ASYMPTOTIC BEHAVIOR OF POISSON KERNELS ON NA GROUPS
DARIUSZ BURACZEWSKI, EWA DAMEK, ANDRZEJ HULANICKI
Abstract. On a Lie group S = N A, that is a split extension of a nilpotent Lie group N by a one-parameter groups of automorphisms A, a probability measure µ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a µ-invariant measure m on N . Properties of m have been intensively studied by a number of authors. The present paper deals with the situation when µ(A) = P(st ∈A), where R+ ∋ t → st ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. This paper deals with the most general operators of this kind. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudo-differential calculus for operators with coefficients of finite smoothness is formulated and applied.
1. Introduction
The present paper is an outgrowth of the study of the Poisson kernels on boundaries of symmetric spaces, on one hand side, and invariant measures for Markov processes obtained by random sampling of affine transformations of Rd, on the other. The fact that the Poisson kernel for a symmetric space S, identified with the solvable group N A in the Iwasawa decomposition, is the invariant probability measure m for a Markov process on the boundary N of S, was discovered by H.Furstenberg over forty years ago. In the present paper, we study the tail behavior of m in the particular case when m is invariant for a diffusion process t → st on S = N A, S acting on S/N on the left:
s : N ∋ x → s · x ∈ N. That is
(1.1) µ˘t∗ m = m, or
Z
Eef (st· x0)dm(x) = Z
f (x) dm(x),
where {µt}t>0 is the semigroup generated by a left-invariant second order subelliptic differential operator L on S. Then m has density m(x) and we study the asymptotic of m(x) as x goes to infinity along a ray. When L admits non trivial bounded harmonic functions then m reproduces them [DH1]via
F (xa) = Z
N
f (xΦa(x)) dm(x).
If N = R and the transformations R∋ x → ax + b ∈ R are sampled according to a probability distribution µ, the tail asymptotic of the invariant measure is quite well understood, see e.g. [DF]
and [GM] for a fairly recent account. For a general measure µ the following conditions are sufficient for the existence and uniqueness of the invariant measure m.
Research partially financed by the European Commission IHP Network 2002-2006 Harmonic Analysis and Related Problems (Contract Number: HPRN-CT-2001-00273 - HARP) and by KBN grant 1 P03A 018 26. D.
Buraczewski was also supported by Foundation for Polish Science (annual stipend for young scientists, 2004) and by the Polish Academy of Science, being on a special position at the Institute of Mathematics there. The final version was prepared while the first and the second named authors were visiting at Department of Mathematics, Universit´e de Rennes and the third named author was visiting at Department of Mathematics, Cornell University. They would like to express their gratitude to the hosts for hospitality.
1
(1.2) Z ¡
| log a| + log+|x|¢2+δ
dµ(xa) <∞,
(1.3)
Z
log a dµ(xa)≤ 0.
Under these assumptions the measure m is a probability measure, if, and only if, (1.4)
Z
log a dµ(xa) < 0 One also assumes that
(1.5) µ is not supported by a line
Suppose that conditions (1.5) and (1.2) hold and, moreover, that (1.6)
Z
aλ dµ(xa) > 1 for some λ.
Then it is easy to verify that there exists the unique α > 0 such that (1.7)
Z
aαdµ(xa) = 1.
This α describes the asymptotic behavior of m.
THEOREM [K, Go]. We have
t→+∞lim tα[t, +∞) = c+, lim
t→−∞tα(−∞, t] = c+ and c++ c− > 0.
In the case when m is unbounded we have
THEOREM[BBE]. Under conditions (1.5) and (1.2), Z
log a dµ(xa) = 0 implies m[tr1, tr2] = L(t) logr2
r1
,
where L is a slowly varying function. So m is a Radon measure with logarithmic tail.
However, our knowledge about the asymptotic behavior of m, for a general measure µ, even in the case N = Rd, d > 1, is very limited. In the present paper we describe it for the measures m invariant under ˘µt, i.e. such that (1.1) holds. Our particular interest has been caused by questions concerning the Martin boundary for harmonic functions on homogenous manifolds with negative curvature [D1]
[DHU]. It is also the only case when asymptotic for the invariant measure in multidimensional case is known. Of course our estimates agree with those quoted above for the case N = R.
Our earlier work [DH2] does not allow to consider semi-groups of measures µt generated by operators with an independent drift in the direction on N and these are the only cases when, for N = R, one of the constants c+or c− can vanish. To understand this phenomenon for the measures m invariant under ˘µt and to formulate appropriate conjectures in the multidimensional case for general measures µ, it is necessary to study the operatorsL on S in full generality.
2. Preliminaries and the Main Theorem The setting of the present paper is as follows.
Let S = N A be a split extension of a nilpotent Lie group N by a one dimensional group A = R+ of dilating automorphisms Φa:
(x, a)(y, b) = (xΦa(y), ab).
LetN be the Lie algebra of N. “Dilating” means that in the Lie algebra A of A there is H0 such that
(2.1) the real parts of all the eigenvalues of adH0 :N → N are positive.
This includes, in particular, semi-direct products of a homogeneous group N (in the sense of E.M.Stein, cf e.g [FS]) with A acting by dilations. In this case,
(2.2) Φa(x) = (ad1x1, ..., adnxn) with dj> 0.
There are three important occurrences of Lie groups S = N A:
(i) the groups of of affine transformations Rd∋ x → ax + b ∈ Rd,
(ii) rank one symmetric spaces G/K identified with the N A part of the Iwasawa decomposition G = N AK, or more generally
(iii) homogenous simply connected Riemannian manifolds of negative curvature. In this case S is the simply transitive solvable Lie group of isometries of the manifold.
