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London Mathematical Society Lecture Note Series. 289

Aspects of Sobolev-Type Inequalities

Laurent Saloff-Coste Corne11 University

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge, CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melborne, VIC 3207, Australia Ruiz de Alarcfin 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Laurent Saloff-Coste 2002 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002

Printed in the United Kingdom at the University Press, Cambridge

.4 catalogue recordfor this bookrs availablefrom theBrilish Library ISBN 0 521 00607 4 paperback

Contents ix

Preface

Introduction 2

1

Sobolev inequalities in R''

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Sobolev inequalities . . . . . . . . . . . . . . . . . . . 1.1.1 Introduction . . . . 1.1.2 The proof due to Gagliardo and to Nirenberg 1.1.3 p = 1 implies p > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Riesz potentials . . . 1.2.1 Another approach to Sobolev inequalities . . . 1.2.2 Marcinkiewicz interpolation theorem . . . . . . 1.2.3 Proof of Sobolev Theorem 1.2.1 . . 1.3 Best constants . . . . . 1.3.1 The case p = 1: isoperimetry . . . . . . . . . . 1.3.2 A complete proof with best constant for p = 1 . . . . . . . . . 1.3.3 The case p > 1 . . . . . 1.4 Some other Sobolev inequalities . . . . . . . 1.1

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CONTENTS

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An abstract lemma . . . . . 2.3 Harnack inequalities and continuity 2.2.3

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3 Sobolev inequalities on manifolds . . . . . . . . . . . . . . . . . 3.1 Introduction . 3.1.1 Notation concerning Riemannian manifolds . 3.1.2 Isoperimetry . . . . . . . . . . . . . . . . . 3.1.3 Sobolev inequalities and volume growth . . . . . . 3.2 Weak and strong Sobolev inequalities . . . 3.2.1 Examples of weak Sobolev inequalities . 3.2.2 (Se,)-inequalities: the parameters q and v . . . . . . . . 3.2.3 The case 0 < q < oo . . . . . . . . . . . . . . . . . 3.2.4 The case q = oo . . . . . . 3.2.5 The case -oo < q < 0 . . . 3.2.6 Increasing p . . . . . . . . . . . . . . . . .

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Ultracontractivity . . 4.1.1 Nash inequality implies ultracontractivity 4.1.2 The converse . . . . . . . . . . . . . 4.2 Gaussian heat kernel estimates . . . . . . . . . . 4.2.1 The Gaffney-Davies L2 estimate . : . . . 4.2.2 Complex interpolation . . . . . . . . 4.2.3 Pointwise Gaussian upper bounds . . 4.2.4 On-diagonal lower bounds . . . . . . 4.3 The Rozenblum-Lieb-Cwikel inequality . . . . . 4.3.1 The Schrodinger operator A - V . . 4.3.2 The operator TV = 0-1V . . . . 4.3.3 The Birman-Schwinger principle . . . 4.1

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vii

CONTENTS

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. Mean value inequalities for subsolutions . . Localized heat kernel upper bounds . . . . . . . . Time-derivative upper bounds . . . . . . . . . . . Mean value inequalities for supersolutions . . . . Poincare inequalities . . . . . . . . . . . . . . . . . . . . 5.3.1 Poincare inequality and Sobolev inequality . . . . 5.3.2 Some weighted Poincare inequalities . . . . . . . . 5.3.3 Whitney-type coverings . . . . . . . . . . . . . . . 5.3.4 A maximal inequality and an application . . . . . . . 5.3.5 End of the proof of Theorem 5.3.4 . . . . . Harnack inequalities and applications . . . . . . . . . . . 5.4.1 An inequality for log u . . . . . . . . . . . . . . . 5.4.2 Harnack inequality for positive supersolutions . . 5.4.3 Harnack inequalities for positive solutions . . . . 5.4.4 Holder continuity . . . . . . . . . . . . . . . . . . 5.4.5 Liouville theorems . . . . . . . . . . . . . . . . . 5.4.6 Heat kernel lower bounds . . . . . . . . . . . . . . 5.4.7 Two-sided heat kernel bounds . . . . . . . . . . . The parabolic Harnack principle . . . . . . . . . . . . . . 5.5.1 Poincare, doubling, and Harnack . . . . . . . . . 5.5.2 Stochastic completeness . . . . . . . . . . . . . . 5.5.3 Local Sobolev inequalities and the heat equation . 5.5.4 Selected applications of Theorem 5.5.1. . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Unirnodular Lie groups . . . . . . . . . . . . . . . 5.6.2 Homogeneous spaces . . . . . . . . . . . . . . . . 5.6.3 Manifolds with Ricci curvature bounded below . . Concluding remarks . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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Index

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ix

Preface These notes originated from a graduate course given at Cornell University during the fall of 1998. One of the aims of the course was to present Sobolev inequalities and some of their applications in the context of analysis

on manifolds -including Harnack inequalities and heat kernel estimatesto an audience not necessarily very familiar with analysis in general and Sobolev inequalities in particular. The first part (Chapters 1-2) introduces the reader to Sobolev inequalities in R7. An important application, Moser's proof of the elliptic Harnack inequality for uniformly elliptic divergence form

second order differential operators, is treated in detail. In the second part (Chapters 3-4), Sobolev inequalities on complete non-compact Riemannian manifolds are discussed: What is their meaning and when do they hold true? How does one prove them? This discussion is illustrated by the treatment of some explicit examples. In the third and last part, Chapter 5, families of local Sobolev and Poincare inequalities are introduced. These turn out to be crucial for taking full advantage of Sobolev inequality techniques on Riemannian manifolds. For instance, complete Riemannian manifolds satisfying a scale-invariant parabolic Harnack inequality are characterized in terms of Poincare inequalities and volume growth. These notes give the first detailed exposition of this fundamental result. We warn the reader that no effort has been made to include a comprehen-

sive bibliography. Many important papers related to the topics presented in these notes are not mentioned. Actually, the literature on Sobolev inequalities is so vast that it would certainly be difficult to list it all. A few of the classical books on the subject have been listed here. Concerning Riemannian geometry, the books [5, 29] and [12, 13] are very useful references and contain some material related to the present text. There is some overlapping between these notes and the monographs [39, 40],

but it may be less than one would think in view of the titles. In particular, the applications presented here and in [39, 401 are different. Some of the techniques from functional analysis used here are developed in greater generality in [21, 72, 87]. Of these three books, the closest in spirit to these notes might be [21], although there is very little direct overlapping and the two complement each other. Grigor'yan's survey article [34] is a wonderful source of information for many related topics not treated in this monograph.

x

PREFACE

It is a pleasure to acknowledge the influence, direct or otherwise, that many colleagues and friends had on the writing of this text. Thanks to A. Ancona, D. Bakry, A. Bendikov, T. Coulhon, P. Diaconis, A. Grigor'yan, L. Gross, W. Hebisch, A. Hulanicki, M. Ledoux, N. Lohoue, M. Solomyak, D. Stroock and N. Varopoulos. Thanks to the students and colleagues at Cornell who attended the class on which these notes are based. They helped me to try to stay honest. Finally, I would like to thank the various institutions whose support over the years has made the writing of this book possible. They are, in no particular order, Le Centre National de la Recherche Scientifique, l'Universite Paul Sabatier in Toulouse, France, the National Science Foundation (grant DMS-9802855), and Cornell University.

I

Introduction This introduction describes some of the main ideas, problems and techniques presented in this monograph. Chapter 1 gives a brief but more or less self-contained account of Sobolev

inequalities in R. The Sobolev inequality in R' asserts that

If

\np/(n-p)

1/P

(x)InP/(n-P)dx

C(n,p) (f IVf(x)WPdx )

(fRfl

that is,

Ilfllq 0

with 1 < p < q to conclude that the strong inequality IIf IIq 5 CIIof IIP holds (with different constants C). Another example is the equivalence between the Nash inequality d f E Co (M),

Ill

II2(1+21")

2 (again with different C's). The Nash inequality is (a priori) weaker in the sense that it is easily deduced from the Sobolev inequality above and Holder's inequality. Chapter 3 gives a rather complete treatment of this phenomenon using elementary and unified arguments taken from [6]. Related results and interesting developments concerning Sobolev spaces on metric spaces can be found in [38]. The equivalence between weak and strong forms of Sobolev-type inequalities turns out to be extremely useful when it comes to prove that a certain

INTRODUCTION

3

manifold satisfies a Sobolev inequality. This is illustrated in the last section of Chapter 3 where some fundamental examples are treated. A basic tool used here is the notion of pseudo-Poincare inequality. Given a smooth function f, let f,.(x) denote the mean of f over the ball of center x and radius r. One says that M satisfies an LP-pseudo-Poincare inequality if, for all f E Co (M) and all r > 0,

Ill - frllp < Cr

llofllp.

For manifolds satisfying a pseudo-Poincare inequality, Sobolev inequalities can be deduced from a simple lower bound on the volume growth. This is more precisely stated in the following theorem.

Theorem Let M be a complete Riemannian manifold. Fix p, v with 1 < p < v and assume that M satisfies an LP-pseudo-Poincare inequality. Then the Sobolev inequality

Vf E Co (M),

Ilfll"p/("-p) < CllVflip

holds true if and only if any ball B of radius r > 0 has volume bounded below by µ(B) > cr".

The idea behind this theorem first appeared rather implicitly in [72] in the setting of Lie groups. It was later developed in [6, 19, 74] and other works. To illustrate this result, we treat in detail the case of unimodular Lie groups equipped with a left-invariant Riemannian metric as well as manifolds with non-negative Ricci curvature and maximal volume growth. The LP-pseudo-Poincare inequality should be compared with the more classical LP-Poincare inequality IVf(y)Ipdy)1/p

d f E C°°(B),

(IB

lf(y) - fBlpdy) 1/p

< Cr (JB

where B = B(x, r) denotes a geodesic ball of radius r and fB = fr(x) is

the mean of f over B. This last inequality may or may not hold on M, uniformly over all balls B = B(x, r), x E M, r > 0. The pseudo-Poincare inequality may hold for all r > 0 in cases where the Poincare inequality does not (for instance on unimodular Lie groups having exponential volume growth). Chapter 4 develops two different but related applications of Sobolev-type inequalities. These two applications have been chosen for their importance and their simplicity. First, we show that Nash inequality is equivalent to a uniform heat kernel upper bound of the form

sup h(t, x, y) < Ct x,YEM

INTRODUCTION

4

where h(t, x, y) denotes the fundamental solution of the heat equation

(cat+A)u=0 on (0, oo) x M, with A = -div o V. In particular, under a Nash inequality, the heat diffusion semigroup (Ht)t>o is ultracontractive (i.e., sends Ll to L°°). This has been developed in the last fifteen years into a powerful machinery which produces Gaussian heat kernel upper bounds. Although this circle of ideas has its roots in Nash's 1958 paper [67], it was only after 1980 that the full strength and the scope of this technique was identified. The books [21, 72, 87] contain different accounts of this topic, various applications and further developments. Here, under the basic hypothesis that

Vt > 0,

sup h(t, x, y) < Ct-v12, x,yEM

we prove that the heat kernel satisfies the Gaussian upper bound

h(t, x, y) S Clt-"/2(1 +

d2/t)"/2e-d2/4t

where d = d(x, y) is the Riemannian distance between x and y. Our proof is somewhat different from those found in the literature. It is adapted from [41] and uses complex interpolation as a main technical tool (and, ironically, no Sobolev-type inequality). The second topic treated in Chapter 4 is a spectral inequality known as

the Rozenblum-Lieb-Cwikel estimate. This inequality was first proved in 1R'ti by Rozenblum in 1972. It asserts that the number of negative eigenvalues of the Schrodinger operator 0 - V is bounded above by C(v) !I V+ 1 1v/2 as soon

as the manifold M satisfies the Sobolev inequality IIf II2v/("-2) < CII Vf 112-

The proof presented here is due to P. Li and S-T. Yau, [55]. A central part of this proof is very close in spirit to Nash's ideas concerning ultracontractivity. It illustrates well what can be done by a skillful use of Sobolev inequality and basic functional analysis.

Despite important examples such as R' and hyperbolic spaces, many Riemannian manifolds fail to satisfy a global Sobolev inequality of the form

Yf E Co (M),

11f 112./(.,-2) S CIIVf1I2

for some v > 2. For one thing, such an inequality implies that the volume of any ball of radius r is at least Cr" for all r > 0, ruling out many simple interesting manifolds such as Sm x RI: (the product of an m-sphere by a k-dimensional Euclidean space). More generally, such a global Sobolev inequality requires too much "uniformity" of the Riemannian manifold M. Fortunately, there is a way to cope partially with this difficulty. The idea

INTRODUCTION

5

is to use families of local Sobolev inequalities instead of one global Sobolev

inequality. For any ball B = B(x, r) on a complete Riemannian manifold, one can find a constant C(B) such that, for any smooth function f with compact support in B, 2/q

If C B


2 are some fixed constants related by 1/q = 1/2 - 1/v. A lot of information is encoded in the behavior of the function B H C(B). The simplest and perhaps most interesting case is when this function is bounded,

that is, supB C(B) = C < oo. This can happen in cases where the global Sobolev inequality J2/g

(JM

2 does not satisfy

any global Sobolev inequality (assuming m # 0) but satisfies a family of local Sobolev inequalities with v = m+ k, q = 2v/(v - 2) and SUPB C(B) = C < oo. In the other direction, the hyperbolic space of dimension n satisfies the same global Sobolev inequality as R" but does not have SUPB C(B) < oo. In fact, as far as many applications are concerned (e.g., heat kernel bounds), a family of local Sobolev inequalities with supB C(B) < oo contains more useful information than a global Sobolev inequality. Chapter 5 develops these ideas and culminates with a complete proof of the following theorem, where V (x, r) denotes the volume of the ball of center x and radius r, and d is the Riemannian distance. For any x E M and s, r > 0, let Q = Q(x, s, r) be the time-space cylinder

Q(x, s, t) = (s - r2, s) x B(x, r). Let Q+, Q- be respectively the upper and lower subcylinders Q+

(s - (1/4)r2, s) x B(x, (1/2)r)

Q- _ (s - (3/4)r2, s - (1/2)r2) x B(x, (1/2)r). We say that M satisfies the scale-invariant parabolic Harnack principle if there exists a constant C such that for any x E M and s, r > 0, and any positive solution u of (O + O)u = 0 in Q = Q(x, s, r), we have

sup{u} < Cinf{u}. Q_

Q+

Theorem A complete Riemannian manifold M satisfies the scale-invariant parabolic Harnack principle if and only if M satisfies the doubling property

V x E M, V r > 0, V (x, 2r) < DoV (x, r)

INTRODUCTION

6

and the scale-invariant Poincare inequality

d B = B(x, r),

jii - fBI2dµ < Por2

p f 2dµ B

where fB denotes the mean off E C°°(B) over the ball B.

In fact, the equivalent properties above are also equivalent to the fact that the heat kernel h(t, x, y) satisfies the two-sided Gaussian estimate V t > 0, V x, y E M,

cl e-cld(x,u)2/t

V(x, vt-)

C2e-c2d(x,g)2/t

M.

Note that each Fi depends only on n - 1 variables, i.e., all coordinates but the irh. Similarly, Fi,,,, depends on either n - m or n - m - 1 variables

CHAPTER 1. SOBOLEV INEQUALITIES IN RN

10

depending on whether i < m or i > m. In particular, for m = n, F2,n(x) _ fn 182 f (y) I dy is a constant function. Now, we can estimate f by If 1:5

(1/2)(F1...

so that

If

In/(n-1)

< (1/2)n/(n-1)

Fn)1/n

(Fl.. .

= pk = n - 1 and induction on

Using (1.1.6) with k = n - 1, pi = p2 = m < n, one easily proves that

f ... f I f(x)I

nnn-1)dx,

Fn)'/(n-1).

... dxm < (1/2)n/(n-1) (Fi,m(x) - -

Fn,m(x))1/(n-1)

For m = it this reads n

Il f lln/(n-1) 2 and real p, 1 < p < n, set q = np/(n - p). Then V f ECU (Rn),

Ill Ilnp/(n-P) C

2(n - p) Th

IIofIIP-

The Sobolev constant given by this theorem (i.e., the constant appearing in front of II Vf IIp) is not the best possible constant. This will be discussed in Section 1.3.1 below.

1.2

Riesz potentials

1.2.1

Another approach to Sobolev inequalities

Sobolev inequalities relate the size of V f to the size of f. In order to prove such inequalities, one may try to express f in terms of its gradient. We now derive such a representation formula. Using polar coordinates (r, 9), r > 0, 9 E Sn-1, in Rn, write

f(x) _ -8rf(x+r9)dr

f

for any f E Co (Rn). Integrating over the unit sphere Sn-1 yields 00

1

AX)

wn-1 Js^-1

49,f (x + r9)drd0 Jo

J Here wn_1 is the (n - 1)-dimensional volume of the unit sphere Sn-1 C Rn. That is, if Stn is the volume of the unit ball, wn-1 = nSZn = 27Cn12/r(n/2)

CHAPTER 1. SOBOLEV INEQUALITIES IN RN

12

where F is the gamma function (F(n) = (n+1)! when n is an integer). Now,

ify=x+r9,wehaver=ly-xI and n

dy = rn-idrd9 and arf (x +r9) = Iy - XI-1 Dyi - x:)aif(y) 1

Hence 1 f(x) = Wn-1

J

In particular I f (x) I < Wn-1

(x-y'Vf(y))dy.

(1.2.1)

Iy -xIn

n

f

W :: X

'

dy.

(1.2.2)

1

In view of this formula, we are led to study the properties of the convolution operator associated with x -+ IxI-n+1 More generally, for 0 < a < n, consider the R.iesz potential operator Ia defined on Co (Rn) by

Iaf(x) =

1 Ca J

Rn

ly

f(y)

-xIn-a

d1.2.3 y

(

)

where ca =7r n/22aI'(a/2)/I'((n - a)/2). By Fourier transform arguments, one verifies that

i0 f = A-a/2 f

where 0 = - En O? f is the Laplace operator. Here, 0-a/2 is defined using

i

Fourier analysis. Namely, for all functions f in Co (Rn),

f = (2irIxl) f, f (x)

= f ef (y)dy

The identity Ia f = 0-a/2 f amounts to the fact that the Fourier transform of ca1IxI-n+a is precisely (2lrIeI)-a in the sense that

C' f IxI-""f (x)dx = n

f

f

a

0 < a < n corresponds to the requirement that both IxI-n+a and IxI-a must be locally integrable for the above identity to make sense. One can show that IaI# = la+p for

a,f3>0,a+f t}) < (A/t)

d t > 0, dx, y E M,

(1.2.5)

.

Again, if r = oo, this must be understood as supx,y l K(x, y) l < A < oo.

Theorem 1.2.3 Assume that Kf(x) =

fM K(x,y)f(y)dµ(y)

where K is a kernel of weak type r for some fixed 1 < r _< oo. Then the operator K is of weak type (p, q) for all 1 < p < oo and p < q < oo such that 1 + 1/q = 1/p + 1/r. Moreover, for each such p, q, there exists a constant B = B(r, p) such that

bf ELp,

(1.2.6)

llKfll9 0. For each t > 0, write K = Kt + Kt where Kt(x, y) = K(x, y)1 {(u,v):x(u,v) 1. There exists a constant B1 such that, for all

t>0andallfELp,

11 Ktf lip
0 and all f E LP, II Ktf Ii

B2t1-r(p-1)/p

_
s})ds

I

00

< tµ({y : K(x, y) > t}) + AJ

s-rds < Bt tt-r

(1.2.7)

because r > 1. Thus 11 Kt f 1100 < Bt tl-r ll f ll,,. and, by duality, 11 Kt f 11, < Bt tt-rll f 111. The duality argument runs as follows. For f E Lt f1 L°°, we have 11Ktf11 = sup 9ELpO II9IIM s})ds 0

< p' A

f

sp'-l-rds = p'(p' -

r)'Atp"

p' < r. It follows that I Ktf I < B2t1-r/p'IIf II,.

