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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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CONTENTS
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An abstract lemma . . . . . 2.3 Harnack inequalities and continuity 2.2.3
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vii
CONTENTS
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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
n, the Holder continuity estimate sup r IfIx(x) - f (01 < cIIo f lip - yll )
x,yERn
holds instead. We discuss a number of different proofs of Sobolev inequalities in Rn. Each yields a different and useful point of view on the meaning of Sobolev inequalities. Of course, this material is covered in greater detail in a number of books and monographs including 11, 30, 60, 79]. The important topic of Sobolev inequalities in subdomains of Rn (see, e.g., [61]) is not treated here. The theory of partial differential equations provides a host of important applications of Sobolev inequalities. Consider for instance the equation n
8;az,3(x)8,u(x) = 0
where the coefficients a2j are real measurable functions such that Ilai,j II,. < Ci and
n
n
t1 x E II$n, d t E Rn,
a:a (x)U.i ? cl
INTRODUCTION
2
That is, consider a divergence form, uniformly elliptic equation in R. Moser's elliptic Harnack inequality [30, 63] states that any positive weak solution u of this equation in an Euclidean ball B satisfies sup{u} < C inf{u} zB
zB
where C depends neither on u nor on B but only on the constants C1, cl above and the dimension n. Moser's proof, presented in Chapter 2, is a striking application of Sobolev inequalities. It also serves as an introduction to our later treatment of parabolic Harnack inequalities on manifolds. In Chapter 3, Sobolev inequalities are discussed in the context of Riemannian manifolds. A number of related functional inequalities are introduced and relations between these inequalities are established. One of the most basic facts is that any Sobolev inequality implies a lower bound on the volume growth of the geodesic balls. In particular, the inequality V f E C0 (M),
IIfIIq
<_ CIIVfIIP
for some fixed q > p ? 1, implies that the volume of any ball of radius r must be bounded below by a constant times r" with v related to p, q by
1/v = 1/p - 1/q. A more technical but very important fact is the equivalence between strong forms and weak forms of Sobolev inequalities. An example of this phenomenon is that it is enough to have the weak Sobolev inequality
b f E C (M), sup {s/2({x: If(x)I > s})1/q} < C'Ilof IIP s>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")
<- CIIof II2IIf II1/"
and the Sobolev inequality `d f E Co (M),
Ill II2"/(v-2) 5 CIIof II2
when v > 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 fB (Ivf
(B)1J
I gdµl
I2 +r -21f 12) dµ
where q, v > 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
/M
(IvfI2)
IfIgd,4
dp
does not hold. For instance, the manifold Stm x Rk, m + k > 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
<_ h(t, x, y)
V(x
Such a two-sided heat kernel bound was first derived for uniformly elliptic divergence form second order differential operators in R" by Aronson [3]. These results are taken from [32, 741 (a more complete discussion is given at the beginning of Section 5.5). The equivalence between the parabolic
Harnack inequality on the one hand and the (more geometric) doubling property and Poincare inequality on the other hand is a very useful tool. Both directions of this equivalence are interesting and this illustrated by a few simple examples. For instance, it follows from the theorem above that the parabolic Harnack principle is stable under quasi-isometries.
Chapter 1 Sobolev inequalities in Rn 1.1
Sobolev inequalities
1.1.1
Introduction
How can one control the size of a function in terms of the size of its gradient? The well-known Sobolev inequalities answer precisely this question in multidimensional Euclidean spaces. On the real line, the answer is given by a simple yet extremely useful calculus inequality. Namely, for any smooth function f with compact support on the line, 1
if (01 S 2
f
+00
(1.1.1)
If (s)Ids. 00
The factor 1/2 in this inequality comes from the fact that f vanishes at both +oo and -oo. In this respect, note that if f is smooth but no other restriction is imposed the inequality above may fail. It is natural to wonder if there is such an inequality for smooth compactly supported functions in higher-dimensional Euclidean spaces. More precisely, for each integer n, can one find p, q > 0 and C > 0 such that `d f E Co (R"),
11fllq _< CIIVfllp?
(1.1.2)
Here and in the sequel Co (RII) denotes the set of all smooth compactly supported functions in R". For f E Co (W'), we set 11f11q =
f
1/q n
If(x)I°dx)
,
IIf1100 = p{Ifl}
and IVf(x)Ipdx)'lp
Ilofllp = U where V f = (81 f, ... , 8" f) is the gradient of f and I V f I = E 1 18= f 12 is the Euclidean length of the gradient. In R", we denote by p, = p the 7
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
8
Lebesgue measure and by µrz_1 the volume measure on smooth hypersurfaces of dimension n - 1. When using coordinates x = (x1, ... , xn), we also write dµ(x) = dx = dx1...dxn.
This question was first addressed in this form by Sobolev in [78] which appeared in Russian in 1938. Fixing a function f E Co (Rn) and replacing x H f (x) by x H f (tx), t> 0, in (1.1.2) yields t-"/411f11q < Ct1_1hIlofII
.
Letting t tend to zero and to infinity shows that (1.1.2) can only be satisfied if the exponents oft on both sides of the inequality above are the same. That is, (1.1.2) can only be satisfied if
np
1
q=p1 -n,1
n-p
For instance, in 1R2, this says that one might possibly have V f E Co (R2),
IIfII. <1R2
Vf(y)I2dy.
The next example shows that this last inequality fails to be true. EXAMPLE 1.1.1: Consider the function
Ax)
log I log IxI I
0
if IxI < 1/e otherwise.
/e
27r but f is not bounded. Of course, f is rp not smooth, but it can easiy be approximated by smooth functions fn such that JJ V fn I I2 - I I V f 112 and fn - f. This shows that that (1.1.4) cannot be true. Then Il Vf II2 = 2ir fo
What is true is recorded in the following theorem.
Theorem 1.1.1 Fix an integer n _> 2 and a real p, 1 _< p < n and set q = np/(n - p). Then there exists a constant C = C(n, p) such that V f E C (]Rn),
JIfII9 <- CIIVfllp
(1.1.5)
This inequality is called the Sobolev inequality although the case p = 1 is not contained in [78]. Note that the case p = n (i.e., q = oo) is excluded in this result as should be the case according to the preceding example. In the next few subsections we will give or outline several proofs of (1.1.5). As it turns out, when p = 1, (1.1.5) has a very simple proof based on
9
1.1. SOBOLEV INEQUALITIES
(1.1.1) and Holder's inequality. This well-known proof (due independently to E. Gagliardo [28] and L. Nirenberg [68]) is presented in the next section. Moreover, as we shall see in 1.1.3 below, the case p > 1 follows from the case p = 1 by a simple trick. We conclude this short introduction to Sobolev inequalities by recording a couple of useful remarks concerning the validity of (1.1.5). First, if (1.1.5) holds for all f E Co (R' ), it obviously also holds for a larger class of functions including for instance all C' functions with compact support or even Lipschitz functions vanishing at infinity. In fact, (1.1.5) holds for all functions vanishing at infinity whose gradient in the sense of distributions is in LP. Second, (1.1.5) restricted to non-negative functions in Co (Rn) suffices to prove (1.1.5) in its full generality. Indeed, (1.1.5) for such functions implies that it also holds true for non-negative Lipschitz functions with compact support and, if f E Co (R'), If I is Lipschitz and satisfies I V If 11 < I Vf I almost everywhere. It then follows that (1.1.5) holds for f E Co (Rn)
1.1.2
The proof due to Gagliardo and to Nirenberg
Recall that Holder's inequality asserts that, for any positive measure µ,
if
<
IllIIPII9IIP'
for all f E LP(,u), g E LP'(p), 1 _< p, p' < oo with 1 = 1/p + 1/p'. By a simple induction we find that
if
j<- IIf1I IP1 IIf2IIPZ
IIfkIIPk
(1.1.6)
for all fi E LP,, 1 < i < k, 1 < pi < oo, 1/p1 + 1/p2 + '+ 1/pk = 1. Now, fix f E Co (][fin). By (1.1.1), for any x = (x1, ..., x,) and any integer 1 < i < n, we have +00
1
If(x)I < 2f
00
(with the obvious interpretation if i = 1 or n). Set +00
Fi(x) = f
I0if(x11...,xi-17t,xi+l,...,xn)Idt 00
and
Fi,m (x) _
... f+: (aif (x) I dxl ... dxm f { f ao f Fi(x)dxl ... dx,n +.Oo
if i < m if i > 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) <_ (1/2)
1/n
rj IIa2f
(1.1.?}
iI1
As (rji a2) 1/n < y: a= for any positive numbers a2 and integer n, we n obtain n IlfIIn/(n-1) < 2n
11a8f(x)H1dx <
(1.1.8)
1
To see the last inequality, use En Ia2fI < ,fn-lof1. This proves (1.1.5) for
P=
1.
1.1.3
p = 1 implies p > 1
Assume that (1.1.5) holds for p = 1, that is,
d f E Co (R'),
IIfIIn/(n-1) < CIIVfIII -
(1.1.9)
Fix p > 1. For any a > 1 and f E Co (Rn), note that If I° is C', has compact support, and satisfies
Iolf1°I = alfl°-'IVfI. Since we can easily approximate If 1° by a sequence (f2) of smooth functions with compact support such that Vf2 --> VI f I°, inequality (1.1.9) holds with f replaced by If 1°. This yields Ilfllan/(n-1)
< Ca I
If(x)l°-'lVf(x)ldx
< Ca (f,f(x)I'ix)
1/P'
1/P
(Jlvf(x)rdx)
1.2. RIESZ POTENTIALS
11
where 1/p + 1/p' = 1. If we pick a = (n - 1)p/(n - p), we find (rather miraculously) that (a - 1)q = n(p - 1)p'/(n - p) = np/(n - p). Thus IIf Ilnp/()-p)
-P)
- C (n - pp II f IInp/(n) p) -P) IIQJ IIP
Finally, (n -1)p/(n - p) - n(p - 1)/(n - p) = 1, so that simplifying the last inequality yields Ilf IInp/(n-p) < C (n
- pp,,V f 11 P.
Thus we have proved the following version of Theorem 1.1.1.
Theorem 1.1.2 For any integer n > 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
Theorem 1.2.1 Fix 0 < a < n, 1 < p < n/a and define q by 1/q = 1/p-a/n, i. e., q = np/(n-ap). Then there exists a constant C = C(n, a, p) such that b f E C0 00(R),
IIIaf 119 < CIIf
IIp.
1.2. RIESZ POTENTIALS
13
This theorem will be proved below in a more general form. As a corollary
of Theorem 1.2.1 and (1.2.2), we obtain (1.1.5) for 1 < p < n. Observe that the case p = 1 is excluded in Theorem 1.2.1. This has to be the case. Indeed, if we had IIIaf II9 c Cllf 111, we could let f E Co (Rn) tend to the Dirac mass. This would imply that the function x " I xI _n+a is in L9 with q = n/(n - a). But this is clearly not the case. Similarly, the case p = n/a must also be excluded. This follows from the case p = 1 by duality, or more directly by the following example. EXAMPLE Consider
Ax)
IxJ-0(log
1/Ixl)-(a/n)(1+E)
for lxl < 1/2 otherwise.
0
Then f E Ln/a if E > 0, but Iaf (0) =
1
Ca
f
Ixl_n(log
IxI)-(a/n)(1+e)
= 00
xI<1/2
when e > 0 is chosen small enough so that (a/n) (1 + E) < 1 (recall that
a/n < 1).
1.2.2
Marcinkiewicz interpolation theorem
Consider a measure space (M, µ). Fix 1 < p, q < oo. A linear operator K defined on L' n LOO is of weak type (p, q) if there exists a constant A such that
Vt > 0, V f E L' n L°°, µ({x : IKf(x)I > t}) < (All flip/t)q. If q = oo, this must be understood as II K f II0
(1.2.4)
All f 11p.
Theorem 1.2.2 Assume that K is of weak types (pl, q1) and (p2, q2) with
1 < pi, qi < oo, pi < p2i ql 34 q2. Then for each 0 < 9 < 1 and 1/p = 9/p1 + (1 - 9)/p2, 1/q = 9/q1 + (1 - 9)/q2i there exists a constant C = Ce such that
IIKfll9 <-CIIfIIp This is an interpolation result due to Marcinkiewicz. It says that boundedness of K at the end points (1/p1, 1/q1), (1/p2, 11q2) implies boundedness along the segment joining these points in the (11p, 11q) plane. See [79, 811
for a proof. Of course, one of the important aspects here is that weak boundedness at the end points is enough to prove strong boundedness in the interior.
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
14
We want to apply this result to the case where K is given by a kernel K(x, y) of weak type r for some 1 < r < oo, that is, which satisfies
µ({z : IK(x, z)l > t}) < (Alt)", µ({z : l K(z, y)l > 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
Without loss of generality we can assume that K > 0. For each t > 0, write K = Kt + Kt where Kt(x, y) = K(x, y)1 {(u,v):x(u,v)
-
Lemma 1.2.4 Fix p > 1. There exists a constant B1 such that, for all
t>0andallfELp,
11 Ktf lip <
Btt-r+1
ll f lip-
Moreover, if p/(p-1) < r, there exists a constant B2 such that, for all t > 0 and all f E LP, II Ktf Ii
B2t1-r(p-1)/p
_<
11 f lip,
To prove the first inequality, observe that 00
fM
l Kt(x, y)I dµ(y) =
fo
jiffy: 1 Kt(x, y)l > 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<_1
f(Ktf)gd.
15
1.2. RIESZ POTENTIALS Moreover,
f(Kf)gd/A = Jf(ktg)dj
where K(x, Y) = K(y, x). From the x, y symmetry of our hypothesis it follows that (1.2.7) also holds for fM IKt(x,y)Idµ(y). Hence
IIKt9Il. <
Bltl-r
and
J(Ktf)gdµ = f f (K19) dµ < IIKt9II.IIfiii < Bltl-rII9lI.IIfII1Thus
II Ktf III = sup gELO°
f (Ktf) 9dµ < Bitl-rllf III.
Ilslloo<_1
We still need to prove that
IIKtfIlp <
Bjtl-rIIfIIp
for 1 < p < oo. To this end, use Jensen's inequality to obtain P -I
I Ktf (x)Ip <_ (fM Kt(x, y)dµ(y))
fm Kt(x, y)I f (y)I pdp(y)
Using the Ll -> Ll bound, we finally get IIKtf llp < Bl t'-rllf IIp as desired. To prove the second inequality of Lemma 1.2.4, write 1/p + 1/p' = 1 so
that p' = p/(p -- 1) and note that 00
JM
IKt(x, y)Iddi(y) = p' f sP'-lµ({z : I Kt(x, z)I > 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 <
Pick t so that
B2t1-r(P-1)/P
B2tl-r(P-1)IP
= s/4. Thus t = (B2
s/4)P/(p+r-r)).
Then
p({z : I Ktf (z) l ? s/2}) = 0 and
p({z : I K f (z)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
<-
f 0
pn({ f >
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
010
1110
1
00
I11{f(.)>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
B1(x)={z:zj=xj for j
=I
g(z) dzi ... dzn i (2)
/(//'
" Bi (x)
g(z)dzi ...
1.3. BEST CONSTANTS
19
Since g > 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
ns ,n
JI V f (x) I dx.
(1.3.3)
(1.3.4)
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
20
Removing the normalization f g(x)dx = 1, we finally obtain l/nIIVfIII
Uflln/cn-
nin
which is exactly (1.3.2).
The same argument gives (1.3.1) if we take f to be the characteristic function of a bounded domain 0 with smooth boundary and if we replace (1.3.4) by
divFdµn =
J
(F,
n is the exterior normal along 9t1.
1.3.3
The case p > 1
The following theorem gives the best constant in the Sobolev inequality for
1
Ilf119
CIIVfI1p
holds with C equal to C(n, p) =
p-1
1/9
(_np
n - p n(p - 1))
fort
n + 1)
r(n/p)r(n + 1 - n/p)wn-1
1(n
1/n
(1.3.5)
1/n
(1.3.6)
n wn-1
These are the best possible constants and the functions
x -, (a + bIx -
yjP/(P-1))1-n/P
a, b > 0, y E Rn, are the extremal functions when 1 < p < n. We only briefly sketch the proof. See [5, 85) for details.
The proof is in two steps. First, one shows that it suffices to treat the case where f is a non-negative radial decreasing function. This follows from the following classic rearrangement inequality. For any function f E Co (Rn), f > 0, let f- be the radial decreasing function such that
µn({z : f* (Z) > t}) = µ-({z : f (z) > 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 <
IIofIIP.
One proof of this theorem uses the isoperimetric inequality (1.3.1) and the co-area formula of Theorem 1.3.1. Talenti [85] gives a nice account. See also [5, Proposition 2.17]. Theorem 1.3.3 reduces the proof of Theorem 1.3.2 to the following 1dimensional statement.
Lemma 1.3.4 Fix 1 < p < n and set q = pn/(n - p). Let h be a decreasing function which is absolutely continuous on [0, oo) and equal to zero at infinity. Then \1/9
(p0
.
Ih(t)Igtn-1dt)
/
o
C'(n,p) =
P-1 ( n-p
h(t)IPtn-1dt) I
r(n + 1)
1/g
n -p n(p - 1)) C(n,
1/P
00
< C'(n, p) (10
1/n
(r(n/p)r(n + 1 - n/p) )
/n
p)wn1-1
Moreover, for 1 < p < n, equality is attained for the functions t
" (a +
btP/(P-1) )1-nlP
See [5, 85] for a proof and earlier references.
1.4
Some other Sobolev inequalities
1.4.1
The case p > n
What can be said about the size of smooth functions with compact support in terms of II V f IIP when p > n? The following theorem gives a partial answer.
Theorem 1.4.1 For p > n there exists a constant C = C(n, p) such that for any set SZ of finite volume we have V f E Co (1),
Ilf lloo <- CJUn(Q)1/n-1/PII Vf IIP
Start with (1.2.2), that is, If (x) I -<
1
n-1
I
J lVf(y) n
IyIn-1
dy.
(1.4.1)
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
22
Define p' by 1/p+l/p' = 1 and note that (n-1)(p'-1) = (n-1)/(p-1) < 1. Let also R be such that p ,,(Q) = µn(B(R)), that is R = (µn(S2)/SZn)'/n Then use the following computation. Write 1 11
Ix
- yl p'(n-1)
1
dy
B(R) Ix - yIp'(n-i) dy R
G wn-1 0
= wn-1(1 - (n - 1)(p' - 1))`1 Rl-(n-1)(P'-1) wn-1(1
- (n -
_
1)(p' -
1))-1
R(P-n)l(P-1)
wn-1/Ln(n)(P-n)/n(P-1)
0(P-n)/n(P-1)(1
(1.4.2)
- (n - 1){p' - 1))
B µn (!Q) (p-n)/n(p-1) .
Now, by (1.2.2), 1
I I ! ll.
1/'
1
yIP,(n-1)dy) (wn-1 Jn Ix < CjLn(D)1/n-1/pllof II,'
IIofIIP
This proves Theorem 1.4.1. One crucial difference between Theorem 1.4.1 and Sobolev inequality
(1.1.5) for 1 < p < n is that the right-hand side of (1.4.1) depends on the set Q on which the function f is supported. As the volume of Il tends to infinity, the term µn(0)1/n-1/P also tends to infinity since n < p. In fact, when n < p, there is no way to control the size of f purely in terms Ilof IIP. To see this, consider the function fr : x --, (r - l xl)+. This function is supported in B(r) and Qn/Prf/P Ilofrllp = P-n(B( ))1/p =
Also,
µn(B(r/2))(r/2) = f2n/q(r/2)n/q+1.
Ilfrll9
For any fixed q, the ratio Ilfrllq/llofllp tends to infinity as r tends to infinity
ifn
Theorem 1.4.1 can be complemented as follows.
Theorem 1.4.2 For p > n, there exists a constant C = C(n, p) such that any function f E C°°(R') with Ilofllp < oo satisfies sup X'VEHn
=#v
with a = 1 - n/p.
If (x) - f (y) I Ix
- yIa
< C l l V f l ip
1.4. SOME OTHER SOBOLEV INEQUALITIES
23
Note that this result does not require that f vanishes at infinity. For the proof we need a localized version of the representation formula (1.2.2).
Lemma 1.4.3 Let B be a ball of radius r > 0. Then,
I
n
V f E C°°(B), V x E B, If (x) - fBI <
IVf(y)I
n -1 B Ix -
yIn-1
dy
where
fB =
µn(B) Ja
f(z)dz.
For x, y E B write
fix-vi Y-x .9 f (x + p (x) -f(Y) = - J Ax)
l y - xI
It follows that
where
if x E B
F(z) = f I Vf (z) I
otherwise.
0
Integrating with respect to y E B yields If (X)
- fBI = f (x) ___ 1 p(B)
<_
1
Stnrn
Jy:Ix--yl<2r)
=
0
Jo 2n wn-1
This proves Lemma 1.4.3.
J
F o (
dy
00
nrn
(y)dy+
Jfd-
1
-
ff
x+
y-x
2r
Jn-1 JO
F (x + p9) sn-ldsd9dp
jF(x+rO)dodr n-1 I V.f (y) I
Iy - XIn-1
dy.
d
ply - xI / p}
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
24
To obtain Theorem 1.4.2, it now suffices to apply Lemma 1.4.3 and the argument of the proof of Theorem 1.4.1 to obtain 1/p
I Qf
f (x) - fB I < Cpn (B) 1/n-1/p
<
I pdpn I
Cµn(B)1/n-1/pll
V f Il p
UB
for all x E B and all balls B C Rn. Thus, for all x, y such that Ix - yl < r, we get If (x)
- f (y)I < 2C 1nr1-n/pll Vf lip <- 2CSinlx - yil-n/pllof lip
This proves Theorem 1.4.2.
The case p = n
1.4.2
To treat the case p = n, we first compute 1
in Ix -
ylr(n-1) dy
when r < n/(n - 1). Actually, this has already been done in (1.4.2) where we have shown that wn-1
1
(1.4.3) 1-(r-1)(n-1) Foranyn
fn
Now, write
If(x)I
1f
<_
wn-1
lof(Y)1 dy Ix - yIn-1 n/q
< wn`1
f Ix - yl r( -1)/q l
I
x l V f (y)
l' x Ix - yl r(f 1)(1-1/n)
dy.
Notice that 1/q + 6 + (1 - 1/n) = 1, and use the Holder inequality (1.1.6) with p1 = q, p2 = 1/6, p3 = n/(n - 1) to get if(x)l
<
1
+
Wn-1 \
f Ix -
lVf{y)ln
dy1/q x
)
ylr(n-1)
b
f
(l
Y Iof(y)Indy)
1-1/n
1
Usupp(f) Ix -- ylr(n-1)dy)
It follows that, if f is supported in St, ill IIq
<
1
wn-1
< wn-1
Ilof Iln/q+' (f1X 2
llof lln
in I x - yl
11q+1 1/n
1
-
ylr(n-1) dy)
r(n-1)
dy /
1/r
25
1.4. SOME OTHER SOBOLEV INEQUALITIES Thanks to (1.4.3), this yields -1+1/r Ilfll9
[1 - (r -
As 1/r= 1+1/q- 1/n, we get
Ilfll9 <
1)]1/Qn/Rµn(Q)1/911pf 11n
[1 - (r - 1)(n n1/9
[1 - (r - 1)(n - 1))1/rwn! i
µn0)1/911pf
Note that 1 - (r - 1)(n - 1) = n(n + 1)/(nq + n - q) > (n + 1)/q because
q > n. Hence, for q > n, we get Ilfll9 < ql+9(n-1)/-wn 9inµn (!Q)Il pf II9
(1.4.4)
It follows that for any integer k = n, n + 1, ... , kn/(n-1)
If () I
St
dx < [kn/(n -
(l;%)
1)]1+kw,,ki(n-1)µ-(S1).
-
For k = 0, 1, . . , n - 1, the left-hand side is easily bounded by Cµ-(12) by Jensen's inequality. Clearly, the series .
akkk
00
0
n
(k -1)!
(n -
n1
1)wl/(n-1)
converges if a > 0 is small enough, e.g., a < (n - 1)w,l,(n-')/en, and for such a we have
nI eXp
a(
If(x}i
IIVf!In)
00
nl(n-1)
dx
1 0
ak kV
If (x)l )kn/(n-1) dx
is, \Ilvflln/
< Cµn(S1).
