Around Nash Inequalities

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Dominique Bakry, François Bolley, Ivan Gentil

To cite this version:

Dominique Bakry

∗ †

, François Bolley

‡

and Ivan Gentil

§

September 10, 2010

Introdu tion

In the Eu lidean spa e

R

n

, the lassi al Nashinequality may be statedas

(0.1)

kfk

1+n/2

2

≤ C

n

kfk

1

k∇fk

n/2

2

for all smooth fun tions

f

(with ompa t support for instan e) where the norms are

omputed with respe t to the Lebesgue measure. This inequality has been introdu ed

by J. Nash in 1958 (see [9℄) toobtain regularity properties on the solutionsto paraboli

partial dierential equations. The omputation of the optimal onstant

C

n

has been

performed morere ently in[6℄.

This inequality may be stated in the general framework of symmetri Markov

semi-groups,where itis asimple andpowerful tooltoobtain estimates onthe asso iated heat

kernel. In this ontext, one repla es

k∇fk

2

2

by the Diri hletform

E(f, f)

asso iated with thesemigroup, andthe Lebesguemeasurebyitsreversible measure. Moreover, the power

fun tion

x

n

in the inequality is repla ed by a more general onvex fun tion

Φ

, and un-der this form it an be valid(and useful) even in innite dimensional situations su h as

those whi h appear in statisti al me hani s. One an also give weighted forms of these

inequalities : they also lead to pre ise estimates on the semigroup, or on the spe tral

de ompositionof the generator.

TheaimofthisshortnoteistoexplainhowNashinequalitiesleadtosu hestimatesin

ageneral settingand alsoto show simple te hniques used to establishthe required Nash

inequalities. There isno laim for originality,most of the materialin luded here may be

found invariouspapers su h as [1, 2, 5,7, 13℄.

∗

Institut deMathématiquesdeToulouse,UMRCNRS5219,UniversitéPaul-Sabatier

†

Institut UniversitairedeFran e.

‡

Ceremade,UMRCNRS 7534,UniversitéParis-Dauphine.

§

Nashinequalitiesbelongtotheverylargefamilyoffun tionalinequalitiesfor

symmet-ri Markov semigroups whi h have led tomany re ent works. Many of these inequalities

ompare

L

p

norms of fun tions to the

L

2

norms of their gradients, whi h in this

on-text is alled the Diri hlet form; this is the ase of the simplest ones, the spe tral gap

(or Poin aré) inequalities. But one may also onsider

L

1

norms of the gradients, in the

area of isoperimetri inequalities, or

L

p

norms, even

L

∞

norms, when one is on erned

withestimates onLipshitzfun tions,for instan e inthearea of on entration ofmeasure

phenomena.

Here, we shall on entrate only on

L

2

norms of gradients. Even in this setting, there

exists awide variety of inequalities,whi hare adapted tothe kind of measure one wants

to study on one side, and to the properties they des ribe on the other. For example,

measures with polynomial de ay are not overed by the same inequalities as measures

with exponential, orsquare exponentialde ay.

The family of Nash type inequalities we present here belongs to the wide family of

the Sobolev type inequalities. Their main interest is that they easilyprovide good (and

sometimesalmostoptimal) ontrolonheat kernels. Startingfromthe lassi alinequality,

we shall show how to extend them rst by the introdu tion of a rate fun tion

Φ

, and thenbythe extraintrodu tionofaweightfun tion

V

(aLyapunovfun tion). As weshall see, the link between Nashinequalities and estimates onthe semigroup spe trumis very

simple and, as usual in the eld, roughly relies on derivation along time and integration

by parts. This is why it is tempting touse itin awide range of situations.

Then, we shall show how to obtain these inequalities in the simplest models on the

realline. Restri tingourselvestothereallinemaybethought aslookingonlyattheeasy

ase. In fa t, by hoosing various measures, one may produ e a lot of dierent model

ases whi h really illustrate what may or may not be expe ted from these inequalities.

Then the extension to higher dimensional situations(like

R

n

ormanifolds) is very often

apure matter of te hni alities, extendingin adire t way the one-dimensional methods.

The paper is organized asfollows. In the rst se tion, we briey present the ontext

of symmetri Markov semigroups, and parti ularlydiusion semigroups. Then, we show

dierent variations of Nash inequalities and how to get estimates on heat kernels from

them. Then, inne, we show howtoprodu e su hNash inequalitiesonthe basi models

onthe real line weare interested in.

1 Symmetri Markov semigroups and difusions

To understand the general ontext of Markov semigroups, we rst onsider a measure

spa e

(E, B, µ)

, where

B

isa

σ

-eld and

µ

is a

σ

-nite measure on it. Although we shall alwaysfo us onexampleswhere

(E, B)

is

R

n

equipped withthe usualBorelsets (orsome

openset init,oranite dimensionalmanifoldwithorwithoutboundaries), itmaybean

aboutthemeasurablestru ture ofthe spa e. Inany ase, oneshouldalwayssuppose that

(E, B, µ)

is a "reasonable" measure spa e : we shall not say in details what we mean by "reasonable",but resultssu hasthede ompositionofmeasure theoremsshouldbevalid,

whi h overs all asesone ouldlook atin pra tise.

Given

(E, B, µ)

asymmetri Markov semigroup is afamily

(P

t

)

t≥0

of linear operators mappingthesetofboundedmeasurablefun tionsintoitselfwiththefollowingproperties:

(i) Preservation of positivity: if

f ≥ 0

, so is

P

t

f

. (ii) Preservation of onstant fun tions :

P

t

1 = 1

. (iii) Semigroup property :

P

t

◦ P

s

= P

t+s

.

