)

X, = dt] + (Bjt) ■ V

where V denotes the gradient with respect to the variable of hidger degree of homogeneity (different than t-variables).

Within this class we would like to indicate a special class of H-type groups defined with r = 1 and a single antisymmetric block B. In this case we have a nice family of reflection maps a, j = 1, ..,n0, satisfying

Xj(f o a,) = -(X,f) o a,

explicitly given by

aj(w) = w • (-2(63ZU}, 0) = (|(-1)5jiti}, xk - 2 ^ BkjitjU)

i

with j,i = 1,..,n0 and k = 1,..,r. These maps are idempotent, i.e. aj o aj = id, and (in general) they do not commute with each other. There are also some other reflection maps given as follows

a- (w) = (((-1)6- u}, -x)

which do commute between themselves. It is interesting to notice that one can also have some other maps which intertwine the fields and are not of order two. To see this consider the following simplest example of such group.

3.3. H1 Group

With w • w' = (x + x',y + y', z + z' + 2a(yx' — xy')), a = 0, and fields

X = dx + 2an2dz

Y = dv — 2ani dz

with ni(w) = x and n2(w) = y, we have the following corresponding reflections

ax(w) = (—x, y,z — 4axy), dx(w) = (—x, y, —z),

and a common one Moreover, for we have

ay (w) = (x, —y, z + 4axy), ây (w) = (x, —y, —z)

à = (—x, —y,z). a+(w) = (y, —x,z ), a-(w) = (—y,x,z)

a- о a+(w) = id = a+ о a-, a°_ = id = a

_o4

„.°2 - o2

a = à = a_ .

We note also the following relations

ax о = âX о ax,

à о ax = ax о à, à о âx = âx о à = ây,

a у о âY = ây о ay, à о ay = ay о à, à о ây = ây о à = âx,

and

a+ о ay = ax о a+, a_ о ay = ax о a_,

à о a

+ =

a+ о à = a_,

a+ о ây = âx о a-, a- о ây = âx о a+,

à о a_ = a_ о à = a

For a+, a-, we have

x(f ° a+)(w) = X(f (n.2, — ni, n3))(w) =

= —(dy f) ° a+(w) + 2an2(w)(dzf) ° a+(w) =

= (—(dyf) + 2ani ° a-(dzf)) ° a+ = —(Yf) ° a+(w)

where we have used n2 ° a- = n1. Similarly

Y(f ° a-)(w) = Y(f (—n2, ni, n3))(w) = —(dxf) ° a-(w) — 2ani(dz f) ° a-(w) =

= (—(dxf) — 2ani ° a+(dzf)) ° a-(w) = —(Xf) ° a-(w)

where we have used ni ° a+ = n2. Finally for the reflection o(w) = (—x, —y,z), one has

X (f ° a) = Xf (—ni, —n2, n3) = —(dxf) ° a + 2an2(dz f) ° a = = —(dxf) ° a — (2an2dzf) ° a = —(Xf) ° a

and similarly

Y(f о à) = —(Yf ) о à.

The Coxeter group generated by ax and aY is of infinite order and all its elements can be describe as follows: We set

Ln = (ax ay )n Lo = id L-n = (vY ax )

ne N

and

Rn+1 = Ln о ax = ^x о Ln,

With this notation we have

and

R-n-1 = L-n о Vy = aY о L-n,

ne N

Ln о Lm Ln+m,

and Rn о Lm = R,

^n+m-

The subgroup {Ln}neZ is abelian and moreover any (group) commutator in the group generated by ax, aY belongs to {Ln}neZ. The group is furnished with the following intertwining structure

a± o Rn+1 = R-n-1 o a± and a± o Ln = I-n o a±

Finally we mention that the fields X and Y are preserved by the following families of translations

Tx (w) = (x + a,y,z + c), ty (w) = (x,y + b,z + c), respectively, with a, b, c G R, as well as the left action of the group.

