NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 37-42

A N OT E O N L IPSC H IT Z FU N C T IO N S , U P P E R G R A D IE N T S ,

A N D TH E P O IN C A R E IN E Q U A L IT Y

J u h a H e in o n e n * a n d P e k k a K o s k e l a

(Received July 1998)

Abstract. We prove the equivalence of two different definitions for a Poincare

inequality in certain metric measure spaces.

1. Introduction

If u is a smooth bounded function in a ball B in Euclidean n-space Mn, then the

following Poincare inequality holds:

/ \u — UB\dm < C(n) diamB / \Vu\dm. (1.1)j b J b

Here dm denotes Lebesgue n-measure, ub the mean value of u over B, and C(n)

a positive constant that only depends on n. For 1 < p < oo, (1.1) implies by way

of Holder’s inequality that

^ J b \n - uB\dm < C(n) diamB ^ |VU|" dm ) . (1.2)

In [6], [7], we postulated what it means for a metric measure space to carry a

Poincare inequality of the type (1.2). To that end, one defines an upper gradient

of an arbitrary real-valued function u in a metric space to be any Borel function

p in the space with values in the extended nonnegative reals [0, oo] such that the

inequality

pds (1.3)

holds for each rectifiable curve 7 in the space with end points a and b. (We used the

term very weak gradient in [6], [7].) Note that p = 00 is always an upper gradient

and that p — |Vu| is an upper gradient of a smooth function u on a Riemannian

manifold.

Let (X, d) be a metric space, and let p, be a Borel regular measure on X that

assigns finite but positive mass on each ball in X . We say that X admits a (1 ,p)~

Poincare inequality, 1 < p < 00, if there are constants Cp > 1 and A > 1 such

that

m h " u b W - Cp diamB ( i m L ' ^ ' (L4)

1991 AMS Mathematics Subject Classification: 46E35, 43A85.

*The first author was supported by NSF grant DMS-9622844.

Dedicated to Professor Vladimir G. Maz’ya on his sixtieth birthday.

38 JUHA HEINONEN AND PEKKA KOSKELA

for each ball B in X , for each continuous and bounded function u on A5, and for

each upper gradient p of u in AB. Here and below, AB = B(x,\r) if B — B(x,r)

is an open ball with center x and radius r.

The above definition was stated in [6], [7], and we emphasize that u is required to

be continuous; this requirement was most convenient for our purposes at the time.

Now there are other natural classes of functions u that one can consider. We say

that X admits a (l,p)- Poincare inequality for Lipschitz functions (respectively,

measurable functions) if (1.4) holds whenever u is a bounded Lipschitz function

(respectively, a bounded (Borel) measurable function) in a ball AB in X and p is

its upper gradient there.

To unify the terminology, we also say that X admits a (l,p)-Poincare inequality

for continuous functions if (1.4) holds as first indicated.

The purpose of this note is to prove that in a large class of metric spaces, one

obtains (1.4) for measurable (a fortiori, for continuous) functions, provided (1.4)

holds for Lipschitz functions:

Theorem 1.1. A proper and quasiconvex metric space X equipped with a doubling

Borel measure admits a (1 ,p)-Poincare inequality for measurable functions if and

only if it admits a (1, p) -Poincare inequality for Lipschitz functions.

We call a metric space proper if its closed balls are compact and quasiconvex

(with constant b > 1) if every pair of points in the space can be joined by curve

whose length does not exceed a fixed constant b times the distance between the

points; a Borel measure p, is called doubling if there is a constant C > 1 so that

p{2B) < Cp{B ) (1.5)

for all balls B in the space.

Theorem 1.1 is quantitative in that all the relevant constants depend only on

each other.

An example in [8] shows that one cannot replace the assumption “proper” by

“locally compact” ; indeed, Theorem 1.1 fails for certain Euclidean domains that are

quasiconvex in the Euclidean metric and on which Lebesgue n-measure is doubling.

We do not know to what extent the assumptions “quasiconvex” and “doubling” are

necessary. (One should note here that the validity of a Poincare inequality in a

space often implies quasiconvexity, cf. [7, 5.8].)

