Fat sets and pointwise boundary estimates for p -harmonic functions in metric spaces

F A T S E T S A N D P O I N T W I S E B O U N D A R Y E S T I M A T E S

F O R p - H A R M O N I C F U N C T I O N S IN M E T R I C S P A C E S

BY

JANA BJ()RN, PAUL MACMANUS AND NAGESWAR! SHANMUGALINGAM

A b s t r a c t . We extend a result of John Lewis [L] by showing that i fa doubling metric measure space supports a (1, q0)-Poincare inequality for some 1 < qo < p, then every uniformly p-fat set is uniformly q-fat for some q < p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation of p- harmonic functions and p-energy minimizers near a boundary point.

I I n t r o d u c t i o n

The main theorem in this paper is a version of Lewis's theorem [L] on p-fat

sets in metric measure spaces.

Definition 1.1. A set E C X is said to be uniformlyp-fat i f there exist constants

Co > 0 and r0 > 0 so that for every point x E E and for all r E (0, ro) we have that

Capp(B(x, r) fq E; B(x, 2r)) (1) ~(z,E,r) := _> Co,

Capp(S(x, r); B(x, 2r))

where Capp is a p-capacity defined on X.

Lewis's theorem states that for every uniformly p-fat set in a Euclidean space

there is an exponent q < p such that the set is uniformly q-fat. The main theorem

of this paper is the following partial generalization of Lewis's theorem to metric

measure spaces. Unlike Lewis, who treated more general capacities, we consider

only capacities o f the first order.

T h e o r e m 1.2. Let X be a proper linearly locally convex metric space endowed

with a doubling regular Borel measure supporting a (1, qo )-Poincar~ inequality for

some qo with 1 < qo < oo. Let p > qo and suppose that E C X is uniformly p-fat.

Then there exists q < p so that E is uniformly q-fat.

339 JOURNAL D'ANALYSE MATHI~MATIQUE. M~I. 85 (2001)

340 J. BJORN, P. MACMANUS AND N. SHANMUGALINGAM

Theorem 1.2 was first proved in the Euclidean setting by Lewis [L], using

Green functions. Unfortunately, we do not have representation formulae associated

with Green functions in the general setting of metric measure spaces. The thesis

of Mikkonen [Mi] gives an alternative proof of Theorem 1.2 in the Euclidean

setting using the Wolff potential. We follow his arguments here. In particular,

we characterize positive Radon measures in the dual N~'P(f~)*: we show that

v E Nol'P(f~) * if and only if for some r > O,

L Wp(x,r)dv(x) < oo,

where W3 is the Wolff potential of v; see Definition 4.1 and Theorem 4.12. While

proving this characterization, we obtain a pointwise estimate for solutions of the

equation - div(iDuiP-2 Ou) = v E N~'P(f~) *, namely

see Proposition 4.9 and Corollary 4.11. We also obtain the following estimate for

the oscillation of p-harmonic functions and p-energy minimizers near a boundary

osc u < osc u + oscu exp - C ~(x, X \ 12, t) U(p-1) �9 ~ N B ( x , p ) O f l n B ( x , 5 r ) Of~

see Theorem 5.1. As a corollary to Theorem 1.2, we obtain the Hardy inequality

results as in [KM2]; see Corollary 6. I. The Hardy inequality gives us information

about the isoperimetric profile of the domain; see [MV].

Let us also mention that we have managed to simplify some of the proofs from

[Mi]. In particular, we have avoided the use of superharmonic functions in the proof of Theorem 4.12.

Acknowledgment. This research was begun while the second and third

authors were at the Department of Mathematics, National University of Ireland at

Maynooth and the first author was visiting the Institute Mittag-Leffler in Stockholm,

Sweden. We wish to thank both institutes for their support. The first author was

also supported by grants from the Swedish Natural Science Research Council and

the Knut and Alice Wallenberg Foundation. We also thank Juha Kinnunen for

pointing out the reference [Mi] and for other useful discussions. We also wish to

thank Andreas Wannebo for interesting discussions related to his work and Juha Heinonen for his encouragement.

inf u+CW~(x,r) < u(x) < C( inf u + W~(x,2r)); B ( x , 2 r ) - - - - \ B ( x , r )

point, cf. [M2],

FAT SETS AND POINTWISE BOUNDARY ESTIMATES 341

2 Notation and preliminaries

We assume throughout that X = (X, d, #) is a proper (that is, closed bounded

sets are compact) metric space endowed with a metric d and a nontrivial Borel

regular measure # which is finite on bounded sets and is doubling, i.e., there exists

a constant C such that for all balls B = B(x0, r) := {x E X : d(x, :Co) < r} in X,

(2B) <

wherc 2B denotes the ball concentric with B and having twice the radius. By

measure, we always mean a positive measure.

In addition, we assume that X supports a (1, q0)-Poincar6 inequality for some

1 <_ q0 < P (see Definition 2.1 below) and that it is linearly locally convex (LLC),

i.e., there exist constants C~ > 0 and rl > 0 such that for all balls B in X with radius

at most r l , every pair o f distinct points in the annulus 2B \ / 3 can be connected by

a curve lying in the annulus 2CIB \ C~- 1 B; see [HeK, Section 3.12]. The definition

o f LLC in [HeK] is weaker than the one assumed here; however, it is shown in

[HeK, Lemma 3,17 and Remark 3.19] that Ahlfors Q-regular spaces supporting a

(1, Q)-Poincar6 inequality satisfy both versions o f LLC.

A Borel function g on X is an upper gradient of a real-valued function f on X

if for all rectifiable paths "r : [0, l~] -~ X, parameterized by the arc length ds,

If(-y(0)) - < f gd .

I f the above inequality fails only for a curve family with zero p-modulus (see, e.g.,

[HeK, Section 2.3] for the definition o f the p-modulus of a curve family), then g

is called a p-weak upper gradient of u. It is known that the LP-closure o f the set

o f all upper gradients o f u that are in L p is precisely the set o f all p-weak upper

gradients o f u that are in /2 ' ; see [KMc].

D e f i n i t i o n 2.1. We say that X supports a (1,p)-Poincar~inequality i f there

is a constant C > 0 such that for all balls B C X with radius r , all measurable

functions f on X and all upper gradients g o f f ,

If - fB[ d# < Cr gPd# , where fB := f dp := #(B) fd#.

In the above definition o f Poincar6 inequality, we can equivalently assume that

g is a p-weak upper gradient in L p see the comments above.

Following [Shl] , we define a version o f Sobolev type spaces on the metric

space X with 1 < p < oo.


Definit ion 2.2. Let

Ilull N' p = lul' d# + inf gV d# , g \ d x

where the infimum is taken over all upper gradients of u. The Newtonian space on

X is the quotient space

N I ' P ( X ) ---- {it : INIIN"p < ~ } / ~ ,

where u ,-. v if and only if ][u - vllN,, = O.

The space NI ' v (X ) equipped with the norm I1" IIN'p is a Banach space and a

lattice; see [Shl]. Cheeger [C] gives an alternative definition o f Sobolev spaces

which leads to the same space; see [Shl, Theorem 4.10]. Cheeger's definition

yields the notion of partial derivatives in the following theorem [C, Theorem 4.38].

T h e o r e m 2.3. Let X be a metric measure space equipped with a doubling

Borel regular measure #. Assume that X admits a weak (1, p)-Poincar~ inequality

fo r some 1 < p < (x).