To prove the first assertion of Theorem 1.2.3, fix t > 0 to be chosen later. Then, for any s > 0 and f E Ll fl LOO with IlfIlp = 1, write

µ({z : I Kf (z)I ? s}) < p({z : I Ktf (z)I ? s/2}) + p({z : I Ktf (z)I ? sl2}).

CHAPTER 1. SOBOLEV INEQUALITIES IN RN

16

By Lemma 1.2.4,

p({z : I Ktf (z)I

- s12})< (2II Ktf IIP/s)P < (2B1t1-r/s)P

and

I Ktf I
s}) < p({z : I Kt f (z) I > s/2}) < (2B1t1-r/s)P

< B.

B. s-Prl

g-PIl-(1-r)P/(7

(P+r-rp)

r-rp)]

= B3 s-9

if 1/q = 1/p+ 1/r -1, that is q = pr/(p+ r - rp). In words, the operator K is of weak type (p, q). This is true for all 1 < p < oc. The last assertion of Theorem 1.2.3 now follows from the Marcinkiewicz interpolation theorem,

i.e., Theorem 1.2.2 with 1 < pl < p2 < oo arbitrary and 1/q2 = 1/p1 + 1/r - 1. This ends the proof of Theorem 1.2.3. This is a typical use of the Marcinkiewicz interpolation theorem. We have turned a weak (p, q) boundedness result into a strong (p, q) boundedness result using the fact that the weak result holds for all p in a certain interval.

1.2.3

Proof of Sobolev Theorem 1.2.1

In order to prove Theorem 1.2.1, note that K(x, y) = IX - yI-n+a is of weak type n/(n - a). Hence, by Theorem 1.2.3, Ia is of weak type (1, n/(n - a)) and satisfies II Ia f II9 5 CII f IIP for all 1 < p < oo with 1/q = 1/p - a/n.

1.3 1.3.1

Best constants The case p = 1: isoperimetry

Let 13 (r) and Sn-1 (r) denote respectively the ball and the sphere of radius r centered at the origin in Rn. Let On = pn(Bn(1)) and wn_1 = Pn-1(Sn-1(1)) The isoperimetric inequality in lfln asserts that among sets having a smooth boundary of given finite (n-1)-dimensional measure, the ball has the largest n-dimensional volume. Namely,

pn(1)

An(Bn(r')) = 1l rn

where r is such that pn-1(010) =

An-1(Sn-1(r))

=

4Jn_lrn-1'

1.3. BEST CONSTANTS

17

that is Hence, for any compact set 0 C 1R' with smooth boundary, [On(g)](n-1)/n

where

nn-1/n Cn ==

=

G Cnµn-1(asl)

(1.3.1)

[F((n - 11)/2)11/n

wn-1

V/7 -r

Indeed, recall that nfln = wn-1 and fln = irn/2/r((n-1)/2). This inequality has been known to geometers for a very long time; in particular, it was known well before Sobolev's work in the 1930's. Apparently, the discovery that (1.3.1) is equivalent to Sobolev inequality (1.1.5) for p = 1 with the same constant, that is, V f E Co (fin), Ilf lln/(n-1)

- CnllOf 111,

(1.3.2)

was only made much later. In fact, in [78], Sobolev only proved (1.1.5) for

p > 1. The case p = 1 is attributed to Gagliardo and to Nirenberg who published the proof given in Section 1.1.2 in 1958 and 1959 respectively. The connection between (1.3.1) and (1.3.2) was made in 1960 by Maz'ja and by Federer and Fleming. See [60]. The fact that (1.3.1) follows from (1.3.2) is rather straightforward. One approximates the function 10 by smooth functions fn so that Ilfnlln/(n-1)

(1l)(n-1)/n

iz

and

IIVfnIll - µn-1(O l).

To prove the other direction one needs the following co-area formula. See, e.g., [60, 1.2.4] and the references therein.

Theorem 1.3.1 For any f, g E Co (Rn),

f

9IVfIdµn =

f

'010

9x)dµn-1(x\

Cff(X)=t

) j dt.

Indeed, with this theorem at hand, for any smooth compactly supported f > 0 we have

f

I f (x) I

n/(n-1)dx



t})(n-1)/ndt

J' /ln-1({f = t})dt

C CnJ -

oo

= CnJ 1VfIdii =I IVfll1

CHAPTER 1. SOBOLEV INEQUALITIES IN R"'

18

To see the first inequality, write

f

00

AX) =

1{f()>t} (t)dt

0

and use the Minkowski inequality

III f(',y)dyl

< a

f

IIf(',y)Il,dy

with q = n/(n - 1) > 1 to obtain

t}IIn/(n-1)dt

o

{f(.)>t}(t)dtlln/(n-1)

00

1

µn({z : f (z) >

t})(n-1)/ndt.

This shows that the isoperimetric inequality (1.3.1) implies the Sobolev inequality (1.3.2) with the same constant Cn.

1.3.2 A complete proof with best constant for p = 1 According to M. Gromov [62], inequality (1.3.2) goes back to H. Brunn's inaugural dissertation in 1887. My understanding is that Brunn proved the celebrated Brunn-Minkowski inequality (for convex sets and without the case of equality due to Minkowski) from which the isoperimetric inequality easily follows. Whether or not Brunn dicussed the isoperimetric inequality is not clear to me. Of course, he did not discuss (1.3.2). As mention earlier, the observation that (1.3.2) is equivalent to (1.3.1) is usually attributed to Maz'ja and to Federer and Fleming. Gromov gives the following beautiful proof of (1.3.2) which he attributes to H. Knothe [48]. As we shall see, this proof yields both a proof of the isoperimetric inequality (1.3.1) and a proof of the (equivalent) Sobolev inequality (1.3.2). Again, as far as I can tell, there is no discussion of (1.3.2) in the work of Knothe. Let g be a non-negative, locally integrable function with compact support S. For X E S, set

Ai(x)={z:zj=xjfor j 0 and A6 C Bi, we have 0 < yi < 1 for all x. That is, y is a map from S to the cube [0, 1]". Obviously, this map is triangular, that is, for each i, yi is a function of x1, . . . , xi only. Clearly, for each i, the partial derivative ayi/axi is non-negative and equal to 8yi

axi

(x) -

fB,+1(x) 9(z)dzi+i ... dzn /fB.(x) g(z)dzi ... dzn

if 1 < i < n if i = n

g(x) /fB.(x) g(z)dz"

at each point x where g is continuous. Thus, at any such point, the Jacobian of x H y is equal to J(x) =

11axi i

= g(x)

/f g(z)dz.

First, apply this construction to the case where g = x is the characteristic function of the unit ball and observe that yx : Bn -p (0, 1)" is invertible. Let z = yX 1 be the inverse map, z : (0, 1)" -- if". Clearly, this is a triangular map with Jacobian equal to On = pn(3") in [0, 1]". The fact that yx is invertible is one of the crucial points of this proof. In fact, yx is invertible as soon as X is the characteristic function of a convex set. We urge readers to check this for themselves. Now, fix f E Co (llt"), f > 0 and set g = We can assume that f g(x)dx = 1. Construct the map y9 as above and consider the map f"/("-1)

F = z o y9 : S = supp(f) -- Bn. By construction, the map F has non-negative partial derivatives aFi/axi and its Jacobian satisfies JF(x) = On 9(x)

for all x in the interior of S. Thus, the divergence divF = Fn aFi/8xi satisfies

n divF(x) > [JF(x)]1l" = [1n9(x)]1l". Furthermore, by the divergence theorem, we have

J f (x)divF(x)dx = -

F(x))dx f Ip f (x) I dx

because IFI < 1. Hence

1 = ff(xYu'('_1)dx = ff(x)9(x)'IThdx

[1(x) divF(x)dx
1

The following theorem gives the best constant in the Sobolev inequality for

1 t}). That is, f*(x) = f*(Ix() where f- (t) = Sup{s : µ.({z : f (z) > S}) > Ontn}, On = wn_1/n being the volume of the unit ball.

1.4. SOME OTHER SOBOLEV INEQUALITIES

21

Theorem 1.3.3 For all f E Co (Rn) and all 1 < p < oo, we have Ilof*IIP
0 . Thus, u is a subsolution if Lu < 0 in the weak sense (2.1.6). A function u is a supersolution if -u is a subsolution. We will prove the following theorem.

Theorem 2.1.1 Let L be as in (2.1.3) with A = (ai,j) satisfying (2.1.4) for some A > 0. For any 6 E (0, 1), there exists C = C(n, A, 6) > 0 such that any positive solution u of (2.1.5) in a ball B satisfies the Harnack inequality susp{u} < C of{ u}. 6B

6

(2.1.7)

Moreover, for any b E (0, 1), there exist C' = C'(n, A, S) > 0 and a = a(n, A, b) > 0 such that any solution u of (2.1.5) in a ball B satisfies the Holder continuity estimate sup { l x,yESa

u(x) - u(y)I

ix -

y1a

< C'r-°ilull.,B

J1 -

(2.1.8)

where r is the radius of B.

Here and in the sequel SB is the ball with the same center as B and radius equal to 6 times the radius of B. The sup's and inf's above must be understood as essential sup's and inf's.

CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY

36

One of the important aspects of this result is that the constants C, C', a do not depend on the function u or on the ball B. The technique that will be used below to prove Theorem 2.1.1 consists in taking advantage of the weak form (2.1.5) of the equation Lu = 0 by

adequate choices of the test function 0. One of the key ideas is to use test functions 0 of the form uQ,02 where u is the studied weak solution of the equation under consideration and -0 E C0 1(B) is a genuine test function. Here, it will be useful to consider positive and negative /3's. This wonderfully simple idea is surprising: using the unknown u to define the test function 0

is a rather bold idea. It requires one to check that u1'i2 is in W01,2 (B). For positive u bounded away from 0 and Q < 1, this does work fine. For ,(3 > 1, this is fine if u is a positive locally bounded function in W1'2(B). However, requiring u to be locally bounded is not a reasonable a priori hypothesis when dealing with solutions (resp. subsolutions) of operators L with nonsmooth coefficients. When one wants, as we do, to deal with the general case of measurable coefficients this difficulty can be resolved in two different ways:

(1) One may regularize the coefficients and show that the solutions of (2.1.5) can be approximated by solutions of similar uniformly elliptic equations with smooth coefficients. These solutions are smooth by local elliptic theory. It then suffices to prove the desired bounds for solutions of equations with smooth coefficients. Implementing this idea does require some work.

(2) Instead, one may work directly with the desired equation and its weak

solutions in W1"2(B). Then, for 8 > 1, one must work with approximate power functions of the form t H 0 for 0 < t < T, and t i- T' 't for t > T, and afterwards pass to the limit as T tends to infinity. Here we will take this second route which is much more direct. Of course, a posteriori, weak solutions (resp. subsolutions) of uniformly elliptic equations are locally bounded functions so that this difficulty disappears. The following lemma is a good simple example of a result which requires the boundedness of the weak subsolution u in order to be meaningful. Indeed, use of this lemma should be avoided if one wants to deal directly with non-smooth coefficients, and we will not use it the sequel.

Lemma 2.1.2 If u is a subsolution of (2.1.5) in B and e _< u < c for some 0 < e < c < oo, then u°` is also a subsolution for all a > 1. We have, for any 0 E Co (B),

E ai,jaiu'ajq5

= a E ai,ju'-18iu8j(0 i,j

2,7

a i, j

37

2.1. ELLIPTIC OPERATORS IN DIVERGENCE FORM

-a(a - 1)

a%jO%uC7ju

U0-2o

%,j

2, and any positive subsolution u of Lu = 0 in a ball B of volume V, 6Bp{up} < C2(1

- 5)-" (V-1 fB updµ)

.

2.2. SUBSOLUTIONS AND SUPERSOLUTIONS

41

We first prove a somewhat weaker version where the constant C2 depends also on p > 2. Fix p > 2 and 0 < b < 1. For each integer i > 0, set pi = phi,

Po=1,

=i

pi=1-(1-b)E2-i, i>1. j-1 Then pi+1 - pi = (1 - b)2-'-', pi+l = pi6, and, by Lemma 2.2.1, upi+ldµ

< C(1 -

(p2f

b)-2 22(i+i)

Jn+1B

up;dpl

J

iB

or

\ 1/pi+1

upi+1dui

< [C(1 -

b)-211/pi+i 22(i+l)/pi+i

pi/p'

UpiB uPi dli

p i+, B

1/pi

fu1

for i = 0, 1.... with C = 2C,2,(1 + 4A 4). This yields upi+idpl1

l

1/pi+1

11/p

< C(n)C(p) [C(1

where

C(n) = Observe that

22(E'.ie-j)

C(P) = e2Eo B-ito$(pei)

pi ---> 6, 00

6-J

= 0-'(1

- 6-')-' = n/2

and

Hence

sup{u} < 6B

(C(n)C(p)Cn(1-

b)n)'lp IIuIIp,B.

This proves Theorem 2.2.2 when B is the unit ball and with a constant C2 which depends on p. In any case, it shows that all positive subsolutions in B are locally bounded. It follows that we can repeat the proof of Lemma 2.2.1 with G(t) = tp-'. If one then uses direct computations instead of the inequality G(s) < sG'(s), one obtains a sharper version of Lemma 2.2.1 that reads

fp'B

e

upedp < C1(P - P')-2V 1_B

t

f updp)

.

\ pB This is sharper because the factor p2 in front of the last integral has been removed. Based on this inequality the argument above proves Theorem 2.2.2 with a constant C2 which is independent of p > 2.

CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY

42

Theorem 2.2.2 can be interpreted as a form of LP mean value inequality

for subsolutions of Lu = 0. From this point of view, it is natural to try to extend the result to the case where the right-hand side is the LP mean of u, 0 < p < 2. In particular, it is natural to ask whether or not the L' inequality

{u} < C(1- 6)_n (V-1 jud) li

a

holds for positive subsolutions. That it holds is a special case of the following

statement.

Theorem 2.2.3 Fix 0 < p < 2. There exists C3 = C3(n, A, p) such that, for any 0 < 6 < I and any positive subsolution u of Lu = 0 in a ball B of volume V,

< C3(1 - 6)n/P V-1

u { u}

1/P.

JB updµ/

This can be obtained from Theorem 2.2.2 by the following similar but different iterative argument. As usual, we assume that B is the unit ball. Fix o, E (1/2, 1) and set p = a + (1 - 0)/4. Then Theorem 2.2.2 yields

Sup{ u} < C(1 -

a)-n/2II

o,B

uIl2,pB

for some C = C(n,.)). Now, as IIu1I2,B 0.

Hence

f *2

a2,ja1wa,wdp < if -rbw E J

,9

By uniform ellipticity, this yields

J v2lvwl2dp
(1 - 6), then

Iu(x) - u(y)I < 2sup{u} < 2(1 - h)-"Ix - yl'sup{u}. B

B

If Ix - yI < (1 - 6) then the ball B' of radius (1 - 6) and center (x + y)/2 contains both x and y and is contained in B. Moreover, x, y are contained in (Ix - yI/(1 - b))B'. Applying Lemma 2.3.2 in B' yields I u(x) - u(y)l 5 21 «(1 - b)-'fix - yI° Sup{u}. B

This proves (2.3.3) and finishes the proof of Theorem 2.1.1.

Chapter 3 Sobolev inequalities on manifolds 3.1 3.1.1

Introduction Notation concerning R.iemannian manifolds

We want now to replace the Euclidean space lilt" by a Riemannian manifold M and consider the possibility of having some kind of Sobolev inequalities. This brings in a whole new point of view. On Euclidean space, we could only

discuss whether inequalities were true or not. In the more general setting of Riemannian manifolds, we can investigate the relations between various functional inequalities and the relations between these functional inequalities and the geometry of the manifold. We can search for necessary and/or sufficient conditions for a given Sobolev-type inequality to hold true. This leads to a better understanding of what information about M is encoded in various Sobolev-type inequalities. Sobolev inequalities are useful when developing analysis on Riiemannian manifolds, even more so than on Euclidean space, because other tools such as Fourier analysis are not available any more. This is particularly true when one studies large scale behavior of solutions of partial differential equations such as the Laplace and heat equations. In the sequel, we will focus on complete, non-compact Riemannian manifolds. For compact manifolds, local Euclidean-type Sobolev inequalities are

always satisfied and the interesting questions have to do with controlling the constants arising in these inequalities in geometric terms. We refer the interested reader to [5, 39, 40] where this is discussed at length. Let us briefly introduce notation concerning Riemannian manifolds. Let M be a Riemannian manifold of dimension n with tangent space TM and co-tangent space T'M. The tangent space TM is the union of the tangent spaces Tz where x E M. 7. is the dual of the n-dimensional linear space

T. and T'M is the union of these spaces. Smooth sections of TM are 53

54

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

called vector fields and smooth sections of T*M are called forms (1-forms). There is a natural pairing TM x T*ll1 (C, 77) '--p ii(e) E R induced by the

natural pairing of T. and Ty , x E M. It is a basic fact that vector fields can equivalently be defined as derivations, i.e., maps C : C°°(M) -* C°°(M) such that C(fg) = fCg+gCf for all f, g E C°°(M). If f is a smooth function

on M, the relation between its derivative df which is a form and f where C is a vector field is given by

ef. Because M is a Riemannian manifold, we are given on each T., a scalar product If f c C°°(M), its gradient is defined as the unique vector field V f such that

d x E M, V E TM, (V f (x), (x))x = df (e)(x). There is a canonical distance function associated to the Riemannian structure of M. We will denote it by (x, y) '--+ d(x, y). It can be defined as the shortest length of all piecewise Cl curves from x to y. The topology of (M, d) as a metric space is the same as that of M as a manifold. See, e.g., [13, §1.6]. There is also a canonical Riemannian measure on M which wy denote by either dx or p. depending on which is more convenient. See, e.g., [13, §3.3]. We denote by V (x, t) the volume of the ball of radius t > 0 around x E M, i.e., V(x, t) = µ(B(x, t)). Thus V (x, t) 'describes the volume growth of M. The divergence dive of a vector field 6 is defined as the unique smooth function on M such that

f divdp =- fM df ()dp.

V f E Co (M), IM

The Laplace-Beltrami operator A on M is the second order differential operator defined by

V f E Co (M), A f= -div(V f ). Note that, with this definition,

df, g E Co (M), JM f Lgdµ =

f

M

so that, in particular, JM

fofdp =

(O f, V f )dp > 0. JM

3.1. INTRODUCTION

55

All the objects introduced above can of course be computed in local coordinates. See, e.g., (12, 13].

We will always work under the assumption that M is complete. Although this could a priori be interpreted to have various metric or geometric meanings (e.g., geodesically complete), the different interpretations turn out to be equivalent. Thus, M is complete means that (M, d) is a complete metric space. In particular, all bounded closed sets are compact. See [13, §1.7].

3.1.2 Isoperimetry On a Riemannian manifold, any smooth (n - 1)-submanifold (i.e., hypersurface of co-dimension 1) inherits a Riemannian measure which we will denote by An-1. The isoperimetric problem on M asks for the maximal volume that can be enclosed in a hypersurface of prescribed (n -1)-volume and for a description of the extremal sets, if they exist. The first part of this problem can be interpreted as the search for some function (depending on M) such that (D(Pn(cl)) < An-l( &I)

for all bounded sets Tl C M with smooth boundary BSZ.

