Nash-type Inequalities and
decay of semigroups.
Patrick Maheux
University of Orléans.
Fédération Denis Poisson.
Workshop in Nanterre . June 4-5-6, 2007.
SETTING
• (X, µ) measure space with µ a σ-finite
measure.
• (Tt)t>0: a symmetric submarkovian semigroup:
•
•
•
(Ttf, h) = (f, Tth),
∀f ∈ L2,
∀f, g ∈ L2
0 ≤ f ≤ 1 ⇒ 0 ≤ Ttf ≤ 1
(Tt)t>0 acts as a contraction on Lp.
• L: infinitesimal generator associated to
the semigroup Tt = e−tL
∂Tt
= −ATtf
∂t
• E: Dirichlet form associated to (Tt):
E(f ) = (Lf, f ),
f ∈ D(L)
Important property
∀f ∈ D(E) ⇒ h = (f ∨ 0) ∧ 1 ∈ D(E)
and
E(h) ≤ E(f )
MOTIVATION
Let λ > 0.
• Spectral decay :
(SG)
−2λt||f ||2
||Ttf ||2
2≤e
2
• Poincaré inequality :
(P )
λ||f ||2
2 ≤ (Lf, f ).
• (SG) can be written as
G ||Ttf ||2
2
−2t
2
≤e
G ||f ||2 ,
||f ||q ≤ 1.
with G(x) = x1/λ.
So the spectral dimension can be seen
as a function namely G .
Main goal
Study the Exponential Functional Decay
( G-decay for short)
Let 1 ≤ q < p < ∞ and G increasing
(EF D)p,q
G ||Ttf ||pp
−pt
p
≤ e G ||f ||p ,
||f ||q ≤ 1
Under some conditions, we deduce:
• a Generalized Spectral Decay:
(GSG)
||Ttf ||pp ≤ φ(t)||f ||pp,
∀t > 0
• Ultracontractivity:
(U LT )
2
||Ttf ||2
2 ≤ ψ(t)||f ||1 ,
∀t > 0
Nash-type inequality
Consider SuperPoincaré inequality
(SP )
||f ||2
2 ≤ s(Lf, f ) + β(s)||f ||1 ,
∀s > 0
What is a Nash-type inequality ?
(N T I)
Θ(||f ||2
2 ) ≤ (Lf, f ),
( Θ increasing)
||f ||1 ≤ 1
What relation between Nash-type inequality (NTI) and SuperPoincaré ? (in L2)
Answer (SP) implies (NTI) and Θ(x) =
supt>0(tx − tβ(1/t)) (and conversely...).
Here we study a more general situation
(N T I)p,q
Θ(||f ||pp) ≤ (Lf, f p−1),
||f ||q ≤ 1.
We assume Θ of the form Θ(x) = x N (ln x)
with N increasing continuous (which often appears...). So
(N T I)p,q
||f ||pp N
log ||f ||pp
≤ (Lf, f p−1)
with ||f ||q ≤ 1.
On the other side, we have
(EF D)p,q
G ||Ttf ||pp
−pt
p
≤ e G ||f ||p ,
||f ||q ≤ 1
• This is equivalent to
pt
p
t → e G ||Ttf ||p
non-increasing.
• Why Θ of this special form ? (p = 2, q =
1)
Assume
||Ttf ||2 ≤ eM (t)||f ||1,
∀t > 0
from Davies-Simon Thm
Z
f 2 ln
f2
||f ||2
2
!
dµ ≤ 2tE(f ) + 2M (t)||f ||2
2
Let 0 ≤ f, ||f ||1 = 1 i.e f dµ is a probability.
Then
2
2
||f ||2
2 ln ||f ||2 dµ ≤ 2tE(f ) + 2M (t)||f ||2
So
||f ||2
2N
log ||f ||2
2
≤ E(f )
with
1
1
ln x − M (t)
N (ln x) = sup
t
t>0 2t
Main result
Theorem
Let L be the generator of a symmetric
submarkovian semigroup (Tt). Let 1 ≤
q < p < ∞ and D be the domain of L.
The two following statements (1) and
(2) are equivalent:
1. There exists N : R −→ (0, +∞) a continuous non-decreasing function such
that for all f ∈ D with ||f ||q ≤ 1,
||f ||pp N
ln ||f ||pp
≤ (Lf, f p−1).
(1)
2. There exits G ∈ C 1((0, ∞), (0, ∞)) an
increasing function such that for all
t > 0 and for all f ∈ D with ||f ||q ≤ 1,
G( ||Ttf ||pp ) ≤ e−pt G( ||f ||pp ).
(2)
Moreover
• (1) implies (2) with
G = exp o F o ln
with the derivative of F satisfiying
F 0 = 1/N
• (2) implies (1) with
G(ey )
N (y) = y 0 y , y ∈ R
e G (e )
Corollary 1 (spectral decay)
• Assume that the semigroup (Tt) satisfies a G-decay for some 1 ≤ q < p < ∞.