Then there exists a countable collection (U,~, X '~) o f measurable sets U, and �9 . X ~ t Lipschitz "coordinate"funct ions X ~' = (X~ , . , k(,)) : X --+ R k(") such that

# ( X \ U,, u,,) = o and,.for all a, the fol lowing hold.

The functions X~, .. . , X"k(,) are linearly independent on Uo and 1 < k(a) < N,

where N is a constant depending only on the doubling constant o f # and the

constants f rom the Poincark inequality. I f f : X --+ ~ is Lipschitz, then there

exist unique bounded vector-valued functions d~ f : U~ --+ R k(") such that f o r

#-a.e. xo E U~,

If(z) - / ( x o ) - d " f ( X o ) �9 ( S ~ - g~'(z0))l lira sup = 0.

r---~O+ zEB(zo , r ) ~"

We can assume that the sets U,~ are pairwise disjoint and extend d~f by zero = d o outside U,~. Regard d " f ( x ) as vectors in R N and let D f ~ f . The differential

mapping D : f ~ D f is linear; and it is shown in [C, p. 460] that for all Lipsehitz

functions f and #-a.e. z ~ X,

(2) ID/(z)l .~. g l (x ) = infl imsup ; gd#, g r-~O+ J B ( x , r )

where gy is the minimal p-weak upper gradient o f f (see [Sh2, Corollary 3.7]),

and the infimum is taken over all upper gradients 9 of f . Also, by [Shl] or [C,

Proposition 2.2], D f = 0 #-a.e. on every set where f is constant.


By [C, Theorem 4.47] and [Sh 1, Theorem 4.10], the Newtonian space N 1 'p (X)

is equal to the closure in the Nl,p-norm of the collection of locally Lipschitz

functions on X with IIflINI.~ < c~; and by [FHK, Theorem 10], there exists a

unique "gradient" Du satisfying (2) for every u E NI 'p (X) . Moreover, if ui is a

sequence in N l ' v ( X ) , then ui --+ u in NI'p(x) if and only ifu~ --+ u in /F(X, #) and

Dui --+ Du in LP(X,/~; RN).

One easily verifies (see also [C, p. 458]) that the "gradient" Du satisfies the

product and chain rules, i.e., ifv is a bounded Lipschitz function on X , u E NI'p(x), and ff : R --+ R is continuously differentiable with bounded derivative, then uv and

�9 (u) belong to NX,P(X) and

(3) D(uv) = uDv + vDu, D(,~ o u) = ~ ' (u)Du.

Defini t ion 2.4. Let E c X be a Borel set. The p-capacity of E is the number

Cp(E) := inf(fx lul" du + fx 'DulP

where the infimum is taken over all u E NI 'P(X) such that u = 1 on E.

Let f~ C X be bounded and E C f~. The relative p-capacity of E with respect

to ft is the number P

Capp(E; fl) := inf J x IDul d/~,

where the infimum is taken over all functions u E NX'v(X) such that u = 1 on E

andu = 0 o n X \ fL

R e m a r k 2.5. The standard definition of p-capacity uses the minimal p-weak

upper gradient 9~, of u instead of IDul (see, e.g., [HeK], [KM 1] or [Sh 1 ]); but by (2),

the above definition of capacity is comparable to the standard definition. Also, by

[KaSh], it is sufficient to consider only Lipsehitz functions u with compact support

in the definition o f p-capacity.

We say that a property holds p-quasieverywhere (p-q.e.) if the set of points

for which it does not hold has p-capacity zero. The p-capacity is the right gauge

for distinguishing between two Newtonian functions. In particular, Corollary 3.3

in [Shl] shows that i f u , v E Nx'P(X) and u = v/~-a.e., then u = v p-q.e.

The following lemma compares the capacities Capp and Cp and shows that they

are in many eases equivalent; for a proof, see Lemma 3.3 in [Bj].

L e m m a 2.6. Let B be a ball in X with radius r and E c B be a B o w l set.

Then f o r each A > i with Ar < �89 there exists Ca > 0 such that

#(E) < Capp(E; AB) < C~#(B) Cxrp - - rp


and

( CA(1 + rp) < Capv(E; AB) < CA 1 + Cp(E).

ln particular, Cp(E) = 0 if and only if Capp( E n B; AB) = O for all balls B c X and some A > 1; and Capp(B; AB) is comparable to r-P#(B), where the comparison constant depends only on the data of X and on A.

To compare the boundary values of Newtonian functions, we need a Newtonian

space with zero boundary values. Let

N~'P(E) = {~ e

Corollary 3.9 in [Shl] implies

a closed subspace of NI'P(X).

NI,v(X). An important consequence

Sobolev inequality; see [HaK,

C' > 0 and ~ > 1 such that for

u E N~"(B) we have

NI'v(X) : u = 0 p-q.e, on X \ E}.

that N~'V(E) equipped with the norm I1" IIN,,~ is Note also that if Cv(X \ E) = 0, then No'V(E) =

of the (1,p)-Poincare inequality is the following

Section 5] and [KiSh, Lemma 2.8]. There exist

all bails B in X with radius r < ~diam X and all

(4) ( fBlul~P d#) l/~P < Cr(fBlDUlP d#) x/p

In the classical situation X --- ~'~ and # Lebesgue n-measure, we have n = n/(n-p) . In order to avoid clumsy notation, we assume that diana X = oo, i.e., that the above

Sobolev inequality holds for all balls. In the opposite case, some of the results in

this paper only hold for small bails whose radius depends on diam X.

To estimate the p-capacity of a set, it is useful to consider the p-harmonic

functions.

Defini t ion 2.7. Let f~ be a domain in X such that the p-capacity o f X \ f~ is

positive. By a p-harmonic function corresponding to boundary data f E NI'p(X)

we mean a function u E NI'P(X) such that u - f E N~'P(f~) and

fx IDul" d. <<_ fx lDvl" d•

for every function v E Nt'P(f~) + f = {v E NI'~'(X) : v - f E N~'P(gt)}. By a

p-energy minimizer we mean a function u E NI'p(X) for which

fxg~ d~ <_ fx g~ d# for every v E N~'P(f~) + u.


R e m a r k 2.8. In [Sh2] and [KiSh], p-energy minimizers were called p-

harmonic functions. Unfortunately, in general such p-energy minimizers need

not satisfy a corresponding differential equation, and we need an elliptic partial

differential equation in order to associate with such solutions a corresponding Wolff

potential; see Section 4. The integral form of the differential equation satisfied by

the p-harmonic functions defined above is

(5) fn IDu[P-eDu " D~d# = 0

for every function ~ in NI'p(O). Moreover, p-harmonic functions as defined above

are p-quasiminimizers in the sense of [KiSh]. Hence the results of [KiSh] apply to

these p-harmonic functions as well.

By [Sh2, Theorem 5.6], for every f E NI'P(X) there exists a unique p-energy

minimizer with the boundary data f . The existence and uniqueness of p-harmonic

functions is proved in the same way. Hence the p-potentials defined below always

exist.

Definit ion 2.9. If K is a compact subset of a domain f~ in X, then by the

p-potential for K with respect to ~2 we mean the function u E NI'P(X) which is

p-harmonic in f~ \ K, u = I p-q.e, on K, and u = 0 p-q.e, on X \ fL

Unless otherwise stated, C denotes a positive constant whose exact value is

unimportant, can change even within a line and depends only on fixed parameters,

such as X, d,/~ and p. If necessary, we specify its dependence on other parameters.

By a ~ b we mean that a/C < b < Ca.

3 S u p e r s o l u t i o n s and R a d o n measures

Defini t ion 3.1. A function u is said to be a p-supersolution in f~ if for every

nonnegative function ~ in NI'p(fl) we have

f lDulV-2Du �9 D~d# > O,

or, equivalently,

(6)

The following

p-supersolutions.

~ lDulP dg <_ ~ ID(u + ~o)lP d#.

lemma illustrates the relationship between p-potentials and


L e m m a 3.2. Let f~ be a bounded domain in X and let u be the p-potential f o r a compact set K c ~. Then u is a p-supersolution in f~.