Remark. In complex representation of Heisenberg group where w = (v,t) one has the following natural reflections a*,±(w) = (v*, ±t), â(w) = (-v,t) = a±(w) where a±(w) = = (^iv,t). Then dx(w) = â o dY(w) with dY(w) = a*,-(w). Finally aY(w) = (v*,t + + 2a^R(-iv2)) and ax = a+ o aY o a-.

3.4. Examples of Higher Order

(i) Let G = R4 and

W • w' = (xi + x'i,x2 + x'2,x3 + x'3 + x'ix2,x4 + x'4 + x'ix2 + 2x2x'3) with dilation b\(w) = (Xxi, Xx2, ~h2x3, \3x4). Consider the following system of fields

Xi = di + x2d3 + x22d4, X2 = d2. See e.g. [15] for such the structure. Define

al(w) = w • (—2xl, 0, 0, 0) = (-x1,x2, x3 — 2x1x2,x4 — 2x1x2). Then we have al o al = id and

Xi(f o ai) = Xi(f (—ni, n.2, n — 2niU2, n — 2nin2,)) =

= —(dif) o ai — 2n2(d3f) o ai — 2n2(d4f) o ai + n(d3f) o ai + n^/) o ai =

= —(Xif) o ai.

Also for

a[(w) = (-x\,x2, -x3, -x4), a = (—x\, -x2, x3, —x4)

one gets

Xi(f ° a'i) = —(Xif) ° ai, and Xi(f ° a) = — (Xif) ° a. In all cases we have id = ai ° ai = ai ° ai = a1 ° ai. For the second field the relation

X2U ° a2) = —(X2f) ° a2 is satisfied with any map of the form

a2(w) = (ax]^, —x2 + fix^^,yx3 + bx"2, ex4 + Zx1x3)

with a, fi,y, b, e, Z E R. Such the map satisfies condition a2 ° a2 = id if additionally one requests (a, b, y, e) = (—1, 0,1,1), (—1, 0, —1, —1) or (a, fi, b, y, e) = (1, 0, 0, —1,1) and (a, fi, b, y, e, Z) = (1,0,0,1, —1,0). For the last choice we obtain a which also reflects the first field. This is a situation which is different from the Hi group. (ii) Let G = R4 and

W • w' = (xi + x'i,x2 + x'2,x3 + x':i + x\x2,x4 + x'4 + 1X2 (xi)2 + X3XD with dilation b\(w) = (\xi, Xx2, X2x3, X3x4). Consider the following system of fields

Xi = di + X2d3 + X3&4, X2 = 82,

see e.g. [15]. Define

ai(w) = w • (—2xi, 0, 0, 0) = (—xi,x2,x3 — 2x1x2,x4 + 2x\x2 — 2xix3). Then we have

Xi(f ° ai) = Xi(f (—ni, n.2, n3 — 2nin2, n.4 + 2n2^K2 — 2nin3)) =

= —(dif) ° ai — U2(d3f) ° ai + (2niU2 — n3)(d4f) ° ai =

= —(Xif) ° ai.

For

a[(w) = (—x1,x2, —x3,x4) or a = (—x1, —x2,x3, —x4)

we get also reflection of the field X1. In all cases we have id = a1 ° a1 = ai ° ai = a ° a. For the second field we have

X2U ° a2) = —(X2f) ° a2

with any map of the form

a2(w) = (ax1, —x2 + fix1,y^3 + bx2i, ex4 + Zx1x3)

with a, fi, y, b, e, Z E R. Such the map satisfies condition a2 ° a2 = id, if additionally one requests (a, b, y, e) = (—1, 0,1,1), (—1, 0, —1, —1) or (a, fi, b, y, e) = (1, 0, 0, —1,1). We remark that in the present case also a reflects X2.

4. Problem of Effective Description of Coxeter Groups

The first emerging general issue is as follows.

(P1.i) Given a basis Xj, j = 1, ..,N, of free nilpotent Lie algebra, as described in [48], describe the Coxeter groups CXj of a given field Xj.