(iv) Symmetry :

P

t

maps

L

2

_(µ)

intoitself and, forany pair

(f, g) ∈ L

2

_(µ)

,one has

Z

E

P

t

f g dµ =

Z

E

f P

t

g dµ.

(v) Continuity at

t = 0

:

P

0

= Id

and

P

t

f → f

when

t → 0

in

L

2

_(µ)

.

Su hsemigroupsnaturallyappearinprobabilitytheoryas

P

t

f (x) = E(f(X

t

)/X

0

= x)

where

(X

t

)

t≥0

is a Markov pro ess. The symmetry property does not always hold and it is equivalent to the reversibility of the pro ess. They also appear in many situations

whenone solves a"heat equation"of the form

∂

t

F (x, t) = LF, F (x, 0) = f (x);

here

L

is ase ond order sub-ellipti (orhypo-ellipti )dierential operator,in whi h ase

P

t

f

isthe solution

F (x, t)

at time

t

when the initialvalue is

F (x, 0) = f (x)

. Letus start with some elementary preliminaryremarks.

(a) Sin e

P

t

is symmetri ,and

P

t

1 = 1

, one gets

R

E

P

t

f dµ =

R

E

f dµ

by taking

g = 1

in the property (iv):

µ

is invariantunder the semigroup.

(b) Sin e

P

t

is linear and positivity preserving,

|P

t

f | ≤ P

t

|f|

. This implies that

P

t

is a ontra tionin

L

1

_(µ)

by invarian eof the measure.

( ) Bythe sameargument,

P

t

is alsoa ontra tionin

L

∞

_(µ)

and therefore,by

interpola-tion,

P

t

is a ontra tion in

L

p

_(µ)

for any

p ∈ [1, ∞]

.

(d) Sin e

(P

t

)

t

is a semigroup of ontra tions in

L

2

_(µ)

, by the Hille-Yoshida theory, it

Formally, any property of the semigroup may be translated into a property of the

generator

L

, and vi e versa. For instan e, the preservation of onstants property translates into

L1 = 0

. Also, the symmetry translates into the fa t that

L

is self-adjoint,that is,

Z

E

f Lg dµ =

Z

E

gLf dµ.

The positivity preserving property is more subtle. In general, it is translated into

a maximum prin iple of the generator. But this requires a bit more than just a

measurablestru ture on the spa e, and weprefer to translatethis intothe positivity

of the arré du hamp operator, see point(k) below.

(e) The measure

µ

being symmetri (or "reversible") is in general unique up to a nor-malizing onstant (it is however a restri tive ondition that su h a measure exists :

see formula(1.3) below). When the measure is nite, we may therefore normalizeit

as tobea probability measure, and we shall always do it. In this ase, the onstant

fun tion

1

isalwaysanormalized eigenve tor,asso iated withthe eigenvalue

0

whi h isthe smallestvalueof thespe trumof

−L

. Inthe innite ase, thereisno anoni al way of hoosing agoodnormalization.

(f) Sin ethemeasurespa e

(E, B, µ)

isa"reasonable"spa e,any su hoperator

P

t

whi h preserves the onstants and positivitymay berepresented as

P

t

f (x) =

Z

E

f (y)P

t

(x, dy),

where

P

t

(x, dy)

is akernel of probability measures, that is, aprobability measure on

E

dependingonthe parameter

x ∈ E

inameasurable way. Thisenables forexample to apply to

P

t

any generi property of probability measures, su h as the varian e inequality

P

t

(f

2

_{) ≥ (P}

t

f )

2

.

(g) Veryoften(andweshallseethatNashinequalitiesprovideauseful riteriumforthis),

this kernel has a density with respe t tothe reversible measure

µ

,that is

P

t

(x, dy) = p

t

(x, y)µ(dy);

here

p

t

(x, y)

is a non negative fun tion whi h is dened almost everywhere (with respe t to

µ ⊗ µ

) on

E × E

. Then the symmetry property (iv) is equivalent to the symmetryofthiskernel

p

t

(x, y) = p

t

(y, x)

. Mu hattentionhas beenbroughtoverthe lastde ades tovariousestimates onthis kernel density (inparti ular in Riemannian

geometry, for heat kernels on Riemannian manifolds, using tools from geometry like

urvature, Riemannian distan e, et ). On e again, Nash inequalities may provide

good su h estimates, as we shall see later on.

(h) When we have su h densities, the semigroup property translates intothe

Chapman-Kolmogorov equation

p

t+s

(x, y) =

Z

E

Hen e, by the Cau hy-S hwarz inequality,

p

2t

(x, y)

2

≤ p

2t

(x, x) p

2t

(y, y).

As the onsequen e, the maximum of

p

t

(x, y)

isalways obtained onthe diagonal. (i) The generator

L

being self-adjoint has a spe tral de omposition with a spe trum in

(−∞, 0]

a ording to(1.2).

(j) It may be the ase that the spe trum is dis rete, and that we have a omplete

se-quen eoforthonormaleigenve tors

(f

n

)

in

L

2

_(µ)

,witheigenvalues

−λ

n

for

L

. Inthis situation,the kernel density

p

t

(x, y)

is given by

p

t

(x, y) =

X

n

e

−λ

n

t

_f

n

(x)f

n

(y).