This inequality is often attributed to N. Trudinger [86] but it appears in an earlier paper of V. Yudovich 1901. See [61]. We record it in the following theorem.
Theorem 1.4.4 There exist two constants Cn, c,,. > 0 such that, for any 0 < c < c,,, for any bounded set 0 C Rn and any function f E Co (S2),
JexpIc'' If(x)I
IlpfIIn
)n/(n-1)
dx < Cnµn( l)
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
26
A more precise result is known [66]. Namely, let an = nwn(i -1) Then, for each bounded domain l C 1[l n,
bf
J exp a iflf (x)I
E Co(1Z),
)fl/(fl_1)) dx < Cµn(Q)
st
for all 0 < a < an whereas, if a > an, dx : f E Co (0), Iloflln = 1} _ 00.
(aIf(x)Innn-1))
sup {f exp
This is proved by reducing the problem to a 1-dimensional inequality (thanks to Theorem 1.3.3) and then studying this 1-dimensional inequality. See, e.g., [5].
1.4.3
Higher derivatives
This short section describes Sobolev inequalities involving higher order derivatives. Most of the results easily follow from the first order case, but some additional arguments are needed to obtain optimal statements. For a function f E Co (Rn), let Okf = (O
... Vikf)(i1,...,ik) fail ...azkfl2,
lVkfl = (il ,...,ik )
and
IIVkfllp =
fn
\ 1/p
(lVkf(X)I'dX ) /
with the obvious interpretation if p = oo. By induction based on the case k = 1 (which is treated in the previous sections), one easily obtains the following statement.
Theorem 1.4.5 Fix two integers n, k, and 1 < p < oo. If 1 _< kp < n and q = np/(n - kp), there exists a constant C = C(n, k, p) such that
V f E C (Rn),
Ilf llq
Vkf lip.
If kp = n, there exist c = c(n, k) and C' = C(n, k) such that for all bounded subsets Il C Rn, b f E Co (SZ),
neap
( If(x)l I
p
/(-1
]dx
27
1.4. SOME OTHER SOBOLEV INEQUALITIES
If kp > n, let m be the integer such that m < k - n/p < m + 1 and set a = k - n/p - m. If a > 0, there exists B = B(n, k, p) such that for all f E Co (Rn) and all m-tuples (i1,.. . , i n), lai,
... aimf(x) - ail ... amf(Y)I
-
< BIIVkf llp-
x,yERsup
ix - Y1 x76Y
The result given above for kp = n, k > 2 is not optimal. The optimal result is as follows.
Theorem 1.4.6 If kp = n, there exist c = c(n, k) and C' = C(n, k) such that for all bounded subsets Sl C Rn,
V f E Co (St), f exp
l nicn-k>
dx < C'µn(S2).
ok flipl
C
s
x) I
(IA11
For the proof, recall the representation formula
Ax) = J(Pn,k(x - y), Vkf (y))d(y) where Pn,k is homogeneous of degree -n + k, 0 < k < n. More precisely, using multi-indices notation, we have
f(x) =
(-1)kk Wn_1
0a0.f(y) f Ra!Ix - ylnF'
dy
^
Ial=k
where 6 = (99)i = (x - y)/Ix - yI. Starting from this higher order representation formula and proceeding as for Theorem 1.4.4, one proves Theorem 1.4.6.
The case p = 1, k = n is a very special case. Indeed, we obviously have I f (x) I <
f +°° ... 00
f
+00
Ial
... anf (y) I dy
00
Hence
11f11.5 IIVnfll1. This is (at last) the higher-dimensional version of inequality (1.1.1).
The statement given above in the case kp > n is also not satisfactory because the case where n/p is an integer (i.e., a = 0) is excluded. For instance the case p = 2, k = 2, n = 2 is not treated. When n/p = Q is an integer, the optimal result is as follows.
CHAPTER 1. SOBOLEV INEQUALITIES IN
28
Theorem 1.4.7 Fix n < p < oo with n/p = £ an integer. There exists a constant C = C(n, k, p) such that, for any f E Co (Rn) and any (P-1)-tuple the function g = a=1 ... 8 _, f satisfies
y) - 2g(x)I
sup j I g(x + y) + g
IxI-
T.YERI l
(1.4.5)
} < CII Vkf IIp.
xoy
This is proved by the technique discussed above with the help of the following inequality.
Lemma 1.4.8 Fix n > 2. For any smooth function in a ball B, there exists a linear function Pf such that, for all x E B,
If(x)-Pf(x)I
C(n) J IIV2
(1.4.6)
(Y)I2dy.
To prove this, for any x, y E B, write y - x = p0 and f(y)
JP9.f(x+sO)ds
- f(x) =
6
P
- f a8 f(x + sO)sds + pas f(x + s6)18,P. As a$ f (x + s9) = (0, V f (x + s9)), this yields
If(y)-f(x)-(y-x,Vf(y))l < fo P IV2f(x+sO)Isds. Setting
F(z) = I IV2f the
Be
and integrating in polar coordinates around x in the ball B gives
12r If (x) - Pf(x)1 <
Seri n
in f 2n
where
Pf(x)
ff
°O
I V2f (Y) I d
Ix - Y J
F(x +
r8)sdspn_ldp
y
rn f[f() - (y - x, Vf (y))]dy.
With (1.4.6) at hand, the argument used for Theorem 1.4.2 yields
Vx E B, V f E C°°(B), If (x) - Pf(x)I < C(n,p)r2-nIPIIV2fI{p where B is a ball of radius r and 2p > n. Note that Pf(x + y) + Pf(x - y) - 2Pf(x) = 0
1.5. SOBOLEV-POINCARE INEQUALITIES ON BALLS
29
because Pf is linear. Hence, for any x, y E B, If (x + y) + f (x -
y) - 2f (x)I : 4C(n, p)r2"n"PIIo2f IIp
For any f E Coo (R) and any x, y such that Ix - yI < r, we can use this estimate in the ball of center x and radius r to obtain If (x + y) + f (x - y) - 2f (x)I <_ 4C(n, p)I x -
vl2-n/PIIo2f IIp
This obviously gives (1.4.5) in the case where k = 2 and n/p is an integer with 0 < n/p < 2, i.e., n/p = 1. The case where k _> 3 follows from this by induction, using the first statement in Theorem 1.4.5.
1.5
Sobolev-Poincare inequalities on balls
1.5.1
The Neumann and Dirichlet eigenvalues
Let St be an open bounded domain in Rn with smooth boundary 00. Classically, one considers the following two eigenvalue problems: (1) The Neumann eigenvalue problem
Au = \u { O=0
on fl on aft,
where n is the exterior normal along 00. (2) The Dirichlet eigenvalue problem
f Au = Au
t u=0
on St
on aft.
The boundary condition in (1) (resp. (2)) is known as the Neumann (resp. Dirichlet) boundary condition. Solutions of these problems are pairs (u, A) with u a smooth function and A a real. In both cases, integrating Au = au against u on H with the normalization fn u2dµ = 1 and integrating by parts, we obtain
A = a j u2dµ =
uLudµ
Jin f Ioul2d, + Jai s
=
f!VuI2dIz0.
anudµn-1
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
30
For the Neumann problem, u - 1, A = 0, is an obvious solution. In view of this, it is natural to set )1N(SZ) = inf I
f11
IDuI2dµ
:
fsu2dµ
u
0,
jud,L=0, u E C°°(cl) } 11
and
{fc 0, uE AD(1l)=infIVtddp:u AN(fj)
(resp. AD) is the smallest real A such that Indeed, one can show that (1) (resp. (2)) admits a non-constant solution. Observe that, by definition,
the inequality AN > c (resp. AD > c) is equivalent to saying that, for all u E C°°(1l) (resp. u E Co (SZ)),
ju2dp LIVuI2d1L. c
Poincare inequalities on Euclidean balls
1.5.2
There are two sets of Poincare inequalities on Euclidean balls, corresponding when p = 2 to the Dirichlet and Neumann eigenvalue problems.
Theorem 1.5.1 Let B = B(z, r) be a Euclidean ball of radius r and center z in JR". For any 1:5 p < oo, we have d f E Cow (B),
(f
1/p
IfIPdµ
1/p
< r f IVfIpdµ
(1.5.1)
and also 1/p
`d f E C°°(B), (JB If - fBlpdµ/}
1/p
< 2"r
(f IV f Ipdµ )
(1.5.2)
where fB = µ(B)-1 fB f dµ is the mean of f over B. Clearly, we can assume that B = B is the unit ball. For the proof of (1.5.1), we use (1.2.2), that is If (x) I s
1 f Iy -
I Vf (y) I
wn-1
xIn-1
dy.
This yields
f
Ifldµ <- Wn1
f MAO (f
Ix
yIn_1)
dy.
31
1.5. SOBOLEV-POINCAR.E INEQUALITIES ON BALLS As
dx
18 IX -
we get
f
_ IxIn-1 dx
< yIn-1
JB
wn-1
IfIdµ < f IVfIdtt,
which is the case p = 1 of (1.5.1). The case p > 1 can be obtained in a number of ways. We will use Jensen's inequality for the measure c(x)-1 Ix yI-n+11B(y)dy where x r= B is fixed and c(x) = fB Ix - yI1-ndy. Observe that c(x) < wn_1. By Jensen's inequality, I f (x)I
IVf(y)Ip d < P < c(x)P-1 y wn-1 JB Iy - xIn-1
IVf(y)IP d
1
.
wn_1 JB ly - xIn1 y
Integrating over x E 3 as in the case p = 1 gives the desired result. The proof of (1.5.2) is similar but uses Lemma 1.4.3 instead of (1.2.2). Let us note that the constants in (1.5.1), (1.5.2) are not optimal.
1.5.3
Sobolev-Poincare inequalities
For any open set Sl and 1 < p < oo, we set 1/p
IIf IIP,cz _
(J If Pdµ) I
With this notation the inequalities of Tkeorem 1.5.1 read V f E Co (3), IIf IIp,B < rII V f II p,B, `d f E C°°(B), Ilf - fBII p,B < 2nrllVf IIp,B
Sobolev inequalities localized in a given ball can be obtained by using the representation formula (1.2.1) and Lemma 1.4.3. Indeed, the kernel
K(x,y) = 1B(x)1B(Y)
1
Ix - yIn-1
is a kernel of weak type (n - 1) as defined in Section 1.2.2. Thus, the proof of Theorem 1.2.1 given in Section 1.2.3 applies here and yields the following inequalities. Note that the case s = p in Theorem 1.5.2 below reduces to Theorem 1.5.1 and only the case s = q needs to be proved. Note also that it suffices to treat the case where B is the unit ball.
Theorem 1.5.2 Fix 1 < p < n and set q = np/(n - p). There exists a constant C = C(n, p) such that for any smooth function with compact support in a ball B C Rn of radius r > 0, i.e., f E Ca (B), we have IIf IIs,B C C
rl+n(1/s-1/P)
Ilof IIp,B
(1.5.3)
32
CHAPTER 1. SOBOLEV INEQUALITIES IN RN
for all 1 < s < q. When f is a smooth function on the ball B which does not necessarily vanish on the boundary, i.e., f E COO(B), we have instead 11f - fBlls,B < C
r1+n(1/s-1/p) IIVf
(1.5.4)
II p,B
for all 1 < s < q. Here, fB is the mean off over the ball B. It is natural to wonder whether the ball B can be replaced by some more general bounded domain. Let S2 be a bounded domain in Rn. On the one hand, there is no difficulty with the case of functions with compact support
in f because any f E Co (1) can be extended to a function in C°(R) by setting f = 0 outside Q. Thus, Jensen's inequality and the usual Sobolev inequality in ][tn, i.e., d f E Co (]R'), IIf IIq < CII Vf IIp with q = np/(n - p), yield
V f E C (1l),
11f 118,0 <
Cµ(Q)11,-1/q
IIofllp,0
(1.5.5)
for all 1 < p < n, q = np/(n - p) and 1 < s < q. If SZ has diameter d, we can bound µ(S2) in this inequality by S2nd. On the other hand, consider the problem of whether or not the inequality V f E Coo (Q),
11f
- fc 11 pp < C(p,f )IIVfllp,n
(1.5.6)
holds true for some finite constant C(p, S2). It turns out that the answer depends in a subtle way on the regularity of the boundary of Q. Inequality (1.5.6) does hold on domains with smooth (or even Lipschitz) boundary but there are domains on which it does not hold. The same is true for Sobolev-
Poincare inequalities of type (1.5.4) with s > p. The book [61] gives an account of what is known and references to the literature. Finally, note that it is easy to treat the case of bounded convex domains by adapting the argument given above in the case of Euclidean balls. All the results described above hold for bounded convex domains with the radius r of B replaced by the diameter d of the domain.
Chapter 2 Moser's elliptic Harnack inequality 2.1
Elliptic operators in divergence form Divergence form
2.1.1
Second order differential operators with possibly non-constant coefficients can be written in a number of different ways. In particular, if
L = - E ai,j(x)aiaj + E ci(x)ai + c(x) i
i, j
we can also write, if the ai,j are smooth functions,
L=-
ai(ai,j(x)8j) +
bi(x)Oi + c(x)
(2.1.1)
where
bi(x) = ci(x) +
aeae,i.
Denote by A(x) _ (ai,j(x))1
n
divXaixi for any smooth map X : RI -> Rn. Then L can be written as
Lf = -div (AV f) + (b, Of) + c. In this case, we say that L is in divergence form. The distinction between operators in divergence form and others is much more important than one would think at first sight. Distinct sets of analytical tools are used and different results are obtained for the two types 33
34
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
of equations. Already, at a naive level, what it means to have no lower
order terms (i.e., no terms of order zero or one) is different: the operator L = - Ei j ai,, (x)8i8j has a lower order term when written in divergence form
L=-
i,j
aiai'jaj +
j
aiai,j (x) aj i
Another way to witness the differences between these two forms of second order differential operators is to consider what happens when the coefficients ai,j are not smooth but simply measurable and bounded. For simplicity, we consider only the case of operators with no lower order terms. Let ai, j (x) be measurable bounded coefficients. On the one hand, it is clear that the operator
- 1: ai,j (x)aiaj i, j
Jf
is well defined on, say, C2 functions. On the other hand, making sense of
-
ai(ai,j(x)aj)
requires some work. Actually, it is hard in general to describe a set of functions on which the latter operator acts in a reasonable sense. The clue is to define E ai (ai,j (x)aj ) i,j
-
in a weak sense by saying that - Ei, j 8i (ai, j aj f) = g for functions f, g locally in L2 with IV f I locally in L2 if and only if
J
ai(x)aif (x)8jcb(x)dx =
or all test functions ¢ in a suitably chosen space, e.g., Co (R' ). In what follows we will study second order differential operators in divergence form satisfying a uniform ellipticity condition.
2.1.2
Uniform ellipticity
The aim of this chapter is to study certain properties of (weak) solutions of the equation Lu = 0 (2.1.2)
on a Euclidean ball B, when
Lf = - E ai(ai,jajf)
ij
(2.1.3)
2.1. ELLIPTIC OPERATORS IN DIVERGENCE FORM
35
is in divergence form and the real matrix-valued function A = (ai,j) is uniformly elliptic. This essential hypothesis means that there exists 0 < A < 1 such that tt Sr
`dx E II8n, dS,
E" {
AICI2
<-
Ei,j a2,3(x)Fisj,
A-'IeIIS1 ? I Ei,; ai,j(x) (jI
(214 )
If (ai,j) is symmetric, this means that its eigenvalues belong to [A, A-1] 0. Observe in particular that (2.1.4) implies Ei,j It is helpful to introduce the following Sobolev spaces. Let W0(B) be the closure of Co (B) for the norm I I f II2 + IIVf IIZ Let W1,2 (B) be the closure of C°°(B) for the same norm. Thus Wo'2(B) C W1"2(B). One can show that W1"2(B) is also the space of all functions in L2(B) whose first order partial derivatives in the sense of distribution can be represented by L2(B) functions. By definition, a (weak) solution of (2.1.2) in the ball B is a function u E W1,2 (B) such that `d 4, E Wo'2(B),
JRn
ai,j(x)c?su(x)8j0(x)dx = 0.
(2.1.5)
_
A (weak) subsolution is a function u E W1"2(B) satisfying
J
1: ai,j(x)Oiu(x)8j4,(x)dx < 0
(2.1.6)
for all 0 E Wo'2(B), 0 > 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
aj,ja%uaj(ua-10).
%, j
Moreover, 0`0 is in Wo'2(B). Indeed, ua-1o E L2(B) and ua_' V
V (u"`1o) = (a - 1)ua-24 Vu +
E L2(B)
since e < u < c. Hence,
j
t,j
aa%uaajgdpaJB
< 0. %j
2.1.3 A Sobolev-type inequality for Moser's iteration In this chapter we will use Sobolev inequalities to study some properties of the positive solutions of the equation Lu = 0 using a technique known as Moser's iteration. We will use the Sobolev inequality in the following form. Let 3 be the unit ball in R. For n > 2, Theorem 1.5.2, (1.5.3), yields `d f E Co (3), IllIIq,B < C.11 Vf II2,B
where q = 2n/(n - 2). We also have the Holder inequality, IIfIIP,B _ IIfIIq,BIIfI12,B
for any 1
1-ry+1-i' p
q
2
For ry = n/(n + 2) this gives p = 2(1 + 2/n). Hence f If12,1+2/n)dp < IIfIIQIIfIl2/n
Combining with the Sobolev inequality above, we get V f E Co (13), f If 12(1+2/n)dp < cn J V f B
B
I
2/n
I2dp J IfI2d/ B
(2.1.9) This inequality is still valid, of course, for all functions with compact support
in B and square integrable partial derivatives, i.e., for f E Wo'2(B). For the case n = 1, 2, the above inequality is still valid (with the same proof) if we replace n in the inequality by any given number v > 2 (see
38
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
Theorem 1.5.2). In particular, in dimension 1 or 2 we can use v = 3 and get the inequality
f If
12(1+2/3)dµ
a
<
C2
(
l (\1s If 12dµ)
2/3
(2.1.10)
\fa I V f I2dµl
for all f E Co (l!$). In what follows, we will implicitly use (2.1.10) instead of (2.1.9) when the dimension (i.e., n) is 1 or 2. The proofs given in the next few sections would work as well if we used the usual Sobolev inequality instead of (2.1.9). However, it is somewhat
interesting to note that one can work with (2.1.9) which, a priori, is a weaker inequality. Moreover using (2.1.9) is technically convenient when one treats parabolic (versus elliptic) equations.
2.2 2.2.1
Subsolutions and supersolutions Subsolutions
Let L be given by (2.1.3). Assume that the uniform ellipticity condition (2.1.4) is satisfied.
Let B be a ball in Rn with volume V = µ(B). In every proof below, we assume without loss of generality that B is the unit ball. Indeed, if B is a ball of radius r and center z then v(x) = u(z+rx) is defined on the unit ball and satisfies L,.v = 0 where L,.v = - Ei'j 8=as,jv with a1,3(x) = a;j (z + rx). This operator L, is uniformly elliptic if and only if L is, and with the same constant A. Note that this would not work if we had to worry about the size of the derivatives of the coefficients.
Let u be a positive subsolution of the equation Lu = 0 in B. All the constants Ci appearing in this section are independent of u and B. Lemma 2.2.1 There exists a constant C1 = C, (n, A) such that, for any
02, LB
< C1(p - p')2V 1B (P2fuPd/.L 9
0
)
with 8=1+2/nifn>2and9=1+2/3forn=1,2. By replacing u by u + e, c > 0, we can assume that u is bounded away from
zero. The desired inequality is then obtained by letting a tend to zero at the end. Let us recall that our hypothesis is that VO E
W01,2
(B), f
< 0.
(2.2.1)
2.2. S UBSOL UTIONS AND S UPERSOL UTIONS
39
Let G : (0, oo) - (0, oo) be a piecewise C1 function such that G(s) = as for large s. Assume also that G has a non-decreasing, non-negative derivative G'(u). Hence, G is non-decreasing and G(s) _< sG'(s). Finally, define
H(u) > 0 by H'(u) =
G'(u), H(0) = 0. Observe that H(u) < uH'(u) as
well.
Let zfi be a non-negative smooth function with compact support in B.
Set 0 = 2G(u). Then 0 > 0 and ¢ E Wo'2(B) (here we use the fact that G(s) = as for large s). Thus we can use 0 in (2.2.1). We have 1: ai,jOiuOj¢ _ ,O2G'(u) E ai,jOiu8ju + 2V)G(u) E ai,,OiuOj,0. i,j +,j i,7 Hence, by (2.2.1), jp2G'(u)
JB
I
ai,jOiuOjudp G 2
JB OG(u)
aijOiuOjbdp i, j
Uniform ellipticity and the inequality G(u) < uG'(u) give 2G'(u)IVuI2dµ <
JB
2A-2 f
*uG'(u)JVuJJV,0Idµ
B
The Cauchy-Schwarz inequality yields 02G'(u)Iou12dµ ID
1/2
< 2A -2 UB 02G'(u)JVuI2dA)}
(
\f
B
\ 1/2
V GJ2dp)
Thus
f '&2G,(u)IVuI2dµ G B
4A-4f u2G'(u)IV0I2dµ. B
As V(iH(u)) =,OH'(u)Vu + H(u)Vip and IViH(u)12 < 2(02IH'(u)I2IVUI2 + H(u)21V 12) < 2(,02G'(u)IVuI2 + u2G'(u)JVtJ2), we obtain fB IV bH(u)I2dp < 2(1 + 4A 4) fB u 2G'(u)IV I2dµ.
(2.2.2)
Now, zliH(u) has compact support in B and is in W3'2(B) (this uses the fact that G(s) = as for large s). Thus we can apply the Sobolev inequality
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
40
(2.1.9) which gives
2(1+2/n)dp
(2.2.3)
lB IbH(u)I 2/n
< Cn (JB IV bH(u)I2dµRbH(u)I2dµs 2/n
2C(1+4A-4)((J
2Iul2G'(u)dt)
B IV
(J. I 1+2/n
-4)IIo,0II2II1pII
< 2C, (1 +4A
u2G'(u)dµ)
/n 00
(2.2.4)
$upp(+G)
Given 0
=tin
p'B, ip = 0 in pB and IV &I <- 2/(p - p'). Then, (2.2.4) yields
J
I H(u)I2' 'dµ < 8Cn(1 + 4A-4)(p
'B
- p)-2 (JPB u2G(u)dp)
(2.2.5)
with 0 = 1 + 2/n. Fix p > 1 and some large N. Set HN(S
Sp/2
= { N(P/2)-1s
ifs < N if s > N.
Computing, we find that this HN corresponds to taking
GN(s) = fos H '(t)2dt p2
4(p -1) l
ifs < N
s p-I
1
Pr-
4(-Np-2(s
- N) + Np'1
For any p > 2, these GN's have all the required properties. Moreover, as N tends to infinity, HN(s) -> sp/2, G'N(s) --> (p/2)2sp-2. Hence, (2.2.5) yields
LB
\e
u"9dp < 2C,2 (1 + 4A-4)(p - p')-2 (p2
u'dµ ] fpB
.
///
This proves Lemma 2.2.1. Our next step is to prove an LP mean value inequality.
Theorem 2.2.2 There exists C2 = C2(n, A) such that, for any 0 < 5 < 1, any p > 2, and any positive subsolution u of Lu = 0 in a ball B of volume V, 6Bp{up} < C2(1
- 5)-" (V-1 fB updµ)
.
2.2. SUBSOLUTIONS AND SUPERSOLUTIONS
41
We first prove a somewhat weaker version where the constant C2 depends also on p > 2. Fix p > 2 and 0 < b < 1. For each integer i > 0, set pi = phi,
Po=1,
=i
pi=1-(1-b)E2-i, i>1. j-1 Then pi+1 - pi = (1 - b)2-'-', pi+l = pi6, and, by Lemma 2.2.1, upi+ldµ
< C(1 -
(p2f
b)-2 22(i+i)
Jn+1B
up;dpl
J
iB
or
\ 1/pi+1
upi+1dui
< [C(1 -
b)-211/pi+i 22(i+l)/pi+i
pi/p'
UpiB uPi dli
p i+, B
1/pi
fu1
for i = 0, 1.... with C = 2C,2,(1 + 4A 4). This yields upi+idpl1
l
1/pi+1
11/p
< C(n)C(p) [C(1
where
C(n) = Observe that
22(E'.ie-j)
C(P) = e2Eo B-ito$(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 <- IIUIIOO,B2llullp/2 p,B for any ball B, we get {
IluhIoo,cB
/ J(1
` Or)-n/2llulloo,pB y
where J = CIIuhIp/B.