(Pl.ii) Describe the crystallographic group corresponding to the basis Xj, j = 1, ..,N. In the second case we ask for more than just the description of the group Vj CXj generated by all CXj, as it could also include intertwiners as well as possibly some other elements. One can illustrate that a full variety of structures can occur by considerig the following class of examples of n fields with linear coefficients in Rn+1

x- = — + Y B-X--

^ ~ dxj дz,

J 1=3

where Bji is a nonsymmetric matrix, with associated reflections given by

aj (w) = ({(—1)bij Xi }i=i,..,n,z — 2^ Bij Xj Xi).

i=j

The commutativity structure can be obtained by constructing a graph with n vertices in which vertices Vi and Vj are connected by an edge e^ iff Bij = 0, and in this case ai and aj do not commute. aj,j = 1,..,n, generate a Coxeter group of infinite order. Additionally in the present case CXj contains reflections

(w) = ({(—1)biJ Xi}i=i,..,n, —z)

and

aj (w) = ({(—1)b- (—1)be- xz}l=i,.,n ,z)

and possibly some others (including also number of intertwiners). Moreover the field Xj is invariant with respect to Coxeter group acting on variables which are not involved in definition on Xj as well as with respect to traslations Tj (w) = ({xi + aj bij }i=1,.,n,z + Yj), with aj, Yj E R, respectively.

Generally we may have a finite Coxeter subgroup and additional infinite component in a given Coxeter group. We remark that in some examples we had a subgroup containing reflections of many fields. One may expect that these data could help to classify the corresponding noncompact Lie groups.

5. Inverse Problem for Coxeter Groups

Suppose one is given generators aj of a Coxeter group on RN, then one can ask what are the fields X which satisfy the reflection relation

(Xf ° aj) = —(Xf) ° aj. (**)

For example one knows (see e.g. [70]) that the Painleve II

1

u = 2u3 + tu + b — -,

represented as a Hamiltonian system with canonical variables

as follows

q = и, p = U + и2 + -

Q = f =P - Q2 - 2

dH

p = - ^ = 2pq + b

admits a Coxeter group of Backlund transformations generated by the following map

^ b) = {q+~^ -b)

o2(q,p, b) = (-q, -p + 2q2 + t, 1 - b).

It is the infinite dihedral group encountered before for the Heisenberg group H1. One can solve the reflection equation with respect to the fields getting

d 1 d d Mf o .1) = -{Xi f) o 01, Xi = Wb + - + .1) ^ + Pi

d d d Mf o 02) = (X2 f) o 02, X2 = + (-2q) ^ + Y2 ^

with the functions a1 o a^ = —a1, p1 o a^ = p1 and y2 o o2 = y2. (It would be interesting to repeat that for other Paileve problems using identification of [40; 61].)

Homogeneous Lie groups on Rra

Consider now the case when the group action is given by

(w • w')k = xk + x'k + Qk (w, w')

with Qk (w, w') = Qk (xj ,xj : j = l,..,k — 1), satisfying

Qk (w,w')\x'.=0:j=l,..,k-l = 0.

(According to Theorem 1.3.15 in [15] such properties would satisfy a composition law for any homogeneous Lie group on Rra.) Then, each map

ffi(w) = w • r(i)(w), with (rw(w))j = —2bijXi, (3)

is a reflection, i.e. Oi о Oi = id, if the inverse to r(l\w) is given by —r(l\w). This is because (rW(w • r(^(w)))j = —2bij(w • r(^(w))i = 2bi,jXi and this defines the inverse element to r(l\w) in our group. (In fact, by Proposition 2.2.22 in [15], for stratified Lie groups in appropriate representation this is always the case.)

In this case one can study the corresponding reflection problem (**) for fields

v f v f(w о ¿i) — f(w) d ff ч

Xif = lim-= —f(w о ¿i )£=o.

e^o e de

We have the following result.

Theorem 5.1. For Oi given by (3) the fields Xi satisfy the reflection problem (**) for, respectively, provided that

d

—Qk ((Oi(w) о ei)i<k-i, {—2bij(Oi(w)i + г))^к-1)

|e=0

d

((oi(w))i<k-i, (bije)j<k-\)

|e=0

Proof. Using the definition of the fields Xi, we have

Y(f \ y f ° Gi(w ° ei) - f ° ai(w)

Mf ° Gi) = lim-

e^0 e

with

(&i(w o ei))k = (w ° ei ° r(l)(w ° ei))k

= (w ° ei)k - 2bik(w ° ei)i + Qk(w ° ei} r(l)(w ° ei)) = (—1)bifc (w ° ei)k + Qk(w ° ei} r(i)(w ° ei)).