Then wehave the tra e formula

Z

E

p

t

(x, x)dµ(x) =

X

n

e

−λ

n

t

_.

On e again, Nash inequalities will provide uniform (or non uniform) bounds on the

densities, and therefore bounds onthe ounting fun tion of the sequen e

(λ

n

)

. (k) Byderivation at

t = 0

the varian e inequality

P

t

(f

2

_{) ≥ (P}

t

f )

2

givesthe inequality

L(f

2

_{) ≥ 2fLf.}

In parti ular

(1.2)

Z

E

f Lf dµ ≤ 0

by invarian e of

µ

. Of ourse, one has to take are about whi h fun tions these do apply. In general, we assume that there is an algebra of fun tions

A

dense in the domain of

L

, for whi h this is valid. In this ase, one denes the arré du hamp as the bilinearform

Γ(f, g) =

1

2

L(f g) − fLg − gLf

.

It satises

Γ(f, f ) ≥ 0

, and in somesense this hara terizes the positivity preserving property of

P

t

.

The Diri hlet form asso iated to

P

t

isnally dened by

The last identity is based on the identity

R

E

L(f g)dµ = 0

and is alled the integration by parts formula. The Diri hlet form is in general dened on a larger domain than the

operator

L

itself (formally, it requires only one derivative of the fun tion to be in

L

2

_(µ)

insteadof

2

for the generator).

The knowledge of the measure and of the arré du hamp (or of the Diri hlet form)

entirelydes ribes the operator

L

(and therefore the semigroup), sin e

L

may be dened from

Γ

and

µ

through the integration by parts formula(see (1.3)).

The basi example of su h semigroups is of ourse the standard heat kernel in the

Eu lidean spa e

R

n

; for

t > 0

, its density

p

t

(x, y)

with respe t to the Lebesgue measure

dy

is

p

t

(x, y) =

1

(4πt)

n/2

exp(−

|x − y|

2

4t

).

Here,

µ(dy) = dy

,

L = ∆

and

Γ(f, f ) = |∇f|

2

.

This orresponds to the ase studied by Nash in [9℄. It is one of the very few examples

whereone expli itlyknows

P

t

,sin eingeneralweonlyknow

L

,andtheissue istodedu e asmu h informationas possible on

P

t

fromthe knowledge of

L

.

Another model ase is the Ornstein-Uhlenbe k semigroupon

R

n

, for whi h

Lf (x) = ∆f (x) − x · ∇f(x), Γ(f, f) = |∇f|

2

, µ(dx) =

1

(2π)

n/2

exp(−|x|

2

/2)dx.

Its density with respe t to the Gauss measure

µ(dy)

is

p

t

(x, y) = (1 − e

−2t

)

−n/2

exp



−

1

2(1 − e

−2t

₎

(|y|

2

_e

−2t

− 2 x · ye

−t

+ |x|

2

e

−2t

)



and it behaves in a very dierent way from the previous example as long as fun tional

inequalitiesare on erned.

In the two previous ases, the arré du hamp is the same (and the semigroups only

dierbythemeasure

µ(dx)

). Observethat

Γ(f, g)

isinboth asesarstorderdierential operator in its two arguments. They both belong to the large lass of diusion Markov

semigroups,whi h are semigroups su h that for all smooth fun tions

φ

Γ(φ(f ), g) = φ

′

(f )Γ(f, g)

orequivalently

Lφ(f ) = φ

′

(f )Lf + φ

′′

(f )Γ(f, f ).

This property is alled the hange of variable formula for

L

and is an intrinsi way of saying that

L

is a se ond order dierential operator. The fa t that

L(1) = 0

says that thereis no

0

-orderterm in

L

. Onemay easily he k that amongall dierentialoperators

on

R

n

Γ(f, f ) ≥ 0

,provided the matrix ofthe se ondorder termsispositive-semideniteatany point.

Non diusion ases are of onsiderable interest sin e they are related to Markov

pro- esses with jumps and also naturally appear when one looks at subordinators. However

weshall on entrate here onthe diusion ase, even thoughthe Nashte hniques may be

used inthe same way inthe general ase.

In general,a se ond orderdierentialoperator without

0

-order termsof the form

Lf =

X

ij

a

ij

(x)∂

_ij

2

f +

X

i

b

i

(x)∂

i

f

has a arré du hamp given by

Γ(f, g) =

X

ij

a

ij

_(x)∂

i

f ∂

j

g = ∇f · A(x)∇g.

Therefore the positivity of

Γ

is equivalent to the fa t that at any point

x

the matrix

A(x) = (a

ij

_(x))

is positive-semidenite. Conversely, when the arré du hamp is given

(on aopen set in

R

n

or onasmooth manifold in lo al oordinates) by

Γ(f, g) =

X

ij

a

ij

(x)∂

i

f ∂

j

g,

withpositive-semidenitematri es

(a

ij

_(x))

having smooth oe ients,and whenthe

ref-eren e measure

µ(dx)

has a smooth positive density

ρ(x)

with respe t to the Lebesgue measure, it orresponds toa unique symmetri diusion operator

L

whi h is

(1.3)

Lf =

1

ρ(x)

X

i

∂

i



ρ(x)

X

j

a

ij

(x)∂

j

f



.

Inotherwords,

Γ

odes forthese ondorderpartof theoperatorwhile

µ

odesfor the rstorderterms. Observealsothatea h

(Γ, µ)

leadstounique symmetri

L

,butpossibly several non symmetri

L

.