Solving the second part of the isoperimetric problem of course yields such an inequality. For instance, if M is n-dimensional Euclidean space, balls are extremal sets for the isoperimetric problem and this leads to the optimal function 4 given by WRn (t) = wn-1

01-1/

1-1/n

n

In view of this fundamental example, it is natural to consider the possibility that a Riemannian manifold M satisfies An(cl)1-1/v < C(M, v)An-1(COSZ)

(3.1.1)

for some constant v > 0 and C(M, v) > 0. Note that this could possibly be satisfied for a number of different values of v and that the set of possible values of v is either empty or an interval.

Theorem 3.1.1 Assume that M satisfies (3.1.1) for some positive finite v and C(M, v). Then V (x, r) > c(M, v)r"

where c(M, v) = [vC(M, v)]-". In particular, if M is n-dimensional and satisfies (3.1.1) then v > n.

56

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

The proof is straightforward if one observes that 0,V (x, r) is the (n - 1)dimensional Riemannian volume of the boundary of B(x, r). Indeed, we then have V (x,

r)1-11"

< C(M, v)BrV (x, r).

That is 8,. [V(x, r)'/"] > [vC(M, v)] -1. This obviously yields

V (x, r) > [vC(M, v)] -"r".

However, this proof is not quite complete because the boundary of a ball need not be a smooth (n - 1)-submanifold (for large radius r). Here, we will ignore this difficulty and refer the reader to [13, §3.3,3.5] for details justifying the computation above. A different proof of this lemma, avoiding this difficulty, will be given later on (see Theorem 3.1.5 below).

In Euclidean space, we noticed the formal equivalence of the isoperimetric inequality with the Sobolev inequality IlfIIn/(n-1) : C IlVf111. The argument, based on the co-area formula (see, e.g., [13, Theorems 3.13 and 6.3]), works as well on any Riemannian manifold. This proves the following important result. Theorem 3.1.2 A manifold M satisfies the inequality (3.1.1) for some positive v and C(M, v) if and only if it satisfies the inequality V f E Co (M),

IIfII"/("-1) < C(M, v)IIofIll.

Let us fix p, v such that 1 < p < v. We say that a Riemannian manifold M satisfies an (LP, v)-Sobolev inequality if there exists a constant C(M, p, v) such that V f ECo (M),

llfIIPv/("-P) :5 C(M,p,v)IIVfIIP

Thus, Theorem 3.1.2 can be interpreted as saying that a manifold M satisfies

an (L', v)-Sobolev inequality if and only if it satisfies (3.1.1). The next result shows that the strength of an (LP, v)-Sobolev inequality decreases as p increases.

Theorem 3.1.3 If M satisfies an (LP, v)-Sobolev inequality then it satisfies an (L4, v)-Sobolev inequality for all p < q < v. Apply the (LP, v) inequality to I f 17 for some ry > 1 to be fixed later. Thus

I I f Ii;/(,_) < 'YC(M, p, v)

If (JM

IP(7-1)

l I Vf l Pdµ J

1/P

3.1. INTRODUCTION

57

Now, apply the Holder inequality

f hgdl.c
0. t>o

We will use the fact that the distance function has gradient bounded by 1 almost everywhere. Indeed, for any fixed x E M and t > 0, consider the function f (y) = max(t - d(x, y)), 0}. Then Ilf (lr

Hence

> (t/2)V (x, t/2)1/r

Ilf 11,

tV(x, t)"

Il Vf lip

V(x, t)'1".

V (x, t)o/P+(1-o)/s > 2-1(t/C)6V (x, t/2)1/r.

If we could ignore the fact that we have the volume of the ball of radius t/2 instead of t on the right-hand side of this inequality, we would get V (x, t) > ct" with v given by 0/v = 0/p + (1 - 0)/s - 1/r. Thus, define v by 0

0

1-0

p s and write the last inequality in the form V

1

r

V (X, t) > (2Ce)-r"/(v+or)torv/("+or)V(x,

t/2)"/("+er).

59

3.1. INTRODUCTION It follows that (2CB)-r>iajtsr-iaj2-6rri(j-1)a'V(x,t/2f)a=

V(x,t) >

(3.1.3)

with a = v/(v + Or). Observe that a < 1 as long as 0 # 0. Moreover, in this case, 00

a? - a( 1

-

-

v a) -1 - gr'

00

(' - 1)aj = a2(1 -

1

a)-2 =

v2 02r2'

1

Furthermore, limt-.o t-nV (x, t) = On. Hence for i large enough, lim inf V (x, t/2s)a` > lim

[Stntn/2n+1]a`

Z-00

= lim 2-t'`a` = 1.

i tend to infinity in (3.1.3), we obtain

V(x, r) >

2-"2/er(2CO)-"/et".

Note that this proof uses the (Riemannian) fact that limt--,o t-nV (x, t) = Stn. Later, we will give another proof avoiding the use of this fact.

It is useful to illustrate Theorem 3.1.5 by a very simple case showing how the volume parameter v abstractly defined by (3.1.2) is computed from 0, r, s, p. Take M = R. Then we have the calculus inequality, V f E Co (1[t),

IlfII. t (note that this is off only by a factor of 2). Applying the calculus inequality above to If I2 and usand thus ing Cauchy-Schwarz, we find that IIf II00 o

(C 11 Vf IIp)e (Ilf

60

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

is satisfied for some r, s, 0 with 0 < s, r < oo and 0 < 9 < 1. Assume also that

-+1-0-1>0. p

r

s

Then

V (X' t) > ct"

with 0 < v < oo defined by 0

0

1-0

1

V

p

s

r

and the constant c given by c = 2-v21er-v1eC-v.

In this theorem, if r = oo, the weak L''-norm must be understood as Ilf II.-

3.2 3.2.1

Weak and strong Sobolev inequalities Examples of weak Sobolev inequalities

Any Sobolev inequality can be used to deduce a priori weaker inequalities through the use of the Holder inequality and related inequalities. For instance, the Sobolev inequality IIfI12v/(v-2)

t})(v-2)12v}

< CIIVf112.

t>O

We have also seen in Section 2.1.3 that the Sobolev inequality IIfII2v/("-2)

< C11VfII2

implies 1I2)"/(v+2)Ilfll2/("+2),

Ilf 112(1+2/") 3. In some sense, this means that, if Nash inequality implies Sobolev inequality, it cannot be in a completely obvious way. In general, one can ask: do various weak forms of Sobolev inequality imply their stronger counterparts? The next few sections show that, somewhat surprisingly, the answer is yes. Moreover, this can be proved by some elementary and widely applicable arguments.

3.2.2

(S88)-inequalities: the parameters q and v

Let us fix the parameter p, 1 < p < oo. We want to discuss functional inequalities of Sobolev type for smooth compactly supported functions under the hypothesis that we can control II V f lip. The weakest type of Sobolev

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

62

inequality we have encountered so far is that of Theorem 3.1.6 which states

that, for all f E Co (M), (I1fIJ.µ(SUpp(f))1/1)1-0 .

sup {tp({IfI > t})1/r}
o

We call this inequality (S,*.,B). Recall that it has a slightly stronger version (see Theorem 3.1.5) which reads

V f E C (M),

t)1/r} < CIIV f Jlp t>o

Finally, when p = 2, (S2,1) is the Nash inequality 11f

112(1+(1-19)/0)
1 and k E Z. This function has the following properties. Its support is contained in If > pk}. Moreover,

{fa,k?(P-1)Pk}={f >Pk+l}

(3.2.4)

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

64

Finally, f ,,,k has the same "profile" as f on {pk < f < pk+1 } and is flat outside this set. In particular, (3.2.5)

IVfp,kl < !VfI.

By (3.2.4), (3.2.5) and our hypothesis applied to fp,k, we have (p - 1)Pkµ(f >- pk+1)1/ro < (CIIVfIIP)Bo ((P - 1)Pkµ(f > pk)1/so)1-e0

Let us set

N(f) = sup {Pkµ(f ? Pk)1/a} . k

Using the definition of q we then get > pk+1)1/ro

P(f < p

k(9o+(1-Bo )/8o)

Pkq/ro




t)1/1} < e(1

+ q/(ro9o))C IVf llp.

(3.2.7)

t>0

This is the weak form (in the sense of weak L' spaces) of the desired (LP, v)Sobolev inequality IIfIIp"/(i'-p) < ACIIVfIIp

where v is given by 1/q = 1/p - 1/v. Thus, to finish the proof of Theorem 3.2.2, it suffices to prove the following lemma.

Lemma 3.2.3 Assume that for some 1 < p < q < oo,

V f E C (M), sup{tp(f > t)1/q} < t>o

CI

11 Vf llp

Then

V f E Co (M), IIf

IIq

< 2(1 + q)C1 ilof iip

65

3.2. WEAK AND STRONG SOBOLEV INEQUALITIES For this lemma, we need to improve (3.2.5) and observe that, actually, IVfkl C

(3.2.8)

IVfI1{Pk pk+1)1/q < C1 II Vfklfp.

Raise this inequality to the power q and sum over k E Z to get

< Ci >

1),

k

k

(r \J{Pk ak/' < (> ak)q/P for any sequence (ak) of nonnegative reals. Also k+2

[pq(Pq -1)]-'q E

E Pkgµ(f >_ pk+l)

JPk+l

tq-1µ(f > t)dt

[PQ(Pp - 1)]-'q 1000 tq-1µ(f

? t)dt

0

_ [PQ(Pg - 1)]-1llfllq. Hence (3.2.9) yields

Ilfllq
s' such that r' _< r and s' < s. For instance,

it is enough to prove (Sr,-, "') for (r, s) E S = {(ro + t, so + t) : t > 0}. Now,

fix any t > 0 and set r = ro + t, s = so + t. Multiply (3.2.10) by qkt to obtain

-

Pkrll(f > pk+1) < (C11V ilipI

ro-so pksµ(f

> Pk)

l

Summing over all k, we get

(CIIVfIIPIr-s

EPkrp(f > pk+l)
pk).

But 'k+2 is-lp(f

Pr(P1- 1)IIfIIr
t)dt < pkrp(f > pk+l)

pk+i

and, similarly, pksµ(f

S

> Pk) 1 and r = ro + t, s = so + t. For any fixed p > 1, this gives the desired inequality for all 0 < s < r < oo. This proves Theorem 3.2.5. We now want to obtain a result that complements Theorem 3.2.5 under the same hypothesis. Observe that we cannot take r = oo in Theorem 3.2.5.

3.2. WEAK AND STRONG SOBOLEV INEQUALITIES

67

Actually, the case of R"' shows that we cannot hope to have an inequality of the form IIfIIcl:! (CIIVf110011f1l.

when q = oo. What we can hope for is some form of local exponential integrability. This can be obtained by looking more closely at the proof given above. By Theorem 3.2.5, we can now assume without loss of generality that (S212) is satisfied. Repeating the argument above and using y(x - 1) < x-1 - 1 < y(x - 1)x"r-1 which is valid for ry > 1, we obtain that there exists a constant C such that < 2"rp3(p 1)-1+8/r(CIIVfIIP)1-s/rllf11s/r 11f

-

11,

for all t > 0, r = 2 + t, s = 1 + t. Let 1 be the support of f . As 1 < s < r, we have 11f 118
2 and s = r -1)-1+81r(ClIVfllp)l-8/rµ(c)1-8/rllfllr/r

< 21/rp3(p

which gives (recall that r - s = 1) Ilfllr
2. Picking p = 1 + 1/r yields 11f 1Ir 1, Vf ECa (Q),

11f 11, 0 and A2 such that for all bounded sets St C M,

d f E Coje/0d (Q),< A2()

The second inequality stated above is an easy consequence of the first. Note

that this is not the sharpest result that can be obtained. Indeed, as in the case of Rn, one can show that (ST18p8o/ro) for some fixed 0 < so < ro < o0 implies the stronger integrability d f E Co (9),

fo

e(a'If1/I1Vf11P)PAP-1' dµ

< A'p(cl)

for some constants A', a' > 0. See [6] for a proof of this sharper result using the same ideas. Interestingly enough, in order to obtain the sharper result one apparently needs to use Lorentz spaces instead of mere L' -spaces.

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

68

3.2.5

The case -oo < q < 0

As in the previous two sections, the number p, 1 < p < oo, is fixed and all Sobolev-type inequalities are relative to II Vf 11p.

Theorem 3.2.7 Assume that (S, °) is satisfied for some 0 < ro, so < 00 and 0 < 00 < 1 and let q = q(ro, so, 9o) be defined as in (3.2.1). Assume that -oo < q < 0 (this forces so < ro). Then all the inequalities (Se8) with 0 < s < r < oo, 0 < 9 < 1 and q(r, s, 9) = q are also satisfied. In particular, there exists a finite constant A such that V f E Co (M),

A(Cllofllp)1/(1-8/9)liflls/(1-9/e)

IIfIIoo
0 and 11f 11,,. 0. Fix also c > 0 small enough and p > 1. Define the functions fp,k by setting

fp,k = (f -

flloo -E-Pk))+A pk-1(p- 1)

(11f

for all k < k(f) where k(f) is the largest integer k such that pk < IIf IlooNote that fp,k is compactly supported if 0 < E < II f Ii pk(f) and that lVfP,kl < IVfI for all k < k(f). Set

-

Ak=llfll.-E_pk.

Observe that fp,k has support in If > .Ak} and that pk-1(p

{fp,k J

- 1)} = if

ak-1}.

Applying (ST(,, so) to f,,k, we obtain

Pk-1p(f ?

Ak-1)1/ro


0 for all k < k(f) and that

lim ak = 00

k-+-oo

-

because p(f >- IIf ll,,. E) > 0 and q < 0. This is actually sthe reason why the parameter e was introduced in the computation above. Because of these observations, it is clear that a= inf ak kA) - (p -

P-1 )

.

Rearranging, we obtain

IlfII. 0, llf Its t})l/r} o

llf Ilr A)T/P 0 such that

for allt>1,

V (t) > c exp(at),

or there exist 0 < c < C < oo and d = 0, 1, 2.... such that for all t > 1 c < t-dV (t) < C. A typical case where the volume of G has a polynomial behavior is when G is a simply connected nilpotent Lie group. In this case, the volume growth function V satisfies Vt > 1, ctd < V (t) < Ctd

where d is an integer given by k

d=

i dim(C9t/92+1) 1

82

CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS

Here, the pi's are subalgebras of the Lie algebra G of G defined inductively

by gi = g, Gi = [9,9j-1), i = 2,.... The integer k is the smallest integer such that ck+1 = {0}. That such a k exists is essentially the definition of G (i.e., 9) being nilpotent. As the topological dimension n of G is n = Ei dim(gi/Cgi+1), it follows that any simply connected nilpotent Lie group G satisfies n < d. In particular, such a group has volume growth bounded below by

dt>0,

V(t)>c,,,tm

for all m E in, d). For instance, the group of three by three upper-triangular matrices with diagonal entries equal to 1 (see (5.6.1)) is a simply connected nilpotent group known as the Heisenberg group and having k = 2 and d = 4. Of course, its topological dimension is n = 3. See [87] for details and references.

3.3.5 Ricci > 0 and maximal volume growth Let (M, g) be a complete Riemannian manifold. The Ricci curvature tensor R. is a symmetric two-tensor obtained by contraction of the full curvature tensor. See, e.g., [13, 29]. Thus, it can be compared with the metric tensor g. Hypotheses of the type 1Z > kg, for some k E R, turn out to be sufficient to derive important analytic and geometric results. For instance, if R > kg with k > 0, then M must be compact. See, e.g., [13, Theorem 2.12]. If k = 0, the volume growth on (M, g) is at most Euclidean, that is, V r > 0, V (x, r) < St,,r". See, e.g., 113, Theorem 3.9]. We want to show that complete manifolds of dimension n having nonnegative Ricci curvature and maximal volume growth, that is, for which there exists c > 0 such that V r > 0,

V (x, r) > cr',

satisfy the pseudo-Poincare inequality (3.3.1).

Theorem 3.3.8 Let (M, g) be a complete manifold of dimension n having non-negative Ricci curvature. Assume that there exists c > 0 such that

d r > 0, V (x, r) > crn. Then

V' "O, 0, V f E Co (M), IIf - frlli < (1l/c) r llVfUIi. Moreover, the Sobolev-type inequalities (Se3(p)) with 1

=8i

1

_ 1 n1+1

-e p s , are all satisfied on M. In particular, r

V f E CO (M), IIf

forall1 0 be a smooth bounded

function on M and set dp(x) = V(x)dx. Then

Q(f,f)= f MIVf(x)12dx, fEW1"2(M)nL2(M,dlt) is a Dirichlet form on L2(M, dµ).

Returning to the general case, any Dirichlet form is associated canonically to a self-adjoint semigroup of operators Ht = e-tA, t > 0, on L2(M, pt) generated by a possibly unbounded self-adjoint operator -A with domain D(A) C D. In terms of the semigroup (Ht)t>o, the domain of A is the space D(A) of all f E L2(M, p) such that lim 1(Htf

t-.o t

87

- f)

CHAPTER 4. TWO APPLICATIONS

88

exists in L2(M, it), and A is given by

-Af

li m

t (Htf - f )-

Very generally, one can show by elementary arguments that if f E D(A) then the function x '-4 Ht f (x) is in D(A) and (t, x) H u(t, x) = Ht f (x) satisfies (4.1.1) (at + A)u = 0, u(0, ) = f. Moreover, D(A) is dense in L2(M, dp). See, e.g., [69]. In the case of a selfadjoint semigroup on L2(M, dµ) as above, one can appeal to spectral theory to show that, for any f E L2(M, dµ), Htf is in D(A) and u(t, x) = Ht f (x)

satisfies (4.1.1).

In terms of the form (Q, D), the generator -A and its domain can be defined by

V f E D(A), Vg ED, Q(f,g) = fM(Af Because Q is a Dirichlet form, Ht takes real functions to real functions and so does A. Hence in most instance it suffices to work with real-valued functions. In the sequel, we always assume that all functions are real-valued unless it is explicitly stated that functions might be complex-valued. On at least one occasion we will indeed need to consider complex-valued functions. In fact, not only does Ht preserve real functions but it also preserves positivity. It follows that Ht extends as a (weakly continuous) semigroup of contractions on L°°(M, p). It also extends as a strongly continuous semigroup of contractions on all L" spaces, 1 < p < oo. A simple important fact is that

t ~- -(atHtf, f) = IIA"2Ht,2f I12 = IIHt,2A112fI12 is non-increasing for all f E D. This implies that 11f 112

Hence II AI12Htf 112
- -t(atHtf, f). (2t)-1/2 11f 112 for all f E D and thus for all f E

L2(M, p). By the semigroup property, it follows that Htf is in the domain of Ak12 for all integers k and all f E L2(M, dµ) and (2t/k)-k12.

(4.1.2)

EXAMPLE In the case of our basic example (i.e., when M is a Riemannian

manifold, Q(f, f) = f I V f I2dx and dp(x) = V(x)dx, V is positive and smooth, and dx is the Riemannian measure), we have A = V-10 and the semigroup Ht leads to solutions of the equation

(at+V-1A)u=0.

4.1. ULTRACONTRACTIVITY

89

Observe that positivity preserving and the fact that Ht contracts L°° can be seen as consequences of the (parabolic) maximum principle satisfied by such equations. The local theory of parabolic equations associated to elliptic

operators implies that any solution u of (8t + V-'0)u = 0 is smooth. In particular, for any f E L' (M, µ), Ht f E C°°(M). It follows that Htf(x) = f h(t,x,y)f(y)dµ(y)

M

where h(t, x, y) is a non-negative smooth function which we call the heat kernel associated to V- 'A on L2(M, dµ). When V - 1, this is the usual Riemannian heat diffusion kernel (heat kernel for short) of M. When M = JR1 and V - 1 then h(t, x, y) = (47rt)n/2

exp(-Ix - y12/4t).

