Optimal heat kernel bounds under logarithmic Sobolev inequalities

OPTIMAL HEAT KERNEL BOUNDS

UNDER LOGARITHMIC SOBOLEV INEQUALITIES

D. BAKRY, D. CONCORDET, M. LEDOUX

Abstract. We establish optimal uniform upper estimates on heat kernels whose generators satisfy a logarithmic Sobolev inequality (or entropy-energy inequality) with the optimal constant of^Rn. O-diago- nals estimates may also be obtained with however a smaller distance involving harmonic functions. In the last part, we apply these methods to study some heat kernel decays for diusion operators on^Rn of the type ^;rrU for some smooth potential ^U with a given growth at in nity.

1. Introduction and main result

Let, for example,

M

be a Riemannian manifold of innite volume, and let

dv

denote its Riemannian volume element. In the study of heat kernel bounds, various functional inequalities have been used in the past years: Sobolev inequalities Varopoulos (1985)] (with

n >

2)

k

f

^k22_n=n^;2

C

^{Z jr}

f

^j2

dv f

²

C

_c¹(

M

)

(S) logarithmic Sobolev inequalities Davies (1989)] or entropy-energy inequali- ties Bakry (1994)]

Z

f

²log

f

²

dv

n

2 log

C

^{Z jr}

f

^j2

dv

f

²

C

_c¹(

M

)

^Z

f

²

dv

= 1

(LS) and Nash inequalities Carlen et al. (1987)]

k

f

^{k2+42 =n}

C

^k

f

^{k41=nZ jr}

f

^j2

dv f

²

C

_c¹(

M

)

:

(N) As shown in these papers, these three functional inequalities are all equiv- alent to a common upper bound on the heat kernel

p

_t(

xy

) on

M

given

by _xysup

2M

p

t(

xy

)

C

⁰

t

ⁿ⁼²

t >

0

:

(1

:

1) In particular, the inequalities (S), (LS) and (N) are equivalent, for possibly dierent

C >

0 but for the same (analytic) dimension

n

(

>

2 in case of (S)).

URL address of the journal:http://www.emath.fr/ps/

Received by the Journal May 17, 1996. Revised May 15, 1997. Accepted for publication September 18, 1997.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by TEX.

This may also be shown directly Bakry et al. (1995)]. Actually, the previous inequalities belong to a whole family of equivalent inequalities of the type

k

f

^k_r

C

^k

f

^k_s^{Z jr}

f

^j2

dv

^(1;)=2

f

²

C

_c¹(

M

)

with 1

r

⁼

s

^{+ 1;}

q

²⁰

1

and

q

= 2

n=n

^;2. The logarithmic Sobolev inequality (LS) corresponds to the limiting case

r

= 2 and

= 1 (cf. Bakry et al. (1995)]).

When

M

=^Rnwith Lebesgue measure

dx

, the best constants in the three inequalities (S), (LS) and (N) are known. Namely

C

=

;

1

2(

n

+ 1)!^j

B

^nj2=n

²

n

(

n

^;2)

where ^j

B

^nj denotes the volume of the unit ball

B

ⁿ in ^Rn, in case of (S) Aubin (1982)],

C

= 2((

n

+ 2)

=

2)^(n+2)=n

n

^{N1 j}

B

^nj2=n

where

^N1 denotes the rst non-zero Neumann eigenvalue of the Laplacian on radial functions on

B

ⁿ, in case of (N) Carlen et al. (1993)]. For (LS), the task is easier Carlen (1991)]. One may simply start with the logarithmic Sobolev inequality Gross (1975)] for the canonical Gaussian measure n

on^Rn with density

'

n(

x

) = (2

)^;n=2exp(^;j

x

^j2

=

2), that indicates that, for every smooth function

g

on ^Rnwith ^R

g

²

d

n= 1,

Z

g

²log

g

²

d

n 2

Z

jr

g

^j2

d

n

:

Set

f

² =

'

_n

g

² so that^R

f

²

dx

= 1. Then

Z

f

²log^;

f

²(2

)ⁿ⁼²e^jxj2=2

dx

2

Z

r

f

+

x

2

f

²

dx:

An integration by parts easily yields

Z

f

²log

f

²

dx

2

Z

jr

f

^j2

dx

^;

n

2 log(2

)^;

n:

Changing

f

into

ⁿ⁼²

f

(

x

),

>

0, which still satises the normalization

R

f

²

dx

= 1, shows that, for every

>

0 thus,

Z

f

²log

f

²

dx

2

^{2Z jr}

f

^j2

dx

^;

n

2 log(2

)^;

n

^;

n

log

:

Optimizing in

, we get that for every smooth

f

on ^Rn with^R

f

²

dx

= 1,

Z

f

²log

f

²

dx

n

2 log

2

n

e

Z

jr

f

^j2

dx

:

(1

:

2)

Since we started from the logarithmic Sobolev inequality for nwith its best constant for which exponential functions are extremal,

C

= 2

n

e

is the best constant in the inequality (LS) on^Rn(with respect to Lebesgue measure). To further convince ourselves that this constant is optimal, one may note that among all functions :^R+!R+such that, for every smooth function

f

on^Rn with^R

f

²

dx

= 1,

Z

f

²log

f

²

dx

Z

jr

f

^j2

dx

(1

:

3)

the function

(

u

) =

n

2 log

2

u n

e

is actually best possible. Indeed, apply (1.3) to

f

²(

x

) =

ⁿ

'

n(

x

),

>

0.

Since^R

f

²

dx

= 1, we get

n

2 log

² 2

;

n

2

n

² 4

:

The claim follows by setting

u

=

n

²

=

4.

Since (S), (LS) and (N) all imply the heat kernel upper bound (1.1), one may wonder if one of these inequalities with the optimal Euclidean constant could yield the optimal Euclidean heat kernel bound

xysup²M

p

t(

xy

) 1

(4

t

)ⁿ⁼²

t >

0

:

(1

:

4) It is possible that this is the case for the three inequalities. The main result of this note is that this is the case with the logarithmic Sobolev inequality (LS). The proof of this result appears as a consequence of the very pre- cise study of uniform upper estimates on the heat kernel under functional inequalities between entropy and energy developed in Bakry (1994)]. We present below this result in the context of abstract Markov semigroups of Bakry (1994)]. We obtain by the same method the sharp bounds on the norm^k

P

t^kpq of the heat semigroup on^Rnas an operator from L^p into L^q. Various comments complement this rst section. In particular, we deduce from our main result that a complete Riemannian manifold of dimension

n

with non-negative Ricci curvature that satises (LS) with the best constant of^Rnis necessarily isometric to^Rn. In Section 2, we discuss the o-diagonal upper bounds on the heat kernel. We namely observe that, again with the methods of Bakry (1994)], we may obtain the optimal estimates provided the distance is replaced by a smaller one that takes into account the struc- ture of harmonic functions. In the last section, we use the tool of functional inequalities between entropy and energy to study diusion operators on^Rn dened by L

f

=

f

^;r

f

^r

U

for some potential

U

. According to the growth of

U

at innity, we obtain various heat kernel estimates that extend

and clarify some of the results of Kavian et al. (1993)]. After this work was completed, we became aware of the recent paper Carlen et al. (1995)] by E. Carlen and M. Loss where the authors obtain optimal estimates for vis- cously damped conservation laws also relying on Gross's method and on the sharp logarithmic Sobolev inequality on^Rn. They do not consider however the general implication we establish in Theorem 1.2 below.

We rst turn to (1.4) and describe, to begin with, the framework in which we will formulate most of our results, refering to Bakry (1994)] for further details. On some measure space (

E

^E

), let L be a Markov generator asso- ciated to some semigroup

P

t

f

(

x

) =^R

f

(

y

)

p

t(

xy

)

d

(

y

) continuous in L²(

).

We will assume that L and ^Pare invariant and symmetric with respect to

. We assume furthermore that we are given a nice algebra ^Aof (bounded) functions on

E

dense in the L²-domain of L, stable by L and

P

t and by the action of

C

¹ functions which are zero at zero. (The stability by

P

t may not be satised even in basic examples such as non-degenerate second order dierential operators with no constant term on a smooth manifold. This assumption is however not strictly necessary and is mostly used for conve- nience in order to unify the treatment of a number of various cases that would require each time a separate standard analysis.) We will mainly be concerned here with the case when

is innite. (Although Bakry (1994)]

is mainly concerned with nite measure spaces, all the results from Bakry (1994)] used here immediately extend to arbitrary measure spaces.) We de- note by ^k

P

t^kpq, 1

p < q

¹, the operator norm of

P

t from L^p(

) into L^q(

). Note that

k

P

t^k11 = sup

p

t(

xy

)

where the supremum is understood in the ess sup sense.