Using (/^(w ° ei))k = -2bik(wi + e) and our assumption about Qk's, we have

wk,

(oi(w о гi))k = { -Wi - г,

for к < г for к = г

wk + Qk ((w о г^1<к-1, (-2bij (wi + г))у<к-1), fork>i

Hence

Xi(f о Oi)(w) = -(dif) о Oi(w)

+ £

k>i

d \

—Qk((w о г^1<к-1, (-2bij(wi + г) \ )

J j<k-l

(dkf) о Oi(w)

|г=0

while we have

(Xif) ° Gi(w) = (dif) ° Gi(w) +

+E

k>i

d

-^Qk((oi(w))i<k-i, (bijг)^<к-1)

|г=0

(дкf) о Oi(w).

Thus to get (**) we need

d

Qk ((Oi(w) о г^1<к-1, (-2bij(Oi(w)i + г))^<к-1)

|г=0

d

Qk((oi(w))i<k-i, (bijг)^<к-1)

|г=0

We remark that in non-homogeneous case one can allow for maps having a component

Xj M —Xj + Qj (Xk, k < j)

with even polynomial qj if the components xk, k < j, just change a sign, as we have seen it in case of Backlund transformation.

Remark. In complex representation of Heisenberg group where w = (v, t) one has also the following natural reflection oY(w) = (v*,t - 2aR(v2)) and ox(w) = (—x,y,t + 2aR(v2)). Then the fields

Y = dy - 2aydz, X = dx - 2axdz

would satisfy

Y(f o oy) = -(Yf) o oY, X(f o ox) = -(Xf) o ox.

It would be interesting to find out for what Coxeter groups one can choose a system of fields such that (**) holds and for which Hormander condition is satisfied.

6. Representations and Co- & In-variant Functions

In [78] where we have studied H1 and introduced

ox(w) = (-x, y,z - 4axy), oY(w) = (x, -y, z + 4axy)

we have provided there a simple representation of the corresponding Coxeter group on the linear span of the following generalised linear functions

x,У, n =x + , Z = y- —

2ay 2ax

as follows

x o ox = -x y o ox = y n o ox = n - 4x Z o ox = - Z x o oY = x y o oY = -y n o oY = -n Z o oY = Z - 4y

To this we add relations coming from additional o's described above as follows

vx(w) = (-x,y, -z), oy(w) = (x, -y, -z), o(w) = (-x, -y, z)

o+(w) = (y, -x, z), o-(w) = (-y,x, z)

for which we get

n o o = -n, Z o o = -Z n o Ox = -n, Z o Ox = Z n o oy = n, Z o oy = -Z

and the following intertwining relation

n o o+ = Z.

The Coxeter group generated by Ox, OY, o± preserves the Kaplan norm, while a group including also generators ox, oY would preserve the set of functions spanned by the following

(w) = ^ ^ ex(z-n~xy)$(z - n—xy))

q neZ 1

defined with q e N and (lq)q = 1, kq = 1, A e R and a function Note that

o ax(w) = ^ l™ elx[z+n^xy)$(z + n—xy)) nez 1

and

ф g о CY(w) = ^ knq elX(z+n4^xy)ф(z + n jxy)).

nez

For higher order examples considered in section 3, besides homogeneous norms, one can find also for example the following homogeneous polynomials invariant with respect to the basic finite Coxeter subgroup (not including o^)

(i) X\X2X3 + X\X4 + x\ + x2, + x\

(ii) X\X2X3 + X2X4 + x\ + x\ + x\

which are very similar. This brings us to the following problems. Problems 2.

(P2.i) Describe representations of the groups Vj CXj on spaces of functions. (P2.ii) Describe the invariant homogeneous polynomials with respect to subgroups as well as fully invariant functions.