(i) and (ii) are special cases of (2.2), as (iii) is not.
We consider a second order left-invariant operator L =
Xm j=0
Yj2+ Y
on S and we assume that Y, Y0, ..., Ymsatisfy the weak H¨ormander condition, i.e. Lie{Y, Y0, ..., Ym} is equal toS, the Lie algebra of S.
Under the canonical homomorphism of S onto A = R+ the image ofL, up to a constant, is equal to
(a∂a)2− αa∂a or − αa∂a.
Here we make an additional assumption: the term (a∂a)2 does appear. If it does not, that is the image ofL is only −αa∂a, then the measures µtviolate either (1.3) or (1.6), depending on the sign of α. Moreover, if α > 0, then we expect very fast decay of m, since in the case of
−a∂a+ (ad∂x)2 on R× R+
the invariant measure has density m(x) = c exp[−dx22] on R. In the case we consider i.e. (a∂a)2 does not vanish, the decay is only polynomial and we attempt to find its is precise asymptotic.
We write the operatorL as L =
n+1X
i,j=1
αi,jWiWj+
n+1X
i=1
αiWi,
where W1, . . . , Wn is a basis of N , Wn+1 = H0, the matrix [αi,j] is semi-positive definite and αn+1,n+1> 0. It follows from elementary linear algebra (the standard procedure of diagonalizing quadratic forms) thatL can be written as
(2.3) L = β1(H0+ eY )2+ Xm i=1
Yej2+ eY0+ β2(H0+ eY ) = H2+ Xm i=1
Yej2+ eY0+ βH
with H =√
β1(H0+ eY ), eY , eY0, eY1, . . . , eYm∈ N acting on functions on S as Yejf (xa) = d
dtf (xa exp t eYj)|t=0. Thus our assumptions are
(2.4) β≤ 0 and H, eY1, . . . , eYm, eY0+ βH generateS as the Lie algebra.
Now we writeL in convenient coordinates. We modify the operator multiplying it by c2: c2L = (cH)2+
Xm i=1
(c eYj)2+ c2Ye0+ c2βH.
Clearly,L and c2L have the same harmonic functions, the semi-group for c2L is rescaled µc2t and the Green function is multiplied by c−2. This modification does not change the sign of β.
The decomposition
S = N ⊕ A
is not unique i.e. there is no canonical choice of A. We choose A as A = exp{tcH : t > 0},
and assume, without no loss of generality, that the real parts of the eigenvalues of adH|N are strictly positive.
Decomposing s∈ S as
(2.5) s = xa = x exp(log a)cH, x∈ N, a ∈ A, we obtain
(2.6) c2L = L−α= (a∂a)2− α(a∂a) + Xm j=1
Φa(Xj)2+ Φa(X0),
where Φa= Adexp(log a)H, a∂a= cH and X0, X1, . . . , Xmare left-invariant vector fields on N . Remark. Notice that this way we can make α = cβ a nonnegative integer and the real parts of the eigenvalues of adcH as large as we wish. In particular, if α and the eigenvalues are rational they can be made large enough natural numbers.
Assumptions. For the rest of the paper, we assume that
(2.7) a∂a, Φa(Xj), j = 0, ..., m generateS,
α is a nonnegative integer, if α = 0,ℜλj≥ ⌊n+22 ⌋ + 3 and if α 6= 0, ℜλj > 4.
The main result is
Let Σ be a sphere around e∈ N. For every x ∈ Σ the limit
alim→∞aQ+αmα(Φa(x)) = c(x)
exists and the function Σ∋ x 7→ c(x) is continuous. If a∂a, Φa(X1), . . . , Φa(Xm) generate S then c(x) > 0.
In a number of papers [D1, D2, DH1, DH2, DHZ, DHU, U] we dealt with measures m on N invariant for the semi-groups of operators likeL only the assumptions on L were stronger. Indeed, instead of (2.4) we assumed that
(2.8) X1, ..., Xm generate the Lie algebra of N .
Under this stronger assumptions the Main Theorem 2.15 was proved in [DH2]. This, however, excluded operators like
(a∂a)2+ a8∂x2+ a4∂y+ a6∂z.
Here we go beyond this restriction developing a new method that not only solves the problem, but also shows an interesting phenomenon: subelliptic estimates uniform when a→ 0.
The plan for the proof follows the pattern of [DHU] and [DH2].
We changeL−α into Lα= a−αL−α(aα·) and we divide it by a2 to introduce the operator Lα= a−2Lα= ∂a2+1 + α
a ∂a+ a−2³Xm
j=1
Φa(Xj)2+ Φa(X0)´ .
Then we observe that if G∗(x, a; y, b) is the Green function of L∗α with respect to the reference measure dxaα+1da on N⊕ R+ and the limit
blim→0lim
a→0G∗(x, a; y, b) exists for x6= y, then
mα(x) = G∗(e, 0; x, 1) and an easy homogeneity argument yields the result. Also
(2.9) lim
b→0G∗(x, 0; y, b) = c(y) exists, see Section 6.
The main point of this paper is to show 2.9
A special case. Suppose that the group is as in (2.2) and that our assumptions (2.7) are satisfied.
Moreover, assume that
(i) d1, ..., dn are positive rational numbers, (ii) α is a positive rational number.