We may introduce, following P.-A. Meyer, the so-called \carre du champ" operator ; as the symmetric bilinear operator on ^{A A} dened by 2;(

fg

) = L(

fg

)^;

f

L

g

^;

g

L

f fg

^2A

:

Note that ;(

ff

) 0. We will say that L is a diusion if for every

C

¹ function on^Rk, and every nite family

F

= (

f

¹

:::f

k) in ^A,

L(

F

) =^r(

F

)L

F

+^rr(

F

);(

FF

)

:

This hypothesis essentially expresses that L is a second order dierential operator with no constant term and that we have a chain rule formula for

;, ;((

f

)

g

) = ⁰(

f

);(

fg

),

fg

^{2 A}. By the diusion and invariance properties,

Z

(

f

)(^;L

f

)

d

=

Z

⁰(

f

);(

ff

)

d f

^2A

:

One basic operator is the Laplace-Beltrami operator on a complete connected Riemannian manifold

M

. In the preceding setting, the \natu- ral" measure

is only determined from the Riemannian volume element

dv

up to a constant

d

=

cdv

. For ^A the class, say, of

C

_c¹ functions on

M

(that is however not stable by the heat semigroup in the non-compact case),

;(

ff

) is simply the Riemannian length (squared)^jr

f

^j2 of the gradient^r

f

of

f

^{2 A}. The previous abstract framework includes a number of further examples of interest (cf. Bakry (1994)]). For example, one may consider L = +

X

where

X

is a smooth vector eld on

M

, or more general second order dierential operators with no constant term. We may also consider innite dimensional examples such as the Ornstein-Uhlenbeck generator on

RnL

f

(

x

) =

f

(

x

)^;

x

^r

f

(

x

).

In this setting, and following Bakry (1994)], one may consider general inequalities between the entropy^R

f

²log

f

²

d

and the energy^R;(

ff

)

d

of a function

f

in ^A with ^R

f

²

d

= 1. Assume that, for some : ^{R+! R+}, for every

f

^2A with^R

f

²

d

= 1,

Z

f

²log

f

²

d

Z

;(

ff

)

d

:

(1

:

5)

In most examples is concave, strictly increasing, and of class

C

¹, which we assume throughout the argument below. Therefore, for every

v >

0, (

u

)(

v

) + ⁰(

u

)(

u

^;

v

), so that (1.5) reads, for every

v >

0 and every

f

(and by homogeneity),

Z

f

²log

f

²

d

⁰(

v

)

Z

;(

ff

)

d

+ (

v

)

Z

f

²

d

where (

v

) = (

v

)^;

v

⁰(

v

). We thus deal equivalently with a family of logarithmic Sobolev inequalities. Now, by the diusion property, changing

f >

0 into

f

^p=2 shows that, for every

f >

0,

v >

0,

p >

1,

Z

f

^plog

f

^p

d

^;Z

f

^p

d

log

Z

f

^p

d

;⁰(

v

)

p

² 4(

p

^;1)

Z

f

^p;1L

fd

+ (

v

)

Z

f

^p

d:

Choose now in this inequality a function

v

^!

v

(

p

)

>

0,

p >

1. According to Bakry (1994)], we make then use of the fundamental argument of L. Gross Gross (1975)]. Namely, if

V

(

t

) = e^;m(t)k

P

t

f

^k_p⁽_t⁾

t

0

the preceding inequality will show that

V

⁰ 0 and

V

is non-increasing as soon as

p

and

m

are chosen so that

p

²(

t

)

p

⁰(

t

) = ⁰

;

v

^;

p

(

t

)

p

²(

t

)

4(

p

(

t

)^;1) and

m

⁰(

t

) = ^;

v

^;

p

(

t

)

p

⁰(

t

)

p

(

t

)²

:

If this is the case, for every 1

p < q

¹,

k

P

t^kpqe^m (1

:

6a) where

t

=

t

pq =

Z q

p ^0;

v

(

s

)

ds

4(

s

^;1) and

m

=

m

pq=

Z q

p ^;

v

(

s

)

ds

s

^{2 (1}

:

6b)

provided we can nd a function

v

for which these two integrals are nite. The optimal choice, that will be used throughout this work (cf. Bakry (1994)]), is given by

v

(

s

) =

s

²

=

(

s

^;1), where

>

0 is a parameter.