7. Linear Operators

Given o e CXj, o°2 = id, we can introduce the following linear operators. DeMazur Operators

A3,o = -o (I -I.)

Ho

with /af = f o o and a differentaible function no satisfying

no o a = —no and Xjno =1,

and a constant Ko = 0. First we can introduce it on smooth functions vanishing on an open set containing the set So = {no = 0}, and later extend by continuity. One can see that Ajo vanishes on o-symmetric functions (i.e. functions satisfying Iaf = f o a = /), and we have the following:

Boundary Operator Property

A2,o = 0.

Using DeMazur operator we define the following 1st order operators: Generalised Fields

T^o == Xj + Aj,o-

Some care is necessary here as we are adding two unbounded operators. In case of free nilpotent Lie group when the reflection has a simple implementation (with group left multiplication by (—2bijXj)), we note that

Aitaf(w) = 2k3i ds(Xjf)

0

with a path j™ connecting w and a(w). Since by choice of our a we have the following. Anticommutation Relations

[X3 , la] = X3la + IaX3 = 0, [A^, la] = 0,

so also

[Tjta, la] = 0.

More generally one could introduce

T3 = Xj + Yj &aAj,a aeCXj

with e R, ^a€CXj =1.

With these extended fields, assuming later on fia, Ka > 0, we introduce the following second order operator

j,a

In this case one could check, similarly as in [78], (see also section 10), that such operator satisfy the following.

Proposition 7.1 (Minimum Principle).

f(wo) = min f =^ £f (wo) > 0 Proof. Since A2a = 0, we have

£f = + [Xi, Aaj}) f.

j

If a minimum point x0 e Ua6a we have

XjAaj f(xo) = -Kh (Xif (Xo) — X3 (f ° Oj )(Xo)) — Ka ^ n* )0) Aaj f(Xo) naj (X0) naj (X0)

and

AaX3f = O—t (Xjf (xo) — (Xjf) ° a3(xo)).

Hence, using reflection property, we get

№ -A"f =¿fe^ ^ - m

The first term on the right hand side is equal to zero at the minimum point. If Xjnaj > 0 as we assume, we have

KajXjnaj (xo ) KajXjnaj (x0)

A., f(xo) = ; (f(c3Xo) - f{xo)) > 0

Ы J Ы

because at the minimum point f(ajx0) > f(x0). In case the minimum point x0 e Ua6a, we need to use limiting procedure necessary to extend definition of Aa to reflection ivariant points. As we pointed out above (in case of nilpotent Lie groups) when X3naj = 1, one gets

[X,,Aa, f](xo) = 2KJXJ2f (xo) > 0

as required.

We note that, as in the case of Heisenberg group, the Coxeter group generated by the reflections of the basic fields can be infinite and may contain infinitely many reflections of the space subordinated to some other fields. For example in the Heisenberg case one generates reflections of the form (axay)nax and (ayax)nay, n e N, which would satisfy the reflection relation with the fields of the form Xk = dx + 2kaydz or Yk = dy — 2kaxdz, for suitable k e Z. In this case we get discrete infinite family of representations of the Heisenberg-Lie algebra and in this case one may also consider the following Markov generator

£ = ^ ekT"j,k,a j,k,a

with ek e (0, to) satisfying J2k ek < to.

Proposition 7.2. Assume the fields Xi satisfy integration by parts formula with a measure v invariant with respect all ai. Let p be a density function with respect to v. If

X, log p = 2Kzn~1 (4)

for all i (almost everywhere), then for all T = X^ + Ai the following generalised integration by parts formula holds

i T(g) fpdv = — i gTi(f)pdv. (5)

Proof. Since for a given ai invariant measure v, we have

J Ai(g)fdv = —J gAi(f)dv + J gK^fdv.

Using this together with the integration parts formula for the field Xi with the measure v, we notice that to have (5), one would need to satisfy the following condition

X, log p = 2Kzn~1 (6)

for all (almost everywhere).

