Поточечные оценки и критерий существования решений сублинейных эллиптических уравнений

DOI:

In [78] we have introduced an infinite Coxeter group on two generators associated to the Heisenberg group H1 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 X j , j = 1,...,N , on a differentiable manifold G, given by a family of maps O k : G ^ G, к E N, satisfying the following condition

X j (f ° O k ) = 52 a jk,u (X i f ) о о                          (1)

i,l

(possibly only on certain sub-space of functions) with some a j_k,_il E R. In particular we introduce the following definition.

Definition 2.1. A map о : G > G satisfying

X j (f ° о) = -(X j f) ° о,     о ° о = id                     (2)

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

(t, x) • (t ' , x ' ) = (t + t ' , x ' + e ^tB x)

Then for

a(w) = (— t, e^{-2t B} x ) and

X = d t + (Bx) • V, where V denotes the gradient with respect to x, we have

X(/ о a) = -(X/) о a

For components d j of V the reflections are simply given by maps X j ^ (—1)^5y X j , j = 1, ..,n. the corresponding set (1) of maps includes also linear maps (with respect to x).

3.2.    K-Groups

Let G = R ” 0 x R ” ¹ x ... x R ” , with n ₀ >  n 1 > .. > n _r , r G N. This is similar to B-groups except that t is replaced by a n ₀ -vector t and the matrix in the definition of composition is lower triangular with the sub-diagonal part consisting of blocks n _s x n _s-1 with rank equal to n _s for each s = 1,..,r. In this framework we consider the following fields (homogeneous of order one with respect to a natural gradation)

Xj = 9t3 + (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 В . In this case we have a nice family of reflection maps a j , j = 1, ..,n ₀ , satisfying

Xj(/ о aj) = -(Xjf) о aj explicitly given by aj(w) = w • (-2{5jiti}, 0) = ( {( И'-ti}, xk - 2 ^ Bkjitjti) i with j,i = 1,..,n0 and к = 1,..,r. These maps are idempotent, i.e. aj о aj = id, and (in general) they do not commute with each other. There are also some other reflection maps given as follows aj(w) = ({(-1)5з'гti}, -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', у + y' ,z + z ‘ + 2a(yx' — xy')), a = 0, and fields

X = д _х + 2ал ₂ д _г ,     Y = d _y — 2an 1 д .

with n 1 (w) = x and n ₂ (w) = y, we have the following corresponding reflections

Yx (w) = (—x,y,z — 4axy),     оx (w) = (—x,y, —z), yy (w) = (x, —y, z + 4axy),     oY (w) = (x, —y, —z)

and a common one о = (—x, —y,zY

Moreover, for

Y+(w) = (y,—x,z), we have о-- о o+(w) = id = 0. о o_,

We note also the following relations

Yx orx = Yx о Ox, о о ox = ox о о, Y о Ox = Ox о Y = Yy, and

Y + о Y _y = Y _x о Y ₊ , Y _ о O _Y = Y _x о Y _ , о о ₊ = ₊ о о = _{_} ,

^Y - ^(w) = ^(—y^,x,z)

o4 • j o4         о 2     ^л o2

Y _ = id = Y + ,    Y _ = Y = Y + .

Y y о & y = ^ y о Y y ,

О о Y _Y = Y _Y о O, Y о Y y = Y y о Y = Y x ,

Y + о r _Y = r _x о Y _ ,

Y _ о r _Y = r _x о Y ₊ , о о _{_} = _{_} о о = ₊ .