Fix 6 E (1/2,1) and set co = 6, Qi+1 = of + (1 - ai)14. Then 1 - of = (3/4)i(1 - 6). Applying the above inequality for each i yields Ilulloo,aiB :5 (4/3)'""2J(1 - 6)-n/2llulloo,o/+,B
Hence, for i = 1, 2, ..., Ilulloo,6B < (4/3)(n/2)E0 17(1-P/2)'[J(1
-
6)-n/2jEo 1(1-P/2)'llull(1-p/2)i
Letting i tend to infinity yields IIUIIoo,6B < (4/3)2n/921(1 - 6)-n/2J)2/P,
that is, lI uhI oo,6B s (4/3)2n/P2C2/P(1 - 6)-n/PII uhIP,B
which is the desired inequality.
2.2. SUBSOLUTIONS AND SUPERSOLUTIONS
2.2.2
43
Supersolutions
We now proceed to derive some inequalities for supersolutions of Lu = 0 with L as in (2.1.3) satisfying the uniform ellipticity condition (2.1.4). The first result we derive below is similar to Theorem 2.2.3 except that it deals with negative powers u-p of a positive supersolution u. The second result of this section deals with small positive powers up of supersolutions. These two results (Theorems 2.2.4 and 2.2.5 below) are proved in a form that describes precisely how the corresponding inequalities depend on the parameter p. This will be crucial later on in our proof of Theorem 2.1.1. In this subsection, we let B be a fixed ball with volume V and we let u be a positive supersolution of Lu = 0. In the proofs, we will always assume as we may that B is the unit ball. The different constants Ci appearing below do not depend on u or B.
Theorem 2.2.4 There exists a constant C4 = C4(n, A) such that, for any 6 E (0, 1) and any p E (0, oo), we have
supfu-PI < C4(1 - 6)-" 1 V
6B
J u-pdµ. B
Before embarking on the proof, let us observe that a slightly weaker form of this result can be obtained by applying Theorems 2.2.2, 2.2.3 to the positive, bounded subsolutions (u+e)-1, e > 0. The improvement offered by Theorem 2.2.4 is that the constant C4 above is independent of p whereas the constant obtained by applying Theorem 2.2.3 would be C3 = C(4/3)21/P which tends to infinity as p tends to zero. Later on we will use the full strength of this improvement to obtain Harnack inequalities through the use of Lemma 2.2.6 below.
By replacing u by u + e, e > 0 and letting a tend to 0 at the end of the argument, we can assume that u is bounded away from 0. For any W01,2 E (B), we have
f > aijaiu8jbdp > 0.
(2.2.6)
S1.7
We set -/3ua-11/i2 with /3 < 0 and Wo'2(B) and 8iq5 =
-2,QuR-1'YBiY'
E Wo'2(B), zli > 0. As 4 is in
- 0(/3 -
we have
-2/3
f iu'-a E i'j
/3()3 - 1)
f
J
*20-2 > aijaiu8judp > 0. i>7
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
44
If we set w = uQ/2, we have OOw = ((3/2)ui, /2i-'a2u and the inequality above reads
-4
J
biw E a2,a1wa 'cbdp -
J *2 E
4(I#QI 1)
i > 0.
Hence
f *2
a2,ja1wa,wdp < if -rbw E J
,9
By uniform ellipticity, this yields
J v2lvwl2dp <
A-2 f iPwlvwllv1ldp
and thus, by the Cauchy-Schwarz inequality,
J
b2`vwi2d/, < a-4
I v*}2w2dp.
J
Finally, we write this in terms of z/'w using V(,Ow) = wV 1'+,OVw and get
f
I V bwl2dp < 2(1 + A-4)
f
(2.2.7)
I V 12w2dp.
This is exactly analogous to (2.2.2), and by the argument used after (2.2.2) we obtain
f
Iw{28dµ < 4Cn2,(1 + a-4)(p B
- P')-2 QB IW
12dp)
e
where 0 < p' < p < 1 and 0 = 1 + 2/n. Returning to our supersolution u, we have e
upedp < 4Cn(1 + a-4)(P -
uadp1 p')-2\ (f pB
JP1B
for all 0 < 0. This is analogous to Lemma 2.2.1, and the iterative steps of the proof of Theorem 2.2.2 yield the inequality stated in Theorem 2.2.4. It is not possible in general to control the sup norm of a positive supersolution in SB in terms of some LP norm over B. Still, one has the following weaker result.
Theorem 2.2.5 Fix 0 < po < 0 = 1 + 2/n (0 = 1 + 2/3 if n = 1, 2). There exists a constant C5 = C5(n, A, po) such that for all 6 E (0, 1) and for all p E (O,po/e),
(1 f V
bB
uPOdp f
/
1/ < 1C5(1 -
S)-2n+271/P-1/PO !
1
(V
f uPdp B
1/P
2.2. SUBSOLUTIONS AND SUPERSOLUTIONS
45
As usual, we can assume that u is bounded away from 0. This time, we take
=
< #:< po/9 < 1 and we set c = 1-(po/9) > 0. We also set w = u,6/2. As in the beginning of the proof of Theorem 2.2.4, we obtain 2,(3
f3uQ-izV l,2 in (2.2.6) with 0
Viua-i
J
a1,ja=u8jbidµ +,(3(3 - 1) J V,2u,6-2 E az,38,uO,udµ > 0. i,j i,j
This yields
1 - p J v)2 Q
ai,j8iwajwdµ J
i,j
Tw
i,j
ai,i aiw 9j dµ
The last inequality, the definition of e, the upper bound on ,Q and uniform ellipticity yield Ef V)ZIOwI2dµ < A-2 f 7wIv t IIOwIdp.
As usual, we use this to deduce that f IV ciwl2dµ < 2(1 + )1-4E-2) f IoV)I2w2dµ.
This is (again) exactly analogous to (2.2.2) and by the argument used after (2.2.2) we obtain
f Iwl2edp < 4C,,(1 + A-4E-2)(p p'B
- p')-2 QB Iwl2dµle)
where 0 < p' < p < 1 and 9 = 1 + 2/n. Returning to our supersolution u, we have
LB
uQedµ < 4CC(1
+ )-4e-2)(p
- p)-2
0 UpBu-Odµ\
(2.2.8)
for all 0 <,3 < po(i + 2/n)-1. This is analogous to Lemma 2.2.1, but the iterative steps that will now be used to finish the proof of Theorem 2.2.5 are somewhat different from those used in the proof of Theorem 2.2.2. Define pi = po9-i where 9 = 1 + 2/n. We first prove Theorem 2.2.5 for these values of p. Thus, fix i > 1, and apply (2.2.8) with /3 = pi9j-1,
j = 1, 2, ... , i, and with p' = aj-1i p = aj, where vo = 1 and of-1 - ut = 2-c(1 - 6). Observe that pi0j-1 < po(1 + 2/n) for j = 1, ... , i as required for the validity of (2.2.8). Hence, for all j = 1, ... , i, up`B'dµ < C(1 Q,B
-
upio-7-1dµl/
b)-222j
(Ii-IB
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
46
where C = 4C,2,(1 + A-4E-2). This yields
LB
l
( udµ 1
up°dµ < 2E0'('-?)e' [C(1
B
/J
Finally, observe that i-1 (n/2)29t-1 < (n/2)3(9:
j)9? <
- 1) = (n/2)3(po/pp - 1),
0
t
Qt=1- E2-' (1-6)>b, and
8t-1
i-1
(n/2)(po/pt - 1).
B' = 9 -1
a
This gives 1/po 111
uPOd/
[22(n/2)3C(1
\
(LB
1/pi
-
b)-2]n(1/p;-1/po)/2
\j u di /
)
B
that is, uPOd11/po
(LB
with
C=
<
1
[C'(1 - 6)-]1/:po UEB uPidµ
1/p
22(n/2)4Cn/2.
To obtain the desired inequality for any p E (0, po/9), let i _> 2 be the
integer such that pt S p < pi-1. Then 1/pt - 1/po < (1 + 9)(1/p - 1/po). Thus, by Jensen's inequality, uP°d)
(LB
1/po
1/pi
< [C'(1 -
/J
(lB
< [C'(1 -
-< [C"(1 -
\1/p
6)-n](1+B)(1/p-1/po) 1
(fn u"dµ/
b)--(14.0)nl1/p-1/po J
with C" = (C')1+e
up:dµl
6)]1/p+-1/po
(i updpl\1/p I\ B
/1
2.2. SUBSOLUTIONS AND SUPERSOLUTIONS
47
2.2.3 An abstract lemma This section presents an elementary but subtle lemma due to Bombieri and Giusti [9) which simplifies considerably Moser's original proof of the Harnack inequality. It replaces the use of the well-known John-Nirenberg inequality (i.e, the exponential integrability of BMO functions). This simplification is even more significant when dealing with parabolic equations. See [65]. Consider a collection of measurable subsets UQ, 0 < o < 1, of a fixed measure space endowed with a measure µ, such that U,, C Uo if a' < a. In our application, Uo will be aB for some fixed ball B C R1.
Lemma 2.2.6 Fix 0 < b < 1. Let -y, C be positive constants and 0 < ao < oo. Let f be a positive measurable function on U1 = U which satisfies Of IIao,Uoi :5 [C(o -
0)-7µ(U)-1]
1/Q-11Qo
IIf IIa.U.,
for all or, o', a such that 0 < b < o' < or < 1 and 0 < a < min{1, ao/2}. Assume further that f satisfies
/'(log f > \) <- CA(U)a-1
for all A > 0. Then Ill IIQ0,u6 < Aj(U)'/Qo
where A depends only on b, ry, C, and a lower bound on ao.
For the proof, assume without loss of generality that µ(U) = 1 and set
?P =Ri(o)=log(IIfi(00,uo), for0 0/2 and where log f < ip/2, we get 11f IIQ,ue
<-
0/2)'10-11aO + e'P/2
11f Ila0.u,µ(log f >
<
e''(2C/')11Q-1100
+ e'P/2.
(2.2.9)
Here, we have used successively the Holder inequality and the second hypothesis of the lemma. We want to choose a so that the two terms in the
right-hand side of (2.2.9) are equal, and 0 < a < min{1, ao/2}. This is possible if
(1/a -1/ao)-' = (2/-i) log(0/2C) < min{1, ao/2}, and this last inequality is certainly satisfied when V)
- A,
(2.2.10)
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
48
where Al depends only on a lower bound on ao (note that one can always take C > 1). Assuming that (2.2.10) holds and that a has been chosen as above, we obtain IllIIQ,u
(2.2.11)
<-
The first hypothesis of the lemma and (2.2.11) yield
Co") < log [(C(0' -
Q')-y)11«-110'0
2eP/2J
(1/a- 1/ao) log (C(o' - a)-ry) +,0/2 + log 2 for b < a' < a < 1. By our choice of a specified above we have
log(C(a - a')-") + 1 + log 2. 2 log(,0/2C On the one hand, if 1' 1 2C3(a - Q)-27,
(2.2.12)
we have
/i(a) < 4?/i+log2. On the other hand, if one of the hypotheses (2.2.10), (2.2.12) made on is not satisfied, we have
&(a') 5 iP< Al + 2C3(a In all cases, we obtain
37*1 + A2(a - Q')-try
(2.2.13)
where A2 depends only on C and on a lower bound on ao. For any sequence
0<S=0'O
V)(ao) < (3/4)'V(aH) + A2 E(3/4)'(aa+l
-
0
and, when i tends to infinity, b(S) < A2 1:(3/4)'(aa+1
- 0,;)-2y.
0
The desired bound follows if we set aj = 1 - (1 + j)-1(1 - S).
2.3. HARNACK INEQUALITIES AND CONTINUITY
2.3 2.3.1
49
Harnack inequalities and continuity Harnack inequalities
The main goal of this section is to prove the Harnack inequality of Theorem 2.1.1. In fact, we prove first a weaker inequality (Theorem 2.3.1 below)
which holds for positive supersolutions. Then, the Harnack inequality of Theorem 2.1.1 immediately follows from this and Theorem 2.2.3. Let L at (2.1.3) be a divergence form operator satisfying the uniform ellipticity condition (2.1.4).
Theorem 2.3.1 Fix 6 E (0,1) and p, 0 < p < 6 = 1 + 2/n (0 = 1 + 2/3 if n = 1, 2). There exists a constant C = C(n, A, 6, p) such that for any ball B and any positive supersolution u of Lu = 0 in B we have
p(5B) aB
updp < Cinf{up}. aB
We assume that B is the unit ball. We want to apply Lemma 2.2.6 to a-lu
and ecu-i where c is a well-chosen constant. Pick p so that 6 < p < 1 (for instance p = 6 + (1 - 6)/2). Set U = B' = pB, V' = p(B'), and set U, = aB' = opB. Theorem 2.2.5 shows that the first hypothesis of Lemma 2.2.6 is satisfied by any constant multiple of u with ao = p < 0. To verify the second hypothesis of Lemma 2.2.6, we apply the L2-Poincar6 inequality in the ball U = B' to the function log u. This yields
f I log u- cI2dµ < C.
J
I V log ul2dp
BI
1
where C,i depends only on n and
c = (log U)BI _'
JB log u dp
is the mean of log u over B'. Now, as u is a positive supersolution, ai,,aiuO B
>0
=,i
0 where for all non-negative test functions ¢ E W01'2(B). Pick .0 = u-12 has support in B, 0 < ip < 1, = 1 in B', IV 'I < 2/(1 - p). Then Y ai;0tuOj4 = -02 i,7
at,,azva v + 2,0 E ai,,(9iva,0 i,J
i,.t
where v = log u. Hence
f
P2
aija=vajV5dp
E a=,AO,vO,vdp < 2 IV) ij
ij
CHAPTER 2. MOSER'S ELLIPTIC HARNACK INEQUALITY
50
Using uniform ellipticity and the Cauchy-Schwarz inequality as in the proof of Theorems 2.2.2, 2.2.4, 2.2.5, we get
f O`Ovi2dp < 2a-4 / `V 12dp < 8Q.(1
- p)-2)\-4 = C(n, A, 6).
Thus
tp(B' n {I logu - cl > t}) < j I logu - cIdp 1/2
<
(/
1 log u - cl2dp)
\ B'
< C(n,A,6). This shows that the second hypothesis of Lemma 2.2.6 is satisfied by f = e-cu (and also by ecu 1, for that matter). Hence, we can apply Lenuna 2.2.6 to f = e"`u and conclude that, for any 0 < p < 1 + 2/n, (2.3.1)
JI Ujjp,SB < Ae`
with A = A(n, A, 6). Likewise, Theorem 2.2.4 and the computation above
show that Lemma 2.2.6 applies to ecu-1 with the same c = (log u)5 as above. In this case, ao = oo. Hence
sup{u-' } < Ae-°.
(2.3.2)
6B
For 0 < p < 1 + 2/n, putting (2.3.1) and (2.3.2) together yields 'IUI+p,6B < A2 6inBf{u}
which is the desired inequality.
2.3.2
Holder continuity
We now derive the last part of Theorem 2.1.1, that is, the Holder continuity estimate Cr_a sup{u} (2.3.3) sup I !u(x) - u(y) I
x,yIz - y'a
JJ1
B
for positive solutions of Lu = 0 in the ball B of radius r. Let us mention that this result was first proved by De Giorgi in 1957, before the present proof was given by Moser in 1961. Also, Nash's paper [67], published in 1958, proves the Holder continuity of the solutions of the related parabolic equation. To prove this estimate, let B be the unit ball and set
M(p) = ma {u}, m(p) = mi {u}, 0 < p < 1.
2.3. HARNACK INEQUALITIES AND CONTINUITY
51
Fix0
max {M - u} < C inf {M - u}, (pl2)B
(pl2)B
max {u - m} < C inf {u - m},
(p/2)B
that is, Hence
(p/2)B
M-m'
M-m+M'-m'
that is,
M'-m'
with a = log[(C + 1)/(C - 1)J. It follows that f w(p) < 2°p°w(1). After scaling, this yields the following lemma,
Lemma 2.3.2 There exists a = a(n, A) such that any solution u of Lu = 0 in a ball B satisfies Vp E (0, 1),
sup {Iu(x) - u(y)I } < 21+'p0 sup{u}. x,yEpB
B
We can now finish the proof of (2.3.3). Assume that B is the unit ball. Fix 6 E (0, 1). For x, y E 6B, consider two cases. If Ix - yI > (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 <
with h = If
IP(7-1) and
IIhII91(9-P)II911q1P
g= I V f I P. This yields 1/p
IP(7-1) I V fIPd/
(JM If C
f
) 1/P-1/4
If I
(f Ivf
I qdA
1/q
Hence,
(f if
7
II f II; /(Y-P) <- 7C(M,p, v)
11p-11q
IP4('Y-1)/(9-P) dµ)
qdµ
(J
IVfI
)
1/4
Picking y = q(n - p)/p(n - q) and computing y - 1 = n(q - p)/p(n - q) yields IlfIIgn/(n-q)
< q(n-p)C(M,p,v)IIVfIIq p(n - q)
as desired.
3.1.3
Sobolev inequalities and volume growth
The next lemma introduces a family of (a priori weaker) inequalities that can be deduced from a Sobolev inequality. Our aim in this section is to show that weak forms of Sobolev inequalities are sufficient to imply a lower bound for the volume growth of M. Lemma 3.1.4 If M satisfies an (L", v)-Sobolev inequality V f E Co (M),
II f II Pv/(v-P) <- C(M, p, v) II V f IIP
then it also satisfies V f E Co (M),
Ilf 11,< (C(M,p, u) II Vf llp)°IIf II!-'
for all 0
r
9(1-1)+(1-B). v p s
Set q = vp/(v-p). Then 1/r = 0/q+(1-0)/s and, by the Holder inequality, Ilf llr -< IIf tI9qIIf II!-B
This yields the desired inequality. The possible range for s is actually (0, vp/(v - p)) and the range of r is (s, vp/(v - p)). Note that when 0 = 0 one must have r = s and the conclusion of the lemma is trivial. The following result generalizes Theorem 3.1.1.
58
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
Theorem 3.1.5 Assume that the inequality V f E Ca (M),
(CIIVfIIp)°IIfhl3-B
is satisfied for some r, s, 8 with 0 < s < r < oo and 0 < 0 < 1. Assume also that
0+1-0-1>0. p
r
s
Then V(x,t) > ct"
with v defined by
e_e 1-0 V
p+
s
1 (3.1.2)
r
and the constant c given by c=
2-v216r-v19C-v.
In particular, if M satisfies an (L', v) -Sobolev inequality for some p, v with 1 < p < v, then n > v and the volume growth function V (x, t) satisfies inff {t"V (x, t) } > 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. <-
(1/2)IlVf111.
Theorem 3.1.5 applies with 0 = 1, r = oo, p = 1 (s is irrelevant when
0 = 1) and yields v = 1 and V(t) > 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 <_ (1/2) II.f Vf Ilfll2 <- (1/2)11Vfl12'2IIflI2'2IIfIIi That is, VCo (R),
IIfII2 5
3.
This is the Nash inequality, in one dimension. In the next section, we will take a closer look at this type of inequality. For now, observe that we can apply Theorem 3.1.5 again, this time with r = p = 2, s = 1, 0 = 1/3. Then v is defined by 1/(3v) = 1/6 + 2/3 - 1/2 = 2/6, i.e., v = 1 again. From the proof of Theorem 3.1.5, it is obvious that one can work under a weaker hypothesis and obtain the following statement.
Theorem 3.1.6 Assume that the inequality V f E Co (M), sup {tIL(I f I > t)1/r} t>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) <- CIIof112
obviously implies the weak Sobolev inequality
sup{tp({IfI >
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/") <- (CIIof
A different use of the Holder inequality leads to Ilf 112 -< (CIIof 112)"/("+2'
11f 11 2/(v+2)
which one often writes 112(1+2/,,) < C2IIof
IIf 1121If JIi/v. This last inequality is called a Nash inequality. It first appeared in the 1958 paper of Nash [67] concerning the regularity of solutions of parabolic uniformly elliptic equations in divergence form in R". Nash did not deduce this inequality from the Sobolev inequality. Instead, he gave the following direct proof.
3.2. WEAK AND STRONG SOBOLEV INEQUALITIES
61
Theorem 3.2.1 There is a constant Cn such that any smooth compactly supported function f on Rn satisfies
,If
II2(1+2/n) <
C, IiVf 112IIf 111/n.
Let f be the Fourier transform of f . Then 1102
= lifll2 =
f{i5R
< SZnRnlIjIl2 00 + R-2
<
J41>R
f f1nR"IIfIII+(2irR)-2IIVfII2
Optimizing in R yields Ill II2
(2
n)
(stn/47r)2/(n+2)
i/(n+2)
II Vf I12n/(n+2) Ill II
which gives the desired inequality with Cn = (2 +
n)1+2/n(On/4ir)2/n
Nash's proof is given here to point out how different it is from the proofs
of the corresponding Sobolev inequality that we have seen. The question naturally arises of whether or not this is yet another proof of the L2-Sobolev inequality in Rn. That is, can we easily deduce the Sobolev inequality IIfII2v/(v-2)
CIIVfII2
from the Nash inequality IIfII2(1+2/v)
maybe with different constants C? There is an interesting twist here because the Nash inequality of Theorem 3.2.1 is valid for all n > 1 whereas the corresponding Sobolev inequality is valid only for n > 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} <
(CJIVfIIp)e
t>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),
<_
(CIIVfIIp)BIIfII8-e.
Ilfllr
(S°5)
In these inequalities, we can think of 0 < r, s < oo, 0 < 9 < 1 as parameters and we would like to understand the meaning of these parameters. For the time being, let us simply observe that (S,1.,8) (the parameter s plays no role when 0 = 1) is the classical Sobolev inequality Of
111< li (IVf IIp-
Similarly, (SS,s) (again, the parameter s plays no role when 0 = 1) is the weak Sobolev inequality sup {tp(I f I > t)1/r} < CIIV f Jlp t>o
Finally, when p = 2, (S2,1) is the Nash inequality 11f
112(1+(1-19)/0) <
C2IIVf1121lflli(1-0)/e
Observe also that, by Theorem 3.1.6, each of these inequalities implies inff{t-"V(x,t)} > 0
where the parameter v is defined by 1/r = 1/p - 1/v for the first two inequalities (assuming p < r) and by 2/v = 9/(1-0) for the Nash inequality. It turns out that one of the keys to understanding these inequalities is to consider yet another parameter, call it q = q(r, s, 9), which belongs to (-oo, 0) U (0, +oo) U {oo} and which is defined by
1-0 r1 _9q+ s
(3.2.1)
Observe that q is related to the parameter v of Theorems 3.1.5, 3.1.6 by 1
1
1
q
p
v
(3.2.2)
It is a fundamental difference between q and v that q can be computed from (r, s, 0) without explicit reference to p whereas v cannot.
3.2. WEAK AND STRONG SOBOLEV INEQUALITIES
63
We will prove below that, roughly speaking, any weak inequality (S*o so) for some fixed ro, so, Bo implies the full collection of strong inequalities
(SB8)for all 0
More precisely, this is correct when 1/q(ro, so, 00) < 0.
When 1/q(ro, so, 60) > 0, one needs to assume also that q > p. The proof will depend on elementary functional analysis arguments and repeated application of the given inequality to functions of the type (f u)+ A v with u, v > 0, u A v = min{ u, v}, u+ = max{u, 0}, where f is a given fixed function in Co (M). Here, we consider the parameter p as fixed once and for all. Later, in Section 3.2.6, we will also consider what happens when p varies. Then the parameter v will play a crucial role. Roughly speaking, any of the
-
inequalities (S* so) for some fixed r0i so, 00 and p0 implies all the inequalities (SB8) for all p > po and where the parameters r, s, 0, p satisfy both (3.2.1) and (3.2.2).