A model ase onwhi h weshall fo us is the ase when

E = R

and

Γ(f, f ) = f

′2

. We

shall look atthe measures

(1.4)

µ(dx) = C exp(−|x|

a

_)dx,

where

a > 0

and

C

isa normalizing onstant. In orderto avoidirrelevant di ultiesdue to the non smoothness of

|x|

at

0

, we shall repla e

|x|

by

√

1 + x

2

. Depending on the

value of

a

, the orresponding semigroups present diverse behaviours.

For

a = 2

, the elebrated Nelson theorem [10℄ asserts that the Ornstein-Uhlenbe k semigroupis "hyper ontra tive", whi h means that

P

t

isbounded from

L

2

_(µ)

to

L

for all

t > 0

and some

q(t) > 2

. This is equivalent to the also famous Gross logarithmi Sobolev inequality[8℄ (1.5)

Z

f

2

log f

2

_{dµ ≤}

Z

f

2

dµ log



Z

f

2

dµ



_{+ CE(f, f).}

When

a > 2

, the semigroup is "ultra ontra tive", whi h means that

P

t

maps

L

1

_(µ)

into

L

∞

_(µ)

for any

t > 0

, while it is not even hyper ontra tive for

a < 2

. Nevertheless, for

1 < a < 2

,ithas adis retespe trumanditisHilbert-S hmidt,andweshallsee below howto get estimates onthe spe trumthrough weighted Nashinequalities.

For

a = 1

,the spe trumisnolongerdis reteandtheonlypropertyleftistheexisten e of a spe tral gap : the spe trum of

−L

lies in

{0} ∪ [A, ∞)

for some

A > 0

, and this property is equivalentto aspe tralgap (or Poin aré) inequality

(1.6)

Z

f

2

_{dµ ≤}



Z

f dµ



2

+

1

A

E(f, f).

When

a < 1

,even the spe tral gap property is lost. Of ourse, one may look at similar models in

R

n

, or on Riemannian manifolds with

density measures (with respe t to the Riemann measure) depending on the distan e to

some point. In this latter ase, one would get more ompli ated results, sin e in general

one has to take intoa ount lower bounds onthe Ri i urvature, and even more if one

works with boundaries (with Neumann boundary onditions). We shall not develop this

here.

2 Nash inequalities

Inthe ontextofDiri hletformsasso iatedtosymmetri Markovsemigroupsasdes ribed

above,a Nashinequality isan inequality of the form

(2.7)

kfk

1+n/2

2

≤ kfk

1

k

C

1

kfk

2

2

+ C

2

E(f, f)



n/4

;

herethe norms

L

p

areof ourse omputedwith respe t tothe reversiblemeasure

µ

and

n

isany positive parameter(that we allthedimension in the Nashinequality,sin e in the

lassi al ase the unique possible value for

n

is really the dimension of the spa e). This inequality should apply for any

f

in the Diri hlet domain, but it is enough to he k it in a dense subspa e of it whi h, in many examples, will be the set of smooth ompa tly

supported fun tions.

It is worth mentioning that when

µ

isa probability measure, then

C

1

≥ 1

(as an be seen by hoosing

f = 1

), while for example in

R

n

with the Lebesgue measure, one may

When

µ

is a probability measure and

C

1

= 1

, we say that the inequality is tight. It thenimpliesaspe tralgap inequality,asonemaysee by applyingthe inequality to

1 + ǫf

and letting

ǫ

go to

0

.

Conversely, if (2.7) is valid with

C

1

> 1

and with

µ

being a probability measure, together with a spe tral gap inequality, then a tight Nash inequality holds (see [1℄ for

example). In general, we say that a fun tional inequality is tight when one may dedu e

from the inequality that

{E(f, f) = 0}=⇒{f =

onstant

}

. Here, when

C

1

= 1

, this is ensured by the equality ase in the inequality

kfk

1

≤ kfk

2

. As we shall see, tightness may beused to ontrol the onvergen e toequilibrium,that is the asymptoti behaviour

when

t → ∞

, while the general inequality is useful to ontrol the short time behaviour. Mostof fun tionalinequalitiesmay be tightenedin presen e of aspe tral gap inequality,

asit isthe ase here.

In the ase of an innite measure, tightness orresponds tothe ase when

C

1

= 0

, as inthe Eu lidean ase.

However, there is a strong dieren e between the forms that the Nash inequalities

may take a ording towhether the measure is nite ornot. We know that atight Nash

inequality holds true in the Eu lidean spa e, but it an be proved that the tight Nash

inequality (2.7) may not be validon a nite measure spa e unless the spa e is ompa t.

More pre isely, whenone has atightNashinequality (2.7)onanite measure spa e, one

maygetaboundontheos illationofLip hitzfun tions,when eaboundonthe diameter

of the spa e (this diameter being measured in terms of an intrinsi distan e asso iated

with the arré du hamp [4℄). This explains why belowwe introdu e the extended Nash

inequalities(2.9)and(2.10),whi hmaybevalidonnitemeasurespa eswithunbounded

support, as we shall see.

When

n > 2

, the Nash inequality (2.7) is one of the many forms of the Sobolev inequality

kfk

2

2n/(n−2)

≤ C

1

kfk

2

+ C

2

E(f, f).

Indeed, observe rst that this and Hölder's inequalitieslead tothe Nashinequality (2.7)

with the same "dimension"

n

and the same onstants

C

1

and

C

2

. The way ba k is a little more subtle: the argument in [3℄ is based on applying the Nash inequality to the

sequen e of fun tions

f

n

= min{(f − 2

n

₎

+

, 2

n

}

, addingthe obtained estimates and using the identity

P

n

E(f

n

, f

n

) = E(f, f)

. This enables tokeep the same dimension

n

,but not the onstants

C

1

and

C

2

.