P r o o f . We shall show that the inequality (6) holds for all nonnegative qo �9

N~'P(f~). We can assume that u + ~ < 1 p-q.e., since otherwise the size o f the

integral on the right-hand side in (6) can be reduced by considering the function

min{u + ~, 1}, as this modified function too will be o f the form u + 0 with 0 < r �9

No l,v(fl), because u _< 1 p-q.e. The inequality u + qo < 1 p-q.e, implies that qa = 0

p-q.e, on K and hence qa �9 Nl'r'(f~ \ K). As u is p-harmonic in 12 \ K, we have

L\K IDuIP d# <~ L\K ID(u + ~,)[P d/~,

from which (6) follows. []

R e m a r k 3.3. The above proof also indicates that i f u is any p-supersolution

in f~, then for every k �9 ~ the truncation uk := min{u, k} is also a p-super-

solution. To see this, consider any nonnegative function qo �9 Nl 'p(ft) . Just as

in the above proof, we can assume that uk + ~ < k p--q.e. Then ~, = 0 on the set

K := {x �9 f~ : u(x) > k}. Hence, as u is a supersolution on ft \ K,

\K \ g

L e m m a 3.4. Letu �9 NI,P(X) beap-supersolution in fl C X. I fu >_ Op-q.e. in X \ fl, then u >_ Op-q.e. in X.

P r o o f . Let qo = m a x { - u , 0}.

max{u, 0}. Thus, we have by (6),

and hence

Then ~ �9 N~'P(f~), qo > 0 in X and u + ~p =

L IDu[ p dp < L IDulPx(">~ dl.t

L IDol p .In IDulPx,t~,<o) d# = f

d# O.

It follows that D e = 0 #-a.e. in X, and the Sobolcv inequality (4) implies ~ = 0

#-a.e. (and consequently p-q.e.) in fL []

Next, we show that there is a one-to-one correspondence between supersolutions

and Radon measures in the dual Nol'P(f~) *. By Cc(f~) (resp., Lipc(f~)) we denote

the space o f continuous (resp., Lipschitz continuous) functions in X with compact

support in fL


Proposition 3.5. Let B C X be a ball. Then for every p-supersolution u in B, there is a regular Radon measure v �9 Nx'v(B) * such that

(7) f lDulP-2Du. D~du=f ~dv whenever ~ �9 N~'V(B).

Proof. Every u �9 N~'P(X) defines a linear functional on N~'V(B) via the

Cheeger derivative as follows:

(8) T,,(~) := fB IDulV-2Ou" D~p d#.

It is easy to see that T~, is a linear functional on NI'p(B); moreover, if u is a p-

supersolution in B, then for nonnegative functions ~ �9 N~'V(B) we have Tu(~) > 0.

Let 0 < ,k < 1 and fix a nonnegative function r / � 9 Lipc(B ) such that 77 = 1 on AB.

I f ~ �9 Lip~(,kB), then 0 < ~§ < r := rlsupxB I~1, hence, by the H61der inequality

and by the positivity ofT,,,

0 < T~,(qo+) < T, , ( r f_ IDulV-2Du. Bed# < f_ IOulV-XlOr J IJ r trt (9) i/v

(/B IOulP) l-lIp (/B ID~gIP) <~ C(r], u ) s ~ g I~DI.

Thus T~, is a positive bounded linear functional on Lip~(AB). Since the space

Lip~(AB) is dense in Cc(AB) equipped with the supremum norm (see Remark 3.7),

T~, extends continuously to a positive linear functional on Cc(AB). By the Riesz

representation theorem, there exists a regular Radon measure t,x such that for every

�9 C~(;,B),

T,,(cp) = / R r dv:~;

see, e.g., [Ru l, Theorems 2.14 and 2.18]. Finally, define a measure v on B by

v(E) := lim v:dE ). ,k---~ 1 -

Then for each ~ E Lipc(B ) and all A sufficiently close to 1, we have

(10) fB dv = f By [Sh2, Theorem 4.8], Lipc(B) is dense in N~'V(B), so for every ~ E N~'P(B) there exist ~k E Lip~(B) converging to ~o both in N~'P(B) and p-q.e, in B; see [Shl,

Corollary 3.9]. Identities (8) and (10) yield that the functions ~k form a Cauchy


sequence in LI(B, v), which converges to ff in L 1 (B, u). Hence a subsequence o f

{~Pk} converges to ~ v-a.e, in B. Lemma 3.8 below implies that ~ = ~ u-a.e, and

T~,(~) = lim Tu(~k)= lim f ~kdv = f ~dv, k ~ oe k--roe JB JB

which finishes the proof. []

R e m a r k 3.6. It is clear from the proof of the lemma that a similar argument

goes through for an arbitrary domain ft. In fact, the measure v is calculated from

u as follows: i f K is a compact subset o f f / , then

v(K) = inf{T~,(f) : f E Lipc(ft ), f lK >- 1},

and u is extended to Borel sets as an inner measure.

R e m a r k 3.7. The density o f Lipr in C~(ft) follows from the S t o n e -

Weierstrass theorem (see, e.g., [Ru2, p. 122]); for the reader 's convenience, we

give here an elementary p roof o f this fact in our situation. Let u E C~(f~) and e > 0

be fixed but arbitrary. We can assume that u _> 0 and let ftk = {x E X : u(x) > k~}. Clearly, only finitely many ftk are nonempty. As X is proper, there exists 5 > 0

such that dist(ftk, X \ f~k-1) > 5 for all k > 1. Let, for x Ef t ,

u,(x) = e E max{1 - ~ - l d i s t ( f ] k , X), 0}, k>l

where we sum only over nonempty ~k. Then u, E Lipc(f~) (with constant e/6), and

]u(x) - u~(x)] <_ e for all x E f~.

Since functions in NI.P(B) are well-defined only up to sets o f p-capaci ty zero,

it is useful to know that such sets have zero measure for every Radon measure in

NI 'p(B) *. This is the content o f the following lemma; see also [Mi, Lem m a 2.4].

L e m m a 3.8. I f u is a Radon measure in N~'P(B) * or a Radon measure associated with T~ as in Proposition 3.5 and E C B is such that Cp(E) = 0,

then u( E) = O.

Proof . Since the p-capaci ty is a Choquet capacity (see [KM 1]), we can assume

without loss o f generality that E is a Borel set. Let K be a compact subset o f E.

Then by Lemma 2.6, we have Capp(K;B) = 0. Using Remark 2.5, we can

construct Lipschitz functions ~o~ with compact support in B such that ~i = 1 on K ,

0 < ~i < 1, and fB [D~~ ~ 0, as i -~ oo. Now i f v E N~'P(B) *, we see that

fB ( f B ) l i p v ( g ) < ~oi dr, < CIl llN0' P/B/ <-- C ID~IPd# -~ O,


where we have used the (1, p)-Poincar6 inequality in the last inequality.

Similarly, if u is associated with the operator T~, as in Proposition 3.5, then

u(K) <_ s ~ i d u = T,(~i) <_ s lDulP-~lD~il d . < C(u) ID~dPd# -40.

As u is a Radon measure, this completes the proof of the lemma. []

The following proposition is a converse to Proposition 3.5; cf. [Mi, Corol-

lary 2.7].

P r o p o s i t i o n 3.9. Let f~ c X be bounded and u E N~'P(fl) * be a Radon

measure on fL Then there exists a unique u E N~'P(~) satisfying the identity (7)

for all ~ E N~'P(fl). Moreover. u is a p-supersolution in ft.

Proof . We shall use the following general result (see, e.g., [Li, p. 171]): I f

Y is a reflexive Banach space and T : Y -~ Y* is a linear map that is coercive

(i.e., (Tyi,yi)/llyillY ~ oo whenever yi is a sequence in Y and IlyillY --}

demicontinuous (i.e., (Tyi, z) --> (Ty, z) for all z E Y whenever yi --+ y in Y), and

such that T(E) is bounded in Y* whenever E is a bounded subset of Y, then for all

f E Y* the equation Ty = f has a solution y E Y.