For y ₊ , y _ , we have

^X(f ^{о Y} + ^)(w) = ^{X (}f ⁽ⁿ 2 , ^{— n} 1 , ⁿ 3 ^))(w) =

= —(d y /) о Y + (w) + 2a n 2 (w)(9 _z /) о o . (w) =

= (—(d y / ) + 2an i о о _ (Э _г /)) о о . = — (Y/ ) о o . (w)

where we have used п ₂ о o _ = n 1 . Similarly

Y(/ о o _ )(w) = Y(/ (—П 2 ,П 1 , n 3 ))(w) = — (З _х / ) о Y _ (w) — 2a n i (d _z / ) о o _ (w) =

= (— (д _х / ) — 2an i о о + (Э _г /)) о o _ (w) = —(X/ ) о o _ (w)

where we have used п 1 о о ₊ = п ₂ . Finally for the reflection O(w) = (—x, —y, z), one has

X(/ о О) = X/ (—П 1 , —П 2 , П з ) = — (д х / ) о о + 2a n 2 (d _z / ) о о =

= — (дх/) о О — (2ап2д,/) о о = —(X/) о о and similarly

Y(/ оо) = —(Y/) о о.

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

Ln = (oxoy)n  Lo = id L-n = (oyoy- )n,    n G N and

R _{n +} 1 = L n о o x = o x о L n ,    and    R _ _n _ 1 = L — _n о o y = oy о L — n ,    n G N

With this notation we have

L n о L _m

= L

n+m ,

and

R n о L m

^R n+m •

The subgroup {Ln}neZ is abelian and moreover any (group) commutator in the group generated by ox, oy belongs to {Ln}neZ. The group is furnished with the following intertwining structure o± о Rn+ 2 = R-n-2 о o±

and o ± о L _n = I _-n о o ±

Finally we mention that the fields X and Y translations

T x (w) = (x + a,y,z + c),

are preserved by the following families of 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 = (u, t) one has the following natural reflections o *,± (w) = (u*, ±t), o(w) = (— u,t) = o ±² (w) where o ± (w) = = (yiv,t). Then d" _x (w) = д о oy(w) with 5" _Y (w) = o *_ (w). Finally oy (w) = (u*,t + + 2a^(— iu ² )) and o _x = o ₊ о oy о o - .

3.4.    Examples of Higher Order

• (i)    Let G = R 4 and

w • w' = (x _i + x' _i ,x ₂ + x 2 , x ₃ + x' 3 + x' _i x ₂ ,x ₄ + x 4 + x' 1 x² + 2x ₂ x ‘ ₃ )

with dilation 5 л (w) = (Ax _i , Ax 2 , A ² x ₃ , A ³ x 4 ). Consider the following system of fields

X 1 = d i + x 2 ^ 3 + x 2 ^ 4 ,    X 2 = 8 2 .

See e.g. [15] for such the structure. Define о 1(w) = w • (-2x1, 0, 0, 0) = (—x1, x2, x3 — 2x1x2, x4 — 2x1x2).

Then we have o" 1 о o" 1 = id and

X 1 (/ о o i ) = X 1 (/(—П 1 , П 2 , П 3 — 2П 1 П 2 , П 4 — 2П 1 П 2 )) =

= — (d i / ) о o i — 2 п 2 (д з / ) о o i — 2 п 2 (д 4 / ) о o i + п (д з / ) о o i + ^( 8 4 / ) о o i =

= ^{— (X} i ^/ ) ^{о o} i .

Also for oi(w) = (—xi, x2, —x3, —x4),     д = (—xi, —x2, x3, —x4)

one gets

^ 1 (/ ° 0 1 ) = — №/) ° ^ 1 ,   a^d  ^ 1 (/ °0) = — (Х ^ / ) о а.

In all cases we have id = о 1 ° о 1 = 0 ’ 1 ° 0 ’ 1 = д 1 ° д 1 . For the second field the relation

^ 2 ^(/ ° ^о 2 ⁾ = ^{— (}^ 2 ^/) ° ^o 2

is satisfied with any map of the form о2(w) = («Ж1, —x2 + в^1, 7x3 + 5x21, ex4 + Zx^)

with а, в, Y, 5, e , Z € R. Such the map satisfies condition o ₂ ° o ₂ = id if additionally one requests (а, 5, y , e) = (—1, 0,1,1), (—1, 0, — 1, — 1) or (а, в, 5, Y, e) = (1, 0, 0, — 1,1) and (а, в, 5, y , e , Z) = (1, 0, 0,1, —1, 0). For the last choice we obtain д which also reflects the first field. This is a situation which is different from the H 1 group.

