Non local Poincaré inequalities on Lie groups with polynomial volume growth

0.2 MB·English

by  Emmanuel Russ

NON LOCAL POINCARE´ INEQUALITIES ON LIE
GROUPS WITH POLYNOMIAL VOLUME GROWTH
0
EMMANUEL RUSS AND YANNICK SIRE
1
0
2

n
a Abstract. Let G be a realconnectedLie groupwith polynomial
J
volume growth, endowed with its Haar measure dx. Given a C2
2 positive function M on G, we give a sufficient condition for an L2
2
Poincar´e inequality with respect to the measure M(x)dx to hold

] onG. Wethenestablishanon-localPoincar´einequalityonGwith
A respect to M(x)dx.
F
.
h
Contents
t
a
m
1. Introduction 1
[ 2. A proof of the Poincar´e inequality for dµ 5
M

1 3. Proof of Theorem 1.4 7
v
3.1. Rewriting the improved Poincar´e inequality 8
5
7 3.2. Off-diagonal L2 estimates for the resolvent of LM 8
0 α/4
3.3. Control of L f and conclusion of the proof of
4 M
. L2(G,dµM)
1 Theo(cid:13)rem 1.4(cid:13) 10
(cid:13) (cid:13)
0
4. Appendix A:(cid:13)Techni(cid:13)cal lemma 16
0
1 5. Appendix B: Estimates for gt 16
j
: References 17
v
i
X
r
a
1. Introduction
Let G be a unimodular connected Lie group endowed with a measure
M(x)dx where M L1(G) and dx stands for the Haar measure on G.
∈
By “unimodular”, we mean that the Haar measure is left and right-
invariant. We always assume that M = e v where v is a C2 function
−
on G. If we denote by the Lie algebra of G, we consider a family
G
X = X ,...,X
1 k
{ }
of left-invariant vector fields on G satisfying the H¨ormander condition,
i.e. istheLiealgebrageneratedbytheX s. AstandardmetriconG,
calleGd the Carnot-Caratheodory metric, isi′naturally associated with X
1
2 EMMANUEL RUSSANDYANNICKSIRE
and is defined as follows: let ℓ : [0,1] G be an absolutely continuous
→
path. We say that ℓ is admissible if there exist measurable functions
a ,...,a : [0,1] C such that, for almost every t [0,1], one has
1 k
→ ∈
k
ℓ(t) = a (t)X (ℓ(t)).
′ i i
i=1
X
If ℓ is admissible, its length is defined by
1
1 k 2
ℓ = a (t) 2dt .
i
| | | |
Z0 i=1 !
X
For all x,y G, define d(x,y) as the infimum of the lengths of all
∈
admissible paths joining x to y (such a curve exists by the H¨ormander
condition). This distance is left-invariant. For short, we denote by x
| |
the distance between e, the neutral element of the group and x, so that
the distance from x to y is equal to y 1x .
−
| |
For all r > 0, denote by B(x,r) the open ball in G with respect to
the Carnot-Caratheodory distance and by V(r) the Haar measure of
any ball. There exists d N (called the local dimension of (G,X))
∗
∈
and 0 < c < C such that, for all r (0,1),
∈
crd V(r) Crd,
≤ ≤
see [NSW85]. When r > 1, two situations may occur (see [Gui73]):
Either there exist c,C,D > 0 such that, for all r > 1,
•
crD V(r) CrD
≤ ≤
where D is called the dimension at infinity of the group (note
that, contrary to d, D does not depend on X). The group is
said to have polynomial volume growth.
Or there exist c ,c ,C ,C > 0 such that, for all r > 1,
1 2 1 2
•
c ec2r V(r) C eC2r
1 1
≤ ≤
and the group is said to have exponential volume growth.
When G has polynomial volume growth, it is plain to see that there
exists C > 0 such that, for all r > 0,
(1.1) V(2r) CV(r),
≤
which implies that there exist C > 0 and κ > 0 such that, for all r > 0
and all θ > 1,
(1.2) V(θr) CθκV(r).
≤
NONLOCAL POINCARE´ INEQUALITIES 3
Denote by H1(G,dµ ) the Sobolev space of functions f L2(G,dµ )
M M
∈
such that X f L2(G,dµ ) for all 1 i k. We are interested in L2
i M
∈ ≤ ≤
Poincar´e inequalities for the measure dµ . In order to state sufficient
M
conditions for such an inequality to hold, we introduce the operator
k
L f = M 1 X MX f
M − i i
−
Xi=1 n o
for all f such that
1
f (L ) := g H1(G,dµ ); X MX f L2(G,dx), 1 i k .