It is easy to see that the map T : u ~ Tu, where Tu is as in (8), is bounded

on Y = N~'P(f~) in the above sense. Moreover, by the Sobolev inequality (4), T

is coercive. Also, i fu i -+ u in N~'P(f~), then Dui ---+ Du in LP(X,N;RN); hence

IDuilP-2Dui --+ IDulP-2Du weakly in LP/(P-1)(X, Ig;]~N), i.e., T is demi-

continuous. Hence by the above mentioned functional analytic fact, we have a

function u E NI'p(f~) so that (7) holds.

Finally, if~o E N~'P(fl) is nonnegative, then Tu(~) = fa~odu > 0 and u is a

p-supersolution in ft. The uniqueness o f u is proved as in [Mi, Lemma 2.8]. []

The following lemma gives an equivalent definition of the p-capacity using

Radon measures and supersolutions; cf. [Mi, Lemma 4.6].

L e m m a 3.10. Let K be a compact subset o f B. Then

Capp(K; B) = sup u(K),

where the supremum is taken over all u E NI'p(B) * such that supp u c K and

u. <_ 1 p-q.e, in B, where u~ denotes the solution o f the equation (7) provided by

Proposition 3.9.

Moreover, i f u g is the p-potential o f K with respect to B and uK is the associated

Radon measure in N~'P ( B ) * given by Lemma 3.2 together with Proposition 3,5, then

uK(B) = uK(K) = Capp(K; B).

350 J. BJORN, P. MACMANUS AND N. SItANMUGALINGAM

P roo f . As the p-potential UK is p-harmonic in B \ K, one easily verifies that

the support of uK is contained in K. Hence, as uK = 1 p-q.e, on K and uK is

absolutely continuous with respect to the p-capacity (by Lemma 3.8), we have

vK(K) = vK(B)= L UK dvK = L [DuKIP d# : Capp(K; B).

Conversely, let u ~ N~'r'(B) * be such that supp u C K and u,, <_ 1 p-q.e, in B.

Let also cp E N~'V(B) satisfy ~p > 1 on K. Then by Lemma 3.8, u~, _< ~p u-a.e, on K

and

i.e., IIIDu.lllL.cm ~ IIID~IIILp(m. It follows that

\1-1/v

and taking infimum over all such ~o gives u(K) <_ Capp(K; B). Finally, taking

supremum over all v completes the proof. []

The following lemma enables us to estimate the capacity of level sets o f super-

solutions; cf. Lemma 5.4.

L e m m a 3.11. Let u E Nl'p(f~) be the p-supersolution associated with the

Radon measure v E N~'V(f~)*. Then for all A > O,

Capp({u > A}; f~) _< Al-Vu(I2).

P roo f . Let ux = min{u/A, 1} E N0a'P(ft). As uA = 1 on the set {u > A}, we

have, by (7),

Capp({u > A}; ft) < L IDualP d# = A ' -p s IDulp-2Du . Du~ dl~

= AI-Ps u), du < A 1 -Pu(fl). []

Given a finite Borel measure on B, we would like to be able to approximate

it by Borel measures in the dual space N~'P(B) *. This is the content o f the next

lemma, which in the Euclidean setting is Lemma 2.12 in [Mi]. The proof is not

as direct as that given in [Mi], since we do not have an almost disjoint dyadic

FAT S E T S A N D P O I N T W I S E B O U N D A R Y E S T I M A T E S 351

decomposition in our general setting. However, a modification o f the proof in [Mi]

using a partition of unity yields the desired result.

We say that the measures vj converge weakly on fl to a measure v if, for all

~, 6 Lipr

~ ~oduj ~ ~ ~ du, as j --~ oo.

L e m m a 3.12. Let u be a finite Borel measure supported in a domain f~. Then there is a sequence o f measures vj 6 N~'P(f2) * such that Vj(~'~) < v(fl) for all j and vj converge weakly to v on f2.

Proof . Fix a positive integer j and, using the Hausdorff maximal principle, �9 1 j cover ~2 by balls B~, of radii 2-J so that the balls ~B k are pairwise disjoint. The

doubling property o f # then implies that for all x E f~,

oc oc 1 j #(2Bk) < C

<_ ,(B(T21-J)) k = l k '= l

and hence the overlap ~ = 1 XB~ is bounded by a constant independent of j ; cf.

[Se, Lemma C.1]. Now choose only those balls B~, k = 1, ..., N(j) , for which �9 I IN(J) B j subordinate t o I B j l N ( j ) The B~ C f~. Let ~jk be a partition of unity on uk=l k t kJk=l "

measure vj is defined as follows. I f E c X is a Borel set, then

NCj) fB~nE ~Ojk d# f uj(E) = ,=,~ fni ~ojk d# Jni ~ojk du.

By Fubini 's Theorem, we have

(11) uj(fl) = Tjk du = ~Ok dv --'~ ~Ojk du <_ u(fl). k = l ~ k = l k = l

Let qo �9 N~'V(f~). Then

Using the bounded overlap property, we can choose ~jk so that ~jk > C > 0 on

1 J" hence i-6 Bk,

NCj)

k=l


where Cj : C Y]~(__~) v(B~)/p(B~) < oo. Thus, by the bounded overlap of the balls

B~, we have

f ~duj < C(j, fl) f [~ldp < C(j, fl) ( f [~lVd#) ~/p,

where we have used the H61der inequality in the last step. Hence uj E N~P(~) ".

Let ~ E Lip~(~) and e > 0. Choose a positive integer j so large that supp~ c u N(3) B3 and such that, denoting the center o f B~ by xjk, we have [~(xjk) - ~(x)l < e k=l for all x E B~. Without loss o f generality, ~ > 0. Then

~odu - (pduj : k=l ~ ~jkcpdu -- k:lZ fB~ ~jk du ~ q~jk~d#

N(j) N(j)

N( j ) p

k~ t J B

where we used inequality (11) to conclude the last step in the above chain of

inequalities. The difference f~ ~ dvj - f~ ~ du is treated similarly and hence, for

sufficiently large j , [f~ ~ duj - fn ~ dr[ < 2v(fl)e, i.e., the measures vj converge to

u weakly on f2. []

4 Wolff potentials and supersolutions

In this section, we identify the Radon measures in the dual N I ' p ( Q ) * using the

Wolff potential (introduced in [HW]); see Theorem 4.12.

D e f i n i t i o n 4.1. Let v be a Borel measure on X, and let r > 0. The Wolff potential on X associated with the measure v is given by

f o r // _ v ( n ( x , t )) "~ 1 / (p- l ) dt (12) W~(x,r) := ~ t~" p(---B~(z, t -~)

One o f the properties o f the Wolf f potential is that it is lower semicontinuous

on X, and hence measurable.

L e m m a 4.2. Let v be a Borel measure on X and r > O. Then the Wolff potential W;( . , r) is lower semicontinuous.


Proof . First, note that for each fixed x E X, there are at most countably many

t > 0 such that #(OB(x, t)) > 0. It follows that for all x,

) W ; ( x , r ) t l/(U(x,t)) 1 / (p - l ) = dr. #(B(x , t))

Let {xj}~= 1 C X be a sequence converging to Xo and define functions

\ [ t ] (B(xj , t ) ) ) 1/(p-1) y j ( t ) = I t - - - - , j = O , 1 , . . . .

p(B(x , t))

We shall show that for all t > O,

(13) Yo(t) < liminf f j( t) . j-~oc

Then Fatou's Lemma implies

W~(xo,r) = fo(t)dt <_ l iminf f3(t)dt 3 ---+ oc

<_ lim3_,~inf f j (t) dt = lijm ~ f W~ (x j, t).