We do not know of many previous results along the lines of Theorem 1.1. Note

that the classical smoothening arguments (or their generalizations as in [14, p. 289-

292], for example) cannot be used here; to wit, if u is an arbitrary measurable

function with Lp -integrable upper gradient p, there is no clear way of utilizing

any smoothing procedures without the knowledge that something like (1.4) already

holds.

Haj^asz and the second author have shown [5] that if H is a family of Hormander

type Lipschitz vector fields (satisfying some natural weak hypotheses) in a Euclidean

domain, and if (1.4) holds for smooth functions u, then (1.4) holds for continuous

functions, too, with p = \Hu\, the horizontal (distributional) gradient of u. The

methods used in [5] are based on smooth approximation and cannot be used in

the present abstract setting. (We remark that, in the setting of [5], the function

p = \Hu\ is an upper gradient of u if u is smooth.)

LIPSCHITZ FUNCTIONS, UPPER GRADIENTS, AND THE POINCARE INEQUALITY 39

Our proof of Theorem 5.1 is based on an old idea of Maz’ya, which relates

Poincare type inequalities to capacity estimates [9] - [13], and on some recent

results of ours [7, Proposition 2.17], which relate different capacities to each other

by curve family arguments. We find it interesting that curve family, or extremal

length, arguments enter the proof of Theorem 1.1.

Maz’ya’s method, which is related to the well known connection between Poincare

and isoperimetric inequalities, has been used, refined, and occasionally rediscovered

by many people. See [3], [5], and [13] for more results along these lines, and for

more references. For further studies of functions with Lp-integrable upper gradi­

ents, and for examples and applications of situations where Theorem 1.1 applies,

see [1], [5], [15], and the references there.

An inequality like (1.4) was called a weak (1 , p)-Poincare inequality in [6], [7],

because of the factor A on the right hand side. For the issue of removing this factor,

see [4].

2. Proof of the Theorem

Let X be a metric space as in Theorem 1.1, equipped with a doubling Borel

regular measure \±. Recall our standing assumption that the /^-measure of every

ball in X is finite and positive. It follows that p is a Radon measure, and hence in

particular that

p(A) = sup{p{K) : K C A compact} (2.1)

for each Borel set A C X such that ^(^4) < oo. See [2, 2.2.2 and 2.2.5]. We shall

need (2.1 ) below.

Let S be any of the following three classes of real-valued functions defined on

measurable subsets of X: Lipschitz, continuous, or measurable. Then the meaning

of the assertion 11X admits a (l,p)-Poincare inequality for the class Sv is clear.

Let 1 < p < oo. We define the (p,S)~capacity between two disjoint closed sets

E and F in an open set U in X to be the number

capp(E,F]U) — inf f pP dp,Ju

where the infimum is taken over all upper gradients p of all functions u in U that

belong to S and satisfy u \ E > 1 and u \ F < 0.

Theorem 1.1 follow from the following two propositions:

Proposition 2.1. The space X admits a (l,p)-Poincare inequality for the class

S if and only if there exist constants c > 1 and A > 1 such that

min{p(E), p(F)} < c(diamB)p cap^ (E : F; XB) (2.2)

whenever E and F are two disjoint compact subsets of a ball B in X . The state­

ment is quantitative in that the relevant constants depend only on each other.

Proposition 2.2. Assume that X is quasiconvex with constant b > 1. If E and

F are two disjoint compact subsets of a ball B in X , then

cap$(E,F\B) < cap^(£,F;46i3),

where C and M. denote the classes of Lipschitz and measurable functions, respec­

tively.

40 JUHA HEINONEN AND PEKKA KOSKELA

Proposition 2.2 follows from the Proof of Proposition 2.17 in [7] after a couple

of easy modifications there. (In [7, (2.22)], take the infimum over all curves 7X that

join F to x in the ball 3bB.)

To prove the necessity part of Proposition 2.1, we first observe that (1.4) can be

promoted (quantitatively) to

by arguments in [4], [5]. Then (2.3) follows by applying (2.5) to a test function w,

0 < u < 1, for the capacity capf(E, F; XB).

The sufficiency part requires a truncation argument, which goes back to Maz’ya.

First we require a lemma, whose easy proof is left to the reader.

Lemma 2.3. Let p be an upper gradient of a function u in a metric space M . If

u is constant on a closed set E, then the Borel function p'{x) — p {x )x m \e (x )

an upper gradient of u in M , where x ■ denotes a characteristic function.