Returning to the general case, as Ht preserves positivity and contracts L°°(M, µ), it follows that for each t, x there exists a measure h(t, x, dy) (often called the transition function of Ht) such that

Htf (x) =

fm

f (y)h(t, x, dy)

and

h(t, x, M) < 1.

This observation can be used to show that Ht contracts LP(M, µ), that is, IIHtDIP_P < 1

for all 1 < p < oo. Indeed, by Jensen's inequality,

IHtf(x)IP < [Ht(IfIP)](x)

Thus, it is enough to show that Ht contracts L'(M, p). This follows by duality from the fact that

L gHgfdp

JM fHtgdp s IIgII,I1fIII

for all f,g E L'(M, p) fl L°°(M,µ). There is no reason, in general, that Ht should send L' (M, µ) into any LP space, p > 1. However, in many important cases, (Ht)t>o has the qualitative smoothing effect that Ht f is bounded for any f E L'(M, dµ) and any t > 0. When this is the case, for any t > 0, there exists a constant Ct such that

V f E L'(M,µ),

IIHtf11. 0,

V f E L1(M,µ),

IIHtfII0 < CtIlf11,

and

sup h(t, x, y) < Ct

V t > 0,

2,yEM

are equivalent. The following elegant result was first proved by Nash [67] in the case where A is a symmetric uniformly elliptic second order differential operator

in divergence form in R.

Theorem 4.1.1 Let Q be a Dirichlet form on L2(M, µ) with associated semigroup (Ht)t>o. Assume that the Nash inequality V f E D,

1121+2'v) < CQ(f, f)II

Ilf

f

114/1,

is satisfied for some v > 0. Then (Ht)t>o is ultracontractive and V t > 0,

11 Ht

111_.,

< (Cv/2t)vi2,

or, equivalently, (Ht)t>o admits a density w.r.t. u which satisfies `d t > 0,

sup {h(t, x, y)}
0. Recall that Htf E D(A) for all t > 0 where -A is the infinitesimal generator. Set

u(t) = IIHtfII2 Then u has derivative

u'(t) = 2(BtHtf, Htf) where the scalar product is in L2(M, µ). It follows that

u'(t) = -2(AHtf, Htf) = -2Q(Htf, Htf). Thus, by the postulated Nash inequality and the fact that 11 Htf 111 < 1,

u(t)1+2/" < -(C/2)u'(t)

4.1. ULTRACONTRACTIVITY

91

Setting v(t) = (v/2)u(t)-2l", this yields v'(t) 2 2/C. Hence v(t) > (2/C)t, that is, u(t) < (C1,/4t)"l2.

This means that II Htf II2 < (0,14t)"1411 f III

for all f E L1 (M, µ) because Ll (M, ti) n L2(M, µ) is dense in L' (M, it). In other words (Cv/4t)v/4

I I Ht II 1--.2
0 by choosing t such that vC""2{I f!{2{{f112

= 4Q(f,

f)t'+"/2

After some algebra this yields I1f1121+2/") o be the corresponding Markov semigroup. The following two properties are equivalent.

(i) The form Q satisfies the Nash inequality

df ED,

Ilf112(1+2/") o satisfies

et > 0,

-.2 < eaat.

Set u(t) = IIHt'1f II2, f E L2(M, µ). Then u has derivative u'(t) = -2(Aa,OHt 'Of, Ht 'mf )

CHAPTER 4. TWO APPLICATIONS

94

Thus, it suffices to show that (4.2.1)

(Aa,mf, f) ? -a2II f II2

for all f E Co (M). For, if this holds, we have u' < a2u, that is, u(t) ea2tu(0) and the desired conclusion easily follows. To prove (4.2.1), write

(Aa,mf, f) _ (e-amA(e' f ), f) =

J

IVol2IfI2d,t

> -a2IIf I12 The last step above uses the hypothesis I V 01 < 1. This proves Lemma 4.2.1.

The following version of Gaffney's lemma dealing with time derivatives is also useful.

Lemma 4.2.2 For all functions 0 E CQ (M) with II V0II00 < 1 and all

aER, forallt>0andall(=e1' EC,TER, with IrI 0.

4.2.2

Complex interpolation

In this section we implicitly work with complex LP-spaces in order to use complex interpolation techniques. Let us start with the following simple lemma.

Lemma 4.2.3 Assume that the heat diffusion semigroup (Ht)t>o satisfies

Vt > 0,

IIHtII2-..

(C/t)v'4.

Then, for all 2 < p < oo, we also have

dt > 0,

II Htfp_

S (C/t)'12p

By classical interpolation theory (see, e.g., [79, 81]), the bounds

IIHt II.--i. s 1 and

IIHtII2- 0,

(C(S)C/t)"/4 exp(ta2(1

0. Consider two sequences of positive numbers (si)1°,

(pi)i° such that 1 = Ei° si, pl = 2, pi / oo and 00

E sipi+l

2 where

1/c=E°i-5,p1=p2=2,pi=2(i-1)2fori>2. Then 00

00

E sipi+1 = 2(1 - c) + 2ecE i-3 < 2(1 + b) 2

1

if c > 0 is chosen so small that cc E0 i-3 < 6. Now, using the semigroup property with t = E0° ti, ti = tsi, write 00

Jj

IIHts''IIPi'P4+1

i=1 00


0,

II

Ht II2_.

2 where 1 /c = ET i-5, p1 = p2 = 2,

pti=2(i-1)2 fori> 2, andtake e=cob with Co = (c'/C) =

E2

i-5

00 2-3' F12

1 C

23

E2

as suggested above. Then E sipi+1 v, t > 1), this bound is hard to use. This can be seen from the following elementary lemma.

Lemma 4.2.7 Assume that M has volume growth V (x, t) with

VxEM,Vt>1, c05 t-aV(x,t) 0, R > 1,

cta/2 exp(-CR2/t) < f

exp(-d(x, y)2/t)dy < Cta/2 exp(---cR2/t).

(x,y)>R

Write 00

e-dv)2/tdy

I (t R) = f4(x,y)>R e-d(xv)2/tdy = fR2kk(t)

This proves the upper bound. For the lower bound, it suffices to restrict the integral to a ball around a point z at distance 2R + f from x and of radius R + -1/t-.

Lemma 4.2.7 shows that, on the one hand, the integral of the Gaussian upper bound of Theorem 4.2.6 over the complement of the ball of radius R is not uniformly bounded as t --> oo when V (x, t) ^_- to with a > v. This should be compared with the fact that fm h(t, x, y)dy = 1! On the other hand, if V (x, t) t°, then we have

J

h(t, x, y)dy < Ce-cR21t

(4.2.5)

(x,y)>R

for all t > 0, R > 1, which is an informative result. Our aim in this section is to prove the following theorem.

Theorem 4.2.8 Fix v > 0. Assume that M satisfies the Nash inequality V f E C0 (M),

Ilf

II2(1+2/v)
0, co < r-"V (x, r) < Co. Then the heat kernel h(t, x, y) is bounded above and below on the diagonal by

ct-"12 < h(t, x, x)

Ct-" l2.

The proof uses the fact that, under the above hypotheses, fm h(t, x, y)dy = 1. This is not obvious and requires a proof, which can be found in Section 5.5.2. Assuming that indeed fm h(t, x, y)dy = 1, (4.2.5) implies

J

h(t, x, y)dy > 1/2 (x,A / )

for all t > 1 and A large enough. By Jensen's inequality, this yields

h(2t, x, x) = Jh(t,x,y)2dy

> J (x,AiJ) h(t, x, y)2dy > V(x,

As)-1

2V (x, Ate)

r

JB{x,Af .

h(t, x, y)dy

CHAPTER 4. TWO APPLICATIONS

102

The theorem follows. We now give a quite different proof of a weaker lower bound, namely

sup h(t, x, x) > ct-"I2. XEM

This lower bound is taken from [18]. It is weaker because only the supremum of h(t, x, x) is bounded from below. On the other hand it requires no assumption except the volume estimate

`dxEM,Vr>0, co cV(ol

v t-)

under the sole hypothesis that

V(o,2f) V(o, f) is bounded above by a constant independent of t > 0.

4.3 4.3.1

The Rozenblum-Lieb-Cwikel inequality The Schrodinger operator A - V

Let M be a complete non-compact Riemannian manifold with LaplaceBeltrami operator

of = -div(Vf). Recall that our convention is that 0 is a positive operator in the L2 sense, that is, (A f, f) > 0 for all f E Co (M). In other words, the spectrum of 0 is contained in the non-negative semi-axis [0, oo). We will denote by dx the Riemannian measure on M and often write

IM

f (x)dx = JM

f.

Consider now the Schrodinger operator

L=0-V where V is a non-negative function. Then L may well have some negative spectrum. However, if V is a nice bounded function vanishing at infinity, one may hope that the essential spectra of L and z coincide. In this case, the negative spectrum of L is a discrete set with, possibly, an accumulation

point at 0 (if 0 is indeed the bottom of the spectrum of O). A natural question is: what condition on V will imply a bound on the number of negative eigenvalues of L? The following result (for M = R', n > 3) was first proved by Rozenblum and is known as the Rozenblum-Lieb-Cwikel inequality.

CHAPTER 4. TWO APPLICATIONS

104

Theorem 4.3.1 Let M be a Riemannian manifold satisfying an (L2, v)Sobolev inequality for some v > 2. Let L = A - V with V E L%(M) and V+ E L'12. Let Nv(A) be the number of eigenvalues of L less than A, counting multiplicity. Then NV(0) < C(v)

JM

V+/2.

(4.3.1)

This bound does not hold, in general, for the number of eigenvalues less than or equal to 0. We will follow a proof due to Li and Yau [55] which is also presented in a more general setting in [52]. See also [56] and the references given in [52].

Before embarking on the proof, let us observe that, if we assume that inequality (4.3.1) holds for all potentials V in Co (M), then L is a nonnegative operator for all V E Co (M) with IIVIIV/2 < C(V)-'-

That is, 0 2 and set

q = 2v/(v - 2). Thus 2/q

V f E Co (M),

\ Cim

IfIq}

Cf

IVfI2.

(4.3.2)

M

Then II Vf 11 2 is a norm on Co (M), and we can take the completion of Co (M)

for this norm. We obtain a Hilbert space H. According to (4.3.2), the map J : Co (M) -> L9 extends as a continuous map from H to L. We want to show that J is one to one, that is, that H can be viewed as a closed subspace of L9 with norm II Vf 112. That this is the case does not immediately follow from (4.3.2). Let F = (f,,,) be a Cauchy sequence (i.e., an element of H) for 9 - IIV91I2 on Co (M). Note that (Vfn) converges in L2 to a certain

vector field X. By (4.3.2), there exists f E LQ such that fn -+ f in Lq. Let f be any bounded domain. As LQ(fl) C L2(1l) for q > 2 and since the restriction of g --+ IIVgII2 to any bounded domain defines a closed form, the restrictions of the fn's to 1 converge to the restriction of f to Sl in the norm g -' II V9I I2,n +119112,n- If f = 0, we must have X = V f = 0 in any bounded

domain Q. Thus lim IIIVfnII2 = 0, that is F = (fn) = 0 in H. Fix V E L"/2. Then, by Holder's inequality with conjugate exponents

and the (L2, v)-Sobolev inequality,

if

If12VI

IIf2II.,/(Y-2)IIVIIY/2

=

II f II2v/(v-2) II V II Z//2

< CIIVIIM/2IIVf II2

(4.3.3)

It follows that the self-adjoint operator Tv associated to the quadratic form fm V If I2 on H is a bounded operator on H. The action of Tv on CO '(M) C H can easily be identified if we assume that V is smooth. Indeed, we then have

f (Vf)9 =

J(A1Vf)g J(V(L1Vf),Vg)

for all f, g E Co (M). That is, Tv = 0-1V.

(4.3.4)

CHAPTER 4. TWO APPLICATIONS

106

Theorem 4.3.2 Assume that M satisfies the Sobolev inequality (4.3.2) with

v > 2. Fix V E L12. Then TV : H - H is a compact operator and NT,, (A) = #{k : Ak > A}, the number of eigenvalues of Tv larger than A (counting multiplicity), is bounded by

(eC/Ay/2M f

NTV

AM

The first step in the proof of this result is to reduce it to the case where V E L'nL°°, V is smooth and V > 0. We will not do this in detail. However, this reduction is quite easy once one interprets the desired inequality as a boundedness inequality for the linear map V --, Tv between L"12 and the appropriate Banach space of compact operators. Thus, from now on, we assume that V E Ll n L°°, V is smooth and V > 0. Let p denote the measure

dp(x) = V(x)dx

on M. Consider the space L2(M, A). By (4.3.3), this space contains H. Consider the quadratic form II V f 112 on L2(M, dµ), with domain H. By (4.3.3) and the definition of H, this form is positive definite and closed. Thus, it is a Dirichlet form on L2(M, dµ). Actually, it is easy to compute the associated infinitesimal generator -K. Indeed,

f (Vf, Vg) = f(if)g = f(Vf)gdp. Hence K = V -10 f . By (4.3.4), this means that

K-1f = Tvf

(4.3.5)

on Co (M) (which is dense in the domain H of Tv!). As V is bounded and M satisfies (4.3.2), we have /q

(JlflQd)2

C IlVjI00

f IVfJ2

where q = 2v/(v - 2). That is, the Dirichlet form Q(f, f) = f IVf 12 on L2(M, dµ) satisfies a Sobolev inequality. As Sobolev inequality implies

Nash inequality, we can apply Theorem 4.1.1 to Q. This shows that the semigroup Kt = e-tK is ultracontractive. In particular, for each t > 0, Kt has a bounded kernel k(t, x, y) w.r.t. dp. As fm dp is finite, it follows that the function k defined by

k(t) =

f

k(t, x, y)2dp(x)dp(y)

4.3. THE ROZENBLUM-LIEB-CWIKEL INEQUALITY

107

is finite. That is, Kt is a Hilbert-Schmidt operator. Therefore, the spectrum of the self-adjoint operator K is discrete. Let A , i = 0, 1, 2, ... be the eigenvalues of K repeated according to multiplicity and in non-decreasing order. Let ui be the corresponding real eigenfunctions normalized in L2(M, dte). The kernel k(t, x, y) can be expressed in these terms as

k(t, x, y) _ > e-°tui(x)ui(y) i

Hence

k(t) = E e-2Ait i By (4.3.5),

NT,(A) = #{i: A, < 1-1} and thus

NT,(A)
0

and all f E L2(M, dµ) and that kk E L2(M, dµ), t > 0. The latter is a consequence of the ultracontractivity of Kt since k(2t, x, x) = IIkt IIL2(M,dµ)

The former is a general fact about analytic semigroups of operators (any self-adjoint (CO) semigroup of contractions on L2(M, dµ) is analytic). See Section 4.1.1 and (4.1.2), (4.1.4). Thus

f

k(t, x, y)2V (y)dy 0, f

If -.fBI2dc cr"

for all r > 0. This rules out many interesting manifolds for which one would expect to be able to produce a global analysis of, say, the heat kernel.

For instance, the Euclidean cylinders R1 x T', n, m < oo, cannot carry a Sobolev inequality of the type above, for any v. It turns out that there is a simple way to deal with this difficulty. It consists in working with localized Sobolev inequalities.

Fix 1 < p < oo. Very generally, we say that a Riemannian manifold M satisfies a (family of) localized LP Sobolev inequality(ies) with constants C(B) and exponent v > p if, for any geodesic ball B and all f E Co (B), 1P/q

(

C

J(fvfr+ r(B)-PIfIP)dµ

Ifl°dµ

(5.2.1)

where q = pvl (v - p), r(B) is the radius of B and p(B) is its volume. It is not hard to see that any complete non-compact manifold M satisfies such a family of Sobolev inequalities. The essential information is then concentrated in the behavior of the constant C(B). The precise value of the parameter v > p plays only a minor role. In fact, the inequality above can be written 1

(,

.

B) f I f Iqdp J

P/9

:5

C(B)r(B)P

(-L f(IVIIP +

f IJ)dp

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

114

which shows that one can always freely increase the value of v, that is, decrease the value of q > p. It will turn out that increasing v will have little impact on the conclusions that we will draw from these inequalities. One of the most interesting cases is when M satisfies a family of localized Sobolev inequalities with SUPB C(B) = C < oo. This condition implies a different kind of control of the volume growth function. Indeed, the argument of Theorem 3.1.5 easily yields the following two results.

Theorem 5.2.1 Let B be a fixed ball of radius r(B) in M. Assume that the Sobolev inequality

r CJ

If

(Y-P)/v

PiI(Y-P)

< C(B)

r(B)P J(IVfIP + r(B)- I f I )dµ

holds in this ball for some v > p and all f E Co (B). Then there exists a constant C1 such that

µ(B)

< G,1

(r(B) )

j.i(B') -

r(B')

for all balls B'CB. Theorem 5.2.2 Assume that M satisfies (5.2.1) with SUPB C(B) = C < 00 and some v > p. Then there exists a constant C1 such that

V(x,T) V (X' t) -

C1

(T)v t

for all x E M and 0 < t < T < oo. In particular, M satisfies the doubling condition

tlxEM, Vt>0, V(x,2t) 0,

Ilf

-ffIIP:5 CoslJVfll,

where f3(x) = V(x, s)-' fB(.8) fdA. Assume also that M satisfies the doubling volume growth condition

dx E M, Vt > 0, V(x, 2t) < DoV(x, t). Then there exists a real v > p such that (5.2.1) holds true with SUPB C(B) _

C < oo, that is, for any ball B, d f E Co (B),

/' J

(v-P)lY IflP-1("-P)d/Al

1

< C r(B)P

/'

µ(B)P/v J

VflPdp.

115

5.2. LOCAL SOBOLEV INEQUALITIES The proof starts with the following two easy lemmas.

Lemma 5.2.4 If M satisfies the doubling volume growth condition

`dx E M, `dt > 0, V(x,2t) < DoV(x,t) then

V(x, T)

(TV0

< Do

t V(y,t) with vo=log2(Do) for all 0 < t < T < oo and alix EM, yEB(x,T).

Let k be such that 2k < T/t < 2k+1. As B(x, T) C B(y, 2T) C B(y, 2k+2t) and thus V (x, T) < V (y, 2k+2t), we then have V(x,T) < Do+2V(y,t) S D02(Tlt)'0V(y,t) Lemma 5.2.5 Under the hypotheses of Theorem 5.2.3, there exists a constant Cl such that, for any ball B C M, V f E Co (B),

Il f IlP (1 + E)p(B).

Indeed, let z be a point at distance 2r(B) from the center of B. Such a z exists because M is complete and not compact. Then z E B(y, 3r(B)) and B(z, r(B)) C B(y, 3r(B)) whereas B(z, r(B)) n B = 0. By Lemma 5.2.4, it follows that V (y, 4r(B)) > µ(B) + V (z, r(B)) > (1 + E) a(B). This shows that 1

V (y, s ) JB(y,s)

(I + 1

(1 +,E)

f(z)dz

JB f(z)dz

1 fBIf µ(B)

(z)lPdz1

llfll

/)1/P = (1 + E)A(B)1/P

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

116

Now, write If (y)

(JB

-

1 1/p fa(y)Ipdy) >

j

If(y)Ipdy)1/p-

\lBIfa(y)lPdy)1/p

(1±e)Ilfllp

Ilfllp

_ cllfllp where c depends only on e. This and (5.2.2) show that Ilf llp < Clr(B)II

Vfllp

as desired. With these two results at hand, the proof of Theorem 5.2.3 is very similar

to that of Theorem 3.3.1. More precisely, fix a ball B C M. Observe that,

by Lemma 5.2.4, foryeB,0<svo, v

f8(y)1

V(1 3)

J

If(z)Idz
0andanyt>0,write p(f > A) < it({I f - ftl > A/2} n B) + µ({ ft > A/2} n B) and consider two cases. Case 1: If A is such that p(B2

A>

) II!II

then pick t < r(B) so that D r(B)' lI f Ill = a/4.