In order to prove (13), fix t > 0 and choose a nonincreasing sequence of numbers

Ej > 0, j = 1, 2 , . . . with l i rnj_~ ej = 0, such that B(xo, t - e~) C B(x j , t) for all j .

Then

~(B(xo, t)) = lim u(B(xo, t - ej)) <_ li m inf u(B(x j , t)). j--+ c~ 3--~oc

Similarly, using the inclusion B(xj , t) c B(xo, t + ej) and the fact that balls have

finite p-measure, we obtain

p( B(xo, t) ) = jli+m p( B(xo, t + ej) ) >_ lira sup p( B(x j , t)), j---too

which together with the previous inequality proves (13). []

L e r n m a 4.3. Let u be a Radon measure on f~ and let u s E N~'P(O) be the measures given by Lemma 3.12 so that uj --+ u weakly. Then there exists a positive

constant C depending only on X and p such that f o r all x E f~ and j satisfying 2 1 - j ~ r,

(x, r) <_ C W (x, 3r).

Proof . Lct B~ C f~ be balls with radii 2 - j and bounded overlap and ~jk a

corresponding partition of unity, as in the proof of Lemma 3.12. We have, for

t > 0 , fB~nB(~,t) ~jk d# [

~jk &', v3(B(x, t)) = ~k fo t ~ j kdp grit


where we only sum over those balls B~, for which the intersection B~ n B(x, t) is

nonempty. I f t > 2-J, then B~. C B(x,3t); hence, as in (11),

(14) u3(B(x,t)) <_ ~ fo ~jkdu < v(B(x,3t)).

On the other hand, i f0 < t < 2 - j , then B~. C B(x,3 �9 2-J), so by the doubling 1 j property o f # and the fact that ~jk > C > 0 on i-6Bk, we have

iB cPjkd# ~ #(BJk) ,~ p(B(x,3. 2-J)).

It follows that

(15)

Cu(B(x, 3.2-J)) uj(B(x,t)) <_ ~ 3 " : - 2 - - - ~ ) ) ~ ~n,(~.t) ~'j~d#

_< Cv(B(x,~_~,5:3 2 ~ ) # ( B ( x ' t)).

Finally, (14) and (15) yield

L 2-~/" _ v(B(x, 3.2-J ) ) l/(v-1) dt ) w;J(~,,.) ~c ~ t . ~ : 2 _ - : ~ ~ -?

+ c f[_~ [,t,, (--~(~,~ / -~'(B(x'at))) ~/('-''~t 7-

<C2_jp/(p_l) (/2(B(x,3. 2--J)) ~ I/(p--I) - t,(-N~(~, 3: 2--~)) /

+ C L 3".2-, tt"~-~--B-~,F) - v(B(x't) ) ) '/("-') d-/ ;

and as r _> 2 l - j , both terms on the right-hand side are majorized by a multiple of

w;(~,3~). []

The following lemma provides us with a Cacciopoli estimate for supersolutions.

For a proof see [HKM, Lemma 3.57]. The key properties needed for the Cheeger

derivatives in the proof are the product and chain rules (3).

L e m m a 4.4. Let u >_ 0 be a p-supersolution in a domain D. If ~7 E Lipc(f~ )

and e > O, then there is a constant C > O, depending only on p and e, such that

L ]Dufu-l-~lrll p d# <_ C L uV-~-~lDrllP dlz.


Next, we prove a weak Harnack inequality for p-supersolutions. As Newtonian

functions are defined only p-q.e., by infE u we mean the p-q.e.-infimum

sup{ �9 s : c , ( { x �9 E : u(x) < = 0},

and similarly for suPE u.

L e m m a 4.5. Let u be a nonnegative p-supersolution in the ball 10B. Then for

all 7 < n ( p - 1), ( ~ ) 1 / 7

u ~ d# < C inf u, 213

where n is the exponent from the Sobolev inequality (4) and C > 0 depends only

on the constants associated with the Poincar~ inequality, the doubling property,

and p.

Proof . We show that - u is in the De Giorgi class DGn(IOB); see [KiSh,

Definition 3.1 ] for the definition of this class. Then by [KiSh, Theorem 7.1 ], there

exists a > 0 such that the conclusion of the lemma holds with 3' and B replaced by

~r and 2B on the left-hand side. An application of Lemma 4.6 below then yields

the required inequality. In order to prove that - u E DGp(IOB), consider the argument in [KiSh,

Section 3], which shows that quasiminimizers are in the De Giorgi class. The

key point in that argument is to look at the function v = - u - r /max{-u - k, 0},

where 7/is a Lipschitz cut-off function supported in B(x, R) such that 77 = 1 on

B(x, p), with 0 < p < R, and show that

fB a. <_ fB gPv d#, (z,p)nI-u>k} (~,R)nI-u>k }

where K > 0 is a constant independent o f x, p, k and R. By (2), the Cheeger

derivative IDfl is comparable to the minimal p-weak upper gradient g / f o r every

function f �9 NI'P(X). Hence u is a quasiminimizer in the class o f functions

{u + ~ : ~ > 0, ~ E N~'P(10B)}. It follows that taking ~ = - u - v, we have, for

all k E R,

Now by the argument in [KiSh, Section 3], we see that - u is in the required

De Giorgi class. []

In the above proof, we needed the following reverse H/51der type inequality

on balls for p-supersolutions. It is proved in the Euclidean setting in [HKM,

356 J. BJORN, E MACMANUS AND N. SHANMUGALINGAM

Theorem 3.58], and the proof goes through in exactly the same way also in our

situation. Note that the proof uses only the chain and product rules (3), the Sobolev

inequality (4) and Lemma 4.4.

L e m m a 4.6. Let u be a nonnegative p-supersolution in 2B and suppose that

0 < a < 7 < tr - 1), where n is the exponentJrom the Sobolev inequality (4).

Then there is a constant C depending only on X, p, a and 7. such that

Combining Lemmas 4.4 and 4.5, we obtain the following result; for a proof see

[Mi, Lemma 3.4].

L e m m a 4.7. Let u > 0 be a p-supersolution in 2B, where B is a ball o f radius

r > 0. / f r / E Lipc(B), then

i . IDuI'-'u'-alDul d#_< C.(.).-"(infu)'-' The following lemma gives us control of oscillations of p-supersolutions; see

[Mi, Lemma 3.5] and the proof therein. Note that while the statements of the above two lemmas and the following lemma in [Mi] assume u to be p-superharmonic,

the proofs go through if u is a p-supersolution.

L e m m a 4.8. Let u be a p-supersolution in 4B, where B is a ball o f radius

r > O. l f u is the Radon measure associated with u in 4B, then

r pu(B) < C ( i ~ f u _ i n s f U ) p- t . # (B) -

S k e t c h o f p roof . By Lemma 4.5, we know that u is bounded from below on

2B; and, without loss of generality, we can assume that inf2n u = 0. Let a = infB u

and v = min{u, a}. Remark 3.3 shows that v is a p-supersolution. Now noting that

the Cheeger derivative respects the chain and product rules (3) and arguing just as

in the proof of [Mi, Lemma 3.5] (which uses Lemma 4.7), we obtain the desired

inequality. []

Lemma 4.8 provides us with a crucial estimate for comparing a p-supersolution

and its associated Wolffpotential. This is the content o f the following proposition.

Note first that [KL, Theorem 4.5] and [Sh 1, Theorem 4.9] imply that p-q.e, x0 E X

is a Lebesgue point of u, i.e.,

(16) lim 1[ ]u(x) - u(x0)r r d#(x) = o, t-+O+ JB(x,r j

where n is the exponent from the Sobolev inequality (4).