M M i i
∈ D ∈ √M ∈ ∀ ≤ ≤
(cid:26) (cid:27)
n o
One therefore has, for all f (L ) and g H1(G,dµ ),
M M
∈ D ∈
k
L f(x)g(x)dµ (x) = X f(x) X g(x)dµ (x).
M M i i M
·
ZG i=1 ZG
X
In particular, the operator L is symmetric on L2(G,dµ ).
M M
Following [BBCG08], say that a C2 function W : G R is a Lyapunov
→
function if W(x) 1 for all x G and there exist constants θ > 0,
≥ ∈
b 0 and R > 0 such that, for all x G,
≥ ∈
(1.3) L W(x) θW(x)+b1 (x),
M B(e,R)
− ≤ −
where, for all A G, 1 denotes the characteristic function of A. We
A
⊂
first claim:
Theorem 1.1. Assume that G is unimodular and that there exists a
Lyapunov function W on G. Then, dµ satisfies the following L2
M
Poincar´e inequality: there exists C > 0 such that, for all function
f H1(G,dµ ) with f(x)dµ (x) = 0,
∈ M G M
R k
(1.4) f(x) 2dµ (x) C X f(x) 2dµ (x).
M i M
| | ≤ | |
ZG i=1 ZG
X
Let us give, as a corollary, a sufficient condition on v for (1.4) to
hold:
Corollary 1.2. Assume that G is unimodular and there exist constants
a (0,1), c > 0 and R > 0 such that, for all x G with x > R,
∈ ∈ | |
k k
(1.5) a X v(x) 2 X2v(x) c.
| i | − i ≥
i=1 i=1
X X
Then (1.4) holds.
4 EMMANUEL RUSSANDYANNICKSIRE
Notice that, if (1.5) holds with a 0, 1 , then the Poincar´e inequal-
∈ 2
ity (1.4) has the following self-improvement:
(cid:0) (cid:1)
Proposition 1.3. Assume that G is unimodular and that there exist
constants c > 0, R > 0 and ε (0,1) such that, for all x G,
∈ ∈
k k
1 ε
(1.6) − X v(x) 2 X2v(x) c whenever x > R.
2 | i | − i ≥ | |
i=1 i=1
X X
Then there exists C > 0 such that, for all function f H1(G,dµ )
M
∈
such that f(x)dµ (x) = 0:
G M
(1.7)
R
k k
X f(x) 2dµ (x) C f(x) 2 1+ X v(x) 2 dµ (x)
i M i M
| | ≥ | | | |
i=1 ZG ZG i=1 !
X X
We finally obtain a Poincar´e inequality for dµ involving a non local
M
term:
Theorem1.4. LetG bea unimodularLiegroup with polynomialgrowth.
Let dµ = Mdx be a measure absolutely continuous with respect to the
M
Haar measure on G where M = e v L1(G) and v C2(G). Assume
−
∈ ∈
that there exist constants c > 0, R > 0 and ε (0,1) such that (1.6)
∈
holds. Let α (0,2). Then there exists λ (M) > 0 such that, for any
α
∈
function f (G) satisfying f(x)dµ (x) = 0,
∈ D G M
f(x) f(Ry) 2
(1.8) | − | dxdµ (y) λ (M)
V ( y 1x ) y 1x α M ≥ α
ZZG×G | − | | − |
k
f(x) 2 1+ X v(x) 2 dµ (x).
i M
| | | |
Rn !
Z i=1
X
Note that (1.8) is an improvement of (1.7) in terms of fractional non-
local quantities. The proof follows the same line as the paper [MRS09]
but we concentrate here on a more geometric context.
InordertoproveTheorem1.4,weneedtointroducefractionalpowers
of L . This is the object of the following developments. Since the
M
operator L is symmetric and non-negative on L2(G,dµ ), we can
M M
define the usual power Lβ for any β (0,1) by means of spectral
∈
theory.
Section 2 is devoted to the proof of Theorem 1.1 and Corollary 1.2.
Then, in Section 3, we check L2 “off-diagonal” estimates for the resol-
vent of L and use them to establish Theorem 1.4.