,

For this t,

µ({ ft > A/2} n B) = 0 and

u(f > A) < p(If - ftl > A/2) < (2/A)pllf - ftllp < (2CotllVf112/A)p p 1Q2p(B)-'1'r(B)IIf

lli/vllofllea-l-

That is Ap(1+1/v)µ(f >- A) 0 such that µ(B(y, r/4)) > cp(B'). It follows that p(B) > µ(B')+µ(B(y,r/4)) > (1+c)p(B'). The desired result easily follows by a dyadic iteration.

5.2.2

Mean value inequalities for subsolutions

The aim of this section is to show how localized Sobolev inequalities imply certain LP mean value inequalities, 0 < p < oo, for subsolutions of the heat equation (at + O)u = 0. Fix a parameter r > 0. Consider x E M, r > 0, s E R. Consider also a parameter b, 0 < b < 1 and set

Q(r, x, s, r) = Q = (s - rr2, s) x B(x, r) Q6 = (s - &rr2, s) x B(x, 6r). In what follows, we denote by dii the natural product measure on JR x M: dµ = dt x dµ.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

120

Theorem 5.2.9 Fix r > 0. Fix the ball B C M of radius r = r(B) > 0 and center x. Assume that the local Sobolev inequality

\J

If

< C(B)

I2v/(V~2)

dp/

/Y

J(IVf

12

+ r(B)

IfI

)d,i (5.2.7)

is satisfied for some v > 2 and all f E Co (B). Fix 0 < p < oo. Then there exists a constant A(r, p, v) such that, for any real s, any 0 < 5 < 6' < 1, and any smooth positive function u satisfying (8t +0)u < 0 in Q = Q(r, x, s, r), we have A(7-,

sup{up} (b

Q6

p, )C(B)'/2

b)

r(B) µ(B) JQa,

updµ.

(5.2.8)

Before embarking on the proof, let us note that the hypothesis that u is smooth can be relaxed. For the proof given below, one only needs u to be locally bounded and in Lz together with its first derivatives in time and space. In this case, the hypothesis (8t + A)u < 0 must be interpreted as meaning (5.2.9)

for all 0 > 0 in L2(Q) with Iv4)I E L2(Q) and such that x H )(t, x) has compact support in B for all t > 0. By a purely local argument (in R'a) any u which is locally in Lz together with its first order space and time derivatives and satisfies (5.2.9) is indeed locally bounded. In fact, the local boundedness of u can be proved by an argument similar to the one developed below with a technical variation as for Theorem 2.2.2.

Another useful remark in applying the result above is that, for any solution u of (8t + 0)u = 0 (possibly in the weak sense), the function v = I ul is a non-negative subsolution, i.e., satisfies (5.2.9). To prove this without having ever to compute Au, i.e., working with weak solutions, the best way is to show that vE = e -+u2 satisfies (5.2.9) and then let a tend to 0. This is useful, for instance, in deriving bounds on I etul when u is a solution of (8t + 0)u = 0. See Section 5.2.4 below. We now proceed with the proof of Theorem 5.2.9. For simplicity, we

assume for the proof that r = b' = 1. We first prove the case p = 2. The case p > 2 immediately follows since, for any smooth positive solution

u of (8t + A)u = 0, v = up, p > 1 is also a smooth positive solution of (8t + 0)v < 0. Indeed, Aup = pup-'Au - p(p

- 1)up-2IVulz.

The case 0 < p < 2 requires an additional argument as indicated below. For any non-negative function 0 E Co (B), we have

f

[4)8tu + V0 Vu] dµ < 0

(5.2.10)

121

5.2. LOCAL SOBOLEV INEQUALITIES

which is just the integrated form of (at+O)u < 0. For q5 _ '2u, V' E C0(B), we obtain

f I'o2uatu + 9l2IVul2]dµ < 2 if u *Vu - °odµl

< 2f IVV)12u2dµ + 2

f &21 Vul2dµ.

After some a algebra, this yields

[2 2uatu + IV(u)I2] dµ 1, the function v = up is also a smooth positive solution of (8t + A)v < 0. Therefore, (5.2.13) yields

o,\

Jf u

e

E(B) (A(rw)-2 Jf

u21)dA)

(5.2.14)

.

o

S et wi = (1 - b)2-i so that Ei° wi = 1 - b. Set also ao = 1, Qi+1 = Qi - wi = 1 - Ei wj. Applying (5.2.14) with p = pi = 9i, or = Qi, a, _ ori+1, we obtain

I

u2e'1dµ < E(B)

Ai+1[(1

-

'+i

b)rj-2

e

f(

2efdJ

JQoi

Hence, e-1-i u20'+1

dµ


2. The case 0 < p < 2 follows from the case p = 2 by the argument used in the proof of Theorem 2.2.3 in Section 2.2.1. The only modification is that, in the present parabolic case, one must work with the cylinders Q, instead of the balls aB. This ends the proof of Theorem 5.2.9.

5.2.3

Localized heat kernel upper bounds

Consider the heat diffusion semigroup Ht = e-t°, t > 0, on M and its smooth positive heat kernel h(t, x, y). We now show how Theorem 5.2.9 together with Lemma 4.2.1 yields certain Gaussian heat kernel upper bounds.

123

5.2. LOCAL SOBOLEV INEQUALITIES

Theorem 5.2.10 Let M be a complete non-compact manifold. There exists a constant A such that, for any c E (0, 1) and any two balls 131 = B(x, rl), B2 = B(y, r2) satisfying (5.2.7) for some v > 2 with constant C1 = C(B1) in B1 (resp. C2 = C(B2) in B2), we have ex r d(x, y)2 + E h(t,, x, y) < 4t - [p(B1)µ(B2)J1/2 p

d(x,

y)

VI-t

for all t > e-2 max{r2, r2}. Let (Ht'4')t>o be defined by Ht'

f(x) = e-am(x)Ht[e'If](x)

where 0 E Co (M) is a function satisfying Ioq51 < 1 and a is a real parameter. See Section 4.2. By Lemma 4.2.1, we have

t''II2

5

ea2t

Fix x, y E M and r1, r2 > 0, and let xl (resp. X2) be the function equal to 1 on B1 = B(x, r1) (resp. B2 = B(y, r2)) and equal to 0 otherwise. Then

f

h(t, ,

r;)e-acmce)-m(sn

d(

X B2

= (xl, Ht'4' 2) < ea2tµ(B1)1/21u(B2)1/2.

Using the fact that 1.01 < 1, we get

JfBx B2 h(t, , ()dkd( [

(Bl)µ(B2))1/2 exp(a2t + a(q5(x)

-.O(y))

+ IaI(rl + r2))

h(s, t, s) is a positive solution of (a8+0)u = 0 in (0, oo) x M, assuming that t > r2 and applying Theorem 5.2.9 with p = 1, we obtain As u : (s, t;)

h(t, e, y)
e-2 max{rl, r2}, we obtain h(t, x, y)
0. Assume that there exist v > 2 and Csuch that, for any ball B of radius less than R, the local Sobolev inequality v/(,-2'

\J If

12v/(v-2)dµ)

< Cµ((B)2/v J(JVf$2 + r(B)-2I f 12)dµ

is satisfied for all f E Co (B). Then there exists a constant A such that for all x, y E M and all 0 < t < R2, h(t, x, y)

/

A

[V (x, tax(,y))V (y,

e

ty

)]1/2

d(x, y)2 4t \\

exp ` -

)

This follows from applying Theorem 5.2.10 with B1 = B(x,r1), B2 = B(y, r2), r1 = r2 = e%fit, e = (1 + d(x, y)/f)-1. Using Theorem 5.2.1, we can deduce from the bound above a slightly less precise but nicer looking estimate, namely, for all x, y E M, 0 < t < R2, h(t, x, y) 0 such that for all x E M and all 0 < t < eR2, h(t, x, x) >

c

V (x' s)

In particular, if there exist v > 2 and Csuch that (5.2.1) holds true for p = 2 with SUPB C(B) = C < oo, then the lower bound h(t, x, x) >

holds true for all x E M and t > 0.

c

V (x, Vit)

127

5.2. LOCAL SOBOLEV INEQUALITIES

We refer the interested reader to [18] for a thorough discussion of on-diagonal

heat kernel lower bounds. Here we simply note that the hypotheses of the above theorem are not sufficient to imply a Gaussian lower bound of the type h(t, x, y) ? V(x,

f)

exp

(Cd(xY)2)

.

(5.2.18)

A counterexample is obtained by gluing two copies of R3 through a small compact cylinder. Let o be a fixed point near this compact cylinder. It is not hard to see that the manifold M obtained in this way satisfies the usual 3-dimensional Sobolev inequality and has volume growth V (x, r) r3 for all r > 0. Thus it satisfies the hypotheses of Corollary 5.2.12 and Theorem 5.2.14. It can also be shown that the Gaussian lower bound (5.2.18) fails in this case when t, x and y are such that d(o, x) . d(o, y) f - oo with x, y each in one of the two different copies of R . In this case, h(t, x, y) is actually of order t-2 instead of t-3/2. See [35]. Heuristically, a Brownian particle trying to go from x to y is much less likely to succeed in M than, say, in R3. This is because, in M, the particle has to go through a fixed compact neighborhood of the central point o in order to pass from one copy of R3 to the other.

5.2.4

Time-derivative upper bounds

The results obtained so far easily yield some time-derivative estimates for positive solutions of the heat equation. This can be seen as follows. Let u be a non-negative subsolution of (Ot + O)u = 0 in a cylinder Q = (t r2, s) x B(x, r). Set B = B(x, r). From (5.2.11), one easily extracts LB IOtul2dµ < A(1 - S)-2r-2

J

lul2dµ.

If u is a positive solution of (at+/)u = 0 in Q, then o9tku is a solution of the same equation in Q and v = is a non-negative subsolution. Moreover, vlatvl = vlo+lul. Indeed, Iatvl and Iae+'ul differ only when eY°u = 0. Thus we can again apply (5.2.11) to obtain Iai +lul2dµ

< A(1- b)-2r-2

aB

lae ul2dµ

JB

It follows that, as long as (1 - 6)k = 1 - or with 0 < a < 1,

LB Iaul2dµ < Ak2k(1 -

u)_2kr-2k

/ lul2dµ. B

(Note that in the argument above one works with subsolutions that are not smooth; see the remark after Theorem 5.2.9.)

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

128

Assuming that (5.2.7) is satisfied and using Theorem 5.2.9 for the subsolution ti'ul, we conclude that sup{ IOO ul2} < A(o,, v,

k)C(B)"/2r_2k-1µ(B)-1

JQ

4o

l ul2dµ.

Here, Q = (t - r2, s) x B(x, r) and Qo = (t - are, s) x B(x, or). Let us apply this to the heat kernel h(t, x, y) in the case where we have a full scale of localized Sobolev inequalities as in Corollary 5.2.12. Using the argument above and the heat kernel bound given by Corollary 5.2.12, we obtain the following result.

Theorem 5.2.15 Let M be a complete non-compact manifold. Assume that there exist v > 2 and C such that (5.2.1) holds true for p = 2 with supB C(B) = C < oo. Then, f o r each k = 0,1, 2, ... , and e > 0, there exists a constant A = A(k, e) such that for all x, y E M and all 0 < t < oo, 1h(t, x, y)

tkV (x

f)

exp

(- 4t(1

2 ,+c))'

If the hypothesis holds only for balls of radius r < R then the conclusion above holds for all x, y E M and all 0 < t < R2. This method is by no means the only way to derive upper bounds for the time-derivatives of the heat kernel. See [22] for a powerful and widely applicable technique.

5.2.5

Mean value inequalities for supersolutions

The main tool used in the preceding section is the mean value inequality for subsolutions stated in Theorem 5.2.9. Supersolutions satisfy similar but different inequalities that are presented below. In the statement below, we assume for simplicity that u is smooth. This hypothesis (which is very unnatural for supersolutions) can be relaxed to the requirement that u is locally in L2 together with its space and time first derivatives without essential change in the proof (u is then a supersolution in an obvious weak sense and one has to perform integration in the time variable sooner in the argument than we will do below, but this poses no difficulty). Let us note here that, unlike subsolutions, supersolutions need not be locally bounded. Fix a parameter r > 0. Consider x E M, r, s > 0 and a small positive parameter 0 < S < 1 and recall the notation

Q(-r, x, s, r) = Q = (s - rr2, s) x B(x, r), Qb = (s - S7-r2, s) x B(x, Sr), dµ = dt x dµ.

129

5.2. LOCAL SOBOLEV INEQUALITIES

Theorem 5.2.16 Fix r > 0. Fix the ball B C M of radius r = r(B) > 0 and center x. Assume that the local Sobolev inequality (5.2.7) is satisfied for some v > 2. Then there exists a constant A(r, v) such that, for any real s, any 0 < S < S' < 1, any 0 < p < oo, and any smooth positive function u satisfying (8t + 0)u > 0 in Q = Q(r, x, s, r), we have A(T, v)C(B)"i2

sup{u-P} < (5' - 6)21,,r(B)2p(B) Q6

u-Pdµ.

(5.2.19)

JQ,,

For simplicity, we assume for the proof that r = S' = 1. For any nonnegative function 0 E Co (Bwe have

j For

[OOru + V¢ Vu] dµ > 0.

(5.2.20)

= p^-P-1, Vi E Co (B), to = u'P/2 this yields

- f [,128tw2 +4p+p 1,,2IVwI2 + 4wV)Vtp Vw]dµ > 0. As 1 < (p + 1)/p < oo, elementary algebra and the inequality labl < (1/2)(a2 + b2) yield

f

[

f

2atw2 + IV( w)I2) dµ

w2dµ

(5.2.21)

uPP(')

B

where A is a numerical constant which will change from line to line. For any 0 < a' < o, < 1, the argument used to obtain (5.2.13) applies here and yields e

(A(ni)y2

w2Bdµ 0 and center x. Assume that the local Sobolev inequality (5.2.7) is satisfied for some v > 2

and set 0 = 1 + 2/v. Fix r > 0. Fix also 0 < po < 0. Then there exists a constant A(po, T, v) such that, for any real s, any 0 < 6 < 6' < 1, any 0 < p < po/9, and any smooth positive function u satisfying (8t + O)u > 0 in Q = Q(T, x, s, r), we have P/PO

I

uPO dµ 6

-

A(po, T, v)C(B)1+" f 1(S' 5)4+2vr(B)2µ(B) J

1-p/Po

-

f

updµ.

(5.2.24)

6 1

We give the proof assuming 6' = r = 1. In (5.2.20), we set 0 = alpl to+1 with E Co (B) and 0 < a < po(1 + 2/v)-1. We also set w = ut/2. This yields

J

[ b28tw2 +

4a

a 11p2IVwI2

a

+ 4wipVii Vw]dtt > 0.

Note that a - 1 is negative and that

la-ll

> 1-po/9=E>0.

This easily yields

- JB [028tw2 + IV(Y'w),2]

d/L:5 AeIIV 'II 00

J uPP(G)

w2dµ.

This should be compared with (5.2.21). The difference with (5.2.21) is the minus sign appearing in front of the first integral above. This difference explains why we have to work with the cylinders Q''5 (which are chopped at the top) instead of Qa (which are chopped at the bottom). Apart from this, the same argument used to obtain (5.2.13) applies here again and yields uaedµ < E(B)

(A(r)_2

rr J JQ,

\e

u" dµ.1

(5.2.25)

for all 0 < a < po(1 + 2/v)-1, 0 < a' < o < 1, with w = Q - a' and E(B) = C(B)µ(B)-2/"r(B)2. Compare with (5.2.22). To finish the proof, it now suffices to repeat the iteration argument appearing after (2.2.8).

5.3

Poincare inequalities

In Section 5.2, applications of local Sobolev inequalities, including heat kernel upper bounds, were developed. We observed there that Gaussian heat kernel lower bounds cannot be obtained from such Sobolev inequalities

131

5.3. POINCARE INEQUALITIES

alone. The crucial missing tool for obtaining Gaussian heat kernel lower bounds is Poincare inequality. We say that a complete manifold M satisfies a (scale-invariant) Poincare inequality if there exist two constants Po and i > 1 such that, for any ball

BCMofradius r(B)>0, V f E C°O(B),

JB

If _ fBl2di < Por(B)2 f I Vf 12dµ KB

where f B is the mean of f over B. It is useful to generalize this definition and introduce two extra parameters 1 < p < oo and R > 0. We say that a complete manifold M satisfies a scale-invariant LP Poincare inequality up to radius R if there exist two constants P0 and k > 1 such that, for any ball B C M of radius 0 < r(B) < R,

jii - fB lpdp < Por(B)v f IOf IPdp

b' f E C°° (B),

KB

where fB is the mean of f over B. One crucial aspect of these inequalities is that they are assumed to hold for all smooth f E C°°(B) instead of merely f E C000(B).

5.3.1

Poincare inequality and Sobolev inequality

The main result of this section is Theorem 5.3.3, which shows that Poincare inequality and the doubling property of the measure imply a family of local Sobolev inequalities. This is one of the key technical points needed to apply Moser's iterative technique under the assumption that Poincare inequality and doubling are satisfied. We start with an easy lemma.

Lemma 5.3.1 Fix I _< p < oo and 0 < R < oo. Assume that there exist two constants P0 and rc > 1 such that, for any ball B C M of radius

0 < r(B) < R,

d f E COO(M), f If - fBI dµ < Por(B)l B

J KB

lVfIPdµ

(5.3.1)

where fB is the mean off over B. Assume also that M satisfies the doubling condition V x E M, V O < t < R, V (x, 2t) < DoV (x, t). (5.3.2)

Then, for any 1 < r < r£ and any K > 1 there exists a constant C depending on Po, rc, Do, r and K such that V f E C°O(M),

JB

If - fBI'dlL < Cr(B)'°

for any ball B of radius less than KR.

f

TB

I V!Ipdp

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

132

We only give the outline of the proof and leave the details to the reader. Fix

x E M and 0 < r < KR and 0 < T < rs. By a well-known argument, one can cover the ball B = B(x, r) by a finite collection of balls of radius 0 = min{(T - 1)r/(100rc), R} with center in B and such that, for any two balls A, A' in this collection, the balls A, A' are disjoint. Moreover, by (5.3.2) a a collection of balls is uniformly $ such and Lemma 5.2.7, the cardinal of

bounded, independently of B. From this and a chaining argument, the desired result follows. The chaining argument alluded to here is a simpler version of what is done below in (5.3.10), (5.3.11). Of course, one of the points of this argument is that the balls 100rcB are all contained in TB.

Lemma 5.3.2 Fix 1 < p < oo and 0 < R < oo. Assume that M satisfies a scale-invariant LP Poincard inequality up to radius R, i.e., assume that (5.3.1) is satisfied. Assume further that M satisfies the doubling condition (5.3.2). Then Ilf - ffIIPf

2Bi

(I f (x) - f4B, Ip + Ifs (x) - f4Bi I P) dµ. (5.3.3)

By the postulated Poincare inequality, we have I f (x) - f4Bi I pdji

I f (x) - f4B; Ipdti i

ClsP

Li

Ef8Bj lV flPdµ

L.