Proposition 4.9. Let u be a p-supersolution in 4B, where B = B(xo,r) and Xo is a Lebesgue point o f u. Let v be the associated Radon measure in 4B given by Proposition 3.5. Then

U(Xo) > infu + CW~(xo,r). - - 2 B

Proof . For each nonncgative integer i, let B, --- 21- 'B and a, = infB, u. Recall

that. by Lemma 4.5 as u is a p-supersolution, it is bounded below on each B,. Now

by Lemma 4.8, we have, for i = 1, 2 , . . . ,

l / ( p - 1 )

( 2 1 - ' r ) p "( C(a i - a i - 1 ) . 1..,( B, )

Hence, by the doubling property o f #,

--/o s �9 \ #(B(xo,t))] t - , =1

oc < C E ( a i - a , _ , ) = C( ,im a• - ao).

i= 1

Finally, as xo is a Lebesgue point of u, we have

u(xo)= lim [ u d#> lira a k , k ---~ ~ ,] B k - - k --r ~ c

from which the lemma follows. []

The following proposition gives us an upper bound for u in terms of the

corresponding Wolff potential.

Proposition 4.10. Let u be a nonnegative p-supersolution in the ball 2B, where B = B(xo, r) and :Co is a Lebesgue point of u. Let v be the associated Radon measure given by Proposition 3.5. Then.for all "~ > p - 1,

u(xo) < C ( ( ~ u~ d#) l /~+ Wp(xo,2r)).

Proof . See [Mi, Theorem 3.13]. The proof is lengthy but the only tools are

the chain and product rules (3) and the Sobolev inequality (4), which we do have

in this setting. Thus, one shows as in the proof of Theorem 3.13 in [Mi] that for

close to p - 1 and sufficiently small ~ > 0, the sequence a0 := 0,

aj := aj-1 + ? (u - a5-1)~ d . , J


with Bj -- B(xo, 2 l-jr), j = 1, 2 . . . . . satisfies

(17) aj <_ 2al +CW~(xo,2r).

Here, for a given function f , f+ means the positive part max{f, 0} o f f .

have

By ( l 5), the first term on the right-hand side tends to 0 as j -+ oo; and the estimate

(17) completes the proof. []

Applying Lemma 4.5 to Proposition 4. l 0, we obtain the following corollary.

C o r o l l a r y 4.11. For u, r, Xo, and B as in Proposition 4.10,

u(xo) <_ C ( i ~ f u + Wp(xo,2r)).

Now we are ready to provc the main result o f this section.

T h e o r e m 4.12. Let ~ C X be bounded and v a compactly supported Radon

measure in ~2. Then the following are equivalent:

(a) v e N~'P(fl)*;

(b) there is a nonnegative p-supersolution u on fl such that equation (7) is

satisfied and fn u dv < oo;

(c) L W~(z,r).dv(x) < ooforsoraer > O.

By supp v wc mean the smallest closed set U such that v(X \ U) = O.

P r o o f . By Proposition 3.9, (a) implies (b).

To see that (b) implies (c), note first that by Lemma 4.2 the Wolff

potential is lower semicontinuous and hence v-measurable. Whenever 0 < r <

�88 0fl), we have by Proposition 4.9 that W~(x,r ) _< Cu(x) for all

x �9 supp v, and hence

N o w w e

fn Wp(x ,r )dv <_ C f n u d v < oo,


which proves (c).

It now remains to prove that (c) implies (a). Assume condition (c), and let

r > 0 be a radius for which the Wolff potential is integrable. Extend u to the

exterior of f/ by defining u(X \ f~) = 0. Let uj be the restrictions of u to the

closed sets Ej = {x E X : W~(x,r) < j}, and B = B(xo, R) be a ball containing

~. Approximate each uj by measures uji E N~'P(5B) * as in Lemma 3.12, so that

uji -~ vj weakly in 5B as i ~ c~, uniformly in j . Note that supp vji C B for

sufficiently large i.

By Lemma 4.3 and the doubling property of #, we have for all x E 5B and

sufficiently large i,

Wpr (x, 4R) < CW;~ (x, 12R)

fl2R ( /j (~-~) ) 1/(p--l) (18) < C W ; ' ( x , � 8 9 t p

- , ( 8 ( x , � 8 9

<_ c (w; , (x, + c'),

where the constants C and C t do not depend on j. Choose xj E Ej such that

d(x, xj) = dist(x, Ej). This is possible since X is proper. Then B(x, t) C B(xj, 2t)

and #(B(x, t)) ~ #(B(xj , 2t)) whenever t >_ d(z, Xj). Hence

fl/ [r Jd(x,xj) \ tP#(B( x, t)) t

fr/2 [ u(B(xj,2t)) ~ l/(P-1) dt t " - - ~ - ~ -- < CWp(xj , r ) < Cj. <_ c t ) ) ) t - -

It follows that W~ j~ (x, 4R) < C j, for sufficiently large j and i.

By Proposition 3.9, there exist p-supersolutions uji E N~'~(5B) satisfying

equation (7) with measures vii on the right-hand side. Lemma 3.4 implies uj~ > 0 (p-q.e.) in 5B, and Lemma 4.5 implies kji = infs uji > 0. By Lemmas 2.6, 3.11

and 3.12, we see that

R-V#(B) ,~ Capp(B; 5B) _< Capp({uji >_ kji}; 5B) < k~TPuii(5B) < k~;Pv(f~),

so kji <_ M for some M < c~ and all j and i. For x e B, we have B C B(x, 2R);

hence

inf uji <_ i~f uji = kji <_ M. B(x,2R) Corollary 4.11 and the fact that p-q.e, x e X is a Lebesgue point ofuj i (see (16))

now imply that for sufficiently large j and p-q.e, x E B,

uji(x) < C(kjl + Wp~'(x,4R)) < C(M + j) < Cj.


Hence, for sufficiently large j and i, we have, using (7) and Lemma 3.1 2,

f5 IDu3~lP d# = fB uy~dvj~ <- Cjuji(B) = Cjvji(gt) <_ Cjvj(f~) <_ Cjv(f~). B Here we have used the fact that the measures us~ are absolutely continuous with

respect to the p-capacity; see Lemma 3.8. It follows that for large j , the sequence

{IDujilP-2Duji}~=l is bounded in the reflexive space LP/(P-I)(hB,#;]~N). The

local weak sequential compactness of reflexive spaces (see [Yo]) implies that

there exist vector-valued functions vj E Lv/(P-~)(5B, #; RN) such that (passing, if

necessary, to a subsequence, again denoted uj~) we have

IDujilP-ZDuj, ~ vj weakly in L p/(p- I)(5B, #; l~ N), as i --~ ec.

Consequently, we have by (7) and Lemma 3.12 for all qa E Lipc(5B),

(19) f5BqOdvj = ili~moofhBq~ i = ili naoofhBIDuj~lp-2Duji'Dqodl ~

= fhB vj .D~pdlz < I Iu j I IL I , / (p - , ) (5B , I .~ ;RN) I I~oI [ Iv , , p (hB) .

Now, by the fact that Lipr is dense in NI'V(5B) (see [Sh2, Theorem 4.8]), for

any function ~ E NI'P(5B) we can find a sequence ~Pk E Lip~(hB) converging to

~a both in N~'P(5B) and p-q.e. (see [Shl, Corollary 3.9]). By (19), the functions

~ak form a Cauchy sequence in LI(5B, uj); hence ~k --+ ~ in LI(5B, uj). Thus, a

subsequence of {~k}~=a converges to ~ vj-a.e, in 5B; and Lemma 3.8 implies that

~a = ~ uj-a.e, in 5B. It follows that (19) holds true for all functions ~ E NI'V(5B),

i.e., uj E N01'P(5B) ".