M
NONLOCAL POINCARE´ INEQUALITIES 5
2. A proof of the Poincar´e inequality for dµ
M
We follow closely the approach of [BBCG08]. Recall first that the
following L2 local Poincar´e inequality holds on G for the measure dx:
forallR > 0, thereexistsC > 0such that, forallx G, allr (0,R),
R
∈ ∈
all ball B := B(x,r) and all function f C (B),
∞
∈
k
(2.9) f(x) f 2dx C r2 X f(x) 2dx,
B R i
| − | ≤ | |
ZB i=1 ZB
X
where f := 1 f(x)dx. In the Euclidean context, Poincar´e in-
B V(r) B
equalities for vector-fields satisfying H¨ormander conditions were ob-
R
tained by Jerison in [Jer86]. A proof of (2.9) in the case of unimodular
Lie groups can be found in [SC95], but the idea goes back to [Var87].
A nice survey on this topic can be found in [HK00]. Notice that no
global growth assumption on the volume of balls is required for (2.9)
to hold.
The proof of (1.4) relies on the following inequality:
Lemma 2.1. For all function f H1(G,dµ ) on G,
M
∈
k
L W
(2.10) M (x)f2(x)dµ (x) X f(x) 2dµ (x).
M i M
W ≤ | |
ZG i=1 ZG
X
Proof: Assume first that f is compactly supported on G. Using
the definition of L , one has
M
L W k f2
M (x)f2(x)dµ (x) = X (x) X W(x)dµ (x)
M i i M
W W ·
ZG i=1 ZG (cid:18) (cid:19)
X
k
f
= 2 (x)X f(x) X W(x)dµ (x)
i i M
W ·
i=1 ZG
X
k f2
(x) X W(x) 2dµ (x)
− W2 | i | M
i=1 ZG
X
k
= X f(x) 2dµ (x)
i M
| |
i=1 ZG
X
k 2
f
X f X W (x)dµ (x)
i i M
− − W
i=1 ZG(cid:12) (cid:12)
kX (cid:12) (cid:12)
(cid:12) (cid:12)
X(cid:12)f(x) 2dµ (x).(cid:12)
i M
≤ | |
i=1 ZG
X
6 EMMANUEL RUSSANDYANNICKSIRE
Notice that all the previous integrals are finite because of the support
condition on f. Now, if f is as in Lemma 2.1, consider a nondecreasing
sequence of smooth compactly supported functions χ satisfying
n
1 χ 1 and X χ 1 for all 1 i k.
B(e,nR) n i n
≤ ≤ | | ≤ ≤ ≤
Applying (2.10) to fχ and letting n go to + yields the desired
n
∞
conclusion, by use of the monotone convergence theorem in the left-
hand side and the dominated convergence theorem in the right-hand
side.
Let us now establish (1.4). Let g be a smooth function on G and let
f := g c on G where c is a constant to be chosen. By assumption
−
(1.3),
(2.11)
L W b
f2(x)dµ (x) f2(x) M (x)dµ (x)+ f2(x) (x)dµ (x).
M M M
≤ θW θW
ZG ZG ZB(e,R)
Lemma2.1showsthat(2.10)holds. Letusnowturntothesecondterm
intheright-handsideof(2.11). Fixcsuchthat f(x)dµ (x) = 0.
B(e,R) M
By (2.9) applied to f on B(e,R) and the fact that M is bounded from
R
above and below on B(e,R), one has
k
f2(x)dµ (x) CR2 X f(x) 2dµ (x)
M i M
≤ | |
ZB(e,R) i=1 ZB(e,R)
X
where the constant C depends on R and M. Therefore, using the fact
that W 1 on G,
≥
(2.12)
k
b
f2(x) (x)dµ (x) CR2 X f(x) 2dµ (x)
M i M
θW ≤ | |
ZB(e,R) i=1 ZB(e,R)
X
where the constant C depends on R,M,θ and b. Gathering (2.11),
(2.10) and (2.12) yields
k
(g(x) c)2dµ (x) C X g(x) 2dµ (x),
M i M
− ≤ | |
ZG i=1 ZG
X
which easily implies (1.4) for the function g (and the same dependence
for the constant C).
Proof of Corollary 1.2: according to Theorem 1.1, it is enough to
find a Lyapunov function W. Define
W(x) := eγ(v(x) infGv)
−
NONLOCAL POINCARE´ INEQUALITIES 7
where γ > 0 will be chosen later. Since
k k
L W(x) = γ X2v(x) (1 γ) X v(x) 2 W(x),
− M i − − | i |
!
i=1 i=1
X X
W is a Lyapunov function for γ := 1 a because of the assumption on
−
v. Indeed, one can take θ = cγ and b = max L W + θW
B(e,R) M
−
(recall that M is a C2 function). n o
Let us now prove Proposition 1.3. Observe first that, since v is C2 on
G and (1.6) holds, there exists α R such that, for all x G,
∈ ∈
k k
1 ε
(2.13) − X v(x) 2 X2v(x) α.