Inthe ontextofDiri hletforms,su hSobolevinequalitiesmayappearunderdierent

forms su h as Energy-Entropy, Gagliardo-Nirenberg, Faber-Krahn et inequalities. We

referto [3℄ for full details. The onne tion between Sobolev (and Nash) inequalities and

various bounds on heat kernels has been explored by many authors, see [1, 7, 11℄ for

example.

Theorem 2.1. Assume that inequality (2.7) holds. Then, (2.8)

kP

t

f k

2

≤ C(t)kfk

1

,

where

C(t) =



_max{2C

1

,

2nC

2

t

}



n/4

.

Conversely, if (2.8) holds with

C(t) ≤ a + bt

−n/4

, then a Nash inequality (2.7) holds

with the same dimension

n

and onstants

C

1

and

C

2

depending only on

n, a

and

b

. Proof  Letus rewritethe inequality under the form

 kfk

2

2

kfk

2

1



1+2/n

≤ C

1

kfk

2

2

kfk

2

1

+ C

2

E(f, f)

kfk

2

1

.

Now, hoose apositive fun tion

f

and apply the pre eding bound to

P

t

f

. We knowfrom invarian eof

µ

that

R

E

P

t

f dµ =

R

E

f dµ

. Letus set

H(t) =

kP

t

f k

2

₂

kP

t

f k

2

₁

=

kP

t

f k

2

₂

kf k

2

1

. Wehave

∂

t

kP

t

f k

2

2

= 2

Z

P

t

f LP

t

f dµ = −2 E(P

t

f, P

t

f ).

Therefore,

H

isde reasing and

H

′

_{(t) = −2}

E(P

t

f, P

t

f )

kP

t

f k

2

1

,

and the Nashinequality (2.7) be omes

H

1+2/n

_{≤ C}

1

H − 2C

2

H

′

.

Now, as long as

H ≥ (2C

1

)

n/2

,one has

H

1+2/n

_{≥ 2C}

1

H

,and we get

H

1+2/n

_{≤ −4C}

2

H

′

,

and this dierential inequality (with the fa t that

H

isde reasing) givesthe result. To see the reverse way, we may observe that, for a general symmetri Markov

semi-group, the fun tion

t 7→ K(t) = log kP

t

f k

2

2

is onvex. Indeed, the derivative of

K

1

(t) =

kP

t

f k

2

2

is

−2

R P

t

f LP

t

f dµ

, whilethe se ondderivativeis

4

R (LP

t

f )

2

_dµ

,and thereforeby

Cau hy-S hwarz inequality one has

K

₁

′

2

_{≤ K}

1

K

1

′′

,

whi hsaysthat

log K

1

= K

is onvex. Thensoisthefun tion

h(t) = log

Now, if we have a bound of the form

H(t) ≤ a + bt

−n/2

, we may plug this upper bound

inthe previous inequality, and then optimisein

t

to get the result. In fa t,havinga bound for

P

t

asan operatorfrom

L

1

into

L

2

,we are very lose from

a uniform bound on the kernel

p

t

. Indeed, if

P

t

is bounded from

L

1

into

L

2

with norm

C(t)

, thenby symmetryand duality, itisalsoboundedfrom

L

2

into

L

∞

with norm

C(t)

,

and therefore by omposition and semigroup property

P

2t

is bounded from

L

1

into

L

∞

with norm

C(t)

2

.

Conversely,bytheRiesz-Thorintheorem,if

P

t

isbounded from

L

1

into

L

∞

withnorm

C

1

(t)

, being bounded from

L

1

intoitselfwith norm

1

,it isalsobounded from

L

1

into

L

2

with norm

C

1

(t)

1/2

. Inthe end, we have obtained the following

Theorem2.2. ANashinequality (2.7)holdswithdimension

n

ifandonlyif

P

t

isbounded

from

L

1

into

L

∞

with norm bounded above by

a + bt

−n/2

.

Of ourse, inthe ase when

C

1

= 0

, whi h orresponds tothe lassi alEu lidean Nash inequality, the equivalen e is valid with a bound of the form

C(t) = at

−n/2

.

Moreover, a verygeneral fa t(validon"reasonablemeasure spa es"

(E, B, µ)

)asserts thatanoperator

K

isbounded from

L

1

_(µ)

into

L

∞

_(µ)

ifandonlyifitmayberepresented

by a bounded kernel density

k

:

K(f )(x) =

R

E

k(x, y)f (y)µ(dy)

. Moreover, the norm operatorof

K

is exa tlythe

L

∞

norm of

k

(on

E × E

).

Sowehaveseen thataNashinequalityisequivalenttoauniformboundonthe kernel

of

P

t

(and also arries the existen e of su h kernel), with very few assumptions on the spa e.

Observe that there is noreason why we shouldrestri t ourselves tothe ase ofpower

fun tionsin Nashinequalities. One may onsider extensions of the form

(2.9)

Φ



kfk

2

2

kfk

2

1



≤

E(f, f)

kfk

2

1

,

validsaywhenever

kfk

2

> Mkfk

1

. Here

Φ

isasmooth onvexin reasingfun tiondened on an interval

(M, ∞)

. (It does not require formally to be onvex in reasing, but it is reallyuseful only inthis ase).