• (ii)    Let G = R ⁴ and

w • w ‘ = (x 1 + x ‘ 1 , x ₂ + Я 2 , x ₃ + x 3 + x ‘₁ x ₂ , x ₄ + x 4 + -x ₂ (x ‘ 1 ) ² + X ₃ x 1 )

with dilation 5_A(w) = (Ax, Ax ₂ , A ² x ₃ , A ³ x ₄ ). Consider the following system of fields

%1 = д1 + Х2З3 + Хз^4,    %2 = ^2, see e.g. [15]. Define о 1(w) = w • (—2x1, 0, 0, 0) = (—x1, x2, x3 — 2x1x2, x4 + 2x2x2 — 2x1x3).

Then we have

^ 1 (/ ° C 1 ) = Х 1 (/(— П 1 , П 2 , П 3 — 2П 1 П 2 , П 4 + 2П 2 П — 2П 1 П 3 )) =

= —(д 1 /) ° о- x — П 2 (д з /) ° о- ₄ + (2П 1 П — П з )(д 4 /) ° th =

=   (-V/) ° ° 1 .

For о Kw) = (—x4, x2, —x3, Ж4) or д = (—x4, —x2, x3, — Ж4)

we get also reflection of the field X 1 . In all cases we have id = 0 1 ° о 1 = 0 1 ° 0 1 = д ° д. For the second field we have

^ 2 ^(/ ° ^о 2 ⁾ = ^{— (}^ 2 У) ° ^о 2

with any map of the form о2(w) = (аX1, —x2 + ex1, Yx3 + 5x2, Ex4 + Zx4x3)

with а, в, Y, 5, e , Z € R. Such the map satisfies condition o ₂ ° o ₂ = id, if additionally one requests (а, 5, y , e) = ( —1,0,1,1), (—1, 0, —1, —1) or (а, в, 5, y , e) = (1,0, 0, —1,1). We remark that in the present case also д reflects X ₂ .

4.    Problem of Effective Description of Coxeter Groups

The first emerging general issue is as follows.

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

In the second case we ask for more than just the description of the group V j CX j generated by all CX j , 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 R ⁿ⁺¹

^Xj = dX j ⁺ ^ ^{B x} g? ¹¹       1=3

where B^ is a nonsymmetric matrix, with associated reflections given by

^C j (^w) = ^({( 1) ³' ^X i ^} i =1 ,..,n , ^{z   2} ^ ^B ij X j X i ).

i=j

The commutativity structure can be obtained by constructing a graph with n vertices in which vertices x _i and X j are connected by an edge e _ij iff B^ = 0, and in this case C _i and C j do not commute. C j ,j = 1,..,n, generate a Coxeter group of infinite order. Additionally in the present case CX j contains reflections

C j (w) = ({(-1) ^{5 i3}' X i } i =i ,..,_n , -z)

and

C j (w) = ({(-1)^s« (-1) ⁵-= » x , } , =i..... , ,z)

and possibly some others (including also number of intertwiners). Moreover the field X j is invariant with respect to Coxeter group acting on variables which are not involved in definition on X j as well as with respect to traslations T j (w) = ({x _i + a j Sy } _i = ₁,..,„ , z + Y j ), with a j , Y j € 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 C j of a Coxeter group on R ^v , then one can ask what are the fields X which satisfy the reflection relation

(X/ о C j ) = -(X/) о C j .                            (**)

For example one knows (see e.g. [70]) that the Painleve´ II u = 2x3 + tx + b - -, represented as a Hamiltonian system with canonical variables

• —             — • ,   2 , ^t

q = и, р = и + и + ^

as follows

2 1

^q др ^{р q} 2

р = - д! = 2рq + b admits a Coxeter group of Ba¨ cklund transformations generated by the following map

^ i (q,p^,b) = ⁽q + ^b,p, ^{— b)} р

^ 2 (q,P^,b) = ^(—q, -р + V + ^t, 1 ^{— b)}.