Of course, (i) concerns the group, (ii) the operator. By a change of variable a→ ar(see 2.5) we can make α a positive integer and dj’s positive even integers greater or equal 4. We then take
L∗α= ∂a2+α + 1
a ∂a+X
j
αja2dj−2Xj2+X
l
βladl−2Wl,
where the summation in the first order term is over distinct eigenvalues i.e. l16= l2implies dl1 6= dl2. In particular on R3× R+ it could be
∂a2+α + 1
a ∂a+ a6∂x2+ a2∂y+ a4∂z
Let
u = (u1, ..., us), s = α + 2 and a = q
u21+ ... + u2s=|u|.
Then, if ∆u= ∂u21+ ... + ∂2us, we have Le∗α=∆u+X
j
αj|u|2(dj−1)Xj2+X
j
βj|u|dj−2Xj
=∆u+X
j
αj|u|2·δjXj2+X
j
βj|u|2·γjXj , where
δj = dj− 1, γj= dj− 2
2 .
For our particular case we have
∆u+|u|6∂2x+|u|2∂y +|u|4∂z,
which clearly satisfies the H¨ormander condition on R3+s. Now, putting k= (k1, ..., ks), kj≥ 0, |k| = k1+ ... + ks, we obtain
|u|2·δj = (u21+ ... + u2s)δj = X
|k|=δj
µδj
k
¶
(uk11...ukss)2, and similarly,
|u|2·γj = (u21+ ... + u2s)γj = X
|k|=γj
µγj
k
¶
(uk11...ukss)2. Therefore,
Le∗α= ∆u+X
j
αj
X
|k|=δj
µδj
k
¶
(uk11...ukssXj)2+X
l
βl
X
|k|=γl
µγl
k
¶
(uk11...ukss)2Wl.
Thus we see that the condition Lie(H, Y1, ..., Ym, Y0) =S implies that the Lie algebra generated by αjXj’s and βlWl’s isN i.e eL∗α satisfies the weak H¨ormander condition. Then eL∗a is hypo-elliptic as an operator on Rs× N an so its Green function eG is C∞ off the diagonal,[Bo]. Going back to the original operator, before the change of variable, we see that
G(e, 0;e ·) = G∗(e, 0;·).
Hence 2.9.
It is tempting to prove the result in the case when Φa have only eigenvalues with positive real parts (not necessarily rational) and are not diagonal, based on the argument above. However, we have been unable to use any procedure of approximation arbitrary real dj by rationals and replace the diagonal action of Φa’s by the general. To prove the main theorem, as it is formulated, we need a uniform subelliptic estimate when a→ 0. Therefore, we go deeper into the proof of the H¨ormender’s theorem and use subelliptic estimates for operators with coefficients of finite smoothness. The proof of the subelliptic estimates 5.1 and 5.3 is long and technical. A big part of it is rather standard so we put it into Appendix. Section 5 contains only what is specific to our operators, Theorem 5.6 being the crucial step.
In a series of papers cf. [BGGK] and [BBG] the Green function for operators like
∆u+|u|k∆t on Rn⊕ Rm
has been studied. The authors give explicit formulae for the Green functions G, if k is an even integer and the change of variable |u| → |u|γ yields immediately continuity of G for arbitrary positive rational k. However, continuity of the Green function of a slightly more general operator (2.10) ∆u+|u|k∆t+|u|l∆s on Rn⊕ Rm⊕ Rj, k, l being independent over Q
cannot be obtained along these lines and, to our embarrassment, we have to go via tedious estimates to obtain the result for operators like (2.10).
Now we are going to introduce the rest of notation and to formulate the main theorems. In N we define a “homogeneous” norm| · |. Let (·, ·) be an arbitrary fixed inner product in N and let
hX, Y i = Z 1
0
(Φa(X), Φa(Y ))da
a , kXk =p hX, Xi.
We put
| exp X| = |X| = (inf{a > 0 : kΦa(X)k ≥ 1})−1. Since for X 6= 0 we have
alim→0kΦa(X)k = 0,
alim→∞kΦa(X)k = ∞, and a7→ kΦa(X)k is increasing, if follows that for every Y 6= 0 there is precisely one a such that
Y = Φa(X), |X| = 1, |Y | = a.
If the action of A on N is diagonal,| · | is the usual homogeneous norm on N.
LetNC be the complexification ofN and let E = {λ1, . . . , λD} denote the set of all eigenvalues of adH|N . For λ ∈ E let NλCbe the coresponding eigenspace. We may assume ℜλ1 ≤ . . . ≤ ℜλD. Then there exists a basis Zjλ of NC, Zjλ ∈ NλC, for λ∈ Λ, j = 1, . . . , d(λ), in which adH|N can be written in the Jordan canonical form, i.e.
adH(Z1λ) = λZ1λ,
adH(Zjλ) = λZjλ+ δZj−1λ , for j = 2, . . . , d(λ), δ = 0, 1.
(2.11)
In the chosen basis one can compute explicitly the action of Φa. Namely we have (2.12) Φa(Zjλ) = aλ³ Xj
k=kj
1
(j− k)!(log a)j−kZkλ´ . The above formula implies that the are m1≥ m2> 4 and C > 0 such that
(2.13) kΦakN →N ≤ C(am1+ am2).
We choose a basis X1, . . . , Xn of left invariant vector fields coresponding to the decomposition N = ⊕ℑλ≥0Vλ,
where
Vλ=N ∩ (NλC⊕ Nλ¯C) i.e.
(2.14) Xj ∈ Vλ(Xj).