Su hinequalities have been introdu edby F.-Y. Wang in[12℄ under the form

kfk

2

2

≤ aE(f, f) + b(a)kfk

2

1

,

alledsuper Poin aré inequalities. Theseinequalitiesmaybeoptimizedunderthe

param-eter

a

togive

kfk

2

2

kfk

2

1

≤ Ψ



E(f, f)

kfk

2

1



with some on ave fun tion

Ψ

, whi h isequivalent toinequality (2.9).

Then, one an write the argument of Theorem 2.1 with (2.9) instead of (2.7) and we

see that the key assumption is

R

∞

₁

Theorem 2.3 (Wang). Assume that an extended Nash inequality (2.9) is valid with a

rate fun tion

Φ

dened on someinterval

(M, ∞)

and su hthat

R

∞

₁

Φ(s)

ds < ∞

. Thenwe have

kP

t

f k

2

≤ K(2t) kfk

1

for all

t > 0

and all fun tions

f ∈ L

2

_(µ)

; here the fun tion

K

is dened by

K(x) =

 pU

√

−1

(x)

if 0 < x < U(M),

M

_{if x ≥ U(M)}

where

U

denotes the (de reasing)fun tion dened on

(M, +∞)

by

U(x) =

Z

∞

x

1

φ(u)

du.

In parti ular, the density

p

t

(x, y)

is bounded from above by

K(t)

2

.

Conversely, if there exists a positive fun tion

K

denedon

(0, ∞)

su h that

kP

t

f k

2

≤ K(t)kfk

1

for all

t > 0

, then the Nash inequality (2.9) holds with

M = 0

and fun tion

Φ(x) = sup

t>0

x

2t

log

x

K(t)

2

,

x ≥ 0.

With the te hniques presented in the next se tion, we may see that su h extended

Nashinequalities with fun tions

Φ

of the form

x(log x)

α

are adapted tothe study of the

the measures

µ

a

des ribed in (1.4) for

a > 2

: as we already mentioned, be ause of non ompa tness, there is no hope inthis ase to have a lassi alNash inequality (2.7) with

apowerfun tion

Φ

.

In the ase when the measure is nite(and therefore aprobability measure), then we

know that

kfk

2

/kfk

1

≥ 1

. For su h a general inequality, tightness orresponds to the fa t that

Φ(x) → 0

when

x → 1

. (Of ourse, this supposes that

M = 1

in the previous theorem).

In this situation,assume that

Φ(x) ∼ λ(x − 1)

when

x → 1

and

1/Φ

is integrableat innity. This is the ase in parti ular for the tight form of the lassi al Nash

inequal-ity (2.7). Then

K(t) ∼ 1 + Ce

−λt

when

t → ∞

. This shows that the kernel

p

t

(x, y)

is bounded from above by a quantity whi h onverges exponentially fast to

1

as

t

goes to innity. This is what may be expe ted, sin e

P

t

f → µ(f)

when

t → ∞

. In the ase of a lassi al tight Nashinequality(whi h an onlyo ur when the measure has abounded

support), then one may also dedu e a lower uniform bound on the kernel

p

t

whi h also goesexponentially fastto

1

, but this requires someother te hniques (see [1℄).

The dierent Nash inequalities introdu ed so far may only arry information on the

the general ase when it is not bounded we may still use this method with the tri k of

introdu ingan auxiliaryLyapunov fun tion

V

and weighted Nashinequalities.

Forus,aLyapunovfun tion

V

issimplyapositivefun tion

V

on

E

su hthat

LV ≤ cV

forsome onstant

c

. Weshallrequire thosefun tions

V

tobein

L

2

_(µ)

and inthedomain

toget interesting results,but it is not formallyne essary.

Being a Lyapunov is not a very restri tive requirement for smooth fun tions in the

examples below, as long aswe do not ask

c < 0

(inwhi h ase it annot be true for any fun tion

V

in the domain).

The weighted Nash inequality takesthen the form

(2.10)

Φ



kfk

2

2

kfV k

2

1



≤

E(f, f)

kfV k

2

1

forallfun tions

f

inthe domainof theDiri hlet formsu h that

kfk

2

2

> M kfV k

2

1

,where the rate fun tion

Φ

is dened on

(M, ∞)

and su h that

Φ(x)/x

isin reasing.

Theorem 2.4 (Wang). Assume that a weighted Nash inequality (2.10) holds with a rate

fun tion

Φ

dened on some interval

(M, ∞)

su hthat

R

∞

₁

Φ(s)

ds < ∞

. Then

kP

t

f k

2

≤ K(2t)e

ct

kfV k

1

for all

t > 0

and all fun tions

f ∈ L

2

_(µ)

, where

K

is dened as in Theorem 2.3. In

parti ular, the kernel density

p

t

(x, y)

satises

p

t

(x, y) ≤ K(t)

2

e

ct

V (x)V (y).

Conversely, if there exists a positive fun tion

K

denedon

(0, ∞)

su h that

kP

t

f k

2

≤ K(t)kfV k

1

forall

t > 0

, thentheweightedNash inequality (2.10)holdswith

M = 0

andratefun tion

Φ(x) = sup

t>0

x

2t

log

x

K(t)

2

,

x ≥ 0.