It might be worthwhile noting that, conversely, the previous bounds on

k

P

t^kpq imply that the corresponding entropy-energy inequality (1.5) holds.

This is a consequence of the following proposition.

Proposition 1.1.Under the previous notation, assume that, for some 1

p <

¹ and every

q

in some neighborhood of

p

, ^k

P

t^kpq e^m where

t

and

m

are dened with (1.6b) through some function . Then, for every non- negative

f

in ^A,

Z

f

^plog

f

^p

d

^;Z

f

^p

d

log

Z

f

^p

d

;⁰(

v

)

p

² 4(

p

^;1)

Z

f

^p;1L

fd

+ (

v

)

Z

f

^p

d:

The proof reduces to check that if, for

f >

0 in ^A,

U

(

"

) = e^;m(")k

P

t⁽"⁾

f

^k_p⁺_"

where

t

(

"

) =

t

pp⁺",

m

(

"

) =

m

pp⁺", then

U

⁰(0)0 amounts to the inequal- ity of the proposition.

The main observation of this work is that the general method leading to the bound (1.6) yields the optimal heat kernel bound (1.4) if we start from a (LS) inequality for the generator L (or rather the carre du champ ;) with the best constant of^Rn.

Theorem 1.2.Assume that for every

f

in ^A with ^R

f

²

d

= 1,

Z

f

²log

f

²

d

n

2 log

2

n

e

Z

;(

ff

)

d

:

(1

:

7) Then,

sup

p

t(

xy

) 1

(4

t

)ⁿ⁼²

t >

0 (where the supremum is understood in the ess sup sense).

If ^Z

f

²log

f

²

d

n

2 log

C

^Z ;(

ff

)

d

^Z

f

²

d

= 1

for some

C >

0, by a simple change of variables,

sup

p

t(

xy

)

n

e

C

8

t

n=²

t >

0

:

It is worthwhile mentioning that the logarithmic Sobolev inequality (1.7) of Theorem 1.2 behaves correctly under tensor product. Namely, if two car- res du champ;¹ and ;² satises such an inequality with dimensions

n

¹ and

n

² respectively, then the product operator ;¹;² will satisfy this inequality with dimension

n

¹+

n

². This immediately follows from the classic product

property of entropy together with the linearized version of (1.7) (the in- equality leading to (1.2)). This stability property is re ected similarly on the heat kernel bound as can be seen from the example of the Euclidean spaces.

Proof.We simply use (1.6) with (

u

) =

n

2 log

2

u n

e

:

Hence ⁰(

u

) =

n=

2

u

and

(

u

) =

n

2 log

2

u n

e²

:

Then^k

P

t^k11 e^m with

t

=

t

(

) =

n

8

Z

1

1

ds s

^{2 =}

n

8

and

m

=

m

(

) =

n

2

Z

1

1

log

2

n

e²

s

²

s

^;1

ds s

²

where

>

0. From the rst equality,

=

n=

8

t

. The second yields

m

=

n

2 log

2

n

e²

+

n

2

Z

1

1

log

s

²

s

^;1

ds s

²

=

n

2 log

2

n

e²

;

n

2

Z

1

0

log^;

r

(1^;

r

)

dr

=

n

2 log

2

n

e²

+

n

with the change of variable

r

= 1

=s

. Since

=

n=

8

t

,

m

=

m

(

) =

n

2 log

1 4

t

which yields

k

P

t^k11 e^m= 1 (4

t

)ⁿ⁼²

and the result. The proof is complete. ^tu

It is clear that the same proof yields upper bounds for ^k

P

t^kpq for every

t >

0 and 1

p < q

¹. Namely ^k

P

t^kpq e^m where

t

=

t

pq(

) =

n

8

Z q p

ds

s

^{2 =}

n

8

1

p

^;1

q

and

m

=

m

pq(

) =

n

2

Z q p log

2

n

e²

s

²

s

^;1

ds

s

²

:

After some calculations very similar to the previous ones, we nd that

k

P

t

f

^k_pq

"

p

^1p;1^;1_p^1;p1

q

^1q;1^;1_q^1;1q

# n

2

"

41

t

1

p

^;1

q

#n

2 (

1

p

; 1

q )

:

(1

:

8) It is less clear however why these bounds should be optimal on ^Rn. The next proposition answers this question positively. These bounds (1.8) and their optimality may also be shown to follow from the work of E. Lieb Lieb (1990)] on Gaussian maximizers of Gaussian kernels.

Proposition 1.3. With the preceding notation, let be such that (1.5) holds and^k

P

t^k11 = e^m where

t

=^{Z 1}

1

⁰

s

²

s

^;1

ds

4(

s

^;1)

and

m

=

Z

1

1

s

²

s

^;1

ds s

² for some

>

0. Then, for every 1

p < q

¹,

k

P

t^pqkpq= e^mpq where

t

pq=

Z q

p ⁰

s

²

s

^;1

ds

4(

s

^;1)

and

m

pq =

Z q

p

s

²

s

^;1

ds s

²

:

The proof of the proposition is easy. By the hypothesis,

k

P

t^1p+t^pq+t^{q 1k11} =^k

P

t^k11= e^m= e^{m1p+mpq+mq 1}

:

But now also, by the semigroup property,

k

P

t^1p+t^pq+t^{q 1k11k}

P

t^1qk1q^k

P

t^pqkpq^k

P

t^{q 1k}q¹

e^{m1p+mpq+mq 1}

so that it is impossible that ^k

P

t^pqkpq

<

e^mpq for some

p < q

. This re- sult clearly applies in the Euclidean case to prove that (1.8) are actually equalities in this case. It moreover implies the somewhat surprising follow- ing observation. For every 1

p < q < r

¹, and every

t

0, there is an unique

s

such that

k

P

t^kpr=^k

P

s^kpq^k

P

t^;s^kqr

:

Indeed, since ⁰(

u

) =

n=

2

u

in this case, one may dene

>

0 by

t

=

t

pr(

) =

Z r

p ⁰

s

²

s

^;1

ds

4(

s

^;1) =

n

8

Z r p

ds

s

²

:

Set then

s

=

t

qr(

) and the claim follows from the preceding argument.

In order to e!ciently use Theorem 1.2, it would be worthwhile to know how to establish (LS) with sharp constant in a Riemannian or abstract set- ting using curvature-dimension hypotheses. Before inspecting this question more closely, let us observe the following.

Proposition 1.4. Let

M

be a Riemannian manifold of dimension

n

and non-negative Ricci curvature. If, for every non-negative

f

in

C

_c¹(

M

),

tlim^!1(4

t

)ⁿ⁼²

P

t

f

=

c

^Z

fdv

, then the heat kernel

p

t(

xy

) on

M

satises

xysup²M

p

t(

xy

)

c

(4

t

)ⁿ⁼²

t >

0

:

Proof. It is an immediate consequence of the Li-Yau inequality Li et al.

(1986)] for Riemannian manifolds with non-negative Ricci curvature, that ensures in particular that, if

f >

0,

;

P

t

f P

t

f

n

2

t

for every

t >

0 (where

P

t denotes the heat semigroup on

M

). In another words, at every point, the function

t

ⁿ⁼²

P

t

f

is increasing in

t

. But then, (4

t

)ⁿ⁼²

P

t

f

increases to

c

^R

fd

so that

k

P

t^k11

c

(4

t

)ⁿ⁼²

:

The proof is complete. Note that by the results of P. Li Li (1986)],

M

is isometric to^Rnif

c

= 1 (cf. the proof of Corollary 1.6). Observe furthermore that the proposition similarly applies to generators ^;rlog

for which the results of Li et al. (1986)] are also available (in which case the hypothesis on non-negative Ricci curvature has to be replaced by the condition ;²(

ff

)

n1(L

f

)² as explained below). ^tu

What we actually conjecture, is that if lim_t

!1

(4

t

)ⁿ⁼²

P

t

f

=

c

^Z

fdv

for ev- ery smooth

f

on a manifold

M

of dimension

n

and non-negative Ricci cur- vature, then the logarithmic Sobolev inequality (LS) holds with its sharp constant from^Rn, i.e.