< C1NosPJM

Pdp. IVf I

(5.3.4)

133

5.3. POINCARE INEQUALITIES By the doubling condition and Poincare inequality, we also have

s

f

1B:

Ifs (x) - f 4B; I pdp
1, there exist v > p and C such that, for any ball B of radius less than KR, `d f E Co (B),

(f If lp°n"-P)diL

< C-(B})/v f IV

(5.3.6)

This follows from Lemma 5.3.2, Theorem 5.2.6 and Lemmas 5.2.7, 5.3.1.

5.3.2

Some weighted Poincare inequalities

The results contained in this section are important technical tools. We show that, if M has the doubling property (5.3.2) and satisfies the Poincare inequality (5.3.1), then one can always take n = 1 in (5.3.1). This fact is due to D. Jerison [46]. The idea is to use a Whitney-type covering of the

ball B. Jerison's proof uses a rather subtle analysis of the covering near the boundary. It was later observed by Guozhen Lu [57] that a simpler argument can be given based on ideas from earlier works of Bojarski [8] and

Chua [16] on Euclidean domains. This argument has been used by many authors in various settings, e.g., [25]. We also produce some useful weighted Poincare inequalities, for which we need the following notation. The weights we are interested in are rather simple. Their introduction appears to be crucial in the last part of Moser's iteration argument for parabolic equations. For R > 0, a E (0, 1), let M(R, a) be the set of all non-increasing functions (0, oo) -> [0, 1] such that:

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

134

inf {s > 0: ¢(s) = 0} = R `d0 < s < R, q5(s + (1/2)[(R - s) A (R/2)]) > aq(s). Two interesting examples of such functions ¢ are:

0(s) = 1 on [0, R] and 0 otherwise, which belongs to M(R, 1).

¢(s) = (1 - s/R)+ for some y > 0, which belongs to M(R, (1/4)").

Thus, functions in M(R, a) may vanish at s = R but, if they do, they do so in such a way that (R - s) (R - t) implies ¢(s) ti ¢(t). Given a function 0 E M(R, a) and x E M, we set 4z(y) _ q5(d(x, y))

Theorem 5.3.4 Fix 1 < p < oo, a E (0,1) and R E (0, oo]. Assume that M satisfies a scale-invariant L" Poincare inequality up to radius R, i.e., assume that (5.3.1) is satisfied. Assume further that M satisfies the doubling condition (5.3.2). Then there exists a constant Pa such that, for all x E M, all 0 < r < R, and all functions ¢ E M(r, a) with 0 < r < R, we have

`d f E C°°(M),

Jii - ftlpWµ < ParP / {V f INMp

where ft = f f 4bdµ/ f Wit and 4D(y) = 0(d(x, y)).

Corollary 5.3.5 Fix 1 < p < oo and R > 0. Assume that (5.3.1) and (5.3.2) are satisfied. Then there exists a constant P such that, for all x E M and all 0 < r < R we have

t/ f E C°°(M), f If - fBIPdµ < Pr' f s

IofJPdp

where fB is the mean of f over B.

The proof of these results will be given below. The main ingredient is a somewhat subtle covering argument that will be expained in detail. Here, let us observe that Lemma 5.3.1 shows that we can always decrease the constant is appearing in (5.3.1) to any specified value strictly larger than 1 at the expense of a larger but still finite Po. What is not obvious but is achieved in Theorem 5.3.4 and Corollary 5.3.5 is that one can in fact take K = 1. We will use Lemma 5.3.1 to simplify the proof of Theorem 5.3.4 by assuming that (5.3.1) holds with r. = 2.

135

5.3. POINCARE INEQUALITIES

5.3.3

Whitney-type coverings

All the balls considered below are open balls, i.e.,

B(x,r) = {z E M : d(x,z) < r}. We will use without further comment the fact that for any two points x, y there exists a distance-minimizing curve joining x to y in M. In this subsection we assume that the doubling condition (5.3.2) is satisfied for some fixed R > 0.

Let us fix a ball E = B(x, r), x E M, 0 < r < R. We claim that there exists a collection F of balls B having the following properties:

(1) The balls B E .F are disjoint.

(2) The balls 2B, BE F, form a covering of E, i.e., E = UBEF2B. (3) For any ball B E .F, r(B) = 10-3d(B, OE). In particular, 103B C E. (4) There exists a constant K depending only on the constant D0 in (5.3.2)

such that sup #{B E .F : 77 E 102B} < K. 'EE

(5.3.7)

We will call F a covering of E (although only the balls 2B, B E F actually cover E). To construct F, start with the collection 7 of all balls B with center in

E and radius r(B) = 10-3d(B, aE). Let us start by noting that for each z E E there exists a ball B E F with center z. Indeed, for any B with center z, the condition

d(B,OE) = 103r(B) is the same as d(z, OE) = (1 + 103)r(B).

Start F by picking a ball Bo in T with the largest possible radius. Such a ball exists by a simple compactness argument; see below. Then pick the next ball B1 in F to be a ball in T which does not intersect BO and has maximal radius. Assuming that k balls Bo, B1,... , Bk_1 have already been

picked, pick the next ball Bk to be a ball in F which does not intersect UU-1 Bz and has maximal radius.

Let us show that such a ball does exist. Let B3 = B(xj, rj), 0 < j < k-1. Let p be the least upper bound of the radii of balls in F that do not intersect

Wk-1 = Uo-1 Bi. Then, there exist two sequences zj E M, pj > 0 and a point z E E such that B(zj, pj) E Y, B(zj, pj) n Wk-1 = 0, z3 --> z, pj -+ p. Consider the ball Bk = B(z, p). By continuity, d(xk, z) > rk. Thus B(z, p) does not intersect Uo-1 B;. For each j, let yj be a point such that d(zj, yj) _ pj and d(B(zj, pj), aE) = d(yj, tE). By extracting a subsequence, we can

136

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

assume that yj -+ y. Then, by continuity d(z, y) = p and d(B(z, p), OE) _< d(y, OE) = 10-3p. To prove that d(B(z, p), OE) > 10-3p, observe that for

any e > 0, B(z, p) c B(z,, p; + E) for all j large enough. It follows that d(B(z, p), OE) > 10-3p; - E for all j large enough. Letting first j tend to infinity and then e tend to zero yields d(B(z, p), OE) > 10-3p. This procedure defines F = {B0, B1,. - -, B,,.. .... } inductively. By construction, properties (1) and (3) are satisfied. Let us show that property (2) is also satisfied. Fix Z E E. By continuity, there exists a p > 0 such that d(B(z, p), OE) = 10-3p. Let k be the largest integer such that the ball Bk E F has radius rk _> p. By construction, we must have B(z, p) fl (Ui p and this contradicts the definition of k. Now, B(z, p) fl (Ui R and

d(1/B, () + d((, OE) > d(B, OE).

Moreover, d(x, () + d((, rla) = d(x, r/B). Hence d(x, rlB) + 2d((, OE) > R + d(B, OE).

As d(x, rlB) < R, this yields 2d((, OE) = 2d('yB, OE) > d(B, OE)

5.3. POINCARE INEQUALITIES

137

which is the desired inequality. Now, for any ball B' E F such that 2B' intersects ryB,

d(B', OE) > r(B') + d(ryB, 8E).

Thus (103 - 1)r(B') > (1/2)103r(B), which implies r(B') > (1/4)r(B). This finishes the proof of Lemma 5.3.6.

Next we introduce an important notation. For any B E F, we now choose a string of balls in F, call it -7:7(B)= (Bo, B1, ... , Bt(B)_1),

joining By to B with Bo = Bx, BI(B)-1 = B and with the property that 2Bi n 2Bi+1 # 0. Let us show that such a string exists. Let to be the first point along rye (starting from x) which does not belong to 2Bo (recall that x E 2Bo). Define B1 to be one of the balls in F such that 2B1 contains 60. Having constructed Bo, B1, . . . , Bk, let Sk be the first point along ryB that does not belong to Uo 2Bi, and let Bk+1 be one of the balls in F such that 2Bk contains' k.

By (5.3.2), Lemma 5.3.6 and the fact that the balls in F(B) are disjoint, there are only finitely many balls B' in .'F that can intersect ryB. In particular, F(B) is finite. It may well be that the last chosen ball in the above construction is not B. In this case, we simply add B as the last ball in F(B). Let us emphasize that the collection F(B) is finite but that we have no precise information on its cardinality £(B). Lemma 5.3.7 For any B E F and any two consecutive balls Bi, Bi+1 in the string F(B),

(1+10-2) -'r(Bi) < r(Bi+1) S (1 + 10-2)r(Bi) and B;+1 C 4Bi. Moreover µ(4B; n 4Bi+1) >- c max{p(Bi), p(Bi+1)}.

As the balls 2Bi, 2Bi+1 intersect each other, one easily checks that Bi, Bi+1

have comparable radii and that Bi+1 C 4Bi (this follows from the fact that each of these balls has radius equal to a small multiple of its distance from the boundary of E). Moreover, if t; E 2Bi n 2Bi+1 and p = min{r(Bi), r(Bi+1)} then B(e, p) C 4Bi n 4Bi+1. Lemma 5.3.7 now follows from the doubling property (5.3.2) which shows that the balls Bi, Bi+1, B(C, p) have comparable volume.

Lemma 5.3.8 For any ball B E F and any ball A E F(B), B C 104A.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

138

Let 77B be the center of B and 77A be the center of A. As A E .F(B), 0. Thus, by Lemma 5.3.6, 2A n ryB

r(A) > (1/4)r(B). Now, let A be a point in 2A n ryB. We have d(iiA, r/B) = d(x, 7/B) - d(x, rl'A) < R - d(x, iA) and

R < d(x, 7)A) + d(77A, OE).

Thus, d(??A, 7/B) < d(7/A, aE) < (3 + 103)r(A) and

d(77A, 7)B) < 2r(A) + d(i7A', /B)

< (5 + 103)r(A). Thus B C (9 + 103)A.

Lemma 5.3.9 Under the hypotheses of Theorem 5.3.4 and assuming (as we may) that (5.3.1) holds with , = 2, there exists a constant C such that for any B E F and any consecutive balls Bi, Bi+1 in .F(B), 1/p

If4B, - f4B,+i

I

C- r(B7

P

(LB.

IVfIPdIL

For the proof, write 1/P

µ(4B2 n 4Bi+1)1/PIf4B, - f4B,+1I = (fB;n4B;t1 If4B: - f4B;+1IPdp 4

1/p


c4)(B)/µ(B) for all Bi E F(B). The desired inequality follows.

5.3.4 A maximal inequality and an application Let f E L' + L°°. The maximal function Mn f is defined by M'-f(x)

sup r(B) 0 and assume that the doubling condition (5.3.2) is satisfied. Then for all 1 < p < oo and K > 1, there exists C = C(p, K) such that for all 0 < r < KR, Vf E Co (M),

IIMrfIIP

s

CIIfIIP.

If (5.3.2) holds with R = oo, then one can take r = oo. This is obvious for p = oo. By the Marcinkiewicz interpolation theorem, it suffices to show that Mr is of weak L' type. Thus it suffices to show that there exists C such that for any f E CO '(M), f > 0, and A > 0 µ(EA) < CA-111f II i

where E,, {x:M,.f(x)>A}. Now, for any x E Ea there exists a ball Bx of radius less than r such that

j If Id4u ? p(Bx). z

140

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

Obviously, the balls B., cover Ea. By a well-known covering argument, one can extract from the family {B,, : x E EA} a sequence of balls (Bi) that are

disjoint and such that (5Bi) covers Ea. See, e.g., [79, page 9]. By (5.3.1) and Lemma 5.2.7, it follows that

0 and assume that the doubling condition (5.3.2) is satisfied. Fix K > I and 1 < p < oo. Then there exists a constant C = C(K, p) such that for any sequence (Bi)i° of balls of radius at most R, and any sequence of non-negative numbers (ai)r, aixxBi ilp

11

aif

MKRc5dµ

B:

C

i

J

E atXB5 (x)MKRc(x)dx a

CIIgIIpIIMKRqIIIq-

As 1 < p < oo implies 1 < q < oo, we can apply Theorem 5.3.11 which yields

f

.

fOdd < C'II9IIpIIOIIq

as desired.

5.3.5

End of the proof of Theorem 5.3.4

We keep the notation introduced in Section 5.3.3. We also assume that the hypotheses of Theorem 5.3.4 are satisfied. Thus, ¢ is a function in M(r, a) with 0 < r < R and -O(y) = 0(d(x, y)) for some fixed x E M. Moreover,

E = B(x, r) and F is a Whitney-type covering of E as in Section 5.3.3. Recall that .P contains a so-called central ball B,, with the property that x E 2B, Any ball B E F comes equipped with a finite string of balls of F which is denoted by F(B) = (Bo, , BeiBi-1) This string has a number of specific properties explained in Section 5.3.3. In particular, Bo = B, Be(B)_1 = B.

Recall that D(B) = fB 4Ddp. As E = UBE7 2B and

has support in E,

we have (5.3.9)

f if - f4B.Ip-tdµ
1u(A) C J AEf

r(A)P

P

1/P

(LA IOf I P'pdµ)

(1.2A

I V f I Pcdµ)

XA

XAdu

< C'r1 E J Io f IP4dµ < C" rJ I V f I AEf

2A

dp

dp.

(5.3.12)

143

5.4. HARNACK INEQUALITIES AND APPLICATIONS

For the last step, we have used (5.3.7) and the fact that 32A C E for all

AE.F.

To conclude the proof of Theorem 5.3.4, it now suffices to use (5.3.10), (5.3.11) and (5.3.12). Indeed, these inequalities yield Jii - f4Bx 11"Ddl.c < Crp J

f I P Ddµ.

dµ, we conclude that

As this implies I fV - f 4 Bx I P f 4dµ < Crp f I V f l

Jii - f4,I p4Ddµ < Crp J I

f I p4Ddµ

as desired.

5.4 5.4.1

Harnack inequalities and applications An inequality for log u

Recall that dµ denotes the natural product measure on R x M: dµ = dt x dµ.

Lemma 5.4.1 Fix 0 < R < oo. Fix T > 0 and 6,,q E (0,1). Assume that (5.3.1), (5.3.2) are satisfied. For any real s, any r with 0 < r < R, any ball B of radius r, and any positive function u such that (at + O)u > 0 in Q = (s - Tr2, s) x B, there is a constant c = c(u, r7) such that, for all A > 0,

µ ({(t, z) E K+ : log u < -A - c})
A -c})< Cr2µ(B)A-' where K+ = (s - rrrr2, s) x bB and K_ = (s - r2, s - rlrr2) x SB_ Here the constant C is independent of A > 0, u, s and the ball B of radius r E (0, R).

For the proof, we assume that r = 1. Note that 6 and 77 play somewhat different roles here. The parameter 6 is used to stay away from the boundary

of the ball B. The parameter 17 is used to define a fixed point s' = s - qr2 in the interval (s - r2, s), away from s - r2 and s. Let us first observe that (by changing 6) we can assume that u is a supersolution in (s - r2, s) x B' where B' is a concentric ball larger than

B = B(x, r). We set to = -log u. Then, for any non-negative function V) E Co (B'), we have

at

ftwd µ < J,2u1Lud,L = J [- 2 Vw 12 + 2 bV w Oi&] dµ.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

144

Using 2f abf < (2a2 + 2b2), we get 8t

Jw + 2 J

IVW120'2

< 2JIV,112 t,(supp(V)).

(5.4.1)

Here, we choose bi(z) = (1 - p(x, z)/r)+ where x is the center of B and r its radius (V) is not smooth, but it can easily be approximated by nonnegative functions in Co (B')). In the notation of Theorem 5.3.4, we have jp2(y) = D(y) with 0(t) _ (1 - t/r)2, t > 0. Thus, the weighted Poincare inequality of Theorem 5.3.4 with weight 4P., =

J

(w - W 12i,&2dp < Aor2

J

2 yields IV

12V)2dp

with

W = Jwb2d,1/fb2d,. Setting

V = A(B), this and (5.4.1) give

8tW+

(A1r2V)_1

LB

Jw - W 12dµ < A2r-2

for some constant A1, A2 > 0. Rewrite this inequality as atW + (A1r2V)-1

J

- W I2dp < 0

(5.4.2)

LB

where

w(t, z) = w(t, z)

- A2r-2(t - s')

W(t) = W(t) - A2r-2(t - s')

with s'=s-ire. Now, set c(u) = W(s'), and

f2t(a)={zE5B:w(t,z)>c+A} Sgt (A)={zEbB:w(t,z) s', w-(t, z) - W (t) > A + c

- W (t) > A

in S1z (A), because c = W(s') and 8tW < 0. Using this in (5.4.2), we obtain atW(t) + (A1r2V)-1 IA + c - W(t)J2 it(1 (A)) _ p(SZe (A))

5.4. HARNACK INEQUALITIES AND APPLICATIONS

145

Integrating from s' to s, we obtain µ ({(t, z) E K+ : w(t, z) > c + A}) < A1r2VA-1

and, returning to - log u = w = w + A2r-2 (t - s'),

µ ({(t, z) E K+ : log u(t, z) + A2r-2(t - s') < -A - c}) :5 Air2V,\-I. Finally,

µ ({(t, z) E K+ : log u(t, z) < -A - c})

< µ ({(t, z) E K+ : logu(t, z) + A2r-2(t - s') < -(A/2) - c}) +µ ({(t, z) E K+ : A2r-2(t - s') > A/2}) < A3r2V A-1. This proves the first inequality in Lemma 5.4.1. Working with 11 instead of Sti , we obtain the second inequality by a similar argument.

5.4.2

Harnack inequality for positive supersolutions

The following theorem states that positive supersolutions satisfy a weak form of Harnack inequality. For any fixed r > 0, 6 E (0,1) and x E M, s, r > 0 define QQ!

(s - (3 + 6)rr2/4,s - (3 - 5)Tr2/4) x 6B

Q+

(s - (1 + 6)rr2/4, s) x 6B.

(s - Tr2,s - (3 - 6)rr2/4) x bB

Recall also that Q = Q(r, x, s, r) = (s - rr2) x B(x, r).

Theorem 5.4.2 Fix r > 0, 0 < 6 < 1 and 0 < R < oo. Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2. Let C and v > 2 be such that (5.3.6) is satisfied with p = 2 (these exist by Theorem 5.3.3). Fix

Po E (0, 1 + 2/v). Then there exists a constant A such that, for x E M, s E R, 0 < r < R and any positive function u satisfying (8t + O)u > 0 in Q = (s - rr2, s) x B(x, r), we have 1/Po 1

µ(Q'_-)

JQ

uPodµ

< Ainf{u}. Q+

For simplicity, we assume for the proof that T = 1. Fix a non-negative supersolution u. Let c(u) be the constant given by Lemma 5.4.1 applied to u with 71 = 1/2. Set v = eu. Set also U = (s - r2,s - (1/2)r2) x B, U, = (s

-

r2,s - (3 - o)r2/4) x oB.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

146

By Theorem 5.2.17, we have 1/Po

1/P-1/PG

A(po, v)Cl+v

P

-

f

[(Q

(ju,

111/P

µj

Q')4+2vii(U)v d

for all0 0, 0 < b < 1 and 0 < R < oo. Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2. Then there exists a

constant A such that, for x E M, s E R, 0 < r < R and any positive function u satisfying (8t + A)u = 0 in Q = (s - rr2, s) x B(x, r), we have sup{u} < Ainf{u} Q+

Q_

where Q_ = (s - (3 + b)rr2/4,s - (3 - b)rr2/4) x bB

Q+ = (s - (1 + 6)rrr2/4, s) x SB.