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

^ i (f ° O i ) = -(^ 1 /) ° O i ,

^ 2 ^(/ ° ^O 2 ⁾ = ^{— (}^ 2 ^/) ° ^O 2 ,

%i =   + (—F + ai) Г + в1F db 2р dq др

W = I + (—2q) 1 ₊ y₂ ^

dq        др db with the functions a1 ° o1 = — a1, в1 ° O = в1 and Y2 ° o2 = Y2- (It would be interesting to repeat that for other Paileve´ problems using identification of [40; 61].)

Homogeneous Lie groups on R”

Consider now the case when the group action is given by

(w • w') k = X k + x' _k + Q k (w, w ' )

with Q _k (w, w') = Q _k (x j , X j : j = 1, ..,k — 1), satisfying

Q k (w,w' ) |_X , ₌ o _;j₌ i ,..,k- i = 0.

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

O j (w) = w • r ( ² ) (w), with (г ^(г) (w)) j = — 2 dij X i ,                   (3)

is a reflection, i.e. о ° o _i = id, if the inverse to r ⁽ⁱ⁾ (w) is given by —r ⁽ⁱ⁾ (w). This is because (r ⁽ⁱ⁾ (w • r ⁽ⁱ ) (w))) j = —25 _i,j (w • r ⁽ⁱ⁾ (w)) _i = 25 _i,j X _i and this defines the inverse element to r ⁽ⁱ⁾ (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

Xif = lim ^{f ( w} ° ^{£ i ) — / ( w )} = ^f (w ° _£ i )„o.

c > o             e                de

We have the following result.

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

С-Q k ((G i ( w ) O E i ) i1, ( —25ij(Gi(w)i + E))_j<_k-1

)

- |— =0

C — ^Qk ^((G i ^(w)) l1, ⁽^ij^-)j1)^      .

Proof. Using the definition of the fields X i , we have

X i (/ o G i ) = lim

— > 0

/ O G i (w O £ i ) - / O G i (w)

ε

with

(G i ( w O - i )) k = ( w O - i O r ⁽ⁱ⁾ (w O - i )) k

= ( w O - i ) k — 25 ik ( w O - i ) i + Q k ( w O - i , r ⁽ⁱ⁾ (w O - i ))

= (—1) ^{5 ife} ( w O - i ) k + Q k ( w O - i , r ⁽ⁱ⁾ (w O - i )).

Using (r ⁽ⁱ⁾ (w о E _i )) _k = —25 _ik (w _i + e) and our assumption about Q _k ’s, we have

W k ,

( G i (w O E i )) k = <

-

W i

^{— -},

^W k + Q k ^{(( w O -} i ⁾ l1, ^{( 2}^ij (^Wi + ^-))j1),

/or к < i /or к = i /or к > i

Hence

X i (/ о G i )(w) = —(d i / ) O G i (w)

while we have

+ ^ ~ — Q k ((W O E i ) i1, ( — Ж) (Wi + -)

k>i

) J

(dk/) O Oi(w)

|—=0

(Xi/) O Gi(w) = (di/) O Gi(w) +

+ ^     , ^Qk⁽(^Gi ^(w))l1, ⁽^ij^-)j1)

k>i l^

(3k/) O Gi(w).

- |—=0

Thus to get (**) we need

[c^Qk ((Gi(w) O Ei)l1, (-25ij(Gi(w)i + e))_j<_k-1)]

—

, ^Qk⁽(^Gi ^(w))l1, ⁽^ij^-)j1)

.“^—                                        J|—=0

.

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

Xj ^ -Xj + qj(xk, к < j)

with even polynomial qj if the components x^, к < j, just change a sign, as we have seen it in case of Ba¨ cklund transformation.

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

Y = dy — 2aydz,    Ji = dx — 2axdz would satisfy

Y(f ° ⁰y) = -(Yf) ° ^Oy,     J°(f ° °_v) = -(J°f) ° °_v.