Z

f

²log

f

²

dv

n

2 log

2

c

²⁼ⁿ

n

e

Z

jr

f

^j2

dv

^Z

f

²

dv

= 1

:

(Proposition 1.4 would then be a consequence of Theorem 1.2.) We have not been able to prove such a result although we strongly conjecture that it must be true. So far, we have only been able to prove the inequality with a constant that misses the optimal one by a factor 1

=

log2. We present this result in the context of the abstract geometry of Markov generators.

Introduce the \iterated carre du champ operator" by setting, for every

fg

in ^A,

2;²(

fg

) = L;(

fg

)^;;(

f

L

g

)^;;(

g

L

f

)

:

We say that L is of curvature

R

^2Rand dimension

n

1 if for every

f

in

A,

;²(

ff

)

R

;(

ff

) + 1

n

^(L

f

)²

:

By Bochner's formula, the Laplace-Beltrami operator on a Riemannian man- ifold of dimension

n

and Ricci curvature bounded below by

R

is of curvature

R

and dimension

n

in the preceding functional sense.

Proposition 1.5.LetL be a diusion generator of curvature 0 and dimen- sion

n

1. Assume that, for every

f

in ^A with^R

fd

= 1,

limsup_t

!1 Z

P

t

f

log^;(4

t

)ⁿ⁼²

P

t

f

d

^;

n

₂ (1

:

9) for some

>

0. Then, for every

f

^2A with^R

f

²

d

= 1,

Z

f

²log

f

²

d

n

2 log

2

n

e

Z

;(

ff

)

d

:

In particular, if (1.9) holds with

= 1, the inequality holds with the optimal constant of ^Rn.

Proof. It is a simple application of the argument developed in Section 6 of Bakry (1994)]. Let

f >

0 in^Awith^R

fd

= 1. By the semigroup properties, for every

t

0,

Z

f

log

fd

=

Z

P

t

f

log

P

t

fd

+

Z t

0

F

(

s

)

ds

(1

:

10) where

F

(

s

) =

Z ;(

P

s

fP

s

f

)

P

s

f d:

Now, in the proof of Proposition 6.7 of Bakry (1994)], it is shown that, under the curvature-dimension assumption on L,

F

⁰(

s

) 2

n F

⁽

s

)²

s

0

:

Hence, Z t

0

F

(

s

)

ds

n

2 log

1 + 2

t n F

⁽⁰⁾

:

Together with (1.10) and the assumption on the behavior at innity of

P

t, the conclusion immediately follows (changing

f

into

f

²). The proof is com-

plete. ^tu

What we expect is that actually (1.9) with

= 1 appears as a consequence of the fact that lim_t

!1

(4

t

)ⁿ⁼²

P

t

f

=

Z

fd

for every

f

in^A. We only checked so far (1.9) with

= log2. Let

f

be non-negative (for simplicity) and such that^R

fd

= 1. It is easy to see, by Jensen's inequality, that, for every

t

0,

Z

P

t

f

log^;(4

t

)ⁿ⁼²

P

t

f

d

=

Z

fP

t^;log^;(4

t

)ⁿ⁼²

P

t

f

d

Z

f

log^;(4

t

)ⁿ⁼²

P

²t

f

d:

In a Riemannian (or concrete) setting, we know (cf. the proof of Proposition 1.4) that (4

t

)ⁿ⁼²

P

t

f

is increasing (to 1) so that in this case

limsup_t

!1

Z

P

_t

f

log^;(4

t

)ⁿ⁼²

P

_t

f

d

^;

n

2 log 2

:

But Proposition 1.5 then only yields

Z

f

²log

f

²

d

n

2 log

1

n

Z

;(

ff

)

d

^Z

f

²

d

= 1

:

Note nally that (1.9) with

= 1 is satised in^Rn. Let

f

be non-negative on^Rnwith compact support and such that^R

fdx

= 1. Then, for every

x

in the support of

f

and every

t

large enough,

(4

t

)ⁿ⁼²

P

t

f

(

x

) =^Z e^;jx;yj2=4t

f

(

y

)

dy

1^; 1 4

t

Z

j

x

^;

y

^j2

f

(

y

)

dy:

Hence,

log^;(4

t

)ⁿ⁼²

P

t

f

(

x

)^;1 4

t

Z

j

x

^;

y

^j2

f

(

y

)

dy:

Therefore,

P

t^;log^;(4

t

)ⁿ⁼²

P

t

f

(

x

)^;1 4

t

Z

e^{;jx;y;zj2=4tj}

z

^j2

f

(

y

)

dy dz

₍₄

t

)ⁿ⁼²

=^;1 4

t

Z

;

j

x

^;

y

^j2+ 2

tn

f

(

y

)

dy:

(1.9) (with

= 1) follows as

t

tends to innity.

We conclude this section with a comparison theorem for Riemannian man- ifolds with non-negative Ricci curvature.

Corollary 1.6. Let

M

be a complete Riemannian manifold of dimension

n

with non-negative Ricci curvature satisfying the logarithmic Sobolev in- equality (LS) with the best constant of^Rn, i.e.

Z

f

²log

f

²

dv

n

2 log

2

n

e

Z

jr

f

^j2

dv

^Z

f

²

dv

= 1

:

Then

M

is isometric to ^Rn.

Proof. Denote by

V

(

xr

) the volume of the ball with center

x

²

M

and radius

r >

0 in

M

. By Theorem 1.2, for every

xy

²

M

,

t >

0,

p

_t(

xy

) 1

(4

t

)ⁿ⁼²

:

(1

:

11)

In particular, by the results of Li et al. (1986)],

M

has maximal volume growth, that is

liminf_r

!1

V

(

xr

)

r

ⁿ

>

0

:

Li's result Li (1986)] then indicates that, for every

xyz

²

M

,

tlim^!1

V

^;

z

^p

t

p

t(

xy

) = ^j

B

^nj (4

)ⁿ⁼²

where we recall that^j

B

^njis the volume of the Euclidean unit ball. Together with (1.11), we thus see that

liminf_r

!1

V

(

xr

)

j

B

^nj

r

^{n 1}

:

(1

:

12) Now, by Gromov's comparison theorem (cf. Carlen (1991)]), for

s < r

,

V

(

xr

)

j

B

^nj

r

ⁿ

V

(

xs

)

j

B

^nj

s

ⁿ

and, in particular,

V

(

xr

)^j

B

^nj

r

ⁿ as

s

^!0. But now, by (1.12), we also get

V

(

xs

)^j

B

^nj

s

ⁿas

r

^!1. Therefore

V

(

xr

) =^j

B

^nj

r

ⁿfor every

x

²

M

and

r >

0. By the case of equality in the volume comparison theorem,

M

is isometric to^Rn. The corollary is established. ^tu

2. Off-diagonal estimates

As is well-known Bakry (1994)], Davies (1989)], an entropy-energy (or log- arithmic Sobolev) inequality of the type (1.5) also leads to o-diagonal up- per bounds on the heat kernel in terms of the distance function dened as

d

(

xy

) = sup

;(ff⁾¹

f

(

x

)^;

f

(

y

)] (the supremum being understood in the ess sup sense).

Let us recall the general procedure of the method due to E. B. Davies (cf.

Davies (1989)]). The idea is to derive from an entropy-energy inequality for the generator L

Z

f

²log

f

²

d

Z

;(

ff

)

d

^Z

f

²

d

= 1

an entropy-energy inequality for L^h(

f

) = e^;hL(e^h

f

), where ;(

hh

)

, which depends only on and

>

0. This leads, via the method pre- sented in Section 1, to a uniform upper bound on the kernel

p

_ht(

xy

) =

p

t(

xy

)e^h(x);h(y) of the semigroup

P

_ht= e^;h

P

t(e^h) with generator L^h. We thus bound in this way

p

t(

xy

) by

C

(

t

)e^h(x);h(y)

:

Applying this result to

h

, ;(

hh

)1, and optimising in

and

h

, leads to some bound

p

t(

xy

)e^V(td(xy))

:

Theorem 5.3 of Bakry (1994)] yields an explicit form of the function

V

in terms of the entropy-energy function of L. Precisely,

p

t(

xy

)e^V(td(xy)) for every

t

T

^;

d

(

xy

)

where the functions

V

and

T

are parametrized by

and

in the following way: if

is dened by

d

(

xy

) =

Z

1

⁰(

s

)

p

s ds

then

T

^;

d

(

xy

)= 12

Z

1

⁰(

s

)

p

s

(

s

^;

)

ds

and if

is dened by

t

_{= 12}^{Z1 0(}

s

)

p

s

(

s

+

^;

)

ds

then

V

^;

td

(

xy

)=

r

⁽

)^;

^;

2

Z

1

(

s

)

p

s

³(

s

+

^;

)

ds

(where we recall that (

s

) = (

s

)^;

s

⁰(

s

)).

Let us use these bounds with the optimal entropy-energy function on^Rn (

u

) =

n

2 log

2

u n

e

u >

0

:

After some computations, we get that

p

t(

xy

)

e 8

n

n=²

d

(

xy

)

t

n

exp

;

d

(

xy

)² 4

t

:

Unfortunately, this does not give the optimal bound on ^Rn. In order to improve this estimate, we introduce a new distance called the \harmonic"

distance dened by

d

^H(

xy

) = sup

;(ff^)1Lf⁼⁰

f

(

x

)^;

f

(

y

)]

:

Note that

d

^H=

d

on^Rn(while

d

^H = 0 on a compact Riemannian manifold).

Proposition 2.1.Assume we have an entropy-energy inequality (1.5). Then

p

t(

xy

)exp

m

^;

d

^H(

xy

)² 4

t

where

t

and

m

are dened in (1.6b) (assumed to be nite).

Proof. The proof follows the same lines as the proof of (1.6a). Fix

h

with L

h

= 0 and ;(

hh

) 1 and consider

P

_t^h = e^;h

P

t(e^h) whose generator is given by L^h(

f

) = e^;hL(e^h

f

),

>

0. Then, for

f

0,

; Z

f

^p;1L^h

fd

=

Z

;^;e^;h

f

^p;1

e^h

f

d

=

Z

;(

f

^p;1

f

)

d

+

(

p

^;2)

Z

f

^p;1;(

fh

)

d

;

^2Z

f

^p;(

hh

)

d:

The second term on the right-hand-side is equal to

p

^;

p

²

Z

;(

f

^p

h

)

d

=^;

p

^;

p

²

Z

f

^pL

hd

= 0

so that

; Z

f

^p;1L^h

fd

^;Z

f

^p;1L

fd

^;

^2Z

f

^p

d:

Therefore, if, for every

v >

0,

Z

f

^plog

f

^p

d

^;Z

f

^p

d

log

Z

f

^p

d

;⁰(

v

)

p

² 4(

p

^;1)

Z

f

^p;1L

fd

+ (

v

)

Z

f

^p

d

then ^Z

f

^plog

f

^p

d

^;Z

f

^p

d

log

Z

f

^p

d

;⁰(

v

)

p

² 4(

p

^;1)

Z

f

^p;1L^h

fd

+

(

v

) +

²⁰(

v

)

p

² 4(

p

^;1)

Z

f

^p

d:

According to the sketch of proof in Section 1, this leads to the upper bound

k

P

_t^hk11e^m+2t where

t

and

m

are given by (1.6b). Equivalently,

p

t(

xy

)e^{m+2t;h(x);h(y)]}

:

Optimizing in

>

0, and then in

h

, leads to the claim. Proposition 2.1 is

thus established. ^tu

As a consequence of this proposition, if (

u

) =

n

2 log

2

u n

e

u >

0

we get that

p

t(

xy

) 1

(4

t

)ⁿ⁼² exp

;

d

^H(

xy

)² 4

t

for every

t >

0 and

xy

, a formula that is of course optimal on^Rn. It would be worthwhile to study further examples where

d

^H =

d

, as well as the various bounds which may similarly be deduced through distances involving harmonic functions.

Note also that if we are looking for a situation such that (1.5) holds together with^k

P

t^k11= e^m in (1.6), then

p

t(

xy

)

p

t(

xx

)exp

;

d

^H(

xy

)² 4

t

for every

t >

0 and

x

such that

p

t(

xx

) = sup_u

p

t(

uu

).