For a multiindex I we write XI = X1i1. . . Xnin and|I| = i1ℜλ(X1) + ... + inℜλ(Xn). Let
Q = X
ℑλ≥0
ℜλ · dim Vλ. In the case of diagonal action 2.2 this means that
Φa(Xj) =adjXj
|I| =i1d1+ ... + indn
and Q is the homogeneous dimension. Now, given all the necessary definitions, the main results of this paper can be formulated as follows
Main Theorem 2.15. Let Σ be the unit sphere in N , then for every x ∈ Σ
alim→∞aQ+αmα(Φa(x)) = c(x)
a→∞lim aQ+α|(Φa(XI)mα)(Φa(x))| = cI(x)
is finite and the function Σ∋ x 7→ cI(x) is continuous. In particular, for diagonal action 2.2
|XImα(x)| ≤ c(1 + |x|)−Q−α−|I|
and for arbitrary action
|XImα(x)| ≤ c(1 + |x|)−Q−α−|I|log(2 +|x|)I0,
where
I0= Xn j=1
ij(dim Vλ(Xj)− 1).
If a∂a, Φa(X1), . . . , Φa(Xm) generateS then c(x) > 0.
Let
V1={(x, a) : |x| ≤ 2, 0 < a ≤1 2} V2={(x, a) : |x| ≤ a, a ≥ 2}
V3={(x, a) : |x| ≥ a, |x| ≥ 2}
and let
h(xa) =
aα (x, a)∈ V1 a−Q (x, a)∈ V2 aα|x|−α−Q (x, a)∈ V3
Main Theorem 2.16. Let U be the density, with respect to the right Haar measure, of R∞ 0 µt. If a∂a, Φa(Xj), j=1,...,m generateS then there are C1, C2> 0 such that
(2.17) C1≤ U (xa)
h(xa) ≤ C2.
If all a∂a, Φa(Xj), j=0,...,m are needed to generateS, then there is C2> 0 such that
(2.18) U (xa)≤ C2h(xa).
Notice that the change of variables (2.6) is indeed of the form (x, a)→ (x, ac) so it only changes the speed of dilations and so both theorems remain valid as far asℜλj> 0. The subelliptic estimate cannot be transferred by this transformation, but the final result can.
3. Evolution run by the Bessel process
We start by describing the evolution that is used later for the construction of the fundamental solution for our operator. Let R+ ∋ t → σ(t) denote the Bessel process with a parameter α ≥ 0, [RY]. This is a continuous Markov process with state space [0,∞) generated by
∆α= ∂a2+α + 1 a ∂a.
Of course, if α is a positive integer, then ∆αis the radial part of the Laplacian on Rα+2. ∆αis an infinitesimal generator of the Bessel semi-group of operators on L2(aα+1da).
Ptf (a) = Z ∞
0
pt(a, b)f (b)bα+1db.
Then for f ∈ L2(aα+1da) we have
t→0limkPtf − fkL2(aα+1da) = 0 and for f ∈ Cc∞
tlim→0kPtf− f
t − ∆αfkL2(aα+1da)= 0.
Also,
pt(a, b)≤ c1t−1−α2e−c2(a−b)24t
and so for γ≥ 0
(3.1) sup
t≤1,a≤1
Ea Z t
0
σ(s)γ ds = sup
t≤1,a≤1
Z
R+
bγpt(a, b)b1+αdb <∞.
For a multi-index I = (i1, ..., in) and a basis X1, ..., Xn of the Lie algebraN we write XI = X1i1· · · Xnin.
For k = 0, 1, ... we define
Ck ={f : XIf ∈ C(N), for X ij ≤ k}
and
C∞k ={f ∈ Ck : lim
x→∞XIf (x) exists for X
ij ≤ k}.
C∞k is a Banach space with the norm
kfkC∞k = X
Pij≤k
kXIfkC(N ).
For a continuous function σ : [0,∞) → (0, ∞) let
(3.2) Lσ(t)= σ(t)−2³Xm
j=1
Φσ(t)(Xj)2+ Φσ(t)(X0)´
As we will see later on Lσ(t) is closely related to a−2Lα. A standard argument (see e.g. [T]) shows the existence of a unique family Uσ(s, t), 0≤ s < t of bounded operators on C∞(N ) = C∞0 (N ) that satisfy
• Uσ(s, t)f = f∗ pσ(t, s), for a probability measure pσ(t, s),
• pσ(s, s) = δe,
• pσ(t, r)∗ pσ(r, s) = pσ(t, s) for s < r < t,
• limh→0kf ∗ pσ(s + h, s)− fkC∞(N )= 0 for f ∈ C∞(N )
• ∂t(f∗ pσ(t, s)) = (Lσ(t)f )∗ pσ(t, s), for f ∈ C∞2 (N ),
• ∂s(f∗ pσ(t, s)) =−Lσ(s)(f∗ pσ(t, s)), for f ∈ C∞2(N ).
For the argument [T] to work the following properties of Lσ(t) are needed.
• For a fixed s, Lσ(s) generates a semi-group µst with the property
h→0lim
°°
°f∗ µsh− f
h − Lσ(s)f°°°C
∞→ 0 for f ∈ C∞2.
• Let τ be a Riemannian distance on N, φ ∈ Cc∞(N ), φ≥ 0. CT = (1 + sup0≤s≤T|σ(s)|)2ℜλD. There is C such that for every T > 0, s1, ..., sn≤ T , t1, ..., tn > 0 and all β > 0
hµst11∗ ... ∗ µstnn, eβ(φ∗τ)i ≤ eβ(φ∗τ)(0)eC(β+β2)CT(t1+...+tn).
The proof uses the same techniques as in Lemma 3.3 below. As a consequence we conclude that for every k≥ 1 there is CK > 0 such that
kUσ(s, t)kC∞k(N )→C∞k(N )≤ CkeCkCT(t−s) for 0≤ s, t ≤ T .