Proof  It is given in detail in [2℄. It follows the proof of Theorem 2.1by repla ing the

fun tion

K(t) =

kP

t

f k

2

2

kfk

2

1

by

K(t) =

ˆ

kP

t

f k

2

2

kV fk

2

1

. Now, the quantity

R P

t

f V dµ

is no longer invariantin time. Butby properties of the Lyapunov fun tionwe have

(2.11)

∂

t

Z

E

V P

t

f dµ = ∂

t

Z

P

t

V f dµ =

Z

P

t

LV f dµ ≤ c

Z

E

V P

t

f dµ,

fromwhi hwe get

Z

E

P

t

f V dµ ≤ e

ct

Z

Using this, we get again a dierential inequality on

K

ˆ

when we apply the Nash in-equality (2.10)to

P

t

f

, and the

L

1

_{→ L}

2

boundedness result follows.

Togetthe(nonuniform)boundonthekernel,itremainstoobservethatifasymmetri

operator

K

satises

kKfk

2

≤ kfV k

1

, the norms being onsidered with respe t to a

measure

µ

, then the operator

K

1

dened by

K

1

(f ) =

1

V

K(f V )

isa ontra tionfrom

L

1

_(ν)

into

L

2

_(ν)

, where

dν = V

2

_dµ

. Moreover,

K

1

is symmetri in

L

2

_(ν)

, and therefore

K

1

◦ K

1

is a ontra tion from

L

1

_(ν)

into

L

∞

_(ν)

. It follows that it

hasa density kernelbounded by

1

withrespe t to

ν

;and this amountstosay that

K

has a density kernel with respe t to

µ

bounded above by

V (x)V (y)

, sin e the kernel of

K

1

with respe t to

ν

is

k(x, y)

V (x)V (y)

, where

k

isthe kernel of

K

with respe t to

µ

.

Observe that Theorem 2.4 produ es non uniform bounds on the kernel. Moreover,

when

V ∈ L

2

_(µ)

,then theoperator

P

2t

isHilbert-S hmidtsohas adis retespe trumand weget anestimateon the eigenvalues

−λ

n

of

L

:

X

n

e

−λ

n

t

_{≤ K}

2

_(t)e

ct

_{kV k}

2

2

.

3 Weighted Nash inequalities on the real line.

As already mentioned, we shall mainly on entrate on model examples on the real line,

and show elementary te hniques toobtain weighted Nash inequalities for measures with

density

ρ

with respe t tothe Lebesgue measure and the usual arré du hamp

Γ(f, f ) =

|∇f|

2

_{= f}

′

2

. These te hniques may be easily extended to the

n

-dimensional Eu lidean

spa e,and with some extra work toRiemannian manifolds.

Let us rst state a universal weighted Nash inequality in the Eu lidean spa e. We

onsider the ase when

Γ(f, f ) = |∇f|

2

and

µ(dx) = ρ(x)dx

. We are mainly interested inthe ase when

µ

is aprobability measure. Re all that in this situation,there maynot existany lassi alNashinequality ( lassi almeanswithapowerfun tionasratefun tion

Φ

) unless the measure is ompa tlysupported.

Here, the symmetri operator asso iatedwith the orresponding Diri hlet formis

Lf = ∆f + ∇ log ρ · ∇f.

We may always hoose

V = ρ

−1/2

: it is not hard to he k that

LV ≤ cV

for some

onstant

c

toget the universal weighted Nash inequality (with respe t to

µ

)

Here

C

n

isthe onstant forthe Nashinequalityinthe Eu lideanspa e withthe Lebesgue measure.

Tosee this, wejust apply theEu lidean Nashinequality (0.1)to

g = f √ρ

, where

f

is asmooth ompa tly supported fun tion, and observe that

Z

R

n

|∇g|

2

_{dx =}

Z

|∇f|

2

ρdx +

Z

R

n

LV

V

f

2

dµ = E(f, f) +

Z

R

n

LV

V

f

2

_dµ,

through integration by parts. Unfortunately, this bound is not very useful sin e

V /

∈

L

2

_(µ)

. Nevertheless, with some are to justify the integration by parts in (2.11), (with

extra hypotheses like uniform upper bounds on the Hessian of

log ρ

), it may lead to an upper-bound onthe kernel density.

Of ourse, this method has nothing parti ular to do with the Eu lidean ase. It

extends a Nash inequality (without weight) with respe t to a measure

µ

to a weighted Nashinequality with respe t tothe measure

ρdµ

with weight

V = ρ

−1/2

, as soonas the

inequality

LV ≤ cV

issatised.

Forexample, one gets with this simpleargument

Corollary 3.1. In

R

n

, with

ρ(x) = (1 + |x|

2

₎

−β

with

β > n

or

ρ(x) = exp(−(1 + |x|

2

₎

a/2

₎

with

a > 0

, there exists a onstant

C

su h that for all

t > 0

and

x, y ∈ R

n

the kernel density

p

t

satises

p

t

(x, y) ≤

C

t

n/2

e

Ct

_ρ

−1/2

_{(x) ρ}

−1/2

_(y).

Butsin e

V /

∈ L

2

_(µ)

,thismayneverprodu eanyboundonthespe trumforexample.

Soone has to look for more pre iseLyapunov fun tions.

This is what we now perform on our model examples on the real line : we write

T (x) =

√

1 + x

2

and onsider the measure

µ

a

(dx) = C

a

exp(−T (x)

a

)dx,

where

a > 0

and

C

a

is a normalizing onstant. We denote by

ρ

a

the density

exp(−T

a

₎

.

Here, the asso iated operatoris

L(f ) = f

′′

_{− aT}

a−1

_T

′

_f

′

_.