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

In [78] where we have studied Hi and introduced ax (w) = (—x, y,z — 4axy),     aY (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,y, n = x +    , z = y — i

2ay         2ax as follows x ° ax = —x y ° ax = y n ° ax = n — 4x Z ° ax = —Z x ° aY = x   y ° aY = —y n ° aY = —n     Z ° aY = Z — 4y

To this we add relations coming from additional ’s described above as follows ax(w) = (—x, y, —z),     oy(w) = (x, —y, —z), a(w) = (—x, —y, z)

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

for which we get n ° a = —n, Z ° a = — Z n ° ax = —n,   Z ° ax = Z n ° Oy = n,     Z ° Oy = —Z and the following intertwining relation n ° a+ = Z.

The Coxeter group generated by a_x, a_Y, a± preserves the Kaplan norm, while a group including also generators a_x, a_Ywould preserve the set of functions spanned by the following

Ф,(w) = £ «, £ t; xA(z-"4?'z ф. — n4^xy)) t  ;^z                q defined with q G N and (^q)q = 1, ^q = 1, Л G R and a function ф.

Note that ф9 о av (w) = ^ ^ e^+n q жУ)ф(2 + n—xy))

and

ф_уо a_v(w) = ^^ e^tX(z+nq ^^ф (: + n—xy)).

q

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 σ₁)

• (i)    X1X2X3 + X1X4 + X^ + X^ + x4

• (ii)    X1X2X3 + X2X4 + x2 + X^ + x1

7.    Linear Operators

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

Problems 2.

(P2.i) Describe representations of the groups Vj CX j on spaces of functions.

Given a G CXj, a°2 = id, we can introduce the following linear operators.

DeMazur Operators

• a₃,. = Kt (i -^z■

ησ with Ia f = f о a and a differentaible function na satisfying

Пи о a = —na    and    Xjna = 1, and a constant Ka = 0. First we can introduce it on smooth functions vanishing on an open set containing the set Sa = {na = 0}, and later extend by continuity. One can see that Aj,a vanishes on a-symmetric functions (i.e. functions satisfying Iaf = f о a = f), and we have the following:

Boundary Operator Property

^Ala = 0.

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

Generalised Fields q3,a

= ^Xj + ^Aj,a.

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 (—25jjXj)), we note that

^Aj,a^{f (w)} = ^2Kj

/ 1 d_S(X,f)(y- )

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

{X3, I.} = XI. + I.X3 = 0,     {A3,., I.} = 0, so also

{T3,.,I.} = 0.

More generally one could introduce

^T3 = ^X3 + E^e.^A3,.

aECXj with в. G R, Ec£cx3 в. = 1.

With these extended fields, assuming later on в., к. > 0, we introduce the following second order operator

^ ETG

3',.

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).

/(wo) = min/ =^ £/ (wo) > 0

Proof. Since A. = 0, we have

£/ = EW + {X3 -A., })/■

If a minimum point x_o/ U.S. we have

^X3^A._j^/(xo) =  ^^ ^(x3^/(xo) ^{- x}3^(/ о .3X^)) ^-^^ ^ ^{(Жо) A}.J⁽xo)

³           П., ^(xo)                                             П., ^(xo)       ³

A.X3$ = ^-x (X3/(xo) — (X3/) о =3(xo)).

³          n., (xo)

Hence, using reflection property, we get

{X3-A„,J}(xo) = ^^.4X3/(xo) - ^к:_i^Xзn^i^^Xo> A.,/(xo). n., (^xo)                  n.₃(^xo)

The first term on the right hand side is equal to zero at the minimum point. If X3n., > 0 as we assume, we have

K.X3n.₃(xo)              к. X3n.₃(xo)

• ^{3     3}     A. /(xo)=   ^{3     3}     (/(.3-xo) -/(xo)) > 0

П2,^(xo)       ³               П2,^(xo)

because at the minimum point /(ojx₀) > /(xo). In case the minimum point x₀E U_oS_o, we need to use limiting procedure necessary to extend definition of A_a to reflection ivariant points. As we pointed out above (in case of nilpotent Lie groups) when Xjn^. = 1, one gets

X,A_C./}(xo)=2KjXj²/(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 (oyo_yz^uy and (ayo_y)ⁿo_y, ^ E N, which would satisfy the reflection relation with the fields of the form X_k = Э_х + 2kat/<9_zor Y_k = d_y — 2kaxd_z, for suitable к 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

^ E =‘TX j,k,o with гк E (0, to) satisfying ^k гк < то.