Here are some further properties of pσ(s, t).
Lemma 3.3. Let
Aσ(0, t) = Z t
0
³σ(s)2m1−2+ σ(s)m2−2´ ds and r > 0. Then there exist c1, c2 such that
(3.4)
Z
τ (z)>r
pσ(0, t)≤ c1e−Aσ(0,t)γc2 , where γ =12 when Aσ(0, t) < 1 and γ = 1 when Aσ(0, t)≥ 1.
Proof. Fix r. Let φ∈ Cc∞(N ) be a positive function, supported in the ball Br4(0) andR
φ = 1. Let η(x) = τ∗ φ(x). There exists a positive constant C = C(r) such that if X1, ..., Xn is a fixed basis of N then
(3.5) |Xjη(x)| ≤ C, |XiXjη(x)| ≤ C, for i, j = 1, ..., n [H]. Moreover,
(3.6) τ (x)≤
Z
(τ (xy−1) + τ (y))φ(y)dy≤ η(x) +r 4, and
(3.7) η(e) =
Z
τ (y−1)φ(y)dy≤ r 4. For a positive integer m let ηm(x) = τm∗ φ(x), where
τm(x) = min{m, τ(x)}.
τm is subadditive and there exists a positive constant C such that for every m, (3.5), (3.6), (3.7) hold with ηmand τminstead of η and τ respectively.
Let
h(t) = eβηm∗ pσ(0, t)(e).
Since eβηm ∈ C∞2 ,
∂th(t) = (Lσ(t)eβηm)∗ pσ(0, t)(e) and by (2.13) there is C = C(r) such that
∂th(t)≤ Ch
(β + β2)σ(s)−2(σm1(s) + σm2(s))2+ βσ(s)−2(σm1(s) + σm2(s))i h(t).
Hence
log h(t)− log h(0) = Z t
0
∂sh(s)
h(s) ds≤ C(β + β2)Aσ(0, t) and
h(t)≤ eβηm(e)eC(β+β2)Aσ(0,t). Letting m→ ∞, by (3.5) and (3.7) we have
(eβτ, pσ(0, t))≤ eβr2 eC(β+β2)Aσ(0,t).
Therefore, Z
τ (z)>r
pσ(0, t)≤ e−βr2 +C(β+β2)Aσ(0,t).
We may assume that C ≥ 1. If Aσ(0, t) ≥ 1, we substitute β = 4CArσ(0,t), if Aσ(0, t) ≤ 1, we substitute β = 1
C√
Aσ(0,t) and the Lemma follows. ¤
4. Heat equation on N × R+ and N× Rα+2
Now we changeL−αintoLα= a−αL−α(aα·) and we divide it by a2. We describe the fundamental solution for a−2Lα− ∂t as well as for its extension to N× Rα+2. Let
Lα= a−2Lα= ∂a2+1 + α
a ∂a+ a−2³Xm
j=1
Φa(Xj)2+ Φa(X0)´ .
The diffusion generated by Lαon N×R+is decomposed into its vertical and horizontal components.
Namely, for f ∈ Cc∞(N× R+) and a≥ 0 we define F (x, a, t) = Ttf (x, a) =
Z
Uσ(0, t)f (x, σ(t))dBa(σ)
= EaUσ(0, t)f (x, σ(t)), (4.1)
where dBais the probability measure on the space C([0,∞), R+) of trajectories of the Bessel process σ(t). Then F is a solution of the heat equation, i.e.
(4.2) LαF (x, a, t) = ∂tF (x, a, t), on N× R+× R+, F is continuous on N × R+× [0, ∞) and
(4.3) lim
t→0F (x, a, t) = f (x, a).
Writing an appropriate maximum principle we prove that a bounded F satisfying (4.2) and (4.3) is unique. Furthermore, the Markov property implies that Tt is a contractions semigroup on L2(dxaα+1da) [DHU]. Finally, for f ∈ Cc∞(N A)
(4.4) lim
t→0
°°
°Ttf − f
t − Lαf°°°
L2(a1+αdadx)= 0 [DH2]. Now let
(4.5) Leα= ∆u+|u|−2³Xm
j=1
Φ|u|(Xj)2+ Φ|u|(X0)´ ,
where ∆u=Pd
j=1∂u2j, d = α + 2 and if u = 0, then eLα= ∆u. For eLα− ∂t we write a convenient formula for the heat semi-group. Let dWu be the Wiener measure on the space C([0,∞)), Rd of trajectories for the Brownian motion generated by ∆u. For ef ∈ Cc∞(N× Rd) let
F (x, u, t) = ee Ttf (x, u)e =
Z Ue|b|(0, t) ef (x, b(t))dWu(b)
= Eu£ eU|b|(0, t) ef (x, b(t))¤ , Then eF is the unique solution of the heat equation
(eLα− ∂t) eF = 0
with the boundary data ef as described in Theorem 4.7 below. Moreover, if f (x, u) = f (x,e |u|).
for f ∈ Cc∞(N A) then
(4.6) Tetf (x, u) = Te tf (x,|u|),
Indeed, if ef is u-radial then so is eF and (4.6) follows from uniqueness. In particular, by the next theorem, Ttf is continuous on N × [0, ∞) × [0, ∞).
Theorem 4.7. Let η =⌊ℜλ1⌋ − 2. If ef ∈ Cc∞(N× Rd), then eF = eTtfe∈ C∞,η,1(N× Rd× R+).