In this ontext, it isnot hardto he k that, for any

β ∈ R

,

(3.12)

V = T

−β

_/

√

_ρ

a

isaLyapunov fun tion. If

β > 1/2

,this fun tionisin

L

2

_(µ

a

)

. The issueisthento hoose

the smallest possible

V ∈ L

2

_(µ

a

)

and still have a weighted Nash inequality with rate fun tion

Φ

su h that

1/Φ

isintegrable atinnity.

Theorem 3.2 ([2℄). If

a > 1

, then for any

β ∈ R

and

V

hosen as in (3.12), there exist onstants

C

and

λ ∈ (0, 1)

su h that

(3.13)

kfk

2

2

≤ C

"

Z

|f|V dµ

a



2

+

Z

|f|V dµ

a



2(1−λ)

E(f, f)

λ

#

for all fun tions

f

. This orresponds to the rate fun tion

Φ(x) =



x

C

− 1



1/λ

, x > C.

Although tra table, the expli it value of

λ

in terms of the parameters

a

and

β

isnot sosimple. The assumption

a > 1

is ne essary, sin e for

a ≤ 1

the spe trum is no longer dis rete(and therefore noweighted Nash inequality ould o ur with any

L

2

_(µ

a

)

weight

V

). What has tobe underlined here is thatthe introdu tionof a weightallows ustoget polynomialratefun tions

Φ

,althoughwe knowthatsu hpolynomialgrowthisforbidden fornon ompa tly supported nite measures in the absen e of weights. Of ourse,toget

thesepolynomialgrowths,onehasto hooseweightswhi harequite losetotheuniversal

weights

1/√ρ

des ribed before. If one hooses mu h smaller weights, the rate fun tion willbe smaller. For example, when

a > 2

, one may hoose

V = 1

, and in this ase one has

Φ(x) = x(log x)

α

.

The argument of Theorem 3.2 is based on a tail estimate of the measure

µ

a

. If

q

a

(x) =

R

∞

x

µ

a

(dy)

,then, for some onstant

C

,one has

(3.14)

q

a

(x) ≤ C

ρ

a

(x)

T (x)

a−1

.

One rst proves a Nash inequality for smooth ompa tly supported fun tions su h

that

f (0) = 0

. We start with

Lemma3.3. Let

a ≥ 1

,

β ∈ R

and

V

givenin (3.12). Forallsmooth ompa tlysupported fun tions

f

su hthat

f (0) = 0

one has

Z

f

2

dµ

a

≤ CE(f, f)

γ

Z

|f|V dµ

a



2(1−γ)

where

γ = 1 − 2

a − 1

3(a − 1) + 2β

∈

1

3

, 1.

The proof isbased on uttingthe integralon

[0, ∞)

(forinstan e) as

Z

∞

0

f

2

dµ

a

=

Z

∞

0

f

2

1l

n

f

kf k 2

≤V Z

−1/2

o

dµ

a

+

Z

∞

0

f

2

1l

n

f

kf k 2

>V Z

−1/2

o

dµ

a

.

for a suitably hosen

Z > 0

. Then both terms are ontrolled by the estimate (3.14), repla ing

f

2

by

2

R

x

0

f (t)f

′

_(t)dt

inthe se ond integral and using Fubini's theorem.

Lemma 3.4. Let

a > 0

,

β >

3−a

2

and

V

given in (3.12). Then there exist

θ ∈ (0, 1)

and

C

su h that

Z

|f − f(0)|V dµ

a

≤ C

"

Z

|f|V dµ

a

+

Z

|f|V dµ

a



1−θ

E(f, f)

θ/2

#

for all nonnegative smooth ompa tly supported

f

on

R

.

Although quite similar, this lemma is more restri tive on the values of

β

than the previous one. Passing from fun tions whi h vanish in

0

to the general ase is indeed the hardstep. Wereferthe readerto[2℄fordetailsonthe proofs. Itremainstoplug together

those inequalitiestoobtain Theorem 3.2.

Corollary 3.5. Let

a > 1

and let

(P

t

)

t≥0

be the Markov generator on

R

with generator

Lf = f

′′

_{− aT}

a−1

T

′

f

′

,

and reversible measure

dµ

a

(x) = ρ

a

(x)dx = C

a

exp(−(1 + |x|

2

₎

a/2

_)dx

.

Then for all real

β

there exist

δ > 0

and

C

su h that, for all

t

,

P

t

has a density

p

t

with respe t to the measure

µ

a

, whi h satises

p

t

(x, y) ≤

Ce

Ct

t

δ

ρ

−1/2

a

(x)ρ

−1/2

a

(y)

(1 + |x|

2

₎

β/2

_{(1 + |y|}

2

₎

β/2

for almost every

x, y ∈ R

.

Moreover, the spe trum of

−L

is dis rete and its eigenvalues

(λ

n

)

n∈N

satisfy the in-equality

X

n

e

−λ

n

t

_≤

Ce

Ct

t

δ

for all

t > 0

.

When

a > 2

,the samete hniques alsolead toaNashinequalityfor

µ

a

with rate fun -tion

Φ(x) = C x (log x)

2(1−1/a)

, and weight

V = 1

. This re overs the ultra ontra tivity resultmentionedearlier. Re allthat when

a = 2

thesemigroup isnolonger ultra ontra -tive,but onlyhyper ontra tive,the Nashinequality withrate

Φ(x) = x log x

orresponds infa t toanother formof the Logarithmi Sobolevinequality.

Referen es

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Institut de Mathématiques de Toulouse,UMR CNRS 5219

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