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

Xi log p = 2Kin-¹

for all г (almost everywhere), then for all Ti = Xi + Ai the following generalised integration by parts formula holds j T (д')/ pdv =

— j gTi (/)pdv.

Proof. Since for a given oi invariant measure v, we have j A^/dv =

-

J gAi(/)dv + J

gKin— 1/dv.

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

Xi log p = 2Kin—¹

for all г (almost everywhere).

If the fields X_i’s generate nilpotent Lie algebra, (6) provides necessary conditions on n_i’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 X_iagain to (6) and summing over i’s one obtains a relation

£^Xi2^)log P = ^ ^2Ki^Xin-1.                       ⁽⁷⁾

i                                   i

In particular if X an = 1 we get, with X • X = (2, X,2), the following

X • X log p = — У^ 2к,п- ².

г

In case when X, are generators of free nilpotent Lie group, we have X, = д, + Sj>, adj with a dependent on the proceeding coordinates, with nj = ^j one can choose

P = |[ " .

г

A possibility of other choices of η’s is discussed later in section 8.2.

Canonical Markovian and Dirichlet Forms

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

— У gC/d_W = ^g • T/d^ = £ У (T,g)T,/d^. г

On the other hand the canonical Markovian form of ℒ is as follows

Г(/) = 1(£/2 — 2/С/) = |X/12 + £ ^ (А,/)2. 2                                2 К г г

If X,n, > 0, then this form is clearly nonnegative. (In particular this is satisfied for generators of free nilpotent Lie groups with η’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

I Г(/)* = I

Tg•T/d^>

У |X/|W

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(p)

Next we remark that for density p and a differentiable function /, we have by Leibnitz rule X(/p^1/2) = (X/)p^1/2+ 2/p^1/2(Xlogp) and hence for d^ = pdv, we get

У |X/|W >

- У ^X(/p^1/2)•^/p^1/2(Xlog p^)dv + У ^/24^|Xlog p^|2d^.

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).

У |X/|²d^.

>//²(4|Xlogp|2 ⁺ |X • Xlog^p)

d^.

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 кг > 1

J^|T/|2^dW > J ^/2 ^^(к2Пг 2 ^{+ к}г^ХгПг 1)^^.

In particular if Xn = 1 one gets

У |T/|²dw > У /²^ Кг(к — 1)Пг 2dw.

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

У/2 log ff^dv < г / |X/^ + C(г) J f 2dv for any г E (0, to) with C(г) ~ const log г for small г. In particular substituting in this inequality a function /p2, we get

/ ²

^{f /}2^dW

J / 2 log

dw + j /²log pdw < г j |X/|²dw + г j /X/ • X log p dw + + 4 У /2 • |Xlog p|2 dw + C(г) У /²dw.

Hence

/ ² f ^/2^dW

I /²log

dw + У /2 log pd^ < г У |X/|²d^ —

-г У /2 (4 |X log p|2 + |X ^ Xlog ^p) dw + +C(г) У /²dw

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 г E (0,1) we have log+ p-1 < C г Q |X log p|2 + | X • X log p) + DC (г)

with some constants C,D E [0, to) independent of /. Then

У /²log -j/^< г У |T/l²d_W + C‘(г) У /²d_W

with C‘(г) ~ const log | for small г.

Using this and [25] we obtain the following important implication.