Furthermore,
(4.8) LeαF (x, u, t) = ∂e tF (x, u, t) on Ne × Rd× R+ and
(4.9) lim
t→0F (x, u, t) = ee f (x, u),
Writing an appropriate maximum principle we can prove that a bounded eF satisfying (4.8) and (4.9) is unique. Notice that here ef is not necessarily radial and we cannot obtain this regularity of F when ue → 0 from subelliptic estimate (5.4). More derivatives with respect to u would be needed.
First we prove that eF is a solution of an integral equation
Lemma 4.10. For fixed (x, u, t), the function eF satisfies the equation (4.11) F (x, u, t) = Ee uf (x, b(t)) +e
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤ ds, where L|b(t−s)| is given by formula (3.2).
Proof. Observe that (4.11) is well defined. Namely, eF is a bounded function and smooth with respect to x. Indeed, writing XI as a combination of right-invariant vector fields we have
(4.12) XIF (x, u, t) =e X
|J|≤|I|
WJ(x) eTt( eXJf )(x, u),e
where for a multi-index J = (j1, ..., jp), eXJ= eXj1· . . . · eXjpis a right-invariant differential operator and WJ’s are polynomials on N .
Furthermore
|L|b(t−s)|F (x, b(te − s), s)| ≤ Cf(1 +|b(t − s)|)2Q(1 +|x|)M, (4.13)
for an appropriate large constant M . Hence
¯¯
¯ Z t
0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤
ds¯¯¯ ≤ Cf(1 +|x|)M Z t
0
Eu(1 +|b(t − s)|)2Qds
≤ Cf(1 +|x|)M Z t
0
Z
Rd
(1 +|w|)2QpGt−s(u− w) dwds, where pGt denotes the classical gaussian kernel on Rd. So (4.11) is well defined.
Now, by the Markov property, we have Eu£
L|b(t−s)|F (x, b(te − s), s)¤
= Eu
£L|b(t−s)|Eb(t−s)[ eU|σ|(0, s) ef (x, σ(s))]¤
= Eu
£L|b(t−s)|Ue|b|(t− s, t) ef (x, b(t))¤
= Eu£
∂sUe|b|(t− s, t) ef (x, b(t))¤ . Whence
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤
ds = Eu£ eU|b|(0, t) ef (x, b(t))¤
− Euf (x, b(t))e
= F (x, u, t)e − Euf (x, b(t)),e
that finishes the proof. ¤
In the proof of (4.8) some more regularity of eF is needed.
Lemma 4.14. The function eF is continuous and for fixed t, eF (·, ·, t) ∈ C∞,η(N× Rd). Moreover, for every I and |J| ≤ η there are constants C = C(I), M = M(I) such that
|XI∂uJF (x, u, t)e | ≤ C(1 + |x| + |u| + t)M.
Proof. First, observe that because of (4.12), eF is smooth w.r. to x. Next, we shall prove that for all x and t, eF (x,·, t) ∈ Cη(Rd). To do so we use Lemma 4.10. Notice, that the function
x→ Euf (x, b(t)) = pe Gt ∗Rdf (x, u),e is smooth. Hence it is enough to show that
(4.15)
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤ ds
has η derivatives. Using that LuF (x, u, s) grows polynomially as a function of x and u, we maye compute
∂ui
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤
ds = ∂ui
Z t 0
pGt−s∗RdL|u|F (x, u, s)dse
= Z t
0
Z
Rd
(∂uipGt−s)(u− w)L|w|F (x, w, s)dwdse The above expression is well defined, because by (4.13) the inner integral can be dominated by
C(t,x,u)
√t−s and C(t, x, u) growing polynomially with respect to t, x, u. Therefore, by (4.11) there is M such that
|∂ujF (x, u, t)e | ≤ C(1 + |x| + |u| + t)M and so
|∂wjL|w|F (x, w, s)e | ≤ C(1 + |x| + |w| + s)M Using that we compute the second derivative
∂ui∂uj
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤ ds =
Z t 0
Z
Rd
(∂uipGt−s)(u− w)(∂wjL|w|F )(x, w, s)dwdse Applying the same argument η times we prove Cη smoothness with respect to u. Notice that at every step we dominate the derivatives by a polynomial and we can put more and more derivatives on L|w|F (x, w, s).e
To prove continuity of eF with respect to x, u, t together, we use again (4.11), elementary prop- erties of pGt and the fact that LuF (x, u, s) grows polynomially.e ¤ Now, we show that Lemmas 4.10 and 4.14 imply Theorem 4.7. First we prove continuity of F , then we apply ∂tand conclude (4.8). Here are the details.
Proof of Theorem 4.7. By Lemma 4.10 we have F (x, u, t + h)e − eF (x, u, t)
h = Euf (x, b(t + h))e − Euf (x, b(t))e h
+ 1 h
Z t 0
³ Eu£
L|b(t+h−s)|F (x, b(t + he − s), s)¤
− Eu£
L|b(t−s)|F (x, b(te − s), s)¤´
ds
+ 1 h
Z t+h t
Eu£
L|b(t+h−s)|F (x, b(t + he − s), s)¤ ds.
Letting h go to 0, by Lemma 4.14, we get a pointwise equality
∂tF (x, u, t)e = ∆uEuf (x, b(t)) + ∆e u
Z t 0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤
ds + L|u|F (x, u, t)e On the other hand, by Lemma 4.10, again,
LeαF (x, u, t)e = L|u|F (x, u, t) + ∆e uF (x, u, t)e
= L|u|F (x, u, t) + ∆e u¡
Euf (x, b(t)) +e Z t
0
Eu£
L|b(t−s)|F (x, b(te − s), s)¤ ds¢
.