• If the measure µ is finite, then
||Ttf ||pp ≤ ψ(t) ||f ||pp
with ψ(t) = µ(X)γ G −1 e−pt G(µ(X)−γ ) and
γ = pq − 1.
Corollary 2 (Ultracontractivity)
• Assume that the semigroup (Tt) satisfies a G-decay for some 1 ≤ q < p < ∞.
• If G is bounded by k, then
||Ttf ||p ≤ φ(t) ||f ||q
with
i1/p
−1
−pt
ke
φ(t) = G
h
How to get Nash-type inequalities ?
• Directly form the example under study.
• From other functional inequalities:
• Ultracontractivity
||Ttf ||∞ ≤ U (t)||f ||1,
∀t > 0
(Through Davies-Simon log-Sobolev inequality for instance or Coulhon’s Thm)
• From Orlicz-Sobolev inequality...
• From a Nash-type inequality itself :
”from L2 to Lp” (p > 2) (f → f p/2)
• From one operator to another:
• Comparison of Dirichlet forms.
• From L to its fractional powers (A.Bendikov
& P.M):
2
||f ||2
2 B(||f ||2 ) ≤ (Lf, f ),
||f ||1 ≤ 1
Then 0 < α < 1, There exists c1, c2 s.t.
α
2
α
c1||f ||2
2 B (c2 ||f ||2 ) ≤ (L f, f ),
||f ||1 ≤ 1
Example 1
• Usual Laplacian on Rn: Nash inequality
2+4/n
c0||f |2
≤ (Lf, f ),
||f |1 ≤ 1
also for fractional operator 0 < α ≤ 1,
2+4α/n
c||f |2
≤ (Lαf, f ),
||f |1 ≤ 1
2
N (y) = c0 exp
y ,
n
i.e.(α = 1)
y∈R
So
!
n −2
G(x) = exp −
x n ,
2c0
x>0
G-decay can be written as : for any f ∈
L1 ∩ L2
||Ttf ||2 ≤ Ht(f ) ||f ||2

Ht(f ) = 1 + c t
!4/n−n/4
||f ||2

||f ||1
implies
||Ttf ||2 ≤ ||f ||2
,
c
||Ttf ||2 ≤ n/4 ||f ||1
t
G-decay can be seen as ” interpolated
inequalities ” between L2-contraction and
ultracontractivity of the semigroup.
Example 2
1+1/γ
• Let N (y) = k y+
. We have
γ
G(x) = exp − [ln x]−1/γ , x > 1
k
This is the case when
γ
c
/t
1
||Ttf ||2 ≤ c e
||f ||1,
∀t > 0
Example 3
1/γ
• Let N (y) = y+(ln y+)+ , γ > 0.
If γ 6= 1,
γ
G(x) = exp
(ln ln x)1−1/γ
γ−1
!
If γ = 1,
G(x) = ln(ln x))
x>e
G is bounded if and only if γ < 1.
x>e
Example 4 (Weak Gross inequality)
• Let N (y) = c y, c > 0.
2
c||f ||2
2 ln ||f ||2 ≤ (Lf, f ),
G(x) = (ln x)1/c ,
G − decay
with α(t) = e−2ct
||f ||1 ≤ 1.
x>1
2α(t)
||Ttf ||2
2 ≤ ||f ||2
.
• Weak SuperPoincaré (weak Nash-T.I)
2α(t)
||f ||2
2 ≤ 2t E(f ) + ||f ||2
,
∀t > 0
with α(t) = e−2t (for instance)
is equivalent to
2
||f ||2
2 ln ||f ||2 ≤ (Lf, f ),
||f ||1 ≤ 1.
Nash revisiting Gross, Nelson, ...:
(proof may be already known )
(Hyper)
||Ttf ||2 ≤ ||f ||p(t),
⇒ (W N T I)
p(t) = 1+e−2t, ∀t > 0
2
||f ||2
2 ≤ 2t E(f ) + ||f ||p(t)
Weak form of interpolated inequality (between Gauss and Poincaré):1 < p < 2
!
2 ≤ ln
⇒ ||f ||2
−
||f
||
p
2
1
ln p−1
= ln 1 + 2−p
p−1
⇒ (Gross)
Z
1
E(f )
p−1
∼ (2 − p) as p → 2−.
f 2 ln
f2
||f 2||2
2
!
≤ E(f )
Example 5 (symmetric Γ-semigroup)
Let
E(f ) =
Z
ln(1 + 4π 2|x|2)|fˆ(x)|2 dx
R
dx: Lebesgue measure (not finite!)
Let
E(f ) = (ln(I + ∆)f, f )
for any 0 < ε < 1 and any ||f ||1 ≤ 1:
2
2
4
2
(1−ε) ||f ||2 ln 1 + π ε ||f ||2 ≤ (ln(I+∆)f, f )
N (ln x) = (1 − ε) ln 1 + π 2ε2x2
2
2
||f ||2 ln ||f ||2 ≤ (ln(I+∆)f, f )
(ε = 1/2)
but dµ(x) = dx is not finite!
We have a so-called defective Gross inequality.
♣