3.2.3
The case 0 < q < oo
In this section the number p, 1 < p < oo, is fixed and all Sobolev-type inequalities are relative to IIV f IIp. The main result of this section is described in the following theorem.
Theorem 3.2.2 Assume that (S*oe 0) is satisfied for some 0 < r0i so < 00 and 0 < 00 < 1 and let q = q(ro, so, 00) be defined as in (3.2.1). Assume that p < q < oo. Then all the inequalities (SB8) with 0 < r, s < oo, 0 < 0 < 1 and q(r, s, 0) = q are also satisfied. In particular, there exists a finite constant A such that V f E Co (M),
IIfIIq < ACIIVfIIp
where C is the constant appearing in (S,'0,000). That is, M satisfies an (L", v)Sobolev inequality with v defined by 1/q = 1 /p - J 1v.
Fix a non-negative function f E Co (M) and set fp,k = (f - Pk)+ A Pk(P - 1)
(3.2.3)
for any p > 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
<
(ClIVflI)°° N(f)(e0)/s0
(C11VAI P
P-1
00
N(f)q(1-80)/80.
)
Thus N(f)91r0
-
< Pq/ro
CllofIIPI80 N(f)q(1-90)/80.
( p-1
Simplifying and using the definition of q again yields q/ro
N(f) 5 p _ 1 C'liof and thus sup{tj (f >_ t)1/q} o
P
(3.2.6)
IIp
1+q/ro
p-1
C'IIVfIIP.
Setting p = 1 + ro9o/q, we get
sup{tpt(f >
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
Now, applying the hypothesis to fp,k yields
(P - 1)Pkµ(f > 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
q/P I V f 1P
(3.2.9)
As q/p > 1, we have > 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 <
P(PQ _
1)i/gClllofllP
(P - 1)
Using (pq - 1)'/Q < p and picking p = 1 + 1/q < 2 yields the desired inequality.
Corollary 3.2.4 For v > 2, the Nash-type inequality d f E Co (M),
IIf
112(1+2/,,) <
C2Ilofll2llf
Ill/Y
implies the (L2, v)-Sobolev inequality
d f E Co (M),
11f
112.,/(.,-2)
< A.CIIVf112
for some constant A. Indeed, the postulated Nash-type inequality is exactly (S21(y+2}) raised to the power 2(v+2)/v. Furthermore, q = q(2, 1, v/(v+2)) is equal to 2v/(v2) which is finite and larger than 2 if v is larger than 2. Thus we can apply Theorem 3.2.2 which yields the desired Sobolev inequality.
66
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
3.2.4 The case q = oo As in the previous section, the number p, 1 < p < oo, is fixed and all Sobolev-type inequalities are relative to IIoflip.
Theorem 3.2.5 Assume that (S 0,30 is satisfied for some 0 < so < ro < oo (this has q(ro, so,1 - so/ro) = oo for q given at (3.2.1)). Then all the inequalities (Sr1, 1') with 0 < s < r < oo are satisfied. Fix a non-negative f E Co (M) and apply the hypothesis to fp,k defined at (3.2.3). Thanks to (3.2.4), (3.2.5), this gives
(duvfIl)
pk+1) <
k)(3.2.10)
Fix 0 < s < r < oo and observe that it is enough to prove the desired inequality for some set S of pairs (r, s), 0 < s < r < oo such that sup{s : (r, s) E S} = oo. Indeed, the Holder inequality shows that, if r > s, then (Sr',,) implies (Ss;,.,) for all r' > 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) <
p-1
k
k
pksp(f > pk).
But 'k+2 is-lp(f
Pr(P1- 1)IIfIIr <
1)
pr(pr
k
> t)dt < pkrp(f > pk+l)
pk+i
and, similarly, pksµ(f
S
> Pk) <- PiP 1IIf11
k
Hence
r+8
IIf
llr
:5 (PS
r _
1)
p 1)(p - 1)r-s (CII of II P)r-81I f ll
8
s
for all t > 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 <
Ilf
llr
,(fj)1/s-1/r11fllr Hence, for
any r > 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 <
2p3r(p-1)-1CIIVfllpµ(Q)1/r
for all r > 2. Picking p = 1 + 1/r yields 11f 1Ir <- 54r CII Vf
Ilpµ())11r
Hence we have obtained the following result.
Theorem 3.2.6 Under the hypothesis of Theorem 3.2.5 there exists a constant Al such that for all bounded sets SZ C M and all r > 1, Vf ECa (Q),
11f 11,
Moreover, there exist two constants a > 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 <
for all 0 < s < oo (recall that q < 0). Here C is the constant appearing in (S*,ea
l
ro,8o )
Fix f E Co (M), f > 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
<
CIIVf lip
( p-1
ea
p(k-1)(1-Bo)p(f
1k)(1`ea)/8o.
Multiply this inequality by p6(k-1) and rearrange to obtain (pro(k-11)(1+6)µ(f
Ak-1))1/ro
<
[ 8o(k-1)(1+6/(1eo))µlf
P-1 (CIIVfI
)00
Now, choose S so that ro(1 + b) = so(1 + 6/(1 - 90)). It turns out that this
is equivalent to ro(1 + S) = q. Setting ak = pQkµ(f > Ak) yields ak/
i
< p-9(1-eo)/eo
CIIofHIP1
( P-1 J
Bo
aki-eo)18o
(3.2.11)
3.2. WEAK AND STRONG SOBOLEV INEQUALITIES
69
for all k < k(f ). Observe that ak > 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 k
is positive. It follows that (3.2.11) implies a'/ro < p-9(1-eo)/so
/
CClloflip
p-1
l
e°
a(1-90)/80
that is, J
a
kinf {PQk1(f ? )k)} ?
(cl)'o.
P92(1-9o)/8o6o
AsMp(f >A) - P
92(1-oo)/soeo
(CflVfII)°°
p-1
Observe here that Ak = Ak (E) depends on the small parameter E. However,
we can let E tend to 0 in the above inequality. Choosing k = k(f) - 1 and observing that for this choice of k Ak(0) > Pk(P
- 1) > p -2(p - 1)IIf IIo,
we obtain 1)'p28+2gp92(1-9o)/0090
IIf II-+911f Il8 >- (p -
P-1 )
.
Rearranging, we obtain
IlfII. <- P2(p - 1)-1
(p9(1-Bo)/8oOoClloflip)
1/(1-8/9)
Ilf118/(1-q/8).
Picking p = 1 + 1/(1 + Iql) (recall that q is negative), we get IIf 11. < 4(1 + Iql)
(e(1-e0)180°°CII
This ends the proof of Theorem 3.2.7.
Vf lip)
l/(1-s/9)
11f
II8/(1-9/8).
70
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
Corollary 3.2.8 Assume that (S*o so) is satisfied for some 0 < ro, so < 00 and 0 < Bo < 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 there exists a constant A such that V f E Ca (St),
If Iloo < ACA(SI)-1/gllofllp
for all bounded domains SZ C M. Here C is the constant appearing in (S*,eo l ro,so)
if f is supported in 0.
Indeed, for any finite s > 0, llf Its <_ Ilf Hence, after simplifications, we have IlfII. < ACp(SZ)-1/Qllofllp
3.2.6
Increasing p
In the last three sections, we studied the inequalities (S*,8) and (SB$) for a fixed value of p. In order to discuss what happens when the parameter p is allowed to vary, let us introduce the notation (S*,B(p)) and (Se8(p))
to refer to these inequalities with 1 < p < oo.
Theorem 3.2.9 Assume that for some 1 < po < oo, 0 < ro, so < 00, 0 < Oo < 1, the inequality (S,*.o 8p(po)) is satisfied. Assume also that qo = q(ro, so, 90) defined at (3.2.1) satisfies 1/qo < 1/po (which is obviously satisfied if qo < 0 or qo = oo). Let v be defined by (3.2.2), that is, 1/qo = 1/po - 1/v. Then all the inequalities (SB8(p))
where 0 < r,s < oo, 0<9
1,po
q(r, s, 9)
_1_1 p
v
are also satisfied.
By the results of the last three sections, we can assume without loss of generality that the inequality (S,,o(po)) is satisfied for some (or all) uo, ao such that q(po, uo, ao) = qo. For any f E Co (M), we can apply this inequality to If I", ry > 1, llfII y
C(M
l o'o/po Qo) J IIfIiuo7
3.2. WEAK AND STRONG SOBOLEV INEQUALITIES
71
By the Holder inequality 1/r
(ffY_1)rUrd/2)
< l i f 11'r1--Y1 11911 r-Y
this yields °o).
11fIlpory
Simplifying, we get that the inequality (SPu(p)) is satisfied where p = 7po,
with a = ao/(ao +7(1 - ao)), u = puo/po. Now, observe that
1-a
-=7 1-a0 01o
It follows that the parameter v defined by 1/v = 1/po - 1/qo satisfies
1_1_1 po
1Oro
E-i
po
qo
1-ao
1
1
ao
Gauo
uo
Go
1
( uo
1-a0
1
1-Q
1
1
'Yu
uo
Po
PO }
7po)
or
u
p
1
1
1
1
1
p-
1-Q
or
p
u
p
q(p,u,a).
In words, v defined by 1/v = 11p - 1/q has not changed when passing from (S;0 ,,,o(po)) to (S;,u(p))
In any case, we have that (SSu(p)) is satisfied for some a, u such that 1
_1
1
p
v
q(p,u,o)
Now, depending on whether this q is positive, infinity, or negative, one of Theorems 3.2.2, 3.2.5, 3.2.7 shows that all the inequalities (Se3(p)), 0 <
r,s
q(r, s, 0)
_
1
1
p
v
are satisfied. This is the desired conclusion.
Corollary 3.2.10 Assume that for some 1 < po < oo, 0 < ro, so < oo, 0 < 0o < 1, the inequality (S,*.o a(po)) is satisfied. Assume also that qo = q(ro, so, 0o) defined at (3.2.1) satisfies 1/qo < 1/po (which is obviously satisfied if qo < 0 or qo = oo). Let v be defined by (3.2.2), that
is, 1/qo = 1/po - 1/v. Then there exists a constant c > 0 such that V (x' t) > ct".
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
72
By Theorem 3.2.9, we can assume that (S ,.(p)) is satisfied with p so large
that q(r, s, 9) < 0 (indeed, 1/q(r, s, 0) = 1/p - 1/v with 0 < v < oo). By Theorem 3.2.7, we then have the inequality V f E Co (M),
IIfII. <
AIIVfII8IIfIIi_e
with 9 = (1 - 1/p + 1/v)-1. Applying this to the function f (y) = (t d(x, y))+ yields t < AV(x, t)°h't1-BV (x,
that is,
t)",
V(x, t)1/,. > At.
Corollary 3.2.10 is the same as Theorem 3.1.6. Note that the proof above does not use the fact that limt_,o t-V (x, t) = R,, > 0 whereas the proof of Theorem 3.1.6 used this fact. Corollary 3.2.10 shows that the volume growth lower bound V (x, t) > ct" can be interpreted as a very weak form (a vestige) of Sobolev inequality.
3.2.7
Local versions
We would like to record here two useful comments about the results obtained
in the previous four sections. The first comment is that we can replace M by an open subset, say U C M, without changing anything in Theorems 3.2.2, 3.2.5, 3.2.6, 3.2.7, 3.2.9, or in Lemma 3.2.3 or in Corollaries 3.2.4, 3.2.8. In other words, we do not need to assume that M is complete in these results. Note that Corollary 3.2.10 does not extend unchanged to the case of open subsets of M. We would like also to mention that the results listed above can be extended to the case where the inequalities (S,'.,,8 (p)), (S, ,(p)) are replaced by their uniform local counterparts (CIIVfIlp+TllfIlp)°(IIfll,µ(supp(f))11'')`-O,
sup{tµ({IfI > t})l/r} o
llf Ilr <- (CIIVf IIp + Tlif IIp)BIlfII B-0
for all f E Co (M), which we denote respectively by (ST,°(p)), (Se,(p)) This is more or less obvious once it has been observed that the quantity Wp(f) = CIIVf llp +TIIfIIp behaves just like IIV f IIp with respect to the transformation f H (f - t)+ A
s = ft', t, s > 0. More precisely, we have Wp(ff) < W (f) and
E W (fp,k) < Wp(f) k
73
3.3. EXAMPLES
for p > 1 and fp,k = (f - pk)+ A pk(p - 1). Note that the ratio C/T is kept constant in this process. Here are a few statements that are easily obtained by implementing this remark.
Theorem 3.2.11 Let M be a complete Riemannian manifold. Assume that the inequality (IlfIlooµ(supp(f))1/30)1-e0,
sup {tµ({IfI > t})1/r0} <- (CIIVflip+TIIfllp)80 t>o
is satisfied for all f E Co (M) for some 0 < ro, so _< oo, 0 < 00 < 1 and 1 < p < 1. Define q by 1/r0 = 8o/q + (1 - Bo)/so and assume that p < q < oo. Then there exists a constant A such that V f E C00 (M),
II f llq < A (CII Vf lip + Tllf lip)
Corollary 3.2.12 For v > 2, the Nash-type inequality V f E CO- (M),
IIf
1121+2/v) <
(Cllof Ill + TIIf 112) Illll1
implies the (L2, v)-Sobolev inequality
Vf ECo (M),
IIfIl2v/(,.-2)
for some constant A,,.
Theorem 3.2.13 Assume that for some 1 _< po < oo, 0 < r0i so < 00, 0 < 8o < 1, the inequality (STO o(po)) is satisfied for some C, T > 0. Assume also that qo = q(ro, so, 80) defined at (3.2.1) satisfies 1/qo < 1/po (which is obviously satisfied if q0 < 0 or q0 = oo). Let v be defined by (3.2.2), that is,
1/qo = 1/po - 1/v. Then there exists a constant c > 0 such that
`dxEM, VtE(0,T-1), V(x,t)>ct`.
3.3 3.3.1
Examples Pseudo-Poincare inequalities
It turns out that a useful and widely applicable tool to prove Sobolev inequalities is a set of inequalities indexed by the semi-axis It > 0) that we call pseudo-Poincare inequalities. For any f E Co (M), set
A(x) = V(x, t)
JB(x,t)
f (y)dl.(y).
74
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
We say that M satisfies a pseudo-Poincare inequality in LP if there exists a constant C such that
Vf ECo (M), Vt > 0, Ilf - ftllp<- CtIIVfllp.
(3.3.1)
Note that the LP norms on both side are taken over the whole space M. This should be interpreted as an inequality concerning the approximation of f by the more regular functions ft, which are averages over balls of radius t. The important fact about ft is that Ill IIoo < V (X1 t)-1llf fli.
(3.3.2)
This obvious fact should be thought of as a quantitative version of the "smoothing" effect of f -> ft. Actually, in the arguments that will be developed below the only things that matter are that ft satisfies both (3.3.1) and (3.3.2).
Theorem 3.3.1 Assume that M satisfies (3.3.1) for some 1 < po < oo and that inf I t-"V (x, t) } > 0 (3.3.3) xEM
for some v > 0. Then the inequalities of Sobolev type (S,", (p)) hold true for
all p > po and all 0 < r, s < oo, 0 < 0 < 1 such that q(r, s, 0) defined at (3.2.1) satisfies 1/q = 1/p - 1/v. The proof is surprisingly simple. It has its origins in [72]. By Theorem 3.2.9,
it suffices to treat the case p = po. By hypothesis there exists a constant c > O such that V(x, t) > (ct)" for all x E M and t > 0. Fix f E Co (M),
f >Oandareal A>0. Foranyt>0,write µ(f >_ \) : µ(I f - ftl > A/2) + jt(ft >_ A/2) and pick t so that (ct)--"11 f 11, = A/4. Then
ii(ft ? A/2) = 0 and
u(If - ftI > A/2) (2/A)pllf - ftilp
< (2Ctllof llp/A)p [(21+2/Pc/c) hf
That is
p(1+1/")µ(f
? A) <
llpa-l-i/"1p
1/1,11
IIf 1I
Vf llp]" .
.
75
3.3. EXAMPLES
Raising this to the power T/p with r = 1/(1 + 1/v) = v/(1 + v) yields 4(C/c)TIIofIIpIIfIII-"
A (f >_ a)T/p < which is (Spy, I (p)). This, together with Theorems 3.2.2, 3.2.5, 3.2.7, proves Theorem 3.3.1. We now state some obvious but important corollaries of Theorem 3.3.1.
Corollary 3.3.2 Assume that M satisfies (3.3.1) with p = 1 and (3.3.3) for some v > 1. Then the isoperimetric inequality <- C111(an)
is satisfied by all bounded sets 1 with smooth boundary. Also the (L1, v)Sobolev inequality
V f E Co (M),
IIf II"/("-1) < C1 II Vf III
is satisfied.
Corollary 3.3.3 Fix 1 < p < oo and v > p. Assume that M satisfies (3.3.1) and (3.3.3) for these p and v. Then the (LP, v)-Sobolev inequality V f E Co (M),
IIf II p"/("-p) <- C2II Vf lip
is satisfied.
Corollary 3.3.4 Assume that M satisfies (3.3.1) with p = 2 and (3.3.3) for some v > 0. Then the Nash inequality V f E CO- (M),
Ill
112(1+2/,,)
Csllof Ilzllf II1
is satisfied.
3.3.2
Pseudo-Poincare technique: local version
The argument developed in the section above can easily be adapted to the case when one only has local hypotheses. As an example, we prove the following result.
Theorem 3.3.5 Fix R > 0. Assume that M satisfies V f E Co (M), `it E (0, R),
Ilf - ftII, < Ct IIVfllp,
(3.3.4)
for some 1 < po < oo and that
inf {t-"V (x, t) } = c > 0
tE(O,R) XEM
(3.3.5)
for some v > po. Then the inequalities
Ilfll, <- [C(v,p)/c] (CIIVflip+R-IIlflip) are satisfied for all po < p < v and 1/q = 1/p - 1/v. The constant C(v, p) is independent of C, c and R.
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
76
The proof is similar to that of Theorem 3.3.1. We can assume that p = po. By hypothesis, V (x, t) > (ct)" for all x E M and t E (0, R). Fix f E Co (M),
f >0andarealA>0. ForanyO A/2). If possible, pick t so that (ct)-"Il f Ill = A/4. This can be done if A > 4(cR)-"11 f
111.
Then, by (3.3.2)
µ(ft>A/2)=0 and
µ(f > A) < µ(I f - ftl > A/2) < (2/A)Pllf - ftllp (2Ct1lVfllP/A)P
= {(21+2/vc/c)IIfIll/"IIVfIIPA-1-1/v'P That is
'\p(1+1/v)µ(f > A)
[(21+2/Rc/c)IIfll/vIlvfp,]).
Now, if A < 4(cR)-"IIf III, simply write
i(f > A)
A-Pllflip
to see that, in this case, AP(1+1/v)µ(f
? A)
[4h1i(cRyuIIfipi/"l,fIIP]P
Thus, in all cases, AP(1+1/v)p(f > A)
[(2 1+21-IC)Ilf III/"(CIIVfIIP + R-'lIfllp]P
Raising this to the power T/p with r = 1/(1 + 1/v) = v/(1 + v) yields
A (f > A)T/P <4(1/c)"(CIIVfIIP+R-1 IIfIIP)TIIfII1-T which is the local version of (Sp;T,I(p)). This, together with the local version of Theorem 3.2.2, proves Theorem 3.3.5. See Section 3.2.7. Since we have not explained in detail the local version of Theorem 3.2.2,
let us note that in the case po = p = 1, the argument above yields
,\µ(f >
A)'
< C1(CIIVf1I1 +
R-1lIflli)TIIflIi-T-
Applying this inequality to regularizations of f = In (where S2 is a bounded set having smooth boundary) with A = 1/2 yields the isoperimetric inequality µ(9)1-1/" < Cl(Cµ(8f2) + R-1µ(f2)). This inequality can then be integrated using the co-area formula to recover the Sobolev inequality llf lly/("-1) < Ci(CIIDf 11, + R-' IIf 111).
3.3. EXAMPLES
3.3.3
77
Lie groups
A connected Lie group of topological dimension n is a manifold G equipped with a product G x GE) (g, h) i-+ gh such that (G, ) is a group and the apgh-' E G is analytic. The fifth Hilbert problem plication G x G D (g, h) was to decide if every connected locally Euclidean topological group is a Lie group. It was solved with an affirmative answer by Gleason, Montgomery and Zippin in 1952. Any locally compact group carries some left (resp. right) invariant measures called left (resp. right) Haar measures. Any two left (resp. right) Haar measures it,, µ2 are related by µl = cµ2 for some c > 0, so that essentially there is only one left (resp. right) Haar measure. Let µ be a left Haar
measure on G. For g E G and A a Borel subset of G, let µ9(A) = µ(Ag). Then it, is another left Haar measure so that there exists m(g) > 0 such that µ9 = m(g)µ. Obviously, m(gh) = m(g)m(h) and m(id) = 1. The function m is called the modular function of G. It can be trivial (i.e., m - 1), in which case we say that G is unimodular, or non-trivial. Obviously, G is unimodular if and only if any left Haar measure is also a right Haar measure.
Let T be the tangent space at the identity element id. By the left action of G on itself, any vector C in T defines a left-invariant vector field Xt on G. Conversely, any left-invariant vector field is determined by its
value at id. That is, the space of all invariant vector fields on G (i.e., the Lie algebra of G) is isomorphic, as a vector space, to T. We can now fix a Euclidean metric on T and turn it into a left invariant Riemannian structure on G. If (Cl, . . . , C,,) is an orthonormal basis of T and Xs = XE, then, at each point g E G, (Xi(g), ... , X,, (g)) is an orthornormal basis of T. for our Riemannian structure. By construction, the Riemannian measure associated to this structure must be left-invariant: it is a left Haar measure. Call it p. Given a function f E Co (M), V f is the vector field (not a left-invariant vector field!) defined by df (Z) = (V f, Z)
for all vector fields Z. Computing in the orthonormal basis (e1,. .. , en) _ (Xi(g), ... , X. (g)) of T9, g E G, we have n
df (Z)(9) _
where Z(g) _
n
zsXif (9) _
z=(Vf (9))i
z;e;. That is, in coordinates,
Vf(9) = (Xif(9),...,Xof(9)) Hence n
MAO= E IXzf(gW
.
(3.3.6)
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
78
We now want to compute the Laplace-Beltrami operator. Any leftinvariant vector field X generates a one-parameter group {¢x(t) : t E R} where Ox : R -- G satisfies O'x (t) = X (Ox (t)). Moreover, for any function f E Co (M), X f can be computed by X f (g) = ii o
f (Ox (t)) - f (9)
Now, observe that if p,. is a right Haar measure,
G fi(90x(t))f2(9)dpr(9) = J
As Ox1(t)
f f1(9)f2(90x1(t))dp,-(9) G
= ¢x(-t), it follows that fXfl(9)f2(g)d/4(9) = - ffi(9)Xf2(g)dPr().
In particular this applies to the X1's, which are thus skew symmetric with respect to µr. Let
Z(g) _
zi(9)X:(9)
be any smooth vector field with compact support on G. Then
f(ZVf)dr
jEZi(X:f)dPr n
Xizi fdµr
fG
This shows that, if G is unimodular, i.e., µr = p, then n
div(Z)(g) = EXizi(9) 1
In this case, it follows that n
0f = - div(Vf) _
f.
(3.3.7)
If G is not unimodular, we can still compute the divergence by using the following trick. Let m be the modular function on G. Observe that M -1'u is a right-invariant measure on G. Call it µr and write
j(Z V f)dll = j(Z Of)mdµr = f
=-f n G
-
fG
G
n
zXifmdµr 1
Xi(mzi) fdµr 1
fm
n
n
Xzi 1
ziXim m-1dµ. 1
79
3.3. EXAMPLES
It is not hard to see that Xim = aim for some constant Ai. This is due to the fact that m is multiplicative. Indeed, for any left-invariant vector field X, we have
Xm(g)
- m(g) = lim m(gq5x(t)) t lim
m(g)m(g5x(t)) - m(g)
t
c-.o
m(g)
l ,m
If
m(Ox (t)) - 1 = Axm(g)
with Ax = Xm(id). Thus, setting Axi = Ai, we get
c (Z, Vf)dµ which gives
-
n
Xizi +
n
aiz;
dµ
n
n
div(Z)(g) = E Xizi(g) + E Aizi(g) and
n
4f
n
-EAX:f.