Corollary 7.6 (Ultracontractivity Estimate). The semigroup P_t = e^t£, with C = T • T, is ultracontractive i.e. for t > 0 the operator P_t : L1(^) ^ L^(^) is bounded. Hence

Pt/ (w) = J /(w)ht(w,wMdw)

with a bounded (uniformly away from t = 0) smooth heat kernel h_t(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 j e^ht(^,w)d^ < C(t,w) < to with d a natural (homogeneous) distance from a given point on the underlying space and some positive constants a, p. Then one can follow [49] as follows: Using Chapman — Kolmogorov property ht(w',w) = I ht/2(w',w)ht/2(w,w)^(dw)

and elementary inequality

в

0 < ——d (w , w) + ;y(d^a(w', in) + d^a(w, w))

for some C E (0, to) (independent of the points w,w',w), together with Holder inequality, one gets ht(w',w) < e-^d“(w‘’w) •

^У ht/₂(w', w)e^eda(w‘^w)p.(dw)^

• (У ht/₂(¹-^u,w)e^eda(w,w)^(diu)^

< e-^d^a(w‘,w)_5t•

^У ht/2(w',w)e

1/2                                                    1/2

eda(w‘’™Mdw) I • I / ht/2(1u, w)eeda(w’^^(div) I 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 Groups8.1.    Nonhomogenous Nilpotent Lie Groups

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

Let Ф and Ф be strictly increasing odd bijective real functions on R with Ф’(0) = 1 = = Ф‘(0). We consider the following Lie group action on Н_ф,_ф= R3

w о w = (Ф^-1(Ф(ж) + Ф(Ж)), Ф^-1(Ф(у) + Ф(у)), z + z + Ф(ж)Ф(у)).

In this setup we have the following (left invariant) fields

X У(w) = 4/(w о (c,0,0))к= =^/

О

У/(w) = т/(w о (0,с,0))1£=0 = (ду ■ ф(ж'дV(w) dc and associated reflections ох(w) = w о (ф 1 ( 2Ф(ж н. 0, 0) = (-ж, у, z)

оу (w) = w о (0, Ф^-1(-2Ф(у)), 0) = (ж, -у, z - 2Ф(ж)у)

for which we have the following fundamental relations satisfied

X(/ о ох) = -(X/) о ох;    У(/ о оу) = -(У/) о оу

= Z = дг 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

^х (f)- к ^/^ф/^фох;

Ау (/) = к

/ - / о ох Ф⁽У)

Now for we have

^Х = ^X + Ах;

^х(/ о ох) = -(Тх/) о ох;

Ту = У + Ау.

Ту(/ о оу) = -(Ту/) о оу.

and with a measure dv = Ф‘(ж)(Ф(ж))2кФ‘(у)(Ф(у))2кижиу^5

the following formulas of integration by parts hold

I (Тх/)gdv = -у / (Тх g)dv,

I (Ту/ )gdv =

- !/(Туg)dv.

Hence one can build up an interesting analysis in this case. We recall that a special case the group with Ф(ж) = sinh(ж), Ф(у) = у one can find e.g. in [15] (Ex.1.2.17). The DeMazur operators in this case could be related to models with sinh^-2interactions (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 Л^’s are necessary in a natural way. We consider generators of the form

X j = bx + у у Bjk-i XkBXl k=j;k

A_jk / =

/ - / ° Qjk κ

Xj - Xk

Next define

^Tj = ^Xj + У^Лjk. k=j;k<m

If we introduce a measure

Av = П  (Xj — Xk)2k AAn, j=k;j,k

There is also a more general conclusion out this example: For any classical crystallographic 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 Л^ 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 G N, there exists an uncountable set of solutions of an equation

Фоп = 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^-1b^ma = bⁿ}.

By a direct computation one can show that such the group can be realised by a = log0'’' op о exp0-7' (t)

b = log^ofcoq о exp^ofc(t)

with к = i + 1. Then a-1 = log0-7' op-1 о exp0- (t)

b^m = log^ofcoq^mо exp^ofc(t).

And for к > i we have a-1bma = log0- op-1 о logo(fc—•7) oqm о exp0^--7') op о exp0- (t).

If we choose к = i + 1. Then we have a-1bma = log07 op-1 о log oqm о exp op о exp07 (t)

= log⁰⁷о log oq^{mp 1}о exp op^-1p о exp⁰-⁷' (t)

= log^ofco(q^mp-1) о exp^ofc(t) = b^mp-1

That is the group is realised for given m E N with n = mp^-1provided 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.