This follows immediately from Theorems 5.4.2 and 5.2.9. Next we give a useful corollary of this inequality.

Corollary 5.4.4 Fix 0 < R < oo. Assume that (5.3.1), (5.3.2) are satisfied. There exists a constant A such that the following inequality holds.

Let y be a continuous curve of length d joining two points x, y E M. Let r be a p-neighborhood of y, p > 0. Let u be a non-negative solution of (at + A)u = 0 in (O,T) x rp, T > 0 and let 0 < s < t < T. Then log

d.

(5.4.5)

The values of r and k are to be chosen later. Let to = s, ti = s + r2i, 0 < i < k. Now choose r to satisfy the following conditions:

(i) r2 = (t - s)/k so that tk = t. Note that this implies t - s > r2. (ii) r < R, r2 < s and r < 2p so that u is a solution of (8t + A)u = 0 in each of the cylinders (ti r2, ti+1) x 2Bi, 0 < i < k - 1.

-

Then, applying Theorem 5.4.3 successively in (ti-r2, ti) x 2Bi, 0 (t - s) max{l/R2,1/s,1/2p}. Thus we can choose k of order

l+t-s+t-s+t-s+ d2 R2

s

t - s'

p2

This gives the desired inequality.

The next result improves the dependence on t/s in the inequality of Corollary 5.4.4 when u is a solution on (0, T) x M and (5.3.1), (5.3.2) are satisfied with R = oo and p = 2. Corollary 5.4.5 Assume that (5.3.1), (5.3.2) are satisfied with R = oo and

p = 2. There exists a constant A such that the following inequality holds. Let x, y E M. Let u be a non-negative solution of (8t + 0)u = 0 in a (0,T) x M, T > 0 and let 0 < s < t < T. Then 109

/

1 such that 2k+1s

and set

< t < 2k+25

ti=2's, 0 t/2 > (t - s)/2. Moreover, by (5.4.6), log

u(ti-1, x) < Ao, u(ti, x)

1 5 i < k,

because t2 < ti+1 = 2t2. Thus log

u(to, x)

< Aok.

u(tk, x) -

149

5.4. HARNACK INEQUALITIES AND APPLICATIONS

As to = s and k is of order log(t/s), it follows that 2

log u(t'

y)

< A C1 + dtx'

+ log S)

)

s

as desired. The next corollary deals with global solutions on (0, oo) x M and follows directly from the previous results.

Corollary 5.4.6 Fix 0 < R _< oo. Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2.

(1) If R = oo, there exist constants A and a > 0 such that, for any x, y E M, any 0 < s < t < oo and any non-negative solution of (8t + A)u = 0 in (0, oo) x M, we have d(x,y)2))

u(s, x) < u(t, y)p l A (1+ ()°ex

s u(s, x) < u(t, y) exp

5.4.4

(A (1 +

tR2s

+ dtx, y)'

.

Holder continuity

One of the important consequences of the Harnack inequality of Theorem 5.4.3 is that it provides a quantitative Holder continuity estimate for solutions of (8t + 0)u = 0.

Theorem 5.4.7 Fix 0 < R < oo. Fix T > 0 and 6 E (0,1). Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2. Then there exist

a E (0,1) and A > 0 such that any solution u of (8t + 0)u = 0 in Q = (s - Tr2, s) x B(x, r), x E M, 0 < r < R, satisfies I MY, t) - u(y', t')1 (Y,t),(

P)EQ6

[It - t'i1i2 + d(y, y')]°`

)

A

< ra

sQ {IuI}

As usual, we assume that T = 1. Let us start with a simple consequence of Theorem 5.4.3. Fix a > 0, p E (0, R), z E M. Set

W = (a - p2, a) x B(z, p), W_ = (a - p2, a - (3/4)p2) x B(z, (1/2)p), W+ = (a - (1/4)p2, a) x B(z, (1/2)p) Then for any non-negative solution v of (8t + O)v = 0 in W 1

µ(W)

I- vdµ < wax{v} < A in{v}. _

(5.4.7)

150

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

Now, given a solution u, not necessarily non-negative, let Mu, mu be the (essential) maximum and minimum of u in W. Similarly, let M, rn,+ be the maximum and minimum, of u in W+. Define also µu

µ( W_ )

vdd.

N'-

Applying (5.4.7) to the non-negative solutions Mu - u, u - mu, we obtain

Mu - pu< A(Mu-Mu) A,u - mu < A(mu - mu). It follows that

(Mu-mu) 1). Now, referring to Theorem 5.4.7, consider (y, t), (y', t') E Qa. We can assume without loss of generality that t > t'. Let p = 2 max{d(y, y'), t - t'}.

Then (y', t') belongs to Wo = (t-p2, t) x B(y, p). For i > 1, define pi = 2pi_1, po = p, and set Wi = (t - pi, t) x B(y, pi). Then, according to the notation adopted for (5.4.7), we have

(W1)+ = W. Thus, as long as Wi is contained in Q, (5.4.8) yields W(u, Wi-1)

9 W(u, Wi)

and

w(u, WO) < 9i W(u, Q).

Consider two cases. If p:5 (1 - b) r, let k the integer such that 2k < (1 - 6)r/p < 2k+1

5.4. HARNACK INEQUALITIES AND APPLICATIONS

151

Then, as (y, t) E Qa, it follows that

Wk = (t - 4kp2, t) x B(y, 2kp) C (t_(1_6)2 r2, t) x B(y, (1 - 5) r)

C (s - r2, s) x B(x, r) = Q. Thus

w(u, Wo) < 9k w(u, Q) < 0-1(1 - b)-°(p/r)' w(u, Q) with a =1092 9. In particular, Iu(y, t) - u(y', t')I [It

t1+1/2 + d(y, y')]°`


(1 - 6) r, then the last inequality obviously holds. This ends the proof of Theorem 5.4.7. Applying Theorem 5.4.7 to the heat kernel yields the following result.

Theorem 5.4.8 Assume that (5.3.1), (5.3.2) are satisfied with p = 2 and R = oo. Then there exist a E (0,1) and A > 0 such that h(t, x, y) - h(t', x, z)I

A

'1/2

(_ti'2

d(y, z)

a

h(2t, x, y)

for all x E M, t, t' >0 and z, y E M, d(z, y)2 h(t, x, x) exp (-A (1 + RZ + Moreover, there exists a > 0 such that, for all x, y E M and 0 < t < R2,

h(t, x, y) ?

a

V (x, )

exp

(_Ad;2).

Apply Corollary 5.4.6(2) to u(t, y) = h(t, x, y) with x fixed and s = t/2. This gives the first stated inequality because h(t, x, x) is non-increasing. The second inequality then follows from Theorem 5.4.10.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

154

5.4.7

Two-sided heat kernel bounds

It might be useful to gather in one place different heat kernel estimates that have been obtained so far in the important case where there exists a constant Do such that the volume growth function V (x, r) has the doubling property (5.4.9) d x E M, V r > 0, V (x, 2r) < D0V (x, r)

and there exists a constant PO such that the Poincare inequality

V x E M, V r> 0,

f

If _ fB12di- < P0r2 (x,r)

JB(x,2r)

IVf )2dp

(5.4.10)

is satisfied.

Theorem 5.4.12 Assume that (M, g) is a complete Riemannian manifold such that (5.4.9), (5.4.10) are satisfied. Then the heat kernel h(t, x, y) satisfies the two-sided Gaussian bound

cl exp (-Cld(x, y)2/t) < h(t, C2 exp (-c2d(x, y)2/t) x, y)
1, V x E M, V t E (0, R2),

h(t, x, x) < Akh(kt, x, x).

This, together with (5.5.2), (5.5.3), shows that the doubling property (ii) holds true. Let us prove now that (iii) implies (i). For this we need to introduce the Laplacian with Neumann boundary condition on any given metric ball B C M. However, metric balls do not necessarily have a smooth boundary so it is better to define this operator without explicit reference to a boundary

condition. We can proceed as follows. Fix a ball B C M. Consider the subspace D°° C C°°(B) of those smooth functions f such that Vg E C°°(B),

J

gL f dA =

B.

Vg V f dp. B

Observe that D°° contains Co (B) and thus is dense in L2(B). Observe also that the operator A with dense domain D°° is symmetric, i.e., d f, g E D°°,

JYLVd/.t =

ffIgd,t.

Also, Qn+(f, f) = fB fEfdµ is non-negative for all f E D°°. It follows that the quadratic form (Q^', p-) is closable and its minimal closure (QN, D)

159

5.5. THE PARABOLIC HARNACK PRINCIPLE

is associated with a self-adjoint extension of A which we denote by L. Fortunately, we will not need to understand better what the mysterious domains D°°, D are. If B has a smooth boundary, then

Vf,gEC°D(B),

jgtfdJ.z

=jv9 Vfd+

jgoi.fd/.Ln_l

s

w here v is the exterior normal along 8B. It then follows from the above

construction that any function f in D°° must satisfy 8 f = 0, that is, f satisfies the Neumann boundary condition. The closed form (QB, D) on L2(B) is a Dirichlet form and the associated semigroup HB'N = e-t°8 is a self-adjoint Markov semigroup on L2(B). Observe that constant functions are indeed in V so that clearly HB'N 1 = 1. For any function f E COO(B), u(t, x) = HB'N f (x) is a solution of the heat equation (8t + 0)u = 0 in (0, oo) x B. In particular, Ha'N admits a smooth kernel (t, x, y) H h'"(t, x, y), (t, x, y) E (0, oo) x B x B.

Note however that there is no known method, in general, to give a uniform bound (upper or lower) on this kernel for a fixed t and all (x, y) E B x B. This would require some analysis of the boundary of B. We say that h' is the Neumann heat kernel in the ball B.

Theorem 5.5.2 Fix 0 < R < oo and assume that condition (iii) of Theorem 5.5.1 holds true. Then the Neumann heat kernel in any ball B of radius

0 < r < R satisfies

V (y,')

< hB'N(t, y, z) -

V (yA.)

for all t E (0, r2), y, z E B(x, r/2) with z E B(y, vt). To prove Theorem 5.5.2, it is handy (although not strictly necessary) to use here the fact that (iii) implies (ii) (this has been proved above). Note first

that Vt E (0, r2),

by E B(x, r/2),

V(y,av)

< hB

N(t, y,

y)
aV-', y, z E (1/2)B

(5.5.4)

for any 0 < r < R and any ball B(x,r) = B with V = V (x, r) = u(B). With this lower bound at hand, for y E (1/2)B and f E COO(B), write HB'NLf

- HB'Nf (y)]2(y)

= JB hB (r2, y, z) I f (z)

-

HB'N(y)I2d11(z)

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

160

f

>

If

1/2)8

> aV-' {1/2)B

('Z) - HB'N(Y)I2dA(Z)

If -

f(1/2)B I2dµ

where f(1/2)B is the mean of f over the ball (1/2)B. The last inequality follows from the well-known fact that the mean of f over S2 realizes the minimum of c --' fn If - cl2dµ over all reals c, when the integral is over a bounded domain S (here f2 = (1/2)B). Integrating over the ball (1/2)B, we get

f

- HB'Nf (y)j2(y)dµ(y)

HB

>f

1/2)B

HB'Nlf

(5.5.5)

- HB'Nf (y)]2(y)dli(y) (5.5.6)

a 1112)B If - f(1/2)Bl2dpl.

But a simple computation shows that

f HB'NIf -

(y)]2(y)dl1(y) = II f II1,2 -

IIHBN fII2

(5.5.7)

and we have r2

IIfll ,2 - II HB'NfIIZ = - f asIIHB'NfII1ds 0

2

f

r2

(AN HB,N f, HB,N f )ds

0

1,2 2

Q N(HB'Nf, HB'N f )ds

< 2r2Q8 (f, f) =

2r2

Js

IV f I2 dµ.

(5.5.8)

To see the last inequality, observe that s QB (HB°N f, HB,N f) is a nonincreasing function. This can be proved by noting that QB (HB'Nf' HB'Nf)

_ ('B HB'Nf, HB'Nf) = II (OB )1/2HB'Nf II2

It follows from (5.5.6), (5.5.7) and (5.5.8) that

f

/2)B

If - f(1/2)BI2d/L < 2a-ire

f

B

1VfI2dµ

This proves that (iii) implies (i) (see Lemma 5.3.1). Let us mention the following result, which complements Theorem 5.5.1.

5.5. THE PARABOLIC HARNACK PRINCIPLE

161

Theorem 5.5.3 Fix 0 < R < oo. Let M be a complete manifold. The heat kernel h(t, x, y) satisfies the two-sided Gaussian inequality

)

cl exp (-d(x, y)2/Clt) V (X'

< h(t, x, y)

(-d(x, y)2/c2t) - C2 expV(x, f)

for all x, y E M and t E (0, R) if and only if M satisfies the conditions (i) and (ii) of Theorem 5.5.1 for the same R.

We only outline the proof in the case R = oo (the case 0 < R < oo is similar). First, one shows that the Gaussian lower bound implies doubling. Indeed, integrating over the ball of radius 2r with r = f yields cle-acl

V (x, 2r )

V (x, r) < 1

because IN h(t, x, y)dy < 1. With this observation, one can use the twosided Gaussian bound to obtain a lower bound on the Dirichlet heat kernel on any ball B. This lower bound is of the form inf hB (t, x, y) >

c

µ(B) for all 0 < t < ar(B)2, with e, a positive but small enough. It follows that the Neumann heat kernel hB (which is always larger than h°) satisfies the same lower bound and this yields the desired Poincare inequality. Details can be found in [77, Proposition 2]. x,yEEB

5.5.2

Stochastic completeness

Let us first explain the title of this section. Associated to the LaplaceBeltrami operator 0 on M is a stochastic process called Brownian motion with the property that the heat kernel h(t, x, y) describes the probability of reaching y at time t starting from x. More precisely, the probability of reaching a neighborhood U of y at time t, starting from x, is equal to fu h(t, x, y)dy. One says that M is stochastically complete if this process stays on M up to any finite time. That is, M is stochastically complete if and only if JM

h(t, x, y)dy = 1.

It is not hard to see that this true for one (x, t) > 0 if and only if it is true for all (x, t) > 0. If this fails, it means that Brownian motion escapes to infinity in finite time. The question of stochastic completness for Brownian motion on complete manifolds has been studied by many people, including Gaffney [27]. An early and very satisfactory criterion in terms of volume growth was obtained by A. Grigor'yan in [31]. See [34] for a thorough review of the problem. The main purpose of this section is to obtain the following

result, needed in the next section.

162

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

Theorem 5.5.4 Fix R > 0. Assume that the complete Riemannian manifold M satisfies the doubling condition

V r E (0, R), V (x, 2r) < D0V (x, r). Then M is stochastically complete, that is,

VxEM, dt>0, fM h(t, x, y)dy = 1. This will be an easy consequence of the following unicity result, which is of independent interest.

Theorem 5.5.5 Let M be a complete Riemannian manifold. Fix T > 0.

Let u be a solution of (at + A)u = 0 in MT = (0, T) x M with initial condition u(0, ) - 0. Fix o E M and suppose that there exists C such that for any ball B(o, r), r > 0,

f

T

Iu(s,x)12dxds
0, T

0

Ju(s, x)12dxds < V (o, r). JB(o,r)

By Lemma 5.2.7, the hypothesis of Theorem 5.5.4 implies the volume upper bound V (o, r) < '(1+r).

Thus Theorem 5.5.5 applies to u and shows that u must be constant equal to 0 in MT, for any finite T. That is, v(t, x) = fM h(t, x, y)dy = 1 for all (t, x) as desired. We now prove Theorem 5.5.5. The proof is taken from [31] but this line of argument is well known. See, e.g., [2]. Let p be a function such that

(Vp!0and set g(t, x) =

x2 4(t

) s) , t # s.

From this definition, it follows that otg + I Vg(2 < 0

(5.5.9)

5.5. THE PARABOLIC HARNACK PRINCIPLE

163

Let 17 be a smooth function with compact support. Then, as u is a solution of (Ot + O)u = 0,

J-aJM

/'

(Ou)u 2 e9 ddt = -

-J

(Au) u 2 e9 ddt

1r. M

which we can rewrite as

-

T_

J

f f (u(22e9O gdpdt = -2 J

T a JM

aM

Ta

M

(1 u)u1)2edydt.

Integrating the right-hand side by parts yields JV u272e9dµ +

JM

V u (2e1)V ij + 12e9Vg) d JM

- 12

(Vul2r)2e9dy M

2

- 2 fm IuI21O71I2e9d

IVul2?)2e9d)l -

2

IM

IuI2IVgl2g2e9dp

JM

> -2 fM IuI2IViiI2e9dp - 2 JM

u2Vg2772e9d).

Thus u2g2e9dp JM

T-a

i_a

IM I u272etgddt

f a fMj

4 fTTa M f

VgI2ii2e9dµdt.

uI2I

T

By (5.5.9), this gives T

IM u2)2e9d< JT-a fM Iu2IVr2e9dpdt. 111-a

We now fix r > 1 and choose 9, with support in the ball B(o, 4r), equal to 1 in B(o, 2r) and such that 0 51) < 1, IoiiI 0 is fixed small enough so that C'ery < e-1 the above inequality implies

that Al > 1/s2 = e2/r2

as desired. One cannot dispense with introducing a small e > 0 in the statement of Theorem 5.5.8. To see this, consider the connected sum of R3 with a unit 3dimensional sphere. In fact, consider a family of such examples, depending on a small parameter rl > 0 which describes the width of the smooth collar between the sphere and the flat space. When 77 = 0, the sphere and R3 are

disconnected. It is possible to prove that, uniformly in g, these manifolds satisfy (5.5.11) and (5.5.13) for any finite fixed R, for instance R = 2ir. This is obvious for (5.5.11). It is less obvious for (5.5.13), but it should be intuitively clear since the result is certainly true when g = 0. Now, it is not possible to lower bound uniformly the Dirichlet eigenvalues of all balls of radius 27r. Again, this should be intuitively clear because, when 77 = 0, any ball of radius 27r centered on the sphere is just the sphere itself and the "Dirichlet" eigenvalue vanishes (there is no boundary).

5.5.4

Selected applications of Theorem 5.5.1.

We present two related applications The first is that the parabolic Harnack principle is stable under quasi-isometries. Let M be a manifold and g, g be two Riemannian metrics on M. One says that these Riemannian metrics are quasi-isometric if there exists c > 0 such that

`dx E M, VX E Ty, cgx(X,X) < g,(X,X) < c-1gx(X,X).

(5.5.16)

Theorem 5.5.9 For any fixed R, 0 < R G be a distance-minimizing curve from the neutral element e to h, parametrized by arc length. In Section 3.3.4, we proved that If (9h)

-

f (g)12

0. As already mentioned at the end of Section 3.3.4, nilpotent Lie groups satisfy the doubling condition µ(2B) < Doµ(B) and thus (2) above applies to these

groups. That is, any nilpotent Lie group equipped with a left-invariant Riemannian metric satisfies the doubling condition µ(2B) < Doµ(B) and the scale-invariant Poincare inequality

Vf E C°°(B), f if - fsI2dµ < Por2 a

f

B

IVfI2dµ.

Thus, by Theorem 5.5.1, such a group also satisfies the scale-invariant parabolic Harnack principle (5.5.1) with R = oo.

Let us describe in some detail the simplest such example, the three dimensional Heisenberg group Hi. This is the group of all three by three upper-triangular matrices with 1's on the diagonal:

1( 1 IEHI =

x z 0 1 y 0 0 1

: x, y, z E H8

Thus we can view H1 as H83 equipped with the product (x, y, z)

(x',y',z') = (x+x',y+y',z+z'+xy').