(3.3.8)
Let us now look at the Riemannian distance function. As the Riemannian structure we consider is left-invariant, the associated distance (g, h)
d(g, h) is also left-invariant, i.e.,
d(g, h) = d(id, g-'h). For all g E C, we set IgI = d(id, g)
so that d(g, h) = Ig-'hl. It follows from left invariance that all balls of a given radius have the same volume, that is,
V(g,t) = V(id,t) = V(t).
3.3.4
Pseudo-Poincare inequalities on Lie groups
This section shows that any unimodular Lie group satisfies the pseudoPoincare inequality (3.3.1). As a corollary, the results of Section 3.3.1 apply.
Theorem 3.3.6 Let G be a unimodular Lie group equipped with a leftinvariant Riemannian metric and the associated Haar measure p. Then
dt > 0, V f E C (G), Ilf - ftllp < tliVfllp for all 1 < p < oo.
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
80
The proof is simple. Let h E G. Let 'yh : [0, t] --> G be a smooth curve in G such that 'yh(0) = id, -yh(t) = h, and Iryh(s)I < 1. Note that IhI is equal to the infimum of all t > 0 such that such a curve exists. Now, for any g E G and f E Co (M), t
f(9h) - f(g) =
f
a3f(97h(s))ds.
0
Thus
f
t
If(gh) - f(g)I <
Idf,,(S)(at9'rh(S))Ids
0
< Jt IVf(97h(S))II7h(s)lds < This yields
Jt
IVf(rh(S))Ids.
t
If(gh) -
f(g)IP <
tP-1
JIVf(g7h(s))I"ds
0
and
fG
tP1
f(gh) - f(g)Idg =
It! I Vf(91'h(S))IPdgds
tP-1
tP
f
Jo
f
c
I V f (9) I Pdgds
IVf(9)IPdg.
G
Note that here we have used the right invariance of the left Haar measure on G, i.e., the fact that G is unimodular. We can optimize over all curves joining the identity to h. This yields j I f (gh) - f (9) I Pdg < I hI P fa I V f (9) I
Pdg.
Integrating this inequality in h over all h E B(t) = B(id, t) yields
ill - PIP
Jo If (9)
- V(t) JB(t) f (gh)dh
V(t) JB Jc
P
If (9) - f (9h) IPdgdh
1 f IhIIIVfllpdh
V (t) B
< tPIIVflip. This is the desired inequality. Theorem 3.3.6 has the following corollary.
dg
3.3. EXAMPLES
81
Corollary 3.3.7 Let G be a unimodular Lie group equipped with a leftinvariant Riemannian metric and the associated Haar measure it. Assume that the volume growth function V (t) satisfies inf t-"V (t) > 0
for some v > 0. Then G satisfies the isoperimetric inequality µn(1l)1-1/" < CC(v)µn-1('911)
for all bounded sets fl C G with smooth boundary. For each 1 < p < v, G sastisfies the (LP, v)-Sobolev inequality V f E Co (G),
IIf
C(p, v)IIVf IIP-
IIp"/("-P)
More generally, for any 1 < p < oo define q(p) by 1/q(p) = 11p - 1/v. Then G satisfies all the inequalities (SO,, (p)) for 1 < p < oo, 0 < s, r < 00 and 0 < 9 < 1 such that 9
1
r
1-9
4'(p) +
s
In particular, G sastisfies the Nash inequality V f E Co (G),
IIf
112(1+21,,) <
C2(v)IIDf II2IIf lIi/"
This corollary is a useful result because the volume growth of unimodular Lie groups is well understood. First, for 0 < t < 1 we have co <_ t-nV(t) < CO
where n is the topological dimension of G. Second, by the work of Guivarc'h [36] (see also [45]), the volume growth function of any unimodular Lie group
G satisfies the following alternative: either there exist c, a > 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
83
3.3. EXAMPLES
Note that the hypothesis V (x, r) > cr" is also necessary for the last conclusion of this theroem to hold. By the results developed in this chapter, it suffices to prove the first assertion of the theorem. Observe that Ilf
- frill
f
=
M
if(x) - V(x, r) fB(x,r) f(y)dy dx
If
ff
MM
(y) dydx. (x) - f (y) 11B(x,r) V (x, r)
Now, consider the integral fB(x,r)
If(x) -- f(y)Idy
To estimate this integral, we use polar (exponential) coordinates around x. See [13, Proposition 3.1]. This gives (in somewhat abusive notation)
f
jr (x,r)
11(x)
If (x) - f (p, 9) I vi(p, 9)d pd9
- f (y) I dy =
:I
fp I acf (t, 9) I
0
ff
9)dpd9
0
r
p
Here, we have simply used the usual trick to control f (x) - f (y) by integrating along the geodesic segment from x to y and used polar exponential
coordinates y = (p, 9) around x. In particular, / (p, 9)dpd9 = dy is by definition the Riemannian volume element in polar coordinates. Strictly speaking, one should avoid the cut locus C(x) of x in this computation. This however is not a problem because C(x) has measure zero and M \ C(x) is star shaped with respect to x. We refer the reader to [13] for a detailed treatment. We now use the hypothesis that (M, g) has non-negative Ricci curvature. By Bishop's theorem [13, Theorem 3.8], the function s '--> J(s, is non-increasing. It follows that rJop 9)dtp"-ldpd9 fB(x,r) 11(x) - f (y)Idy < lVf(t, 9)/sn-'
J
o
< nf r" n
rIVf(t,9)Itl-"./g-(t,9)dtd9
Iof(y)I dy.
B(x,r)
d(x, y) n i
Using the hypothesis of maximal volume growth, we get
fMfM
If
11B(x,r) (y) (x) - f (y) V (x, r) dydx
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
84
1
<
cn
fJ
IVf(y)I d dx y
(x,r) d(x, y)"
1cn fm IVf(y)I fB(U,r) d(x, y) n-1 dxdy. 1
Finally, by Bishop's theorem again, /(t, 8) < tn-1. It follows that JB(y,r)
y)n-1
d(x
dx < wn-1r.
This yields
Ilf-frill<_W 1rIVI1. Recalling that fin = w,,_1/n, we see that this is the desired inequality. A variation on this proof yields a similar inequality in L' norms. The above proof of the pseudo-Poincare inequality (3.3.1) for manifolds having non-negative Ricci curvature uses the additional hypothesis that the volume growth of M is maximal. One may wonder whether (3.3.1) holds without this additional hypothesis. The answer is yes.
Theorem 3.3.9 Let (M, g) be a complete manifold of dimension n having non-negative Ricci curvature. Then
Vr>0, dfEC0 (M),
Il.f-fr111<4nrHVfll1.
To prove the desired inequality it suffices to bound
f f If (x) -f(y)I
1BV
by C r II V f III. To make things more symmetric in x and y, note that 1B(-") (Y) < 2n
V (x, r) -
This uses the fact that V (y, r)
1B(x,r) (y)
V (x, r)V (y, r)
V (x, 2r) < 2nV (x, r) if d(x, y) < r. The
inequality V (x, 2r) < 2nV (x, r) follows from the celebrated Bishop-Gromov comparison theorem (see, e.g., [13, 29]). Hence it suffices to bound
MJM fm
I f (x) - f (y) I
1B(x,r) (y)
VV (Y, r)V (x, r)
dydx.
Now, let yx, (t) be the geodesic from x to y parametrized by arc length (as in the proIM ofof Theorem 3.3.8, we can ignore the cut locus). Then, dydx
JM If(x)-f()l
V (y, r)V (x, r)
d(x,y)
JMIM J
l
V
f(7x,y(t))Idt
16{x,r) (y)
V (y, r)V (x, r)
dydx.
85
3.3. EXAMPLES
By symmetry with respect to x, y, we can restrict integration in t to the interval (d(x, y) /2, d(x, y)) so that we obtain
f
(y) dydx V (y, r)V (x, r)
f J If (x) - f (y) I
1B(x,r)
M
d(z,y)
2
IVf(`Yx,y(t))I1B(=,r)(y)
L fM Jd(s,y)/2
V (y, r) V (x, r)
dtdydx.
Now, by Bishop's theorem [13, Theorem 3.8], the Jacobian of the map y i-4 z = yx,y(t) is larger than
2-n+1
(see Lemma 5.6.7 for details) and
< 2n1B(.,r)Of.,y(t))
1B(=,r)(y)
V (x, r)
,\/V (y, r)V(x, r) Hence
fM f IfM(x) - f (y)1
1B(x,r) (y)
V (y, r)V (x, r)
< 22n Im < 22n r
dydx
I V f (z) I1 B(=,r) (z)
Im
f
or
V(x,r)
dtdzdx
IV f (z) I dz.
M
This yields the desired pseudo-Poincare inequality. See also Theorem 5.6.6.
3.3.6
Sobolev inequality in precompact regions
Let (M, g) be a complete, non-compact, manifold of dimension n. It is often useful to invoke the fact that, if St is an open precompact subset of M, the usual Rn (local) Sobolev inequality Ilf Ilpn/(n-p) s C(IlVf IIp + Iif lip) holds for smooth functions with compact support in D, with a constant depending on Q. Here we outline a direct proof of this fact.
Theorem 3.3.10 Let (M, g) be a complete manifold of dimension n. For
any open precompact region f C M and any 1 < p < n, there exists a constant C(SZ, p) such that
b f E C o (.),
-
IIf Ilpn/(n-p) < C(cl, p)(I)V f II p +
11f IIp).
We only need to prove the case p = 1. Now, the neigborhood U = {x E M : d(x, SZ) < 11 of Il is also precompact. Clearly, the volume function V (x, t) satisfies inf t-"V (x, t) > 0. tE(0.1)
86
CHAPTER 3. SOBOLEV INEQUALITIES ON MANIFOLDS
Moreover, the volume element in polar coordinates around x, which we denote by Xfg-y (t, 0) as in the previous section, satisfies
t1xE U, VtE (0, 1),
c
(This must be understood outside the cut locus of x. In fact, one can avoid the cut locus completely in this argument by considering only t E (0, E) with e small enough). See [13]. With these observations we can use the proof technique of Section 3.3.5 to show that, for all 0 < t < 1 and all f E C000(11), IIf - f4h <_ C(U) t II Vf II I.
Applying Theorem 3.3.5 yields the desired result. In general, one cannot dispense with the IIf lip term appearing on the right-hand side in Theorem 3.3.10 (to see this consider constant functions on a compact manifold). However, if 1 is a relatively compact set in a complete non-compact manifold then the technique of Section 3.3.5 can be used to show that, for all f E Co (0) IIfIIp C'(1,p)IIofIIp (a Poincare inequality). Thus, one has the following result.
Theorem 3.3.11 Let (M, g) be a complete non-compact manifold of dimension n. For any open precompact region St C M and any 1 < p < n, there exists a constant C(0, p) such that V f E Co (H),
IIf
IIpnl(n-p) !5 C(1l>P)IIVf lip-
Chapter 4 Two applications 4.1
Ultracontractivity
This section presents a few basic applications of Sobolev inequalities to the study of the heat diffusion semigroup Ht = e-t°. We show how Nash inequality implies that Ht acts as a bounded operator from L' to L°° with an explicit time dependent estimate. Basic references for this material are [67] and [11, 21, 87].
4.1.1
Nash inequality implies ultracontractivity
For the purpose of this section, let M be a locally compact, second countable, Hausdorfl space equipped with a Borel measure µ. Let Q be a Dirichlet form on L2(M, dµ) with domain D. We refer to [21] for a short introduction and to [26] for a detailed treatment. The reader who does not want to learn about Dirichlet forms can think of the following basic example. EXAMPLE Let M be a complete Riemannian manifold with Riemannian measure dx. Let W1,2(M) C L2 M, dx be the completion of COI(M) C L2(M, dx) for the norm 111f II2 + II V f I12. Let V > 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 <
- (Htf, f) >- -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. <- CtIIf11,
CHAPTER 4. TWO APPLICATIONS
90
and one says that Ht is ultracontractive. Obviously, this is equivalent to saying that the measure h(t, x, dy) is absolutely continuous with respect to µ and has a bounded density. In this case, we will write h(t, x, dy) = h(t, x, y)dµ(y).
With this notation, the properties Vt > 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)} <
(Cv/2t)v12.
x,yEM
Let f E L1(M, p) n L2(M, µ), Ilf 11, = 1. Then IIHt f 111 < 1 for all t > 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 <
As Ht is self-adjoint, it follows by duality that I
I Ht II2-oo C (Cv/4t)v/4.
By the semigroup property, II Ht II
II
Ht/2 JI 1-+2II Ht/2II2-o,
and we obtain I l Ht ll l_.m < (Cv/2t)"12
as desired.
For later applications, let us observe that the ultracontractivity of the semigroup (Ht)t>o implies that the time derivatives of the kernel h(t, x, y) are also well behaved. Namely, by (4.1.2), under the hypotheses of Theorem 4.1.1, we also have IIB Ht II1-
0
<_ Ckt-"l2-k
(4.1.3)
and thus sup { IOh(t, x, y)I } <
Ckt-"/2-k.
(4.1.4)
x,yEM
4.1.2
The converse
Carlen et al. [11] observed that Nash's result admits the following converse.
Theorem 4.1.2 Let Q be a Dirichlet form with associated Markov semigroup (Ht)t>o. Assume that there exists v > 0 such that Vt > 0, IIHtIl1-.2 < (C/t)"14 Then the Nash inequality
Vf E D,
Ilfll2(1+2/")
is satisfied with
C" = 2(1 + 2/v)(1 + v/2)2l".
CHAPTER 4. TWO APPLICATIONS
92
Let f E D fl L' (M, µ). Then we have t
IllII2 = IIHtf112- fO3IIH5f(Ids.
IIfII2= IIHtfII2+2
f
t
(AHBI,H8f)ds.
0
Now, we observe that s " (H8 f, AH8 f) is a non-increasing function of s because (H8f,AHBI) = IIA112H8f112 = IIHBA"2f112
Hence 1If 112 <_ IIHtf II2 + 2t(f, Af) <_ (C/t)"1211f Ill + 2tQ(f, f )
Optimize in t > 0 by choosing t such that vC""2{I f!{2{{f112
= 4Q(f,
f)t'+"/2
After some algebra this yields I1f1121+2/") <_
2(1 + 2/v)(1 + v/2)2h'CQ(f,
The result of Carlen et at and Nash's semigroup estimate prove the equivalence between Nash inequality and a specific heat kernel bound as stated in the following theorem.
Theorem 4.1.3 Let Q be a Dirichlet form on L2(M, µ) and let (Ht)t>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/")
(ii) The semigroup (Ht)t>o is ultracontractive and its kernel h(t, x, y) satisfies
b't > 0, sup{h(t, x, y)} <
C2t-"12.
sly
This equivalence is a very useful tool. It shows a certain stability of the decay condition supp,y{h(t, x, y)} < Ct-""2 under perturbations of the gen-
erator that preserve the size of Q(f, f) and the size of the measure µ. As a very simple example consider a complete manifold M and two measures dp = V(x)dx and dµ'(x) = V'(x)dx with V, V positive and smooth. Let Q(f, f) = f IV f (x)l2dx and consider Q as a Dirichlet form on either L2(M, dµ) or L2(M, dµ'). This yields two semigroups and two heat kernels
h and h'. Assuming that 0 < c < V/V' < C < oo, if sup.,,{h(t,x,y)} < Ct-"/2, the same must be true for h'(t, x, y). Quasi-isometric changes of metric can be treated similarly. Proving such stability without the help of a result such as Theorem 4.1.3 is not easy.
4.2. GAUSSIAN HEAT KERNEL ESTIMATES
93
Gaussian heat kernel estimates
4.2
This section presents in the Riemannian setting some Gaussian estimates which complement the ultracontractivity bounds of Section 4.1.1. In the presentation given below, Nash-type inequalities enter the argument only through the use of the ultracontractivity bound of Theorem 4.1.1. This approach is adapted from [41].
4.2.1
The Gaffney-Davies L2 estimate
Let M be a complete non-compact Riemannian manifold equipped with its Riemannian measure p. Let A be the Laplacian on M and let h(t, x, y) be the heat diffusion kernel on M. That is, for each x E M, h(t, x, y) = u(t, y) is the minimal solution of (at + 0)u = 0 u(O, y) = bx(y)
Equivalently, h(t, x, y) is the kernel (with respect to p) of the semigroup Ht = e-t° associated to the Dirichlet form Q(f , f) = f I V f I2di with domain W2 (M), the closure of CO(M) for the norm V111 f II2 + II Vf I2For any function 0 E Co (M) with IIVOII0 : 1 and any complex number a, consider the semigroup defined by
H' f(x) = e-00(x) J h(t, x, y)eam(v)f (y)dy = e-aO(x)Ht(ea`f)(x) It is clear that this is a well-defined semigroup of operators on the spaces LF(M, µ). Its infinitesimal generator is given by -Aa,mf = -e-'OA(ea0 f ).
When a is real this semigroup preserves positivity but there is no reason that it contracts L"(M, µ), for any 1 < p < oo. It is not self-adjoint but its adjoint is simply Hta'O. The next lemma estimates the norm of these semigroups on L2(M, µ). It is due to Gaffney [271 and was later rediscovered by Davies, who turned it into a powerful tool to derive pointwise Gaussian estimates under ultracontractivity hypotheses. See [21] and the references therein. Lemma 4.2.1 For any function ¢ E Co (M) with IIVOII,,. <_ 1 and any real number a, the semigroup (Ht '")t>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
< e02(1+e)t
In particular, there exists a constant C such that Ckk!(et)-xea2(1+E)t. ,'II2,2 S II
(4.2.2)
Ht
For the proof, we need to consider complex-valued functions (because the complex time semigroup does not preserve real functions). We denote by t(z) the real part of any complex number z. For two complex-valued functions f, g, we set (f, g) = f f g dµ. Let ( = cos r+i sin r E C with Irl < it/4. For any complex-valued f E Co (M), set u(t) = IIHas,41f112 Then
u'(t) = -2R ((AH'f,
Moreover, (Aa,mf, f)
(e-a4'
(e"If ), f ) r V(ea1f) V(e-amf
f
IVf12dL-a2J
)dµ IVol2IfI2d/_,
+a+ J fV0.Vfdp-J IV0.Vfdµ j
95
4.2. GAUSSIAN HEAT KERNEL ESTIMATES
The last term of this sum is purely imaginary and has modulus bounded by
a2
f
IV
I2IfI2dµ+
( IVf12dit. J
Using 1V4 j < 1, we obtain
-2R (((AQ,mf,f)) <
-2(cosr - Isin rl)
f
IVf12dµ
+2(cos7- + IsinrI)a2 f If12dµ
< 2(1 + ITI)a2
f
I f 12dµ.
From this it follows that u' < 2(1+e)a2u and the first conclusion in Lemma 4.2.2 follows. The bound (4.2.2) is then obtained by applying Cauchy's formula to the holomorphic function z -> Hz ,m f on a circle of radius et/10 centered on the real axis at t > 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- <- (C/t)1,14 imply IIHtIIp_
<
(C/t)°Y/4
if 1/p = 8/2 + (1 - 9)/oo, i.e., 0 = 2/p. This is the desired result. We will now need a more sophisticated version of the classical RieszThorin interpolation theorem. In what follows, i2 = -1, complex numbers are written z = a + bi, a, b E R and we set N(z) = a. Let Tz be an analytic family of linear operators defined on CO -(M) for all complex z with
CHAPTER 4. TWO APPLICATIONS
96
0 < R(z) < 1 (the analyticity hypothesis made here is that z H f gTz f d t is analytic in the strip 0 < t(z) < 1 and continuous in 0 _< R(z) < 1, for all complex-valued functions f, g E Co (M)). Assume that IITi,4 Mi and IITi+ti Ilp2-.q2 < M2 for all reals b and some fixed 1 < pl, p2, ql, q2 < oo. Then the interpolation theorem of E. Stein (which generalizes the classical R.iesz-Thorin interpolation theorem to analytic families of operators) gives the bound YO E [0, 11,
where 1
1-0
pe
P1
IlTellpe-iqe <_ Mi-0M2 0
1
1-0
P2
qe
ql
+ -1
0
+ -. q2
See [80, 81, 82].
Now, we consider the family of operators Ht z'16 where a, QS, t are fixed,
a is real and z is complex with R(z) E [0, 1]. Because multiplication by a complex function of modulus 1 is a contraction operator on LP(M, p), it is easy to see that IIHi z'0II2-.2 < IIHi"c=),'112-.2
In particular, by Lemma 4.2.1,
i
IIH
(1+bt),"12--2
< ea2t
Also,
IIHi
00-00
1.
By Stein's interpolation theorem [81], we conclude that II Ht
a/p,m
llp_P < e2a2t/p
for all p > 2. Changing a to ap/2 yields
IIHi
'4'IIp_p < epazt/2.
(4.2.3)
The following lemma will be proved by a similar argument.
Lemma 4.2.4 Assume that there exists C and v > 0 such that Vt > 0, IIHtII2-.. S
(C/t).'/4.
Then (C/t)"(1/p-11q)12 ega2t/2 IIHi for all q > p > 2, t > 0, a E R and 0 E Co (M) with 110011,0 < 1.
Again, consider Ht Z'0 where a, 0, t are fixed, a is real and z is complex with R(z) E [0, 1]. As above, we have for any p E [2, oo],
IIHi
(1+,j),mllp--p
<
epa2t/2
97
4.2. GAUSSIAN HEAT KERNEL ESTIMATES and, by Lemma 4.2.3, bi,,1IIP-
IIHi
.<
(C/t)"/2P.
Thus, e'O
IIHi IIP_,q :5 with 1/q = (1 - 8)/00 + 8/p. That is, IIHi
(P/q),mIIP-q
(C/t)"(1-B)/2Pepa29t/2
(C1t)"(11P-11q)12ea2P2t/2q.
Changing a to pa/q yields the desired inequality.
Lemma 4.2.5 Assume that (Ht)t>o satisfies V t > 0, IIHi II2_.00 <-
(C/t)v14
for some C, v > 0. For any 6 > 0 there exists a finite constant C(b) such that for all a E R and 0 E CO '(M) with IIVOIIO < 1, we have
Vt > 0,
(C(S)C/t)"/4 exp(ta2(1
<_
IIHi ''II2...
+ b)).
Let us fix a small 6 > 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(1 + b)
1
00
E pt 11og(1/si) = log[2C(b)] 1
with C(6) < oo. For instance, take s1 = 1 - e, si = eci-5 for i > 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
<
11
(C/ti)"(1/Pi-1/Pi+1)/2e«2Pi+1t:/2
i=1
00
`00
< (C/t)"/4 exp (ta2(Pi+isi/2) + 2 (L,pi 1 log(1/si)) 1
< (C(b)C/t)"/4 exp(ta2(1 + 6)). This proves Lemma 4.2.5.
CHAPTER 4. TWO APPLICATIONS
98
Pointwise Gaussian upper bounds
4.2.3
We can now prove the following theorem.
Theorem 4.2.6 Assume that the manifold M satisfies the Nash inequality V f E Co (M),
IIlII2(1+2,v) < CIIvfII2II.fIIi'Y.
Then, for any S > 0 there exists a finite constant C(S) such that the kernel h(t, x, y) of the heat diffusion semigroup Ht = e t°, t > 0, satisfies h(t, x, y)
(C(S)l t)"12 exp
(d(xY)2) 4(1
6)t
.
By Theorem 4.1.1, the hypothesis implies that there exists a constant C such that V t > 0,
II
Ht II2_.
<_ (C/t)"14.
Thus, by Lemma 4.2.5, for any 0 E Ca (M) with IIo.0IL a, we have
0
< I and any real
(IHa'"I(2-,oo <_ (C(6)C'/t)"/4 exp(ta2(1 + S)).
As the adjoint of Ht" is Ht "' we also have, by duality, IIHt ''IIi_2 <_ (C(b)C'/t)""4 eXp(ta2(1 + b)).