If the fields X/s generate nilpotent Lie algebra, (6) provides necessary conditions on •qi's for solvability, (as applying other fields and forming linear combination and repeating this many times one can generate commutators of sufficiently high order which have to vanish by nilpotency condition). Also applying Xi again to (6) and summing over i's one obtains a relation

(^X^)log p = ^ 2KzXzn~1. (7)

i i

In particular if Xini = 1 we get, with X ■ X = (^¿X2), the following

X ■ Xlog p = — ^ 2m-2. (8)

In case when X. are generators of free nilpotent Lie group, we have X. = 8. + J2j>i Ujdj with a dependent on the proceeding coordinates, with m = Xj one can choose

p = nx?Kl. (9)

A possibility of other choices of n's is discussed later in section 8.2. Canonical Markovian and Dirichlet Forms

When the generalised integration by parts formula holds with measure dv = pdv, we have

— J gC fdv = j Tg ■ Tfdv = (Ti9)T.fd^. On the other hand the canonical Markovian form of £ is as follows

r(f) = 1(Cf2 — 2fCf) = \Xf\2 + £ (Aif)?.

2 2 Ki

If Xn > 0, then this form is clearly nonnegative. (In particular this is satisfied for generators of free nilpotent Lie groups with n's given by corresponding coordinate functions.) In this situation the right hand side of (7) is nonnegative.

If the generalised integration by parts formula holds, we have

J r( f) dv = J Tg ■ T f dv > J \X f \2dv (10)

with the left hand side satisfying evidently positivity and contraction property of a Dirichlet form on a suitable dense space of functions. Thus, after closure, it defines a Markov generator in L2(v).

Next we remark that for density p and a differentiable function , we have by Leibnitz rule X(fp1/2) = (Xf)p1/2 + \f p1/2(Xlog p) and hence for dv = pdv, we get

J \Xf \2 dv >— j X( f p1/2) ■ fp1/2(X log p) dv + J f21 \X log p\2 dv. (11)

From this, by integration by parts in the first term on the right hand side, we arrive at the following.

Proposition 7.3 (Hardy type inequality I).

J \Xf\2dv > j 1 \Xlog p\2 + 2X ■ Xlog p^jdv. (12)

Using the condition (6)-(7) we thus obtain.

Proposition 7.4 (Hardy type inequality II). For generators of nilpotent Lie groups with corresponding p given by (9) with к > 1

J |Tf I2d» > J f2 Y,(K2n~2 + кгХгП-1)d». (13)

In particular if Хщг = 1 one gets

J |T f I2d» > J f2 £ кг(кг - ï)n-2d». (14)

Since the set of Hormander fields satisfy Sobolev inequality, one also has the following log-Sobolev type inequality satisfied

/ f2 loë JJ2^dy - г j Ixf I2(iy + c(£) j f2(iy

for any г G (0, то) with С (г) ~ cons tlog 1 for small e. In particular substituting in this inequality a function fp2, we get

i f log + i f2 log pd» - г i XI2d» + г i fXf • Xlog p d» +

Jf2d»

f2 •IXlog p|2 d» +С(e) f2d».

Hence

/ f2 log + / f2 log pdV ^ \xf V —

— zj f2(4 \X log p\2 + 1X ■ X log p) dv +

+C(z) J f2dv

Using Hardy inequality together with quadratic form bound (10) one can dress this inequality up and obtain, (as in [78]), the following result.

Theorem 7.5 (Log-Sobolev Coercive Bound). Suppose Hardy Inequality (12) holds and suppose for any z e (0,1) we have

log+ p-1 -Сe ( 1 |Xlog p|2 + 2X • Xlog p) + DC(e)

with some constants C,D e [0, to) independent of f. Then

j f log jf2^ < z j \Tf\2dv +C'(z) f f2dv (15)

with C'(z) ~ constlog 1 for small z.

Using this and [25] we obtain the following important implication. Corollary 7.6 (Ultracontractivity Estimate). The semigroup Pt = etc, with C = T ■ T, is

ultracontractive i.e. for t > 0 the operator Pt : Li(v) ^ is bounded. Hence

Ptf (w) = j f(w)ht(w,w)v-(dw)

with a bounded (uniformly away from t = 0) smooth heat kernel ht(w,w).