Another application of Proposition 3.9 yields solutions uj E N~'P(5B) of the

equation (7) with measures uj on the right-hand side. It is now shown as before

that for all j and p-q.e, x E B,

inf uj < i~f uj <_ M < oo; B(x,2n)

and Corollary 4.1 1 together with Lemma 3.8 then implies, as in (1 8),

+ C fn(vvp"(x, r) + C') dr(x) < M1 < oo. < CMv(Ft)

Finally, by the monotone convergence theorem and the fact that v(f~ \ Uj Ej) = 0,


we have, for all nonnegative ~o E N~'V(f~),

~ ~&' = lim f ~XEj du = lim f ~duj = lim ~ IDujlv-2Du~ " J~ J---+~ Jf't j--~oc

lijm+s~p ( ~ ,DltjlPdlz) l-lIp ( ~ ID~[Pd#)I/P< M:-l/PllqoI,Nl.~(f~).

Consequently, If. d.I <- If. § + If -d"l <- CII IIN,.(.) for all E N~'P(ft), i.e., u �9 Nol'"(a) *. []

5 A boundary regularity r e s u l t

The main result of this section is a version of [HKM, Theorem 6.18] in the

setting of the metric measure spaces considered here; see Theorem 5.1. It gives

us control over the oscillation of p-harmonic functions at boundary points of the

domain of harmonicity. In particular, it provides control over the oscillation of the

p-potentials considered in the proofofLewis ' s theorem. In the Euclidean case, this

estimate was first proved by Maz'ya in [Ml] (p = 2) and [M2] (p > 1). It implies

the sufficiency part of the celebrated Wiener criterion [Wi]. For an informative

survey on the history of this subject, we refer the reader to the notes to Chapter 6 in

[HKM]. The results of this section remain valid if the condition ofp-harmonicity

is replaced by the condition of p-energy minimization throughout this section, and

hence may have a broader range of applications.

For any real-valued function f defined on a set A C X, let

osc f = sup f - i~f f. A A

Recall also the definition of ~(x, E, t) from equation (1).

T h e o r e m 5.1. Let [2 C X be a bounded domain, and v E NI'p(X) A C(~).

Consider a p-harmonic function (or p-energy minimizer) h on f~ such that v - h E

N~'P(f~). I f xo E Oft and 0 < p < r, then

( i (1) osc h < osc v § oscv exp - C ~o(xo,X \ f ~ , t ) 1 / ( p - 1 ) �9 "~f]B(xo,p) -- OCt~B(xo,5r) Oft R e m a r k 5.2. Note that if the complement of f2 is p-fat at x0, then the

exponential in (1) becomes (p/r) 5 for some 5 > 0 depending only on the fatness

constant.

The proof of Theorem 5.1 uses the following lemmas. First, we need a Harnack

inequality on spheres. It is in this lemma that we need X to be linearly locally

convex (LLC).

362 J. B J O R N , P. M A C M A N U S A N D N. S H A N M U G A L I N G A M

L e m m a 5.3. Let X be an LLC space (with constants C1 and rl) and assume

that positive p-harmonic functions (or positive p-energy minimizers) satisfy the

Harnack inequality on balls. Let B = B(xo,r) be a ball, 2Clr < rl, and u E

NI 'p (x ) a positive p-harmonic function (or positive p-energy minimizer) in the

annulus 5C~B \ B. Then for ever), p such that 2Clr < p < 4Ctr, the function

u satisfies the Harnack inequality on the sphere S = {x E X : d(x, xo) = p},

i.e., there is a constant C depending only on X, p and C1, such that

sup u < C inf u. S S

Proof . Let x and y be two distinct points on the sphere S and 7 c A a curve

connecting them in the annulus A = 4C~B \ 2-B. Using the Hausdorff maximal

principle, cover the annulus A by balls B1, BN with radii x . . . , 7r so that the balls

�89 j = 1 , . . . , N, are pairwise disjoint. By construction 5C2tB C 20C~Bj and the

doubling property of the measure implies

N '

N#(5C~B) < C E # ( ~ B j ) < C#(5C21B). j = l "

Hence, the number of balls N is bounded from above by a constant independent of

r; cf. [Se, Lemma C.1].

Consider only those balls which have nonempty intersection with the curve 3'-

Their union is connected, and we can find a connected chain of balls B~, . . . , B~,

of length k < N such that x E B~ and y E B~. Applying the Harnack inequality

([KiSh, Corollary 7.7]) to each of the balls Bj., we obtain

u(x) < supu <_ Cinfu < Csupu < -.- < C k-1 sup < Ckin fu < CNu(y). B; n~ B~ B' B' k k

Taking the supremum over all x E S and the infimum over all y E S finishes the

proof. []

One of the crucial ingredients in the proof of Theorem 5.1 is the following

estimate for p-capacity. It is proved in [HKM, Lemma 6.19], but the proof there

relies heavily on the differential equation (5). As we want this section to be valid

for p-energy minimizers as well as for p-harmonic functions, we need to use the

proof from [LM, Lemma 3.6 and Remark 3.7], which uses variational methods

rather than the equation (5).

L e m m a 5.4. Let f~ be a bounded domain in X, and let K C f~ be compact, l f u

is the p-potential for K with respect to f~ and K- r = {x E f~ : u(x) > 7}, 0 < 3' < 1,

then

-~P-lCapp(K.y; f~) ~ Cap~,(K; f/).

FAT SETS AND POINTW1SE BOUNDARY ESTIMATES 363

P r o o f . Let fL r = {x E f~ : u(x) > 7}. One easily verifies that the functions

Ul = min{u/7, 1} and u2 = (u - 7u l ) / (1 - 7) are p-potentials for K, r in f~ and

K in f~-~, respectively. Testing the definition of Capv(K; f~) with the function

aul + (1 - a)u2 gives for all 0 < a < 1,

(21) Capv(K; f~) _< aPCapp(KT; f~) + (1 - a)PCapp(K; ft-r)

and, for a = 7, since 7ul + (1 - 7)u2 = u,

(22) Cap2o(K; f~) = 7PCapp(K.r; Ft) + (1 - 7)PCapp(K; Q'r).

Note that Capv(K; fLr) > Capv(K; f~). Hence by (22), we get

Capp(K-r; ~) _< 1 - (17 p- 3,)P Capp(K; f~) <_ __~._1Capp(K; ~),

which proves one half o f the lemma. Conversely, setting

in (21) yields

a = Capp(K~; f~)- t / (p-n

Capp(K.r; f t ) - I / (P-U + Capp(K; ~.r)-l/(P -1)

1 1 > Capp(K; f2)t/(P -1) - Capp(K.y; f~)~/(p-t)

1 + Capp(K; ~"~.r ) 1/(p-l) "

This, together with Capp(K; fLr) <_ Capp(K; f l ) / ( 1 - 7 ) p (which follows from (22)),

gives

Capp (K; 12) l/(n- l) Capp(K~; ~-,~)1/(p--1) __~ i "~ ~_~I~ --~

from which the lemma follows.

(p - 1)Capp(K; ~'-~)l/(p--1) P7

[]

In order to simplify notation, we shall assume that C1 = 1 and rl = cr in the

LLC property. For C1 > 1, one needs to consider the p-potential and the p-capacity

with respect to the ball 5C~B; for rl < oc the results hold only for small balls (the

radii o f which depend on rl). Note, however, that due to the following lemma, the

ball 5B (or 5CIB) in the conclusion o f Lemma 5.6 can easily be replaced by 2B;

see also the proof of Lemma 5.7

L e m m a 5.5. Let B = B(zo,r) be a ball in X and E c B. Then there exists

C > 0 such that for 1 < s < t with tr < �89

Capp(E;tB ) _< Capp(E;sB) _< C 1 + ~(s Capp(E;tB).