2 | i | − i ≥
i=1 i=1
X X
1
Let f be as in the statement of Proposition 1.3 and let g := fM2.
Since, for all 1 i k,
≤ ≤
1
1 3
Xif = M−2Xig gM−2XiM.
− 2
Assumption (2.13) yields two positive constants β,γ such that
k
(2.14) X f(x) 2(x)dµ (x) =
i M
| |
i=1 ZG
X
k
1
X g(x) 2 + g2(x) X v(x) 2 +g(x)X g(x)X v(x) dx
i i i i
| | 4 | |
i=1 ZG(cid:18) (cid:19)
X
k
1 1
= X g(x) 2 + g2(x) X v(x) 2 + X g2 (x)X v(x) dx
i i i i
| | 4 | | 2
i=1 ZG(cid:18) (cid:19)
X (cid:0) (cid:1)
k
1 1
g2(x) X v(x) 2 X2v(x) dx
≥ 4 | i | − 2 i
i=1 ZG (cid:18) (cid:19)
X
k
f2(x) β X v(x) 2 γ dµ (x).
i M
≥ | | −
i=1 ZG
X (cid:0) (cid:1)
Theconjunctionof(1.4), which holdsbecause of(1.6), and(2.14)yields
the desired conclusion.
3. Proof of Theorem 1.4
We divide the proof into several steps.
8 EMMANUEL RUSSANDYANNICKSIRE
3.1. Rewriting the improved Poincar´e inequality. By the def-
inition of L , the conclusion of Proposition 1.3 means, in terms of
M
operators in L2(G,dµ ), that, for some λ > 0,
M
(3.15) L λµ,
M
≥
where µ is the multiplication operator by 1 + k X v 2. Using a
i=1| i |
functional calculus argument (see [Dav80], p. 110), one deduces from
P
(3.15) that, for any α (0,2),
∈
Lα/2 λα/2µα/2
M ≥
which implies, thanks to the fact Lα/2 = (Lα/4)2 and the symmetry of
M M
Lα/4 on L2(G,dµ ), that
M M
α/2
k
f(x) 2 1+ X v(x) 2 dµ (x)
i M
| | | | ≤
ZG i=1 !
X
2 2
α/4 α/4
C L f(x) dµ (x) = C L f .
M M M
ZG(cid:12) (cid:12) (cid:13) (cid:13)L2(G,dµM)
The conclusion o(cid:12)f Theorem(cid:12) 1.4 will follow(cid:13) by es(cid:13)timating the quantity
(cid:12) (cid:12) (cid:13) (cid:13)
Lα/4f 2 .
L2(G,dµM)
(cid:13) (cid:13)
(cid:13)3.2. Off(cid:13)-diagonal L2 estimates for the resolvent of L . The
M
crucial estimates to derive the desired inequality are some L2 “off-
diagonal” estimates for the resolvent of L , in the spirit of [Gaf59] .
M
This is the object of the following lemma.
Lemma 3.1. There exists C with the following property: for all closed
disjoint subsets E,F G with d(E,F) =: d > 0, all function f
⊂ ∈
L2(G,dµ ) supported in E and all t > 0,
M
(I+tL ) 1f + tL (I+tL ) 1f
M − L2(F,dµM) M M − L2(F,dµM) ≤
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) 8e−C √dt k(cid:13)fkL2(E,dµM). (cid:13)
Proof. We argue as in [AHL+02], Lemma 1.1. From the fact that L
M
is self-adjoint on L2(G,dµ ) we have
M
1
(L µ) 1
k M − − kL2(G,dµM) ≤ dist(µ,Σ(L ))
M
where Σ(L ) denotes the spectrum of L , and µ Σ(L ). Then we
M M M
6∈
deduce that (I+tL ) 1 is bounded with norm less than 1 for all t > 0,
M −
and it is clearly enough to argue when 0 < t < d.
NONLOCAL POINCARE´ INEQUALITIES 9
In the following computations, we will make explicit the dependence
of the measure dµ in terms of M for sake of clarity. Define u =
M t
(I+tL ) 1f, so that, for all function v H1(G,dµ ),
M − M
∈
(3.16) u (x)v(x)M(x)dx+
t
ZG
k
t X u (x) X v(x)M(x)dx =
i t i
·
i=1 ZG
X
f(x)v(x)M(x)dx.