(5.6.1)

174

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

Observe that, up to the choice of a multiplicative positive constant, the Haar measure on Hi is just the Lebesgue measure in 1!83, which is both leftand right-invariant. If we let X, Y, Z be the left-invariant vector fields with

X(O) = a

,

Y(0) = av, Z(O) = ez

then, in the g = (x, y, z) coordinate system, X (g) = 8., Y(g) = ab + xaz, Z(g) = az.

We can now turn lHli into a Riemannian manifold so that at each point g, (X (g), Y(g), Z(g)) is an orthonormal basis of the tangent space. The Riemannian measure is then exactly the Lebesgue measure, the length of the gradient of a function f is given by IVfl2 = IXfI2 + IYfI2 + IZf12

and the Laplace-Beltrami operator is A f = -(X 2 f + Y2f + Z2f) = -(fi + fly + (1 + x2)f z + 2x fyz).

We claim that the volume growth function V (g, r) = V (r) satisfies

V(r).

r33 T

if0 0. This curve has length at most 4s in our Riemannian metric (in fact it has length 4s). However, the end point of this curve is (0'0'S2). So, in time s, we can add s2 to the z coordinate! Using this observation one can show that (up to multiplicative constants) the ball of radius r > 1 contains and is contained in rectangular boxes of dimension r x r x r2 and thus has volume comparable to r4. By Theorem 5.6.1, Hi equipped with the Riemannian structure above satisfies the Poincare inequality V f E C°°(B),

isa If - fBl2dIt :: Por2 JB

IVf2d,L

where r is the radius of the ball B and P0 is independent of B. Thus, by Theorem 5.5.1, it also satisfies the scale-invariant parabolic Harnack principle (5.5.1) with R = oo.

5.6. EXAMPLES

175

5.6.2 Homogeneous spaces Let us now consider a simple but non-trivial application of Theorem 5.5.10.

Consider a connected Lie group G and a closed subgroup H C G and let M be the (left-)coset space M = H\G = {x = Hg, g E G}. Let p be the canonical projection. Then M is a manifold and p, of course, is surjective and smooth. Let us assume for simplicity that both G and H are unimodular. Up to a multiplicative constant, there exists a unique measure p on M which is invariant under the right action of G on M. Given a leftinvariant Riemannian structure on G, specified by a basis of left-invariant vector fields (X1, . . . , X, ), we can define a Riemannian structure on M such that p is a Riemannian submersion (see, e.g., [29, 2.28]). It is interesting to note that one can compute the length of the gradient and the LaplaceBeltrami operator on M purely in terms of the vector fields Y = dp(X2). Indeed, for any smooth function f on M, n n =Ely fl2, of=>Y2f. Iofl 1

1

Of course, setting f = f o p, we have n

n

OOf = ->X?f = ->(Y2f) op = (AMf) op I

1

Thus, Theorem 5.5.10 applies in this case. Theorem 5.6.2 Let C be a unimodular connected Lie group equipped with a left-invariant Riemannian structure. Let H be a closed unimodular subgroup

of G. Let M = H\G and equip M with the unique Riemannian structure such that the canonical projection p is a Riemannian submersion.

(1) For any fixed Ro > 0, there exist P0 and Do such that the Poincare inequality

Vf E C°°(B), f if - fBl2dµ < Por2

jIp f I2dp

and the doubling property

µ(2B) < Dop(B) are satisfied for all 0 < r < Ra and all balls B C M of radius r. (2) If the group G satisfies the global doubling property

b B C G, µ(2B) < Dop(B),

p being the Haar measure on G, then the scale-invariant Poincare inequality and the doubling property are satisfied uniformly for all balls

in M. In particular, in this case, the Riemannian manifold M = H\G satisfies the parabolic Harnack principle (5.5.1) with R = oo.

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

176

It is possible to give a more direct proof of this result. This is useful, for instance, in deriving the LP versions, 1 < p < oo, of the above L2 Poincare inequality on homogeneous spaces. See [59].

Again, let us describe in some detail a more concrete example. Let G =1H11 be the Heisenberg group described at (5.6.1). Let H be the closed subgroup

H=

1

0 0

0

1

00

y I:yER 1

Then 111/H can be identified with R2 = {(x, z) : x, z E R2} equipped with the Riemannian structure for which

X =(9

,

Z= 1+x2O

is an orthonormal basis at (x, z). The Laplace-Beltrami operator is

0 = -(O + (1 + x2)o). It is a good exercise to prove by hand that the volume growth is doubling on this Riemannian manifold. Details can be found in [59]. By Theorem 5.6.2, R2 equipped with this Riemannian structure satisfies the Harnack principle (5.5.1) with R = oo.

5.6.3

Manifolds with Ricci curvature bounded below

Following the work of S-T. Yau in the 1970s, a large number of analytic results have been obtained under lower bound hypotheses on the Ricci curvature tensor. We will not go into the details here but present some basic results regarding volume growth and Poincare inequalities. For background on the curvature tensor, see, e.g., [12, 13, 29]. The Ricci tensor R of (M, g) can be considered as a symmetric two-tensor. As such, it can be compared with the metric tensor g. Manifolds with Ricci curvature bounded below are manifolds for which there exists a constant K such that

R > -Kg.

(5.6.2)

That is,

VxEM, bX ETT, Rz(X,X)>-Kgx(X,X). One of the basic consequences of (5.6.2) is a control of the volume growth

on M. Namely, if (5.6.2) is satisfied, the volume of a ball of radius r in

M is at most that of a ball of radius r in the model space having the same dimension and R = -Kg. Here, the model spaces are spaces of constant sectional curvature: spheres (K < 0), Euclidean spaces (K = 0), and hyperbolic spaces (K > 0). See [29, 3.101] or [13, Theorem 3.9]. In particular, one has the following estimates.

177

5.6. EXAMPLES

Theorem 5.6.3 Assume that (M, g) is a complete manifold of dimension n satisfying (5.6.2) with K > 0. Then

V(x,r) 0. Here, Stn is the volume of the Euclidean ball of radius 1 in Rn. M. Gromov observed that this result can be strengthened in a crucial way as follows.

Theorem 5.6.4 Assume that (M, g) is a complete manifold of dimension n satisfying (5.6.2) with K > 0. Then V (x, r) < V (x, s) (r/s)n exp (Vr(-n -

1)K r)

for all x E M and r > s > 0. In particular, (M, g) satisfies the doubling condition

p(2B) < 2n exp (Jh- 1)K R) µ(B) for each 0 < R < oo and all balls B of radius r E (0, R). If K = 0, then p(2B) < 2"µ(B). As it turns out, these manifolds also satisfy scale-invariant Poincare inequalities. This is a result of P. Buser [10]. See [13, Theorem 6.8].

Theorem 5.6.5 Assume that (M, g) is a complete manifold of dimension n satisfying (5.6.2) with K > 0. Then for each 1 < p < oo, there exist Cn,p and Cn such that

I

If - fBIPdµ S Cn,p

IV fIPdµ

T

s

for all balls B C M of radius 0 < r < oo. The proof of this result in [10] is elementary but quite subtle and intricate. We will prove an a priori slightly weaker statement which suffices to imply Theorem 5.6.5 by Corollary 5.3.5.

Theorem 5.6.6 Assume that (M, g) is a complete manifold of dimension n satisfying (5.6.2) with K > 0. Then for each 1 < p < oo, there exist Cn,p and Cn such that

J If - fBIPdµ< for all balls B C M of radius 0 < r < oo.

CnprPeCnV'K-- r

2BI Vf Ipdµ

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

178

For any pair of points (x, y) E M X M, let 1i,y : [0, d(x, y)] -' M, t i-"-t.,y(t)

be a geodesic from x to y parametrized by arc length. Except for a set of it 0 µ measure zero, this geodesic is unique and yy,x(t) = yx,y(d(x, y) - t). Let us prove the theorem above for p = 1 (the same proof works for any other finite p >- 1). Fix a ball B of radius r and write

1, If -.fBIdIA

IK(d(x,y))

180

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

for all s E (0, d(x, y)). Finally,

''

Kl sinh

IK(t)

L

-

to-1

ifK>0 ifK = 0.

Thus, for 0 < s < t,

()n_1 IK(t) 1

exp

(_

Jx,B(y) ? 2n-1 exp -

(n --1) K r>

for all x E M, all y not in the cut locus of x and all s E [d(x, y)/2, d(x, y)J. This proves Lemma 5.6.7.

5.7

Concluding remarks

In this last section we briefly indicate some further developments that emphasize some of the most basic features of the techniques presented in this monograph. In Chapters 3, 4 and 5, we developed in the classical context of Riemannian manifolds a number of techniques based on Sobolev, Poincare and other similar inequalities which allow us to study some of the fundamental properties of solutions of the heat equation (at + 0)u = 0,

in particular Harnack-type inequalities. Beside Sobolev-type inequalities, we mostly based our analysis on a control of the volume growth of the manifold. In fact, we made no explicit use of the Riemannian structure. For instance, we did not place conditions on the curvature tensor except to show that some of the main results that we obtained do apply under certain curvature conditions. It is indeed one of the advantages of the techniques presented in this monograph that they are immediately applicable outside the scope of Riemannian geometry, in particular in the context of "sub-Riemannian ge-

ometry". The simplest and most natural setting for an introduction to sub-Riemannian geometry is that of analysis on Lie groups. Let us identify

the tangent space at the neutral element e of a Lie group G with the Lie algebra 0 of G. Picking a (vector space) basis in t amounts to picking a left-invariant Riemannian structure on G. However, from an algebraic point of view, it is natural to consider not necessarily a linear basis but a family

5.7. CONCLUDING REMARKS

181

of vectors (X1, ... , Xk) which generates 0 as an algebra, i.e., X1,. .. , Xk

together with their brackets of all orders span the vector space 0. This corresponds to the celebrated Hormander subellipticity condition for the left-invariant differential operator L = - Ei X, . See, e.g., [24, 44, 87]. To give an explicit example, consider the Heisenberg group 1H11 at (5.6.1).

In this case, it is easy to see that the vector fields X = ax and Y = ay + xaz generate the Lie algebra since the bracket [X, Y] equals Z = i%. Thus one is led to consider the subelliptic operator L = -(X2 + Y2). This operator is, in many ways, more canonical than -(X2 + Y2 + Z2). For instance, L is homogeneous of degree two with respect to the natural dilation structure (x, y, z) --> (tx, ty, t2z). Using this, one easily shows that the coresponding volume growth function is V (r) = cr4, r > 0. Analysis on Lie groups admitting a dilation structure is treated in [24]. Going back to a general Lie group, the techniques of this book (see in particular Theorem 5.6.1) easily yield a self-contained proof of the fact that the heat diffusion equation (8t + L)u = 0 associated with a subelliptic operator L as above on a unimodulas Lie group G has a continuous strictly positive fundamental solution h(t, x, y) such that h(t, x, x) ti t-d/2 for small t where d is a certain integer that can be computed in terms of the family of left-invariant vector fields {X1, ... , Xk}. All that is needed, in addition to what has been explained in this text, is control of the volume growth function in such a subelliptic situation. The courageous reader will find details and much more in [72, 87]. In fact, one natural setting for the development of the techniques presented in this text is that of a manifold M equipped with a measure µ and a second order differential operator L which is symmetric with respect to p, that is such that fm f Lgdi = fm gL f dp for a large enough class of compactly supported functions. The length of the gradient of f can be defined in this context by setting IVfI2

= -2(Lf2 - 2fLf)

(assuming enough functions f in the domain of L are such that f2 is also in the domain of L). At a formal level, one can also define the so-called intrinsic distance d(x, y) associated to L by setting d(x, y) = sup{ f (x) - f (y) : f such that to f 1 < 1}.

Observe that, if L is the Laplace-Beltrami operator of a Riemannian manifold and a is the R.iemannian measure, then the intrinsic distance d is indeed the Riemannian distance. In general, whether or not the formula above really gives a genuine distance function which defines the topology of M is an interesting and deep question whose answer of course depends on certain assumptions made on L. See [25, 47, 74, 76, 87). Even more generally, one can consider the geometry associated with strictly local regular Dirichlet

182

CHAPTER 5. PARABOLIC HARNACK INEQUALITIES

forms. See [6, 51, 76, 83, 84] for pointers to the literature in this interesting direction. Another important aspect of the methods used above is their great robustness. For instance, we proved the stability of the parabolic Harnack

principle under quasi-isometry. This can be pushed further to treat stability under the so-called "rough isometrics" which preserve the large scale geometry but not the local geometry or topology, a notion that has received great attention as it is central in some of the work and ideas of Gromov. See, e.g., [13, Section 4.4], [20] and the references therein. The simplest setting where this is useful is that of coverings of compact manifolds. Suppose N is a compact Riemannian manifold and M a Riemannian cover of N with deck transformation group F. This means that the finitely generated group

F acts on M by isometries and M/F = N. A typical result that can be proved by using rough isometry techniques and the methods of this book is that, if r is a nilpotent group, then M satisfies the doubling condition and Poincare inequality, uniformly at all scales. Thus, such a manifold satisfies the parabolic Harnack principle at all scales. It is interesting to note that this sort of result does not seem to be attainable by techniques based on curvature lower bounds. Finally, it may be useful to recall that Moser's iteration technique applies to a host of other linear and quasi-linear equations. See, e.g., [4, 75, 76] and the references given there. In particular, it applies to the p-Laplacian div(I O f Ip-20 f) associated to the "energy functional" fm I V f ]pd/,t. For instance, if a manifold M satisfies the doubling property and a scale-invariant LP Poincare inequality for some p > 1, then any non-negative p-harmonic function (i.e., solution of div(jV f ]1' 2Vu) = 0) must be constant. This applies to Lie groups of polynomial volume growth, to manifolds with nonnegative Ricci curvature, and to coverings of compact manifolds with nilpo-

tent deck transformation groups. When p = n is the topological dimension of M the study of the p-Laplacian is relevant to the theory of quasiconformal (or quasi-regular) mappings. For developments and pointers to the literature in this direction see for instance [43, 42].

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Index Giusti (Enrico) 47 gradient 7, 11, 54, 175, 181 Green function 154 Gromov (Misha) 18, 182

Aronson (Donald) 6, 156 Birman-Schwinger principle 109 Bishop's theorem 83 Bishop-Gromov theorem 84, 179 Bombieri (Enrico) 47 Brownian motion 161 Brunn (Hermann) 18 Brun-Minkowski inequality 18

Haar measure 77, 172 harmonic function 111, 151 Harnack inequality(principle) 145, 146

elliptic 2, 35, 49, 111, 151 gradient 113 parabolic 5, 111, 155, 164,

co-area formula 17, 21, 56, 76 covering (Whitney type) 133, 135, 138, 141

168, 175

covering manifold 182

heat diffusion semigroup 4, 87, 95, 98, 122 heat equation 4, 112, 119, 127, 146 heat kernel 89, 93, 101, 122, 128, estimate(s) 92, 154 on-diagonal lower bound 125, 127, 152, 165 upper bound 3, 130 Heisenberg group 82, 173 Holder continuity estimate 35, 50,

deck transformation group 182 De Giorgi (Ennio) 50 dilation structure 181 Dirichlet boundary condition 29 eigenvalue 167 eigenvalue problem 29 form 87, 106, 182 heat kernel 161, 166 semigroup 166 divergence 19, 54 divergence form 2, 33, 49, 68 doubling property 5, 112, 117, 119,

149

Holder inequality 9, 45, 57, 60, 68, 66

Hormander condition 181

127, 139, 154, 155, 173, 175

intrinsic distance 181 isoperimetric inequality 16, 21, 56,

Faber-Krahn inequality 164 Federer (Herbert) 17 Fleming (Wendell) 17

81

problem 55

Gagliardo (Emilio) 9, 17 Gaussian estimate 93

John-Nirenberg inequality 47

Knothe (Herbert) 18

lower 127, 131 two-sided 6, 154, 161, 184

Laplace-Beltrami operator 54, 78,

upper 4, 99, 101, 122, 165

103, 161, 175, 176 189

INDEX

190

Laplace equation 151 Lie group 77, 172, 180 Liouville property(ies) 151 Lorentz spaces 67

Rozenblum (Grigori) 103 Rosenblum-Lieb-Cwikel inequality 4, 103

Marcinkiewicz (Jozef) 13 Marcinkiewicz theorem 16, 139 Maximal function 139 Mazja (Vladimir) 17 mean value inequality(ies) 40, 42,

Serrin (James) 156 Sobolev (Serguei) 8, 17 Sobolev inequality(ies) 1, 11, 19, 20, 53, 56, 60, 62, 75, 76, 81, 85, 104, 156

119, 165

modular function 77, 78 Moser (Jiirgen) ix, 2, 155 Moser's iteration 45, 133, 156, 157, 182

Nash (John) 60, 90, 92 Nash inequality 2, 59, 60, 70, 75, 81, 87, 90, 98, 99, 101 Neumann boundary condition 29, 158, 159 eigenvalue problem 29

heat kernel 159, 161 nilpotent 81, 82, 173, 182 Nirenberg (Louis) 9, 17 p-Laplacian 182

Poincare inequality(ies) 3, 5, 30, 49, 112, 131, 154, 155, 156, 173, 175, 177 weighted 133, 144

polar coordinates 11, 83 pseudo-Poincare inequality 3, 73, 79, 82, 84, 114, 125, 132 quasi-isometry(ies) 6, 168, 182

rearrangement inequality 20 representation formula 11, 23, 27, 31

Ricci curvature 3, 82, 113, 176 Riemannian manifold 53 Riesz potential 12 Riesz-Thorin theorem 95 rough isometry 182

Schrodinger operator 4, 103

local(ized) 5, 113, 128, 130, 131, 156, 165, 174 type 53, 61, 63, 66, 74, 82 weak 2, 60, 70, 71, 74, 76, 82

Sobolev-Poincare inequalities 32 stochastic completness 161 sub-Riemannian geometry 180 subsolution 35, 38, 119 supersolution 35, 43, 49, 128, 145 Stein (Elias) 96 Talenti (Giorgio) 21 tangent space 53, 77 Trudinger (Neil) 25, 156

ultracontractive(ity) 4, 90, 92, 93, 106, 108

unicity (Cauchy problem) 162 uniform ellipticity 2, 35, 38, 49, 68

unimodular Lie group 3, 77, 78, 79, 80, 81, 172, 175 vector field 54

left-invariant 77, 78, 172, 175

volume growth 2, 54, 57, 72, 81, 100, 161

maximal 3, 82 weak type 13, 31 Yau (Shing-Tung) 176 Yudovich (Victor) 25

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES L-diledbN

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with the assistance of S. Donkin (London) 1. Fesenko (Nottingham) 1. Roe (Pcnnsv/vania) E. Sufi ((hfonl) The London tlutltemutic ul Six ie'tt is incur ,vrai al under Ro. ad Charter

Aspects of Sobolev-Type Inequalities Laurent Saloft=Coste

This book focuses on Poincarc. Nash and other Sobolrv-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds. Applications covered include the ultracontracti%ity of the heat diffusion semi-group. (.iaussian heat kernel hounds. the Rozcnhlum-L.ieh-Cwikei inequality and elliptic and parabolic liarnack inequalities. L.mphacts is placed on the role of families of local Poincarc and Soholes inequalities. The text pros ides the first self-contained account of the equivalence between the uniform parabolic Harnack inequality. on the one hand and the conjunction of the doubling volume property and Poincare s inequality on the other. It is suitable to he used as an adsanced graduate textbook and will also be it useful source of intorination titr graduate students and researchers in analysis on manifolds. geometric differential equations. Browman motion and diffusion on manifolds, as well as other related areas.

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