Generally it is an interesting challenge to find estimates on the heat kernel. Since our generator include jump type part, a nice method of [25] may not work. In [78] we proposed a strategy for Heisenberg group based on estimating the moments of coordinate functions and arguments of A. Grigor'yan [49]. It should be possible to generalise that to a class of free nilpotent Lie groups. First of all Gussian exponential bounds in horizontal variables could be achieved via Aronson arguments and one may hope that the bounds in other directions could be reduced to the former via a technique involving generalised integration by parts formula. Given estimates of moments one can find possibly optimal bounds for exponential function and with this obtain a bound of the form

f epd"ht(-,w)d^ < C(t, w) < m

with d a natural (homogeneous) distance from a given point on the underlying space and some positive constants a, |3. Then one can follow [49] as follows: Using Chapman — Kolmogorov property

ht(w',w) = J ht/2(w',w)ht/2(w,w)v*(dm) and elementary inequality

0 < -1da(w', w) + P(da(w', w) + da(w, w))

C 2

for some C E (0, <x) (independent of the points w,w',w), together with Holder inequality, one gets

С

ht/2(v/,w)e

fid01 (w',w)

ht(w',w) < e-?d"(w'w) •

\ 1/2 /л \ 1/2

< e-bd"(w',w)cr 1/2 ч 1/2 ht/2(w'W)eed"(w'w)MdW)) • / h/2(W,w)eed"(w,w)<

^ ht/2(w', w) e ^ dw)j ■ ^J h*/2(w,w)e ^ ^^ dwu)^

where ct is the constant from the ultracontractivity estimate. Given such an upper bound one could possibly use arguments of [9] to obtain a lower bound. One may conjecture that at least in the case of free nilpotent Lie groups, the bounds should have a Gaussian character in a suitable distance (possibly away from reflection hyperplanes). It would be an interesting question what is possible in more general cases and how to quantify the corresponding heat kernel bounds. Finally, is it possible to obtain sharpened version at least with a tight exponential factor on both sides of the sandwich.

8. Further Examples with Nilpotent Lie Groups

8.1. Nonhomogenous Nilpotent Lie Groups

In this section we present an example of the structures discussed above but in case of nonhomogeneous Lie group.

Let $ and ^ be strictly increasing odd bijective real functions on R with (0) = 1 = = W(0). We consider the following Lie group action on = R3

w ow = ($-1($(x) + $(x)), V-1(V(y) + V(y)),z + z + $(x)ty(y)). In this setup we have the following (left invariant) fields

d 1

Xf (w) = - f(w o (s, 0, 0)),=o = -^xfxf (w)

1

Yf(w) = - f(w o (0, s, 0))li=a = (^8y + $(x)dz)f(w)

and associated reflections

ox(w) =w o ($-1(-2$(x)), 0,0) = (-x, y, z) oY(w) =w o (0, ^-1(-2^(y)), 0) = (x, -y, z - 2$(x)y) for which we have the following fundamental relations satisfied

X(f o ox) = -(Xf) o CX; Y(f o oY) = -(Yf) o oY

= Z = dz

and the following operator play a role of dilation operator

D = sinh(x)X + sinh(y)Y + 2zZ. In the present case we propose to consider the following DeMazur operators

Av m = K f-fo °x ■ ^ m = K f-fo °x Ax (f) = K $(x) ; Ay (n = K *(y) .

Now for

Tx = X + AX; TY = Y + Ay.

we have

Tx (f o ox) = -(Txf) o ox; TY (f o oY) = -(TY f) o oY. and with a measure

dv = (x)($(x))2K$' (y)($(y))2Kdxdydz the following formulas of integration by parts hold

J (Tx f )g dv = -J f(Txg) dv, J (TY f)g dv = -J f(TYg) dv.

Hence one can build up an interesting analysis in this case. We recall that a special case the group with $(x) = sinh(x), ^(y) = y one can find e.g. in [15] (Ex.1.2.17). The DeMazur operators in this case could be related to models with sinh-2 interactions (for the theory of completely integrable systems see e.g. in [37; 65]).