ZG
Fix now a nonnegative function η (G) vanishing on E. Since f
∈ D
is supported in E, applying (3.16) with v = η2u (remember that
t
u H1(G,dµ )) yields
t M
∈
k
η2(x) u (x) 2 M(x)dx+t X u (x) X (η2u )M(x)dx = 0,
t i t i t
| | ·
ZG i=1 ZG
X
which implies
k
η2(x) u (x) 2 M(x)dx+t η2(x) X u (x) 2 M(x)dx
t i t
| | | |
ZG ZG i=1
X
k
= 2t η(x)u (x)X η(x) X u (x)M(x)dx
t i i t
− ·
i=1 ZG
X
k
t u (x) 2 X η(x) 2M(x)dx+
t i
≤ | | | |
ZG i=1
X
k
t η2(x) X u (x) 2 M(x)dx,
i t
| |
ZG i=1
X
hence
(3.17)
k
η2(x) u (x) 2 M(x)dx t u (x) 2 X η(x) 2 M(x)dx.
t t i
| | ≤ | | | |
ZG ZG i=1
X
Let ζ be a nonnegative smooth function on G such that ζ = 0 on E,
so that η := eαζ 1 0 and η vanishes on E for some α > 0 to be
− ≥
chosen. Choosing this particular η in (3.17) with α > 0 gives
eαζ(x) 1 2 u (x) 2 M(x)dx
t
− | | ≤
ZG
(cid:12) (cid:12)
(cid:12) (cid:12)
10 EMMANUEL RUSSANDYANNICKSIRE
k
α2t u (x) 2 X ζ(x) 2 e2αζ(x)M(x)dx.
t i
| | | |
ZG i=1
X
Taking α = 1/(2√t max X ζ ), one obtains
i i
k k
∞
1
eαζ(x) 1 2 u (x) 2 M(x)dx u (x) 2e2αζ(x)M(x)dx.
t t
− | | ≤ 4 | |
ZG ZG
Usin(cid:12)gthefact(cid:12)thatthenormof(I+tL ) 1 isboundedby1uniformly
(cid:12) (cid:12) M −
in t > 0, this gives
eαζu eαζ 1 u + u
t L2(G,dµM) ≤ − t L2(G,dµM) k tkL2(G,dµM)
1
(cid:13) (cid:13) (cid:13)(cid:0) eαζu (cid:1) (cid:13) + f ,
(cid:13) (cid:13) ≤ (cid:13)2 t L2(G(cid:13),dµM) k kL2(G,dµM)
therefore (cid:13) (cid:13)
(cid:13) (cid:13)
eαζ(x) 2 u (x) 2 M(x)dx 4 f(x) 2 M(x)dx.
t
| | ≤ | |
ZG ZG
We choose(cid:12)now ζ(cid:12)such that ζ = 0 on E as before and additionnally that
(cid:12) (cid:12)
ζ = 1 on F. It can furthermore be chosen with max X ζ
i=1,...k i
k k ≤
C/d, which yields the desired conclusion for the L2 norm of (∞I +
tL ) 1f withafactor4intheright-handside. SincetL (I+tL ) 1f =
M − M M −
f (I + tL ) 1f, the desired inequality with a factor 8 readily fol-
M −
− (cid:3)
lows.
3.3. Control of Lα/4f and conclusion of the proof of
M
L2(G,dµM)
Theorem 1.4. T(cid:13)his is n(cid:13)ow the heart of the proof to reach the conclu-
(cid:13) (cid:13)
sion of Theorem 1(cid:13).4. The(cid:13)following first lemma is a standard quadratic
estimate on powers of subelliptic operators. It is based on spectral
theory.
Lemma 3.2. Let α (0,2). There exists C > 0 such that, for all
∈
f (L ),
M
∈ D
(3.18)
2 +
Lα/4f C ∞t 1 α/2 tL (I+tL ) 1f 2 dt.
(cid:13) M (cid:13)L2(G,dµM) ≤ 3 Z0 − − M M − L2(G,dµM)
(cid:13) (cid:13)
(cid:13) We n(cid:13)ow come to the desired estimate.
(cid:13) (cid:13)
(cid:13) (cid:13)
Lemma 3.3. Let α (0,2) . There exists C > 0 such that, for all
∈
f (G),
∈ D
∞t 1 α/2 tL (I+tL ) 1f 2 dt
− − M M − L2(G,dµM) ≤
Z0
(cid:13)(cid:13) f(x) f(y) 2 (cid:13)(cid:13)
C | − | M(x) dxdy.
V ( y 1x ) y 1x α
ZZG×G | − | | − |