Thus
II Ht ''IIi, < (2C(6)C'/t)"/2exp(ta2(1 + 6)).
That is, h(t, x, y):5 (2C(S)C'/t)"12 exp (ta2(1 + b) + a[O(x) - O(Y)j).
For fixed x, y, 0, take a = -(O(x) - 0(y))/2(1 + 5)t. The bound becomes
h(t, x, y) < (2C(S)C'/t)"12 exp _ (
[fi(x) - 4 (y)J2 4(1 + S)t
)
'
We can now optimize over all allowed 5's for each fixed x, y E M. The condition IIo0IIO < 1 shows that ¢(x) - q(y) is at most d(x, y). Moreover, one can find 0 E Co (M) with IVOI < 1 and such that ¢(x) - 0(y) is as close as we wish to d(x, y). Thus, h(t, x, y) < (2C(S)C'/t)""2 exp as desired.
(- 4(1x+
)2
2)
99
4.2. GAUSSIAN HEAT KERNEL ESTIMATES
One can refine this result as follows. In the proof of Lemma 4.2.5, take
s1 = 1 - e, si = eci-5 for i > 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 <_ 2(1 + b). Now, repeating the same argument more carefully, we get IIHt ""112-..
-
00
11 i=1 00
II((7h/ti)"(1/p,-1/p,+,)/2ea2P++lt,/2 i=1 2 <_
(C'/6t)"14 eXp
t2
00
°O
Epi+isi + 2
log(i5/c ) (:2-'
1
< (C."/bt)"14 exp(tr 2(1 + b))
where C" does not depend on 5. This leads to the improved Gaussian upper bound (2C"/bt)"12
h(t, x, y) S As
d(x, y)2
d(x, y)2
"
exp
d(x, y)2
4(1 + b)t )
bd(x, y)2
4(1 + 6)t 4t 4(1 + 5)t picking b = (1 + d(x, y)2/t)-1 yields the bound h(t, x, y) < 2 {
2C"
t
)
"/2
(i+ d(x y)2 ) v/2 eXp 1
_ d(x4ty)2
I
.
This argument als\o givess Gaussian upper bounds for the time derivatives of h(t, x, y). Namely, using (4.1.3) and Lemma 4.2.2, we obtain that under the assumption of Theorem 4.2.6, jOt h(t, x, y) I <
4.2.4
Ckt-"/2-k (1
+ d(x, y)2)
"/2+k exp
/
\
{ -- ( 4t )2 j
.
(4.2.4)
On-diagonal lower bounds
Here we want to discuss briefly an application of the Gaussian upper bounds derived above. Let M be a Riemannian manifold of dimension n satisfying the Nash inequality
t/ f E Co (M),
IIf
II2(1+2/")
CUV f
jjzjjf jj 11"
CHAPTER 4. TWO APPLICATIONS
100
for some v > 0. This implies the volume lower bound
Vt>0, V(t)>ct" which in turn implies v > n. If all we have is this information, the Gaussian upper bound of Theorem 4.2.6 is not extremely useful. In particular, if the
volume growth is actually faster than t' (e.g., to with a > 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 = fR2k
Thus 00
2(k+l)ae-R222k/t
I(t, R) < CoRa 0
< CORa
2(k+l)ae-R222k/t 1
k:R222k
+ 1 2(k+1)ae-R222k/t k:R222k>t
Let k(t) be the smallest integer k such that R222k > t (such an integer always exists). Then R222k < t/4 for k < k(t). On the one hand, 2(k+1)ae-R222k/t < Ca2ak(t) k
and
CORa >
2(k+1)ae-R222k/t < CoC0Ra2ak(t) < Cata/2.
k
On the other hand,
E k>k(t)
2(k+1)ae-R222k/t
<
e-12ae-R2/t2ak(t)
2a(k-k(t))e 22(k-k(t)) E k>k(t)
101
4.2. GAUSSIAN HEAT KERNEL ESTIMATES from which it follows that
C01r E
Catn/2e-R2/2t
2(k+1)ae-R222k/t
k>k(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) <
CjjVf lI2IIf 114/v
and the volume growth condition
V x E M, d r > 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
although this is true under the above volume hypothesis. We start with the following computation. Fix f E L2(M) with Of 112 = 1. Then one can check that
AIIHtfll2 ilHtf 112
- -2IIVf112
(4.2.6)
Indeed, the right-hand side is the value of the left-hand side at t = 0. Computing the derivative of the left-hand side, one gets (at UHtfII2)IIHtfIl2 2 - (atllHtfII2)2 2 IIHtfII4
The numerator is equal to
4 ((i2Htf, Htf) (Htf, Htf)
- (AHtf, Htf)2)
which is positive by the Cauchy-Schwaiz inequality. This proves (4.2.6). Now, (4.2.6) implies exp (-211 Vf II2 t) < II Htf
III 2,
As
sup h(t, x, x) = sup Ilh(t/2, x, .)112 =
tEM
IIHtlI2_. =
XEM
we get, for any function f E Co (M), sup h(t, x, x) >
IIfII2
zEM
exp
(_2t)
(4.2.7)
1102
For any fixed t, consider a function f supported in a ball B = B(0, 2VI-t) for some fixed o E M. Then, by Jensen's inequality I I f II2/IIf IIi > V (o, 2 f)-1. Specialize f to be equal to ft(y) = (d(o, y) - OF)+ Then IIVftll22 <
V(o,2y),
IIftI12 2 >_ tV(o,
Vt-).
4.3. THE ROZENBLUM-LIEB-CWIKEL INEQUALITY Hence, sup
(
'p
103
2f)
> V(o,2f) ( V(0, OF) ) Now, the desired inequality follows from the fact that V (o, r) ^ r". In fact, this proves the bound EM1h(t,x,x)I
sup h(t, x, x) > 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 <-
f
IVfl2
- f VIf12
for all such V and all f E Co (M), i.e., sup
{fvIfI2} <
of2VECO (M)
f
IIVL/251/C(i)
As the dual of L"/2 is `d f E C000 (M),
L"/(i-2),
(f
this yields (v-2)/v
< C(v) f IVf12.
IfI2v/("-2))
In words, the Rozenblum-Lieb-Cwikel inequality (4.3.1) with parameter v > 2 easily implies that M satisfies an (L2, v)-Sobolev inequality. According to Theorem 4.3.1, the Rozenblum-Lieb-Cwikel inequality (4.3.1) can thus be seen as yet another form -the strongest form, in some sense- of the Sobolev inequality IIf 1112.1/(Y-2) < CII Vf
112-
Let us us note here that the problem of computing the best constant C(v) for the Rozenblum-Lieb-Cwikel inequality (4.3.1) in IR", n = v, is an important
open problem (e.g., n = 3). It is known that this best constant is strictly larger than the best constant in the corresponding Sobolev inequality. The smallest known constant C(v) in (4.3.1) was obtained by Lieb [56]. See also [73].
4.3. THE ROZENBL UM-LIEB-CWIKEL INEQUALITY
4.3.2
105
The operator TV = 0-1V
Assume that M satisfies an (L2, v)-Sobolev inequality with v > 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) <
E e2(1-A,a)t i:A A<1
e2tk(,\t)
(4.3.6)
for all A, t > 0. This reduces Theorem 4.3.2 to a suitable bound on k(t) as function of t, which is given by the following lemma. Lemma 4.3.3 Assume that II V I I v/2 = 1. Then the function k(t) satisfies
k(t) < (-Ck'(t)/2)v/(v+2). In particular
k(t) <
(Cv/4t)v/2.
Here C is the constant appearing in (4.3.2). Using this in (4.3.6) with t = v/4 yields Corollary 4.3.4 Assume that IIV II v/2 = 1. Then NT, (A) < (eC/A) '/2.
Let us prove Lemma 4.3.3. Observe that
2=
2v
+
4
v+2 v+2
and that the Holder conjugate of 4/(v + 2) is (v - 2)/(v + 2). Then write
f k(t, x, y)2V(y)dy k(t, x, y)2v/(v+2) [k(t, x, y)V (y)(v+2)/4]4/(y+2)dy (v-2)/(v+2) <_
J k(t, x,
f
y)2v/(z-2) dy)
2v/4(v+2)
k(t, x, y)Qdy)
(
\f
4/(v+2) k(t, x,
y)V
(Kt[v(v-2)/41(x))4/(i+2)
(y)(,+2)/4dy)
CHAPTER 4. TWO APPLICATIONS
108
Note that the last factor is indeed Kt [V("-2)/4] (x)
= =
f f
(y)(,,-2)/4V(y)dy
k(t, x, y)V k(t, x, y)V
(y)("+2)l4dy.
Let us consider the factor (f k(t, x, y)gdy)2/g. We want to estimate this factor by using the Sobolev inequality (4.3.2), that is, we want to write 1/Q
Y
k(t, x,
SC
y)Qdy)
f
I V kt(x, y)I2dy = CQ(kt ,
k,-)
where kk (y) = k(t, x, y). This is legitimate if we can show that kt E H. Fortunately, this easily follows from the facts that Ktf E H for all t > 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 <- [CQ(kt ,
kt )]"l(,,+2) (Kt[V
("-2)/4](x))4/(,,+2)
.
(4.3.7)
Observe also that
Q(kt , kt)
= - J(atkt-(y))ktx(y)V(y)dy
so that
f Q(kt , ki )V(x)dx = - f (8tk(t, x, y))k(t, x, y)V(x)V(y)dxdy
= -tat f k(t, x, y)2V(x)V(y)dxdy
_ -1 k'(t). The change in the order of derivative and integral is easily justified by the ultracontractivity of Kt and (4.1.4) (i.e., the analyticity of Kt on L2(M, dp)). We now integrate (4.3.7) against du(x) = V(x)dx and use that (v+2)/v and (v + 2)/2 are Holder conjugate exponents to get
k(t) =
J
f
k(t, x, y)2V (y)V (x)dydx
[CQ(kt , kt)V(x)dx]"/("+2) (Kt[V(v-2)/4](x))4/("+2) V(x)dx
109
4.3. THE ROZENBLUM-LIEB-CWIKEL INEQUALITY v/(L+2)
<
(cfQ(ic" , k, )V (x)dxV (y)dy)
(f
x (( (Kt(V(M-2)/4](x))2V(x)dx)
<
t
2/(v+2)
[-(C/2)k'(t)]"1("+2) (J(Kt [V '2V4] (x) )2V(X)dX) [-(C/2)k'(t)]Ml(,,+2)
r
(x)(t'-1)/2V(x)dX)
2/(v+2)
(J V CJ
< [-(C12)k'(t)]vl(v+2)
2/(+2)
V(x)"/2dx
[-(C/2)k'(t)j"/(,,+2)
)
Here, we have used that Kt is a contraction on L'(M, dp), for all 1 < p < oo, in particular for p = 2, and also
f V(x)(v-1)12V(x)dx = IIVIIYi2 =1. This proves Lemma 4.3.3.
4.3.3
The Birman-Schwinger principle
That the estimate of Theorem 4.3.2 is equivalent to the estimate of Theorem 4.3.1 is known as the Birman-Schwinger principle. More precisely, for V > 0, we have (4.3.8) Nv(0) where Nv(0) is the number of negative eigenvalues of A - V. To see this observe that the Rayleigh ratio for 1 - V on L2(M, dx) is
RL(1) =
f(IVf(x)12 - V(x)If(x)I2)dx f If (x) I2dx
whereas the Rayleigh ratio for K on L2(M, dp), dp(x) = V(x)dx, is
RK(f) = f IVf(x)I2dx
f If (x) I2V(x)dx
and
f(IVf(x)12 - V(x)If(x) 12)dx f If(x)12V(x)dx f IVf(x)I2dx f If(x)I2dx f If(x)I2V(x)dx f If(x)I2dx It follows that the number of eigenvalues of K less than 1, that is NTv(1), is equal to the number of negative eigenvalues of A - V, that is Nv(0). Indeed, a classical argument shows that Nom, (A) = sup{ dim(F) : F C Co (M), RK(f) > )t,0
f E Co (M)}
110
CHAPTER 4. TWO APPLICATIONS
and
Nv(A) = sup{ dim(F) : F C Co (M), RL(f) < A, 0 # f E C0 (M)}. Note that, because we are working with functions in Co (M) on the righthand side, the above formulas only hold for counting eigenvalues strictly larger (or smaller) than a given value A. This proves (4.3.8) and ends the proof of the Rozenblum-Lieb-Cwikel inequality, i.e., Theorem 4.3.1.
Chapter 5 Parabolic Harnack inequalities The aim of this chapter is to characterize those manifolds that satisfy a scale-invariant parabolic Harnack principle. This is achieved in Section 5.5.
5.1
Scale-invariant Harnack principle
Let us start by considering the elliptic version of the Harnack principle since it is easier to grasp. Given a complete Riemannian manifold M, we say that M satisfies a scale-invariant elliptic Harnack principle if there exists
a constant C such that, for any ball B in M and any positive solution u of Au = 0 in B, we have sup{u} < C inf Jul. 21 B
2B
Here 1B is the ball with the same center as B and radius half that of B. Note that this inequality is uniform with respect to: (1) the center of B, (2) the radius of B, (3) the harmonic function u. We now describe the parabolic version of this scale-invariant Harnack
principle. For any x E M and s E R, r > 0, let Q = Q(x, s, r) be the cylinder
Q(x, s, r) = (s - r2, s) x B(x, r). Let Q+, Q_ be respectively the upper and lower sub-cylinders Q+ Q-
E-i
(s - (1/4)r2,s) x B(x, (1/2)r)
FMM
(s - (3/4)r2,s - (1/2)r2) x B(x, (1/2)r).
We say that M satisfies a scale-invariant parabolic Harnack principle if
there exists a constant C such that for any x E M, s E R, r > 0 and any positive solution u of (8t + 0)u = 0 in Q = Q(x, s, r) we have
sup{u} < CinfJul. Q_
Q+
111
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
112
Thus, the main difference between the elliptic and parabolic cases is that the cylinders Q-, Q+ are disjoint whereas, in the elliptic case, the sup and
It is easy to see the necessity of the inf are taken over the same ball lapse of time separating Q_ from Q+: consider the fundamental solution 2B.
n/2
1 U'V
(t'x
(4-irt)
exp r_ ` Ix -4t
yI2
of the heat equation on Rn. Letting IIyfl =,r, one easily computes the sup and inf of u over the ball B = B(0, r) with r < r. One finds that SUPBfUY(t1-))
inf
)} =
exp
\4t [(r
+ r)2 - (r - r)2] ] = exp (-i-).
Taking r = f and T > vrt one sees that this ratio is unbounded as t, ,r tend to infinity.
It turns out that this scale-invariant Harnack principle contains a lot of information. To give only one of the striking consequences that will be developed below, the scale-invariant Harnack principle implies that the fundamental solution h(t, x, y) of the heat diffusion equation (at + O)u = 0 satisfies the two-sided Gaussian estimate
V(xcia Cid(xv)'/t < h(t, x, y) <
)
V(x
e-cZe(x,b)ZJt.
In fact this two-sided bound is equivalent to the parabolic Harnack principle and holds if and only if the following two properties are satisfied:
there exists a constant Do such that the volume growth function V (x, r) has the doubling property
V x E M, V r > 0,
V (x, 2r) < DoV (x, r);
there exists a constant Po such that the Poincare inequality
VxEM, dr>0, f
If -.fBI2dc
IVfI2dt B(x,2r)
is satisfied.
That these two properties are sufficient to imply the scale-invariant parabolic Harnack principle was proved independently by A. Grigor'yan [32] and the author [74] by two different methods. The necessity is proved in [74], building on an idea of Kusuoka and Stroock [50]. See the discussion at the beginning of Section 5.5. In order to study parabolic Harnack inequalities we will have to develop a number of techniques which are of interest in their own right.
5.2. LOCAL SOBOLEV INEQUALITIES
113
The parabolic Harnack principle above follows easily from the type of gradient estimates obtained by P. Li and S-T. Yau in [55] under Ricci curvature lower bounds; see also the treatment in [21]. The approach developed below is very different in spirit and can be used to treat cases where no gradient bound can possibly hold, e.g., in the case of equations having non-smooth coefficients or manifolds with non-smooth metrics. Moreover it seems to be a difficult problem to characterize manifolds on which a uniform gradient Harnack estimate holds true whereas we will be able to characterize those manifolds satisfying a parabolic Harnack inequality.
Local Sobolev inequalities
5.2 5.2.1
Local Sobolev inequalities and volume growth
Recall that the validity of a Sobolev inequality such as V f E Co (M),
IIfIIq
< C'IlofII2
with q = 2v/(v - 2) implies the volume lower bound V (x, r) > 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)
Theorem 5.2.3 Assume that M is complete, not compact, and satisfies the pseudo-Poincare inequality
df ECo (M), ds>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 <_ Clr(B) Il Vf11p
Fix a ball B and s = 4r(B). Then, for all f E Co (B), 1/P
Sllf-f81lP
(fBIf(y)-ff(y)lPdy)
(5.2.2)
Observe that, for y E B, B(y, 4r(B)) D B. Moreover, there exists a constant f > 0, independent of B and y, such that
V(y,4r(B)) > (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 <
p(B)s
IIfII1.
(Y,8)
For any Co (B)i) f>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) <-
P. [Q2tt(B)-'1-r(B) li f lli/vll0f Ilp]p
117
5.2. LOCAL SOBOLEV INEQUALITIES Case 2: If A is such that
2
A < µ(0 Ilf III then simply write (using Lemma 5.2.5) µ(f ? A) < a-PIIf IIp < Cia-Pr(B)PIIVf IIP. It follows that AP(i+1/")µ(f
?)<
f lli "11VfIIP]P
In both cases, we have AP(1+1/")µ(f
? A) <
1C3jz(B)-'1'r(B)IIfII1/"IIVfIIPI P
with C3 = max{C2i C2}. Raising this inequality to the power r/p with
r = 1/(1 + 1/v) = v/(1 + v) yields (C3µ(B)-1/"r(B))T
aµ(f >- A)T/P <
IIVfIlpllf 11"
This, together with Theorem 3.2.2, proves Theorem 5.2.3 (see also the remark in Section 3.2.7). Note that one needs to take v > p in order to be able to apply Theorem 3.2.2, i.e., in order to have the crucial parameter q in Theorem 3.2.2 be such that 0 < q < oo. The same proof yields the following local result. which is
Theorem 5.2.6 Assume that M is complete, not compact. Fix R > 0. Assume that M satisfies the pseudo-Poincare inequality
V f E Co (M), VO < s < R, IIf - ffIIP < CorllVfllp where fa(x) = V (x, s)-1 fB(x,a) f dµ. Assume also that M satisfies the doubling volume growth condition
VxEM, `d0 2 and C such that `d f E Co (B), C
f
if I
P"/("-P)dµl
C r(B)P µ(B)PIM J
I V E IPdµ
for all balls of radius at most R/4. We end this section with a couple of remarks concerning doubling-type conditions.
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
118
Lemma 5.2.7 Assume that there exist R > 0 and Do such that M satisfies the local doubling condition
d x E M, d r E (0, R), V (x, 2r) < DoV (x, r). Then there exists C = C(Do) such that for any x, y E M and 0 < r < R, V (x, r) < eCd(x,y)/rV(y, r).
Moreover, there exists D1 = Dl (Do) such that
Vx E M, VT > R,
V(x,T) < DiInV(x,R).
In particular, there exists C = C(D0) such that V x E M, `d t, T E (0, oo), V (x, T) < (T/t)"° exp(C(1 + T/R) )V (x, t) where vo = loge Do.
Fix 0 < r < R. Let B1, B2 be two balls of radius r/4 that intersect each other. Then, B1 C 4B2. Hence µ(B1) < D021z(B2). By symmetry, µ(B2) < Doµ(B1). It easily follows that there exists D1 = D1(Do) such that any two balls B, B' with radii s, s' E [r/8, 2r] and at distance at most 10Nr from each other satisfy
Di N/(B) 5 ,a(B') < D'y(B).
(5.2.3)
In particular,
p(B(x, r)) < exp (Cd(x, y)/r) µ(B(y, r)) for all x, y E M, 0 < r < R. This proves the first assertion of Lemma 5.2.7. To prove the second assertion it suffices to show that
V x E M, VT->R,
V (x, T+ R/2) < D112V (x, T).
(5.2.4)
Indeed, this will then give
V x E M, d N = 1, 2, ... ,
V (x, (N + 1)R) < Di V (x, R).
Consider a maximal set X of points in B(x, T - R/2) at distance at least R/2 apart. By definition, the balls B(y, R/2), y E X, are disjoint and all contained in B(x, T). Thus,
E V (y, R/2) < V (x, T). yEX
(5.2.5)
5.2. LOCAL SOBOLEV INEQUALITIES
119
As X is maximal, the balls B(y, R), y E X, cover B(x, T - R/2). It follows that the balls B(y, 2R), y E X, cover B(x, T + R/2). Thus
V (x, T + R/2) < E V (y, 2R).
(5.2.6)
yEX
By (5.2.5), (5.2.6) and the doubling hypothesis, V (x, T + R/2):5 DoV (x, T).
This proves (5.2.4). The last inequality in Lemma 5.2.7 then follows easily by Lemma 5.2.4. The next lemma gives a lower bound on the volume growth of noncompact manifolds satisfying the doubling property.
Lemma 5.2.8 Assume that M is a complete non-compact manifold which satisfies the doubling volume growth condition
`d x E M, d 0 < t < R, V (x, 2t) < DoV (x, t). Then there exist y > 0 and C such that
`d x E M, VO < t < s < R,
V(x, s) < C(s/t)_'V(x, t).
By our previous lemma, we can assume that the doubling condition holds for all 0 < r < KR for some large fixed K. Now, consider some r E (0, KR)
and the ball B = B(x, r), B' = B(x, r/2). Since M is connected and not compact, there exists a point y at distance 3r/4 from x. Thus the ball B(y, r/4) is contained in B and does not intersect B'. Moreover, by the doubling property, there exists c > 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 <_ AIlo ll fuPP(+G) u2dt
(5.2.11)
where A is a numerical constant which will change from line to line in the argument developed below. For any smooth function x of the time variable t, we easily get
at
(jExu]2d) + x2 f IV(u)I2dp B
< AX (XIIVII2 +
IIX'II.)
f
u2dµ.
uppOP)
We choose iii and x such that
0 < *< 1, supp(0) C aB, 0 = 1 in o'B, IV 'I < 2(rr)-1, 0-<X:51, x = 0 in (-oo, s - ar2), x = 1 in (s - a'r2,00) , Ix'I < 2(rr2)-1,
where 0 < a' < or < 1 and w = a - a'. Setting Ia = (s - are, s) and integrating our inequality over (s - r2, t) with t E IQ,, we obtain
p {JB bu2dµ} +
f Jr,,xe IV('u)I2dµ <
A(rw)-2
ff
u2dµ. (5.2.12)
Qo
Let E(B) = C(B)p(B)-2/"r(B)2 be the Sobolev constant for the ball B given by (5.2.7) and set q = v/(v - 2) where v > 2. Thanks to the Holder inequality f w2(1+2/") dµ::5
(!
w2q dµ
1/q
(Jw2dp)2/v
(5.2.7) gives
f w2(1+2/") dµ
2/"
< (Jw2 dµ)
E(B) f [I Vwl2 + r-2Iw12] dµ
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
122
for all wEC0 (B). Returning to the subsolution u, the above inequality and (5.2.12) yield
ff
u20dp < E(B) (A(rwy2
Jf
o,
(5.2.13)
Qo
with 9 = 1+ 2/v. Now, for any p > 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
<
AE('+1)e-'-'E(B)E°-'-'[(1-6)r1_2Ee-'
f
J fQ
u2 dµ
where all the summations are taken from 1 to i+1. Letting i tend to infinity, we obtain (5.2.15) sup{u2} < AE(B)v/2[(1 6)rj-2-"IIuI12,Q.
-
Qa
As E(B) = C(B),u(B)-2/"r(B)2, this yields (5.2.8) when p > 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) <
AC1 2
t
!-(1/4)rl
h(s,
()d(ds.