¤ 5. Subelliptic estimates
In this chapter we deal with subelliptic estimates for eLα− ∂t harmonic functions. It can be easily seen that the operator satisfies the H¨ormander condition on N × (Rd\ {0}) × R+ and so, if the vector fields|u|−1Φ|u|(Xj), |u|−2Φ|u|(X0) happen to be smooth, this also is true on the whole of N× Rd× R+. In general they are not smooth but at least they are C2. So we still can get a subelliptic estimate for u-radial functions.
Let U be an open set in N , K a compact subset of U and Br(u0) the ball in Rd with the center u0and the radius r. For x0∈ N, u0∈ Rd and t0< t2< t3< t1 we consider
Ω(x0, u0) = x0U× B23(u0)× (t0, t1) and
K(xe 0, u0) = x0K× B1(u0)× [t2, t3].
Theorem 5.1. If on Ω(x0, u0), |u0| ≥ 74, (eLα − ∂t)F = 0 in the sense of distributions, then F ∈ C∞(Ω(x0, u0)). Moreover, there are C, M > 0 such that for every x0∈ N, u0∈ Rd and every F satisfying (eLα− ∂t)F = 0 on Ω(x0, u0) we have
(5.2) kF kL∞( eK)≤ C(1 + |u0|)MkF kL2(Ω)
with C independent of x0. If the vector fields, |u|−1Φ|u|(Xj), |u|−2Φ|u|(X0) are smooth, then the conclusion holds for all u0.
Assume now that ℜλj > 4, j = 1, ..., D and so the vector fields in eLα− ∂tare C2. We consider the space
W (Ω(x0, 0)) that consists of functions F on Ω(x0, 0) such that
F, ∂tF ∂uβF, ∂βxF, ∈ L2(Ω(x0, 0)),|β| ≤ 2.
Clearly, (eLα− ∂t)F ∈ L2(Ω(x0, 0)) and we have the following theorem Theorem 5.3. Let
• F ∈ W (Ω(x0, 0))
• (eLα− ∂t)F = 0 on Ω(x0, 0);
• for fixed x and t the function F (x, ·, t) depends only on |u|, then
(5.4) kF kL∞( eK)≤ CkF kL2(Ω),
with C independent of x0. If ℜλ1 > ⌊α+n+22 ⌋ + 3 then (5.4) holds also for F that are non-radial w.r. to u .
Below we present a proof of Theorems 5.1 and 5.3 trying to separate what is specific for our operator from what is a rather standard pseudodifferential calculus. Let
Leα,u0− ∂t= Xd j=1
∂u2j +|u0+ u|−2 Xm j=1
Φ|u0+u|(Xj)2+|u0+ u|−2Φ|u0+u|(X0)− ∂t
= Xd j=1
∂u2j + Xm j=1
Yj2+ Y0− ∂t, where
Yj =|u0+ u|−1Φ|u0+u|(Xj), j = 1, . . . m and
Y0=|u0+ u|−2Φ|u0+u|(X0).
In this notation eLα,0− ∂t= eLα− ∂t. In order to keep the subelliptic constant in (5.2) under control we consider the family of operators eLα,u0 on one set
Ω = Ω(e, 0) = U× B32(0)× (t0, t1) instead of the operator eLα− ∂t on the family of sets Ω(x0, u0). Let
Fx0,u0(x, u, t) = F (x0x, u0+ u, t).
Clearly,
(5.5) (eLα,u0− ∂t)Fx0,u0 = 0 on Ω, if, and only if,
(eLα− ∂t)F = 0 on Ω(x0, u0).
We identify N× R+ with Rn× R+ and for f ∈ Cc∞(N× Rd× R+) we define a partial Sobolev norm kfk2s=
Z
Rd|fu|2sdu,
where fu(x, t) = f (x, u, t) and |fu|s is the s-Sobolev norm in Rn× R. For φ, ψ ∈ Cc∞ we write ψ≻ φ, if ψ = 1 on a neighborhood of the support of φ.
Our procedure is going to estimate derivatives of the solutions of of (5.5) and to keep the subel- liptic constant under control. Since the operators have more smoothness in the x, t-direction than in the u-direction, taking partial Sobolev norms seems to be very convenient. The crucial estimate is contained in Theorem 5.6 below. It requires only C2 vector fields which is guaranteed by the assumptionℜλj> 4. Indeed, then|u|λj−2logβ|u| ∈ C2, first and second derivatives being equal to 0 at 0.
For a positive number ε let Pε be the set of smooth vector fields X on Ω with the following property: there is M , that for any ψ, φ∈ Cc∞(Ω) such that ψ≻ φ there exists a constant C that for any f ∈ C∞(N× Rd× R+) and any u0∈ Rd
kX(φf)k2−1+ε≤ C(1 + |u0|)M(kψ(eLα,u0− ∂t)fk2L2+kψfk2L2).
Theorem 5.6. Assume that ℜλj > 4, j = 1, ..., D. Then there is ε such that every smooth vector field on Ω is an element ofPε.
Example 5.1. Since the proof is rather technical, we look at an example first. Consider R2× R with
adH =
µ 512 1
−1 512
¶
and the operator
Leα= ∆u+|u|−2Φu(∂x)2+|u|−2Φu(∂y)2. Clearly,
Φ|u|=|u|112
µ cos log|u| sin log|u|
− sin log |u| cos log |u|
¶
and so
|u|−1Φu(∂x) =|u|92(cos log|u|∂x+ sin log|u|∂y)
|u|−1Φu(∂y) =|u|92(− sin log |u|∂x+ cos log|u|∂y) First we prove that
∂uj,|u|−1Φu(∂x),|u|−1Φu(∂y)∈ P1. But since we take only the partial Sobolev norms then also
(sin log|u|)|u|−1Φu(∂x) + (cos log|u|)|u|−1Φu(∂y) =|u|92∂y ∈ P1.