Now we prove the sufficiency part of Proposition 2.1. Let u be a function from

the class S , defined on a ball AB, and let p be an upper gradient of u there. Pick

a real number t such that both p({x G B : u[x) > t}) and p,({x 6 B : u(x) < £})

are at least p(B)/2. It suffices to prove the (l,p)-Poincare inequality with \u — u b \

replaced with \u — t\, and by replacing u with u — t, we may assume that t = 0. Next,

we observe that p is an upper gradient of both u+ = max{u, 0} and u~ — min{w, 0},

and hence we may assume that u > 0.

For each integer j define

Fj = [x G XB : 2j < u(x) < 2j+1} .

By applying Lemma 2.3 twice, we obtain that the function

P j(x ) = p {x )x u {x ) (2.4)

is an upper gradient of the function

uj(x) = min {2J , max{0, u(x) — 2^}}

in XB if U is any open set in XB such that AB\U can be written as a union of two

closed sets K® C {x G XB : u(x) < 2J} and K j C {a: G XB : u(x) > 2J+1}.

It suffices to prove the inequality

2Jpp({x G B : U j(x ) > 2J}) < C(diam5)p f pPdp, (2.5)

LIPSCHITZ FUNCTIONS, UPPER GRADIENTS, AND THE POINCARE INEQUALITY 41

for some C > 1 depending only on c and A in (2.2). Indeed, we then obtain

m ( m L ud - L uPd =K L BuPdl1j — - OO 3

oo

< 2 p 23’W n 5 )J = - o o

oo

< 2P 2JP/i{rr € B : Wj(x) > 2J}

j= - o o

oo «

< C2p(diamP)p Y '' /

j=-ooJ Fi

= C2p(diam B)p ( pP dfi,J XB'XB

as desired. To verify inequality (2.5), we first observe that by property (2.1) of fi.

it is enough to show (2.5) in the form where F3 is replaced by an open set U D Fj

of the form described above before (2.5). Fix E C {x e B : Uj{x) = 0} compact

such that

^ M x € B : U j ( x ) = 0}) < n(B), (2.6)

and fix F C {a: € B : U j ( x ) = 2J } compact such that

\ »({xeB- .Uj(x) = V}) < H(F). (2.7)

Then

2~j Uj | E < 0, 2~j Uj | F > 1,

and the function 2_Jpj is an upper gradient of 2~3u3 in AS, where pj is given in

(2.4) and U — AB\(E U F). By assumption (2.2),

min{fi(E), n(F)} < c(di&mB)p f 2~ipppdp. (2.8)Ju

On the other hand, f i(E) is not less than \ ^{B) and inequality (2.5) therefore

follows from (2.7) and (2.8).

This completes the proof of Proposition 2.1, and thereby that of our main The­

orem 1 .1 .

3. Concluding Remarks

Write C for the class of continuous functions. As in Proposition 2.2, we have

that

capc(kE,F-B) < c ap^ (£ ,F ;2B ),

where E and F are two disjoint compact sets in B , provided X is (^-convex as

defined in [7, 2.15]. It follows as above that a proper and (^-convex metric space

equipped with a doubling measure admits a (1, p)-Poincare inequality for measur­

able functions if and only if it admits a (l,p)-Poincare inequality for continuous

functions.

42 JUHA HEINONEN AND PEKKA KOSKELA

Theorem 1.1 admits a local version: Let X be proper and 6-quasiconvex, equipped

with a doubling measure, and let U C X be open. If (1.4) is satisfied by all bounded

Lipschitz functions in all balls B such that AB is precompact in U, then (1.4) is

satisfied by all bounded measurable functions in all balls B such that 46XB C U.

In Proposition 2.1, one need not assume that X be proper. The result holds if

X is an arbitrary quasiconvex metric space equipped with a doubling measure.

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Rev. Mat. Iberoamericana, to appear., , TT . Pekka KoskelaJuha Heinonen „, , ^, , ,, ,. ^ , Mathematics DepartmentMathematics Department TT . . . . . . . . . .TT . University of JyvaskylaUn.vers.ty o FIN-40351Ann Arbor, MI 48109 _tt o A Jyvaskyla

FINLANDjuha@math.lsa.umich.edu , , _

pkoskela(Smath.jyu.n