B2
Thus
f
1
h(t, , y)d <
A jt&(Bj 1/2 p(B2) 1/2
eXp(a2t + a(O(x) - OW) + I a I h + r2))
By the same token, working with the variable 1; and assuming t > ri, we get
2C
h(t, x, y)
µ(B1) C182)]1/2
exp(a2t + ca(o(x) - 0(y)) + IaI(rl + r2)).
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
124
Taking a = -(,O(x) - q5(y))/2t and assuming that t > e-2 max{rl, r2}, we obtain h(t, x, y) <
A2C1C2
[µ(Bl)µ(B2)]1/2
(O(x) _ 0(y))2 + E I5(x) - O(y)1
exp
Jt
Taking (as we may) a sequence of cb E C0 (M) with IV I < 1 and Oi(x) - O: (y) - d(x, y) finally gives A C1C2 [µ(B1)µ(B2))1/2
h(t, x, y)
exp I -
d(x, y)z d(x, y) 4t + e
which is the desired result.
Corollary 5.2.11 Let M be a complete non-compact manifold. Fix R > 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) <-
A(1 + d(x, y)l ,1t_)"
eXp
[V(x, v)V(y, f)]1/2
(-d(x, y)2) 4t
.
(5.2.16)
Using Lemma 5.2.7, one also obtains that for any e > 0 there exists Af such
that h(t, x, y)
for all x,yEM,0
V(x
exp ( )
4(1x+ ))t)
(5.2.17)
125
5.2. LOCAL SOBOLEV INEQUALITIES
Corollary 5.2.12 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 Then there exists a constant A such that for all SUPB C(B) = C < oo. x,y E M and all 0 < t < oo, A
h(t, x, y)
(,
Vx
t V t+d(x,y)) (y'
d(x, y)21 4t J
exp t )11/2 t+d(x,y)
Moreover, one also has h(t, x, y)
d{x, y)2)
y)/_ It-)2v
A(1 + d(x,
V(x, )
exp
l(
4t
J
The first bound follows readily from Corollary 5.2.11. To obtain the second bound, we use the first bound and the fact that
l
v
r) V (y, d(x, y) + r) >_ c (r + d(x, y) / V (X, r) y) + for all x, y E M, r > 0. See Theorem 5.2.2. Such precise Gaussian upper bounds lead to an optimal on-diagonal lower bound for h(t, x, x) as explained in Section 4.2.4. For this, one needs the next elementary lemma. V (y, r) > c (d(x,
Lemma 5.2.13 Assume that M is complete, not compact, and satisfies the doubling volume growth condition
`d x E M, V t E (0, R), V (x, 2t) < DOV (x, t). Then there exist C, c > 0 such that, for all x E M, t E (0, R2) and r E (0, R),
cV(x, f
)e-3r2/t <
f
e-d(=,u)2/tdy
< CV(x, )e_r2/(2t).
(x,y)>r
For the proof, fix x E M, t E (0, R2), r E (0, R), and set
I (x t)
f
e-d(x,y)2/tdy.
d(x,,)>r
For the lower bound, consider a point z at distance 2r + Vt- from x. Then
I (x t) >
J (z,r+/)
e-d(x,y)2/tdy
> c1V (z, r + /)e-3r2/t f))e-3r2/t > c2V(z, 2(r + > C3V (x, V
3 t)e-r2/t .
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
126
For the upper bound, observe that it suffices to consider the case when
v < r because e-d(x,y)2/tdy < V (X, vi). JB(Z,%fl)
Assuming v < r, write
(d(xY)2) dy
00
I(x, t) <
exp 2k-1
k=1 00
< J V (x
,
r2k) exp (
k=1
-
r2 22(k-1) t
\\\
V (x r2 k) 00
r222(k-1) exp \-
/
k=1 V (X' V t-)
r2k
00
< V (x, s)
E exp (C \
k=1
r222(k-1) \ J
t
.
/
The last inequality follows from Lemma 5.2.7 which yields, under the volume hypothesis of Lemma 5.2.13, V (x, T) < exp (CT/s) .
VxEM,VsE(0,R),VTE(s,oo), Since
00
exp
(Cr2k
-
r222(k-1)
t
)
V (x' s) -
< exp
r2
l J
k=1
the desired result follows.
With the help of Lemma 5.2.13, the argument of Section 4.2.4 can be adapted to prove the following theorem.
Theorem 5.2.14 Let M be a complete non-compact manifold. Fix R E (0, oo]. Assume that there exist v > 2 and C such that, for any ball B of radius less than R, the local Sobolev inequality "/(v-2)
CJ
V
12v1(v-2)d/1l
I
< Cµ (B 2/v f (jvf j2 +
O
r(B)-2lf
12)d
is satisfied for all f E Co (B). Then there exist two constants c, e > 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 <_
Jf
E(B) o w2dµ) with 0 = 1 + 2/v, w = a - a' and E(B) = C(B)µ(B)-2/"r(B)2. In terms of the supersolution u, this reads
ff
uedµ < E(B) (A(rw)-2u-Pdµ f f/`
\
(5.2.22)
Q
for any 0 < p < oo. From here, the iteration used to prove (5.2.15) yields
sp{u-P} < AE(B)/2[(1
-
5)r]-2-"
fj u-Pdµ
(5.2.23)
which is the desired result.
In order to state the next result we need to introduce the following notation. Given x E M and reals r, s, r, 6 with r, r > 0 and 0 < S < 1, we set
Q'6(r, x, s, r) = (s
- rr2, s - (1 - 6)rr2) x SB.
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
130
Theorem 5.2.17 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
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 - ffIIP<_CsIIVfllp
for all f E COI(M) and all 0 < s < R/4. That is, M satisfies an LP pseudo-Poincare inequality.
By Lemma 5.3.1, we can assume that rc = 2. Fix 0 < s < R/4. Let Bj be a collection of balls of radius s/2 such that Bi n B; = 0 and M = 1 Ji 2Bi. Such a collection always exists (using Zorn's lemma). The doubling condition implies easily that the overlapping number
N(z) = #{i: z E 8Bi} satisfies
sup N(z) = No < oo. zEM
Now, write
Ill - flip
If (x) - f8(x)Ipdp
< 21'>f
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 <
2B{
e
< C2
If (y) - f4B, Pdy] dx I
V (x, s) [JB(xs)
µ(B:)
JaBi J4B;
If - f4B; I Pdµdµ
< C38P E I IVfIpdµ 2
r8B,
< C3Nos?J IVfIpdp.
(5.3.5)
M
By (5.3.3), (5.3.4) and (5.3.5), the desired inequality 11f -
f. 11P
< Cs IIVflly,
follows.
We can now state the main result of this section.
Theorem 5.3.3 Fix 1 < p < oo and 0 < R < oo. Assume that M satisfies a scale-invariant LI 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, for any K > 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<),Bi) * 0 because, if not, Bk+1 must have radius rk+1 > p and this contradicts the definition of k. Now, B(z, p) fl (Ui<
Let us now fix a covering F of E having the properties (1)-(4) above. There is a ball B. E .F such that the center x of E belongs to 2Bx. We call By the central ball of F. Fix a ball B E Y. Let 77B be the center of B and fix a distance-minimizing curve ryB joining x to 71B.
Lemma 5.3.6 For any B E Y, d('yB, OE) > (1/2)d(B, OE) _ (1/2)103r(B).
In particular, any ball B' in F such that 2B' intersects 'yB has radius bounded below by
r(B') > (1/4)r(B). Indeed, let ( E ryB be such that d((, OE) = d('B, OE). Then
d(x, () + d((, (9E) > 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
<
I f - f4B; IPdµ)
+
pdiz\ 1/p
(f
(J4,6.n4Bj+j
4Bin4Bi+1
I f - f4Bi+1 I
4B
1
1/p
1/p
If - f4B, IPdµ)
)
+
If - f4Bi+1 IPd1) (fBi+1
(f,
\ 1/p
< Cr(Bi) (18Bi
I V f IPdµ !
/
+ Cr(Bi+1) (
f
\ 8B,+1
1/p
IV f {Pd i)
/
.
(5.3.8)
The desired conclusion thus follows from (5.3.8) and Lemma 5.3.7 which shows that r(Bi) r(Bi+1), Bi+1 C 4Bi and µ(4B2 n 4Bi+1) ` µ(B2). The next lemma extends Lemma 5.3.9 to the case of a non-trivial weight We will use the following notation. For any 0 E M(r, a) and any x E M,
we let E = B(x, r) and consider the Whitney covering F of E as above. Moreover, we set 4 (y) = 4(d(x, y)) as in Theorem 5.3.4.
139
5.3. POINCARE INEQUALITIES
Lemma 5.3.10 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 B1, B:+i in.F(B), 1/P
1/P
r(B;)
I f4B; - f4B;+, I C µ(B)4)(B)/l
(LB.
IOf IP4idiz
where 4?(B) = fB 4idµ.
By Lemma 5.3.9 and the fact that 4) is essentially constant on 32B if B E F we have 1/p
If4Bi - f4B;+1I
)1/P
(LB.
IOf IP4;dp
By Lemma 5.3.6 and the properties of functions in M(r, a), there exists a
constant c > 0 such that $(B1)/µ(B=) > 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)
j(B)
f Ifldp B
Theorem 5.3.11 Fix R > 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
<_ Ep(5Bi)p
p(EA)
< CA-'>J Ifidµ B;
CA-' f If I dp This finishes the proof of Theorem 5.3.11.
Lemma 5.3.12 Fix R > 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
<_ C11
aiXB. it p.
i
i
Set f = Ei aiXKB,, 9 = Ei aiXBB. It suffices to show that for any 0 E Co (M),
f
f4dp 5 CIIgIipII0II4
where 1 = 1/p + 1/q. Write
f f g5dp =
ai f XKB2 (x)cb(x)dx
a:u(KB:) p(KBi)
< C E aip(Bi) i
p(KBi)
J
OdA B:
IKB,
Now, for any x E Bi, 1
p(KBi) KB,
¢dp 5
Hence
p(KBi) JKBj Ldp
[MKRd1.
p(Bi) Jet
Odµ.
141
5.3. POINCARE INEQUALITIES It follows that
J
f ¢dµ < C > 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 <
I f - f4Bx I pI'd,L
BEY2B
< 2p
f (if - f4BI p +If4B - f4Bx I
BEY 4B
< 2p
f
BEB
f - f4BIdµ +
If4B - f4B I
(4B). (5.3.10)
BEY
is essentially constant on the balls 4B, B E.P. Thus
By hypothesis, (5.3.1) implies
- f4B I pdµ LB
Por (4B)fB 8
I Vf I
dµ.
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
142
Hence
f If - f4BIP4Dd1A < Po BE.F
B
BEf
r(4B)P 8B f
f
< 4PPorPJ IVfIP-,Ddµ
(5.3.11)
B
where the last inequality uses (5.3.7) and the fact that 8B C E for any B E .F. We now have to bound
I=
I f 4B ` f4B2IP41(4B) < C 1 BE.F
BE.
f
If4B - f4B= I P 4)
XBdIL.
B)
J
and, using the string of balls To this end, recall that Bx = Bo, B = ,F(B) = (Bo, BI,-, BP-1) and Lemma 5.3.10, write 4'(B) I f4B
- f4BoI
f(B)-1
1/P
/Bl
1: If4B, - f4Bi+iI
(/L(B))
L(B)-1
1/P
< C E r(Bi p(Bi) lP UUBi
I V f IP-dµ0
By Lemma 5.3.8, the ball B is contained in 104Bi for any Bi E .1(B). Thus, the last inequality yields
//B) P/,/A)
IK pl A)lP (12A IQ XB < C AEf
I A9 - f4Bo I l
f ('
l
d/A
1/p
X104AXB
}
As EBE. XB < 1 (the balls in F are disjoint), we get I f4B - f4BoI' T(B) XB :5 C
1:
BEf
E
1/p
T. (A)
,4(A)11P
(LA
IV f I P44µ /
/
X10 5A
By Lemma 5.3.12 and since the balls in F are disjoint, this yields
f
f'
If4B - f4B0I P
(B) XBdµ
BE
< C' f
r(A) AEs p(A)1/P
>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}) <
Cr2µ(B)A-1
and
ji ({(t, z) E K_: logu >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)) <_ 0 or, equivalently,
-Air2V 8t (IA + C - W(t)I-1) >_ 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<(5
µ(log v > A)< CAW)Thus we can apply Lemma 2.2.6 to conclude that fu, vPodµ < Alµ(U), that is
1/po µ(1Q') J(ecv)a0d/)
< A'1.
(5.4.3)
--
Set now v = e -cu- 1, where c = c(u) is the same constant as above, given by Lemma 5.4.1 applied to u with 17 = 1/2. This time, set
U = (s - (1/2)x2, s) x B, Ua = (s - (1 + Q)r2/4, s) x aB. By Theorem 5.2.16, we have
p{v"} <
(Q -
Q')2+vf(U) J, vPdi2 A(v)Ct/2
U.0
forall0 A) < CA(U)Thus we can again apply Lemma 2.2.6 to conclude that supuo {v} < A2µ(U),
that is sup{(eau)-1} < A2.
(5.4.4)
Q+
Multiplying (5.4.3), (5.4.4) together, we obtain 1/po
1
(Q'-)
J
uPOdµ
< Ai f{u}
which is the desired inequality.
5.4.3
Harnack inequalities for positive solutions
In this section, we describe different forms of Harnack inequalities for positive solutions of the heat diffusion equation
(at+0)u=0. The basic assumption is that there exists 0 < R < oo such that (5.3.1), (5.3.2) are satisfied with p = 2, that is, M satisfies a scale-invariant Poincare inequality (5.3.1) with p = 2 and the doubling condition (5.3.2), up to radius R.
147
5.4. HARNACK INEQUALITIES AND APPLICATIONS
Theorem 5.4.3 Fix -r > 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
u(s,x)
R2
u(t, y)
s
p2
t - s}
Connect the points x, y by a string of k balls Bo, ... , Bk_1 of radius r and centers xo, ... , xk_1, with xo, ... , xk-1 E 7, xp = x, and x¢+1 E Bi, 0 < i < k - 1 with Xk = y. This is possible as soon as
kr > 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 <_ i <_ k-1, we obtain
u(to, xo) < Aou(tl, x1) < Aou(t2, x2) < .. . Aou(tk, xk),
148
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
that is, u(s, x) < Aau(t, y).
Now, (5.4.5) is satisfied if k > d2/(t - s) because kr = k(t - s) by (i). Similarly, (ii) is satisfied as soon as k > (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
/
u(t' x) ( ,y)
ts f
If s < t < 4s, Corollary 5.4.4 yields log
u(s,
x)
u(t,y)
(
1+
d(x, y)2
t-s )
(5.4.6)
If t > 4s, consider k > 1 such that 2k+1s
and set
< t < 2k+25
ti=2's, 0
Then, by (5.4.6), log u(tk, x) < 2A
u(t, y)
(1 +
d(x, y)2
t-s
because tk < t < 2tk and t - tk > 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)
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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)
w(u,W) = Mu-mu W(u, W+) = Mu - mu of u over W and W+ satisfy w(u, W+) < 9 W(u, W)
(5.4.8)
with 9 = 1 - 1/A E (0,1) (note that A > 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')]°`
<
Aa
ra Q
as desired.
The second case is trivial: if p > (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
5.4.5
Liouville theorems
Let M be a complete Riemannian manifold. One says that M has the strong Liouville property if any solution u of the Laplace equation Du = 0 on M which is bounded below (or above) is a constant. Usually, the weak Liouville property refers to the same result for bounded solutions. In the classical case of harmonic functions in ]Rn, the strong (and weak) Liouville property is satisfied and one can also prove that a harmonic function satisfying limn... r supB(o,,,){Iul} = 0, i.e., having sublinear growth, must be constant. These Liouville properties are somewhat subtle properties. In particular, T. Lyons [58] proved that the strong and weak Liouville properties are not stable under quasi-isometric changes of metrics. It is well known that Liouville-type properties follow from Harnack-type inequalities.
Indeed, assume that any non-negative solution u of Du = 0 in a ball B satisfies the elliptic Harnack inequality
sup Jul < C inf Jul (1/2)B
(1/2)B
where C is independent of u and B. Let u be a solution of Au = 0 in M which is bounded below. Let m(u) = infM{u}. Applying the Harnack
152
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
inequality above in the ball 2B and to the function v = u - m(u), we find that susp{u - m(u)} < Cin {u - m(u)}. B
As the radius of B tends to infinity, infB{u - m(u)} tends to zero and we conclude that u = m(u) is constant. Of course, the parabolic Harnack inequality of Theorem 5.4.3 implies the elliptic version used in the argument above. Thus we have proved the first assertion of the following theorem.
Theorem 5.4.9 Assume that M is a complete Riemannian manifold satisfying (5.3.1), (5.3.2) with p = 2 and R = oo. Then M has the strong Liouville property. Moreover, there exists an a > 0 such that any function u such that Au = 0 in M and lim
1 sup {IuI} = 0
r-.oo ra B(o,r)
must be constant. Here o is some fixed reference point in M.
Only the second assertion still needs to be proved. Let a be as given by Theorem 5.4.7. Let u be a function such that Au = 0 in M and lim
1 sup {Iul} = 0.
r- oo ra B(o,r)
Fix some x E M and y such that d(x, y) < 1. Applying Theorem 5.4.7 to u in a ball BR = B(0, R) with R so large that x, y E (1/2)BR, we find that
tu(x) - u(y)I 5 CR-°`sup{juj}. BR
As this holds for all R large enough, we can let R tend to infinity to obtain
that Ju(x) - u(y) I = 0. As x, y with d(x, y) < 1 are arbitrary and M is connected, u must be constant.
Let us conclude this section by pointing to a recent advance in this direction due to T. Colding and W. Minicozzi II [17]. They proved that, for complete Riemannian manifolds as in Theorem 5.4.9 above and for any a > 0, the linear space of solutions of Au = 0 with the growth property
that sup
1
sup { lul} < oo
1 B(o,r) r>1 7*
is finite dimensional. See also [53, 54].
5.4.6
Heat kernel lower bounds
The Harnack inequalities of Section 5.4.3 easily yield heat kernel lower bounds. First, we have the following on-diagonal lower bound.
5.4. HARNACK INEQUALITIES AND APPLICATIONS
153
Theorem 5.4.10 Fix 0 < R < oo. Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2. Then the heat kernel on M satisfies h(t, x, x) >
c
V (X' V1)
for all xEM and all 0 < t < R2. Fix 0 < t < R2. Let B = B(x, f). Let 0 be a smooth function such that
0<¢<1,4 =1onBand0=OonM\2B. Define u(t, y) _ 5 Ht¢(y) 0(y)
if t > 0 if t < 0.
Obviously, this function satisfies (at + O)u = 0
on (-oo, +oo) x B. Indeed, by local estimates, one has
ioHtot -
=-L¢=0
pointwise in B. Applying Theorem 5.4.3, first to u, and then to the heat kernel (s, y) --+ h(s, x, y), we get Au(rt/2, x)
1 = u(0, x)
=A
h(t/2, x, y)q5(y)dy
J < A / B h(t/2, x, y)dy < A2tt(2B)h(t, x, x).
This gives h(t, x, x) > A-2V (x, y L)-1
as desired.
Theorem 5.4.11 Fix 0 < R < oo. Assume that (5.3.1), (5.3.2) are satisfied for this R with p = 2. There exists a constant A such that, for any x, y E M and any 0 < t < oo the heat kernel h(t, x, y) satises
d(x,y)2)) h(t, x, y) > 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) <
V (x, s)
V (x, V't)
Moreover, for any integer k, Idt h(t, x, y)I <
Ak eXPk (-c2d(x, y)2/t)
tV(x,f)
and
IOt h(t, x, y)
- Ot h(t, x, z)I < Ak
a/2 (d(Yz)) , exp (-c2d(x, y)2/t) t k V (x,
for all xEM,t>0and z,yEM,d(z,y)< ft. If the hypotheses above are relaxed so that one only assumes that (5.4.9), (5.4.10) hold for 0 < r < R, for some fixed R > 0, then the same conclusions hold with the restriction that 0 < t < R2.
Corollary 5.4.13 Let (M, g) be a complete manifold. Assume that (5.4.10) and (5.4.9) are satisfied. Then (M, g) admits a positive symmetric Green function if and only if f °° V (x, /)-ldt < oo. Moreover, if this condition holds, the Green function G(x, y) satisfies c
f
00
dt
dt
< G(x, y) S C fd(X,Y)l
V (x, V0
d(x,y)2 V (x, V'1)
and, for some positive a, I G(x, y) - G(x, z) I < C, d(y, z)a
for all x, y, z E M, x
°°
dt
Jd(x,y)2 t°`I2V (x,
y, d(y, z) < d(x, y)/2.
N/ t)
155
5.5. THE PARABOLIC HARNACK PRINCIPLE
To see this, let us first observe that the Green function, if it exists, is the kernel of A-' = f °° e-'t°dt. See, e.g., [34]. Moreover, it is not hard to argue that the Green function exists if and only if the integral f o' h(t, x, y)dt is finite for all x , y and that G(x, y) =
fh(tx)dt.
Now it suffices to apply the bounds of Theorem 5.4.12 and note that d2
h(t, x, y)dt < Cd2/V (x, d)
1 and
00
h(t, x, y)dt > cd2/V (x, d) d2C
when d = d(x, y). Finally, we state a more explicit bound on G under an additional assumption.
Corollary 5.4.14 Assume that (5.4.10) and (5.4.9) are satisfied. Assume that the complete manifold (M, g) satisfies
Yx E M, Yr > 0, Vs E (0,r),
V(x,r) V (x, s) > c
(r
2
`s I
Then M admits a Green function G which satisfies d(x,y)2 G(x, y) < cV(x, d(x, y)) <
Cd(x,y)2 V(x, d(x, y))
and, for some positive a, y)2-,,
IG(x, y) - G(x, z) I < C, d(x, d(y, z)a - V (x, d(x, y))
for all x, y, z E M, x # y, d(y, z) < d(x, y)/2.
5.5
The parabolic Harnack principle
In the two sections above, we saw how a scale-invariant Poincare inequality and the doubling condition imply a scale-invariant parabolic Harnack inequality. It is quite remarkable that, in fact, Poincare inequality and doubling are equivalent to the validity of this parabolic Harnack principle. This yields a characterization of the parabolic Harnack principle as announced at the beginning of this chapter. As we will point out below,
156
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
both directions of the equivalence between "Poincare inequality, doubling" and "parabolic Harnack inequality" can be useful. Let us comment on the history of parabolic Harnack inequalities. In [64], J. Moser refers to works of Hadamard [37] and Pini [70] from 1954 concerning the case of constant coefficients in R's. At the end of his celebrated 1958 paper [67], J. Nash states a parabolic Harnack inequality for positive solutions of parabolic uniformly elliptic second order differential equations in divergence form. His approach is to derive the Harnack inequality from bounds on the fundamental solution. However, Nash's statement is incorrect. A complete (and correct) implementation of Nash's ideas in this direction was later given in [23]. In 1961, Moser published his famous iterative argument [63], giving a proof of the elliptic Harnack inequality for positive solutions of uniformly elliptic equations. In [64], published in 1964, Moser adapts his own iterative argument to the case of parabolic equations. An interesting account is given in [71]. Moser's iteration has been used and adapted in hundreds of papers. Let us mention in particular the works of Aronson, Aronson and Serrin, and Trudinger. Some references are in [4, 30, 71, 75, 76].
One of the reasons for the success of Moser's technique is that it only depends on a small number of functional inequalities: essentially, Poincare and Sobolev inequalities. It can thus be used in many different situations.
Another very important feature of Moser's iteration is that it is a local technique. This is what makes it most useful in the Riemannian context as illustrated in the previous sections. To understand this fully, the reader should compare it with Nash's ideas as developed in [7, 23]. Nash's ideas require global hypotheses to be implemented successfully. This point was not completely understood until recently. In fact, the use of Moser's iteration in the context of Riemannian geometry has often been restricted by the incorrect belief that it would require a global Sobolev inequality to yield global results. See for instance [88, page 202] and the use of Moser's technique in [14, 15] and [89]. See also the introduction of [32]. The observation that a suitable family of local Sobolev inequalities yields good large scale results in a Riemannian context appears in [75]. This idea however is implicit in several papers from the 1980s concerning subelliptic operators. See [47, 25] for pointers to this literature. In the subelliptic context, only small scales are usually considered but the point of a good localization technique as developed above is to make all scales, small or large, look alike. It is interesting to observe how this simple idea enhances some of the techniques developed in [14].