By the same principle |u|10∂y ∈ P1. Now taking a few commutators of the kind [∂uj,|u|10∂y] we arrive to ∂y. With every step the constant increases and grows polynomially with u.
Proof. To get the above subelliptic estimate we have to deal with two obstacles: the first order term being Y0− ∂t instead of Y0 and degeneracy of eLα when u → 0. First we prove that ∂uj, Yj ∈ P1 (Lemma A.1) and Y0− ∂t∈ P12 (Lemma A.2). The proof is rather a standard argument that uses two special properties of our operator:
• coefficients as functions of u are elements of C2,
• multiplication by functions of u commutes with the action of vector fields Y0, . . . , Ym. Here and in what follows we think of u ∈ B32(0) as a variable and of u0 as a parameter. The constant that appears in front of the subelliptic estimate is a power of the C2-norm of the coefficients
|u0+ u|λlogβ|u0+ u| (see (2.12)). Hence the conclusion. The only dependence on u0is reflected in the factor (1 +|u0|)M in front of the subelliptic estimate.
Now we estimate the brackets. Finitely many brackets of the vector fields H, Φa(X0), ..., Φa(Xm)
spans S as a linear space. Since a bracket of two left-invariant vector fields is left-invariant and [S, S] = N , all the brackets to consider are of the form Φa(W ), W ∈ N . Therefore, taking a finite number of brackets we generate vector fields Φa(W1), ..., Φa(Wp) that together with H form a basis ofS. In particular, W1, ..., Wp is a basis of N .
The first step is to prove that there is ε such that
Φ|u+u0|(Wj)∈ Pε, j = 1, ..., p For that we consider brackets of vector fields
Xd j=1
(u + u0)j∂uj, Φ|u+u0|(Xj), Φ|u+u0|(X0)− |u + u0|2∂t
By Lemmas A.1 and A.2, they belong either to P1 or to P12. Taking brackets of vector fields, we substitute Pd
j=1(u + u0)j∂uj, in place of H and Φ|u+u0|(Xj) in place of Φa(Xj), j = 1, ..., m and also Φ|u+u0|(X0)− |u + u0|2∂tin place of Φa(X0). In this way we obtain
(5.7) Φ|u+u0|(Wj), or Φ|u+u0|(Wj)− 2k|u + u0|2∂t
Indeed,
[Φ|u+u0|(W ),|u + u0|2∂t] = 0
and
[Φ|u+u0|(W ), Φ|u+u0|(Z)] = Φ|u+u0|([W, Z]).
Moreover,
[ Xd j=1
(u + u0)j∂uj,|u + u0|2∂t] = 2|u + u0|2∂t
and, if [a∂a, Φa(W )] = Φa(Z), then
[ Xd j=1
(u + u0)j∂uj, Φ|u+u0|(W )] = Φ|u+u0|(Z)
The second case in (5.7) corresponds only to the bracket with Vl=Pd
j=1(u + u0)j∂uj for all l’s. A subsequent application of Lemmas A.3, A.4 and A.5 shows that there is ε > 0 such that for every element W ofN we have
(5.8) Φ|u+u0|(W )∈ Pε, or Φ|u+u0|(W )− cW|u + u0|2∂t∈ Pε Next, for λ∈ Λ and j = 1, . . . , d(λ) let Wjλ and Vjλ be elements of N such that
Wjλ+ iVjλ= Zjλ, where Zjλ are as in (2.11).
Then, (5.8) shows that for all λ∈ Λ and j = 1, ..., d(λ)
Φ|u+u0|(Wjλ)− cjλ|u + u0|2∂t∈ Pε, Φ|u+u0|(Vjλ)− djλ|u + u0|2∂t∈ Pε, (5.9)
for some constants cλj, dλj.
Now applying Lemma A.5, we are going to deduce from (5.9) that there is ελj such that Wjλ, Vjλ∈ Pελj,
which implies that every W ∈ N is an element of Pε0 for ε0= min{ελj}.
We proceed by induction. Fix λ∈ Λ and assume that W1λ, V1λ, ..., Wj−1λ , Vj−1λ ∈ Pεfor some ε.
Then, by (2.12), we have
Φ|u+u0|(Wjλ) + iΦ|u+u0|(Vjλ) = Φ|u+u0|(Zjλ) =|u0+ u|λ³Xj
k=kj
1
(j− k)!(log|u0+ u|)j−kZkλ´
=|u0+ u|ℜλ³
cos(ℑλ · log |u0+ u|) + i sin(ℑλ · log |u0+ u|)´
·³Xj
k=kj
1
(j− k)!(log|u0+ u|)j−kZkλ´
=|u0+ u|ℜλ Xj k=kj
³ 1
(j− k)!(log|u0+ u|)j−k¡
cos(ℑλ · log |u0+ u|)Wkλ− sin(ℑλ · log |u0+ u|)Vkλ
¢´
+ i|u0+ u|ℜλ Xj k=kj
³ 1
(j− k)!(log|u0+ u|)j−k¡
sin(ℑλ · log |u0+ u|)Wkλ+ cos(ℑλ · log |u0+ u|)Vkλ
¢´.