8.2. Nilpotent Lie Groups with more general T

Here we provide an example where number of Aa's are necessary in a natural way. We consider generators of the form

Xj = dXj + £ £ Bjk;i xkdxi

k=j;k<m m+l<l<n

with some constants Bjk;i = 0, for j = 1,... ,m and all k = j, I = m + 1,... ,n. Then each Xj admits a number of classical reflections , k = j, (i.e. reflections defined by matrices with entries ±1) which change the sign of j and k coordinates. In this case a function Xj — xk is antisymmetric and can be used to define the following DeMazur operator

Aikf = k L—lm. xj xk

Next define

T3 =Xi + yj Ajk.

k=j;k<m

If we introduce a measure

dv = JJ (xj — xk)2k d\n,

j=k;j,k<m

with density involving VanDerMonde determinant. Then all Tj, j = 1,...,m, satisfy integration by parts formula. One can redefine these operators by adding suitable DeMazur operators Aj corresponding to nonclassical reflections and defined with nonsymmetric functions xj, provided we modify the measure by factors x^K.

There is also a more general conclusion out this example: For any classical crystal-lographic group one has specific weights (as described e.g. in [36]) and a corresponding Dunkl type theory. One can use this theory for coordinates in the first strata (of coordinates of homogeneus dimension 1) and extend it to a nontrivial nilpotent Lie group case (possibly multiplying weight by suitable product if one would like to add Aa with non-classical reflection). This point of view suggest an interesting classification of nilpotent Lie algebras for which restriction of Cox to coordinates in the first strata conicides with a given classical crystallographic group (and for any classical crystallographic group there exists infinitely many extensions).

9. Appendix: Functional Representation of Groups

An interesting problem is how to represent groups in terms of classes of functions with operation of composition of functions and how to classify such representations.

Such framework is rich enough to accommodate infinite discrete groups. For example, one knows that for any n e N, there exists an uncountable set of solutions of an equation

= I

in terms of continuous functions mapping a unit interval into itself, (cf. [53]).

To illustrate the possibilities we provide here a functional representation of the celle-brated Baumschalg — Soliltar group which is defined as folows, (cf. [18])

{a, b : a-1 bma = bn}.

By a direct computation one can show that such the group can be realised by

a = log0-' op o exp07 (t) b = log0 k o q o exp0k (t) with k = j + 1. Then

a-1 = log0-7 op-1 o exp°(t) bm = log0 k oqm o exp0 k (t). And for k > j we have

a-1 bma = log0 op-1 o log0(k-j) oqm o exp°(k-j) op o exp°j(t).

If we choose k = j + 1. Then we have

a-1 bma = log0,7 op-1 o log oqm o exp op o exp0^(t)

= log0,7 o log oqmp 1 o exp op-1p o exp0:>(t) = log0 k o(qmP-1) o exp0k(t) = bmp-1 That is the group is realised for given m E N with n = mp-1 provided p devides m, although it is possible that the noninteger values of n could be realised in the framework of functional equation theory.

Acknowledgements: The author would like to acknowledge hospitality of various institutions visited while thinking of this project, including : ETH (Zurich), Toulouse (Paul Sabatier, CNRS), IMA (Minneapolis), CNRS-PAN Institute (Krakow). He would also like to acknowledge the support of Royal Society RMA.

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КРИСТАЛЛОГРАФИЧЕСКИЕ ГРУППЫ ДЛЯ ПОЛЕЙ ХЕРМАНДЕРА

Богуслав Зегарлински

Профессор, Факультет естественных наук, Кафедра математики,

Imperial College London

[email protected]

South Kensington Campus, London SW7 2AZ, UK

Аннотация. Это предварительная статья о кристаллографических группах полей Хермандера. Мы описываем картину, возникающую в анализе расширенных групп. В частности, мы вводим понятие и приводим примеры кристаллографических групп, связанных с системой полей Хермандера, а также обсуждаем некоторые связанные вопросы анализа.

Ключевые слова: расширенные группы Ли, некоммутативные операторы типа Данкла, полугруппы Маркова, оценки ядер уравнений теплопроводности и энтропии.