A crucial step towards a better understanding of the geometric meaning of parabolic Harnack inequalities was made independently by A. Grigor'yan [32] and the author [74] (Grigor'yan's work was done and published earlier): it is Poincare inequality, not Sobolev inequality that is crucial for a full parabolic Harnack inequality to holds. This is not so obvious from previous
157
5.5. THE PARABOLIC HARNACK PRINCIPLE
works on the subject which tend to emphasize the role of Sobolev inequality. See for instance the discussion of Sobolev and Poincare inequalities as well as the discussion concerning Moser's iteration in [89]. The papers [32, 74]
each contain a different proof of the fact that a scale-invariant parabolic Harnack principle holds as soon as a scale-invariant Poincare inequality and the doubling volume condition are satisfied. The proof in [74] is based on Moser's iteration. It amounts to proving that Poincare inequality and doubling imply a family of local Sobolev inequalities. See Theorem 5.3.3. The approach taken in [32] is different and more original, avoiding the explicit
use of any form of Sobolev inequality altogether. See the introduction of [321,
What makes the contribution of [32, 74] remarkable is that the scaleinvariant Poincare inequality and the doubling volume condition are not only sufficient but also necessary for a scale-invariant parabolic Harnack principle to hold (details are given in Section 5.5.1 below). Both [32] and [74] noticed that the doubling condition is necessary. As observed in [74], the scale-invariant Poincare inequality is also a necessary condition, thanks to an argument due to Kusuoka and Stroock [50].
5.5.1
Poincare, doubling, and Harnack
The next theorem is one of the major results presented in this monograph. Half of it has already been proved in the two previous sections.
Theorem 5.5.1 Fix 0 < R < oo and consider the following properties:
(i) There exists P0 such that, for any ball B = B(x, r), x E M, 0 < r < R, and for all f E Coo (B),
Ja If -fel2dp
IVfI2dµ.
(ii) There exists Do such that, for any ball B = B(x, r), x E M, 0 < r < R,
µ(2B) < Dop(B). (iii) There exists a constant A such that, for any ball B = B(x, r), x E M, 0 < r < R and for any smooth positive solution u of (8t + O)u = 0 in the cylinder (s - r2, s) x B(x, r), we have
sup{u} < Ainf{u} Q_
Q+
with
Q. = (s - (3/4)r2, s - (1/2)r2) x B(x, (1/2)r) Q+ = (s - (1/4)r2, s) x B(x, (1/2)r).
(5.5.1)
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
158
Then the conjunction of (i) and (ii) is equivalent to (iii). Although it is not strictly necessary to use this fact in the proof given below,
it is often helpful to remember that t - h(t, x, x) is non-increasing. That (i) and (ii) imply (iii) is the content of Theorem 5.4.3. That (iii) implies (ii) can be seen as follows. The proof of Theorem 5.4.10 shows that (iii) implies the on-diagonal heat kernel lower bound (note that one cannot use (ii) here)
h(t, x, x) > aV (x, 2f)-1.
`d x E M, `d t E (0, R2),
(5.5.2)
Inequality (5.5.2) can be complemented with a matching upper bound as follows. Applying (iii), for all x, y E M, t E (0, R2), with d(x, y) < (1/2) f , we have
h(t, x, x) < Ah(2t, x, y).
Integrating over B(x, (1/2)x) yields
V(x, (1/2)')h(t, x, x) < A
J
h(2t, x, y)dµ(y) < A.
Thus V x E M, `d t E (0, R2), h(t, x, x) < AV(x, (1/2)V't-)-1.
(5.5.3)
Finally, a last application of (iii) yields that, for any fixed k > 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) <
V(y,A f).
This can be proved by the argument used for (5.5.2), (5.5.3). Once this has been established, (iii) easily yields the two sided inequality of Theorem5.5.2. What we really need from Theorem 5.5.2 is the lower bound
hB(r2, y, z) > 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 <
ec(1+r)2
B(o,r)
0
Then u = 0 in MT.
Let us first check that Theorem 5.5.4 follows from Theorem 5.5.5. Set v(t, x) = fm h(t, x, y)dy and u = 1 - v. Then 0 _< u < 1 and u(0, x) = 0. Thus for any ball B(o, r), r > 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!<1,fix s>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 <_ 1/r. This yields
f
u2(r - a,
u2(r, x)e9(T,z)dx < J (o,r}
x)e9(r-a,x)dx
(o,4r)
+ 42 T-a L(o,4r)\B(o,2r) We also choose
p(x) = inf{d(x, z) : z E B(o, r)}, s = r + a
IuI2e9dµdt.
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
164
so that p(x)2 < 0 T+a-t -
g(t, x) _
for t < -r and g(t, x) = 0 if x E B(o, r). It follows that the factor e9 can be dropped in the two first integrals. Moreover, for x E B(o, 4r) \ B(o, 2r) and T - a < t < T,
p(x)2 < ,r+a-t
g(t,x
r2 2a
Hence
f
u2(r, x)dx < JB(o,4r) U2(7- - a, x)dx (o,r) 4e-r2/2a
+
fr
IB(o,4r) Iu(t, x) I2dxdt.
2
r
-a
B y the hypothesis made on u in Theorem 5.5.5 we can now choose a small
enough so that T7a
IU12dµdt < e-(r2/2a)+c(1+r)2
e-r2/2a
< 1/4.
B(o,4r))
For this choice of a, we obtain J
u2(r, x)dx <
..11 B(o,r)
r
U
JB(o,4r)
Finally, fix 0 < s < T, R > 0. Pick m large enough and apply the above
mtimes with r=4kR,a=s/m,Tk-Tk_1=a, To=0,k=1,...,m. This yields
JB(o,R)
u2(s, x)dx < f
u2(0, x)dx + R2
4-2k <
R
(o,4'^R)
Since this holds for all R, it follows that u(s, ) = 0 as desired.
5.5.3
Local Sobolev inequalities and the heat equation
The aim of this section is to clarify the role of local Sobolev inequalities in the study of the heat equation presented in this chapter. The interested reader should compare the results below with those contained in [33], which are based on a different but equivalent set of functional inequalities called Faber-Krahn inequalities. Also, the results below should be compared with Theorem 5.5.1. Fix 0 < R < oo, v > 2, and consider the following properties:
165
5.5. THE PARABOLIC HARNACK PRINCIPLE
For any ball B of radius 0 < r(B) < R and for any f ECo (B),
(B2 I (Iof12 +r(B) -21f 12) dµ
Y If12"i("-2)dµ1(v-2)/v < C'i
l
(5.5.10)
For any concentric balls B, B' of radius 0 < r < r' < R,
µ(B') < C2(r'/r)"µ(B).
(5.5.11)
For any 0 < t < R2, any ball B of radius f and any positive solution u of (at + A)u = 0 in Q = (t/2, t) x B, sup{u2} < where Q+
C3 fu2dp t(B)
(5.5.12)
(t/2, t) x (1/2)B.
ForanyxEMand0
C4 V (X' Vat)
(5.5.13)
Foranyx,yEMandany0
v)-1
exp
(_d(xi)2).
(5.5.14)
There exists e > 0 such that, for any 0 < t < eR2 and any x E M,
h(t,x,x) >
V(x
f)
(5.5.15)
Theorem 5.5.6 Let M be a complete non-compact manifold. (1) The local Sobolev inequality (5.5.10) implies the inequalities (5.5.11), (5.5.12), (5.5.13), (5.5.14).
(2) Assume that (5.5.11) holds true. Then the local Sobolev inequality (5.5.10), the mean value inequality (5.5.12), the on-diagonal heat kernel upper bound (5.5.13), and the Gaussian heat kernel upper bound (5.5.14) are equivalent properties.
(3) Assume again that (5.5.11) holds true.
Then the local Sobolev inequality (5.5.10) (or any of the equivalent properties listed in 2 above) implies the on-diagonal lower bound (5.5.15).
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
166
The proof of the first assertion is contained in Section 5.2.1, Theorem 5.2.2, Theorem 5.2.9 and equation (5.2.17). Let us prove the second assertion. Clearly (5.5.14) implies (5.5.13). Let
us show that (5.5.12) implies (5.5.13). Indeed, applying (5.5.12) to the positive solution u(t, y) = h(t, x, y) in the cylinder Q = (t/2, t) x B where B is the ball of radius f around x, we get h(t, x, x)2 <
C3
tV(x, V) JQ
h(s, x, y)2dyds
C3 tV(x, f) / 2 It
< <
tV (x, f) Jft/2 C3
c
C3 2V (x, v Ft)
f
h(s, x, y)2dyds
h(2s,x,x)ds
h(t, x, x)
where we have used successively the facts that h(s, x, y)2dy = J h(s, x, y)h(s, y, x)dy = h(2s, x, x) N
and that s -+ h(s, x, x) is a non-increasing function. Simplifying by h(t, x, x) we get (5.5.13) as desired. Now it suffices to show that (5.5.13) implies the local Sobolev inequality (5.5.10). In doing so, we will use our hypothesis that (5.5.11) holds true.
Fix a ball B of center x and radius r < R. By (5.5.11) and (5.5.13), we have
h( t, x, x) <
C4r(B)"t-.12
- µ(B)
for all 0 < t < r(B)2. Let us introduce the Dirichlet semigroup H$'D which is associated with the minimal closure of the form fM I Vf 12dµ with domain Co (B) in L2(B, du). Let hZ(t, x, y) be the corresponding Dirichlet
heat kernel. We need to use a classical fact: the Dirichlet heat kernel is always bounded above by the full heat kernel h(t, x, y) (this follows from the maximum principle: h - hB is a solution of the heat equation in (0, oo) x B and is positive on (0, oo) x OB because, on the boundary, h > 0 and hg = 0). We will use this fact to prove the following lemma.
Lemma 5.5.7 Assume that (5.5.11) and (5.5.13) hold true for some R > 0. Then there exists a constant C such that, for any ball B of radius r(B) less than R, the Sobolev inequality d f E C000 (B),
is satisfied.
11f ll2vi(v-2) <- Cµ(B)21v
f ({of 12 +
r(B)-2{
f l2) dµ
167
5.5. THE PARABOLIC HARNACK PRINCIPLE
Setting r = r(B), the hypothesis yields the estimate a-t/r2
V y E B,
h°a(t, y, y)
Cp(B)-lr"t-"/2.
e-t/,2h(t, y, y) <
Applying Theorem 4.1.2 to the semigroup e-t/12HB,D, we obtain the Nash inequality
,f il2
1+2/")
Cp.(B)-2/"r2 (fivi I2 + r_2I r j2)dµ) Ilf 11i/"
<
The desired result then follows from the equivalence between the (local) Nash inequality and the (local) Sobolev inequality. See Corollary 3.2.12 and Section 3.2.7. This finishes the proof of the second assertion of Theorem 5.5.6.
Finally, the last assertion of Theorem 5.5.6 follows from Theorem 5.2.14 and this finches the proof of Theorem 5.5.6. For completeness, we note the following result. The Dirichlet heat kernel admits a discrete spectral decomposition h' (t, x, y)
c*
e_tAiOi(x)oi(y)
where Al < )'2 < . are the eigenvalues in non-increasing order repeated according to multiplicity and the ¢i are associated real eigenfunctions normalized by 114112 = 1 (in L2 (B, dµ)) . In particular,
e-t.1 <
fh(t,xx)dx.
This can be used to derive an estimate of the Dirichlet eigenvalue Al (B).
Lemma 5.5.8 Assume that (5.5.11) and (5.5.13) hold true for some R > 0. Then there exists e > 0 such that, for any ball B of radius smaller than ER, the Dirichlet eigenvalue al is bounded below by A1 > c/r2. Here e > 0 depends on the constants appearing in the two hypotheses (5.5.11) and (5.5.13).
Fix the ball B of radius r < ER. Let B' be a concentric ball of radius s E (r, R) to be chosen later. We have z, z)dz - fB hD(s2, B
e-' a1 < and
V z E B',
hD(s2, z, z) < h(s2, z, z) <
It follows that
e`'2al < C4 p(B)
p(B').
IL(BI) B') 9
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
168
By Lemma 5.2.8, there exists ry > 0 such that
µ(B)µ(B,)<- C(r/s)7 Choosing s = r/e (which indeed belongs to (r, R)), we get e-8 gal < C'ery.
If e > 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 <- oo, the parabolic Harnack inequality (5.5.1) is stable under quasi-isometry. That is, if (M, g) and (M, g) are two complete Riemannian structures on M such that g and g are quasiisometric, then (5.5.1) holds on (M, g) if and only if it holds for (M, g).
This immediately follows from Theorem 5.5.1. Indeed, the conditions (i) and (ii) are obviously stable under quasi-isometry since (5.5.16) implies
cg(wf, V f) <_ g(of' Of) < c 'g(V f, Of )
5.5. THE PARABOLIC HARNACK PRINCIPLE and
169
e12 < dµ < C-n/2 d.t -
where n is the topological dimension of M. It is worth pointing out here that, because of the nature of the proof of Theorem 5.5.1, the above result stays valid even if the metrics g, g are not smooth but merely continuous or even measurable. Next, we consider the situation where we have two complete Riemannian manifolds (M, g), (M, g) and a smooth map 0 : M -+ M such that:
(a) ¢ is onto, i.e., 4)(M) = M. (b) For any u E C°°(M), the function v = u o 0: M --> R satisfies AV = (Du) o
Theorem 5.5.10 Assume that (M, g), (M, g) are two complete manifolds and that 4) : M - M is a map satisfying (a) and (b) above. Fix 0 < R < oo. Assume that conditions (i) and (ii) of Theorem 5.5.1 are satisfied on (M, g), that is, assume that there exist Po1 Do such that for any ball B = B(x, r),
xEM,0
`d f E C- (b),
f
1 f - fBI2du < Po r2 J I0f 12dµ
and
µ(2B) < Doµ(B). Then (M, g) also satisfies these two properties.
Stated as it is, this theorem is rather non-trivial. However, it will easily follow from Theorem 5.5.1. To see this, we need to show that for any metric ball b = B(x, r) C M, 4)(B) = B(q5(x), r).
(5.5.17)
Assuming that (5.5.17) is true, let u be a positive solution of (at + O)u = 0
inQ=(s-r2,s)xB(x,r),xE_M,sER,0
Q+
170
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
with Q_ = (s - (3/4)r2, s - (1/2)r2) x B(x, (1/2)r), Q+ _ (s - (1/4)r2, s) x B(i, (1/2)r). Using (5.5.17) again, it follows that scup{u} < A i Jul with Q_ = (s - (3/4)r2, s - (1/2)r2) x B(x, (1/2)r), Q+ (s - (1/4)r2, s) x B(x, (1/2)r). That is, (5.5.1) is satisfied on (M, g). Thus, the conditions (i) and (ii) of Theorem 5.5.1 are also satisfied. Before proving (5.5.17), let us state some further properties of 0 which follow from conditions (a),(b). Let f, h E C°°(M). Set f = f o 0 and define h similarly. Then (a) and the formula
O(fh) = f1h+ hLf + g(Vf, Vh) together with the similar formula on (M, g), imply that
Oh}(x) = g(Vf, Vh)(x),
(5.5.18)
for all x, x such that O(x) = x. Fix a local frame (ei) on M around x E M so that the metric g has the canonical form (i.e., gz = Si,,) at x. For each i, we can find a function hi such that Oihi(x) = 1,O,,hi(x) = 0 if j # i. It then follows from (5.5.18)
that df o dO(Oh(x)) =Oaf (x).
Since this is true for all f, it follows that dq'(Vhi(x)) = ei(x)-
That is, do is surjective. In other words, ¢ is a submersion. In fact, by (5.5.18), it is a Riemannian submersion. In particular, 0 is open. Note that this shows that it is enough to assume that (M, g) is complete since the hypotheses (a),(b) above then imply that that (M, g) is also complete. See, e.g., [29, 2.109].
Let us now prove (5.5.17). In order to give a proof that would easily work in other settings, we will use as little Riemannian geometry as possible. A proof more in the spirit of Riemannian geometry is given in [29, 2.109]. Let us first show r)) C B(q(x), r). (5.5.19)
Recall that the distance d on M can be computed by the formula d(x, y) = sup{ f (x) - f (y) : f E CO°(M), IV f I < 1}.
(5.5.20)
Of course, d can be computed from a similar formula. Let f E C°° (M) and set f = f o 0. By (5.5.18), we have (D.f fi(x) = IVf 1 0 0(x).
(5.5.21)
171
5.5. THE PARABOLIC HARNACK PRINCIPLE From (5.5.20) and (5.5.21) it follows that
d(x,y)>r=d(x,y)>r for all x, y E M, x, y E M such that O(x) = x, 0(y) = y. The desired inclusion (5.5.19) follows. We now want to prove
r)) i B(O(x), r).
(5.5.22)
To this end, we want to show, roughly, that paths can be lifted from M to M in an appropriate way. First let us note that it will be enough to work locally because of our assumption that both M and M are complete manifolds.
_
Fix x E M, x E M, such that ¢(x) = x. As 0 is open, we can find connected neighborhoods U of Y and U of x such that .0(U) = U. We can also assume that in U we have a moving orthonormal frame, say (X1)i , where i is the topological dimension of M. We let Xi = dc(Xi), 1 < i < n. Note that the Xi's span the tangent space at any point x E U. Moreover, aiXi by (5.5.18) and the definition of the Xi's, for any vector field X in U we have n
ai =9(X,X)
g(X,X) where J f-
Ee ai o OXi.
Now, let
y: [0,s]->M be a smooth distance-minimizing curve parametrized by arc length and joining x to y, with y in some neighborhood of x contained in U. Let us write, as we may, n
Ot'y(t)
_
ai(t)Xi('y(t))
with smooth coefficients ai. As ry is parametrized by arc length, we mus have
n
Vt E [o,sl, g(at-y(t),at7(t)) =1= Now, define =y : [0, s] -> M
to be the solution of
{
%y(0)
= X.
ai(t)12.
CHAPTER 5. PARABOLIC HARNACK INEQUALITIES
172
By uniqueness of 7 (as a solution of a Cauchy problem), we must have d t E [0, sj,
q5(ry(t)) = 7(t).
In particular, y = y(s) is a point such that d(Y, y) _< s = d(x, y) and 0(y) = y. This shows that 0(B(x, r)) D B(x, r) for r small enough. The general case follows by the triangle inequality.
Examples
5.6
We describe below a few examples where the results developed in this chapter apply.
5.6.1
Unimodular Lie groups
Let us start with connected Lie groups. Let G be a connected Lie group and (X1,.. . , X,,) be a basis of left invariant vector fields. See Section 3.3.3 for notation and details. Let us assume that G is unimodular so that (X1, ... , X,) yields a Riemannian structure such that A = - Ei Xq , V f = (Xl f, ... , X.f ). The Riemannian measure is a Haar measure d1t on G.
Let g, h E G and let 'Yh : [0, th] --> 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
f 0
th
IVf(97h(s))I2ds
for any smooth function f and all g, h. Let B be the ball of radius r around e. Then we have
j j If (9h) - f (9)I21B(9)1B(gh)dgdh
t < 2r Jc
th
Jc Joth JV f (91'h(s))j21B(9)1B(gh)dsdgdh
< 2r ffJ Bo
IV f
< 2rJ J th \(J < 2r
i
(97h(s))12128(9'Yh(s))12B(h)dsdgdh
IVf(97h(s))I212B(97h(s))d9) 12B(h)dsdh
fth (LB I V f
< 2r2p(2B) (LB
1 (9)I2dg) 12B(h)dsdh
f (9)
I2)
173
5.6. EXAMPLES This computation makes sense for all f E C°°(2B). It follows that
j
IVfI2dp.
Let us note that (a variation of) the proof above shows that a similar Poincare inequality holds in 1/, 1 < p < oo. By Lemma 5.3.1, we thus have proved the following result. Theorem 5.6.1 Let G be a unimodular connected Lie group equipped with a left-invariant Riemannian structure as above.
(1) For all 0 < r < 1 and all balls B of radius r, the Poincare inequality
Vf E C°°(B), f
If
- fB12dµ < Por2 f IV f 12dµ
is satisfied.
(2) If there exists Do such that 1A(2B) < Doµ(B) for all balls B, then the Poincare inequality Vf E C°°(B),
f
a
If _ fs l2di < Por2 f
s
I
Vf 12dµ
is satisfied for all balls B of radius r > 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 1.
Observe that such volume functions have indeed the doubling property. We refer the reader to [87] for details and other references. For small r, the result follows from the fact that 1H[i is viewed here as a homogeneous Riemannian manifold so that the volume of small balls must be uniformly comparable to the Euclidean 3-dimensional volume. To understand what happens for large r, consider the broken curve y which starts at 0, stays
tangent to X, then to Y, then to -X, then -Y, each for the same lapse of time s > 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
<-
IL(B)
B)
µ(B)
I ff1 s
If (x) - f (y) I dxdy d(x,y)
IDf('Y',($))Idsdxdy d(x,y)
a is L(,)/2
Vf ('Yx,y(s))I dsdxdy. I
To obtain the last equality we break the set {(x,y,s) : x,y E B,yx,y(s) E (0,d(x,y)} into two pieces {(x, y, s) : x, y E B, yx,y(s) E (d(x, y)/2, d(x, y)} and
{(x, y, s) : x, y E B, yx,y(s) E (0, d(x, y) /2)},
write the second piece {(x, y, s) : x, y E B, yy,x(d(x, y) - s) E (0, d(x, y)/2)},
and use the (x, y, s) -' (y, x, d(x, y) - s) symmetry. This trick is crucial in obtaining the desired result by this method. It is taken from [49]. Now, suppose we can bound the Jacobian Jx,8 of the map (,,x,8 : y i-* ^fX'V(s)
from below by
V x, y E B, Vs E [d(x, y)/2, d(x, y)], J.,. (y) > 1/F(r)
where r is the radius of the ball B. Then d(x,y)
ff1
8 3d (x,y)/2
I V f (-y.. (s)) I dsdxdy d(x,y)
lB lB L(X,)/2
IOf
(5.6.3)
179
5.6. EXAMPLES d(x,y)
< F(r) fB 1B F(r)
r(x,y)/2
JJ,8(y)dsdxdy I
'BIB 1o I V f (,(y))I J,8(y)dsdxdy T
jr
< F(r)
[J lB I V f
(.t x,8(y))I Jx,8(y)dxdy ds
r
fT
< F(r) I
I Vf(z) I dz
<
F(r)I
dx ds
$x,e(B)
B
T[lB LB
< F(r)rµ(B)J
I Vf(z)ldzdx ds
IVf(z)Idz. B
Hence
fii - fBIdp < 2r F(r) LB IVf(z)Idz.
Thus we are left with the task of proving that (5.6.3) holds with
F(r) < C. exp (C'kr)
(5.6.4)
Lemma 5.6.7 Let M be a Riemannian manifold satisfying (5.6.2) for some
K > 0. Then Vs E B, V s E (d(x, y)/2, d(x, y)), JJ,8(y) > cn exp (-CnvrKr)
for all y E B not in the cut locus of x. This is a consequence of the basic ingredient of the proof of the BishopGromov Theorem [13, Theorem 3.9]. Namely, given x and y (y not in the cut locus of x) let t; be the unit tangent vector at x such that a8ryx,y(s) I8=o = Let I (x, s, ) be the Jacobian of the map (s, Zj) '-- exp., (sC). Then
dµ = I(x, s, e)dsd where dtj is the usual measure on the sphere. Moreover, if we let IK(s) be the analog of I (x, s, ) on the corresponding model space of Ricci curvature -K, then [13, Theorem 3.8] shows that s
I (x, s, .)
IK(s)
is non-increasing. It follows that Jx,s(y) =
I(x, s, e)
IK(S)
I(x,d(x,y),) > 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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[75] Saloff-Coste L. Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geoin. 36, 1992, 417-450. [76] Saloff-Coste L. Parabolic Harnack inequality for divergence form second order differential operators. Potential Analysis 4, 1995, 429-467.
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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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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.
CAMBRIDGE UNIVERSITY PRESS
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