Proof . The first inequality is obvious. As for the second inequality, let

u E N~'P(tB) be a function admissible in the definition of Capp(E;tB). Let v = w?, where 77 is a Lipschitz function with Lipschitz constant 1/(s - 1)r such

that 77 = 1 on B and ~7 = 0 outside sB. Then v E N~'P(sB) is admissible in the

definition of Cap~,(E; sB), and hence

Cap,(E;sB) < f~B lDvlP du < C(f~B lDul" du + fB (s_ l)Prp d~ )"

Thc last integral is estimated using the Sobolev incquality (4) on tB. Thus we

obtain tp

Capp(E; sB) < C ( I + -'--~i-~) L 'Du'P (s

Taking the infimum ovcr all u admissible in the definition of Capp (E; tB) finishes

the proof. []

The following lemma is a generalization of Lemma 6.21 from [HKM].

L e m m a 5.6. Let B = B(xo, r) be a ball in X, r < ~diam X, and K a compact subset o f B. Let u be the p-potential for K with respect to 5B. Then for every x E B, we have

u(x) > C - (Capp(K; 5B) 1/(p--l)

- Capp(B; 5-~ ]

P roof . By thc strong maximum principle applied to u on 5B \ K, we have u > 0

on 5B \ K (see [KiSh, Corollary 6.4] for a proof of the strong maximum principle;

note that p-harmonic functions are quasiminimizcrs in the sense of [KiSh]). Let

M = sup{u(x) : d(x, x0) = 3r}. By the (1,p)-Poincar~ inequality, every sphere of

radius less than half the diameter of X is nonempty. Hence there is a point x such

that d(x, xo) = 3r, so, M > 0. Moreover, M < 1. By Lemma 5.4,

Capp(K; 5B) ~ MP-XCapp({x E 5B: u(x) >_ M}; 5B).

The strong maximum principle applied to u in 5B \ 3B yields

{x E 5B :u(x) > M} C 3B.

Therefore, by Lemma 5.5,

Capp(K; 5B) _< CMp-lCapp (3-B; 5B) ~ Mp-ICapp(B; 5B),

i.e., (Capp(K;5B)'~ '~(P-a)

M _> C k, Capp(B; 5B) ,]

FAT S E T S A N D P O I N T W I S E B O U N D A R Y E S T I M A T E S 365

The Hamack inequality on spheres (Lemma 5.3) implies

m := inf{u(x) : d(z , zo) = 3r} _> C M .

Finally, the strong maximum principle applied to 3B \ K implies inf3B u > m,

yielding the desired inequality. []

The following lemma is a generalization o f Lemma 6.25 in [HKM].

L e m m a 5.7. Let f~ be a bounded domain in X a n d Xo E 0~ . Fix r > O, a n d let

u be the p-potent ia l f o r B(x0, r) \ f~ with respec t to B(zo, 5r). Then f o r 0 < p < r

and x E B(xo , p), we have

1 - u ( x ) <_ exp - C ~(xo, X \ ~, t) 1~(p-x) .

P r o o f . Let B = B(xo , 5r) and Bi = 5- iB . Let ui denote the p-potential for

B~ \ fl with rcspcct to Bi -1 . Note that Ul = u. Let

Capp(Bi \ gt; Bi -1 )

~oi := Capp(Bi; Bz-t)

By Lemma 5.6 and the inequality 1 + t <_ e t, t E R, we have

f , l / ( p - - l ) ~ l ] ( p - l ) (23) ui(x) > (-,~, >_ 1 - e - ~ '

whenever x E B~.

Let Vl = 1 - Ul and define functions vi, i = 2, 3 , . . . , recursively by

/ _ \ i - 1

(24) vi :=eC~:/-(~P-t)vi-i = e x p ( C ~ - ~ ) / ( P - a ) ) v l . " j = l

We know by (23) that vl <_ e -C~I/( ' -~) ~ 1/(~-~) on/~1. Suppose that vi _< e - ' ~

on/~i- Then by the definition of vi+l, we have vi+l _< 1 on Bi; and, as Vl = 0 on

the set B1 \ f~ (which contains B-i \ f/), we also have vi+1 = 0 on Bi \ fL Also,

1 - ui+~ = 1 outside Bi. As both Ui+l and Vi+l take on values only between 0

and 1, we have vi+l < 1 - ui+t on the boundary O(Bi N f~). Now both vi+l and

ui+l are p-harmonic in B i n f/, so the comparison principle [Sh2, Theorem 6.4] l / ( p - l )

together with (23) yield vi+l < 1 - Ui+l _< e -~'~+~ on Bi+l. We now conclude

by induction that for all i = 1, 2 , . . . , vi < e - C ~ / ( € on B~. Consequently, by

(24),

k / k

(25) a - u = 1 - - U l = V l <_exp(-CZ99:/(P-1) ) off B--k. k / j = l

366 J. BJORN, P. MACMANUS AND N. S H A N M U G A L I N G A M

Fix p > 0 so that p _< r and choose an integer k so that 5-kr < p < 51-kr.

the definition of qoj and Lemma 5.5, we have

~i _> C Capv(B(xo, t) \ gt; B(xo, 2t))

Cap~( B(xo, t); B(xo, 2t))

whenever 5-it < t < 51-it. Therefore,

= C~(xo, x \ ~, t)

k k r - i r Z - ' / ( P - I ) > C E T ~ ~~176

gai _ _, "-t i=1 i= l o r

/; > c ~(xo, x \ f~,t) ~/(~-uat.

By

and the inequality (25) finishes the proof.

Theorem 5.1 now follows from Lemma 5.7 and the comparison principle

[Sh2, Theorem 6.4] in exactly the same way as Theorem 6.18 in [HKM].

6 P r o o f o f T h e o r e m 1 . 2

In this section, we prove the main result of this paper, Theorem 1.2. The proof

is similar to that given in [Mi, Theorem 8.2]. However, we have introduced some

simplifications and so present it here.

Proof of Theorem 1.2. Let E c X be as in the theorem, and r0, Co be

the constants associated with the uniform p-fatness of E. Consider x0 c E and

0 < r < r0, and let B = B(x0, r). By the assumption that X is proper and by the

construction in [L] or [Mi], there is a compact set F C E M B such that Xo E F and

F is uniformly p-fat with constants Co and r.

Let u be the p-potential for F relative to 5B and v the Radon measure associated

with u in 5B as in (7) (note that by Lemma 3.2, u is a p-supersolution). Let

1 Since is a doubling measure, we have t-P#(B(x,t)) ~_ xEOFandO<p< ~r. #

p-Pp(B(x, p)) whenever p < t _< 2p. Thus, the Wolff potential o f v satisfies

L2P (l /(B(x, t)))I /(p-I)dt > C (u(B(x,p), ) 1 / ( p - l ) w;(x ,2p) > \ t - ~ ) ) 7- - \p-p~,(B(x,p))

Therefore, by Proposition 4.9 and Theorem 5.1 (Remark 5.2) with a suitable

function v such that OSCB(,,2op)V = 0, we have

C (v(B(x,p)) ) 1 / ( p - l , C(u(x) inf u ) < C ( p ) a \p_, lz(B(x ' p)) <_ W~(x, 2p) <_ - a(=,4v)

F A T S E T S A N D P O I N T W l S E B O U N D A R Y E S T I M A T E S 367

for some 6 > 0 independent o f x and p. Using this inequality, we now construct a

new Borel measure on X which yields a q-supersolution for some q < p. From the t last inequality we have, for all x E OF and 0 < p < gr,

Cp~(p-1)-p (26) u(B(x,p)) <_ ra(p_l ) #(B(x,p)).

Choose q _> q0 so that p - 3(p - 1) < q < p. Here q0 is the exponent in the Poincard

inequality. Let e = q - (p - 6(p - 1)) > 0, and define a measure f, on X by

[/ := r P - q u .

Then by (26),

(27) ~(B(x, p)) < Cr- 'p ' -q#(B(x, p))

1 Since supp P C OF, the estimate (27) holds whenever x E OF and 0 < p < gr. trivially i f B(x, p) N OF is empty�9 Moreover, i f y E 2B \ OF and x E B(y, p) n OF, then (27) and the doubling property o f # imply

f,(B(y, p)) < D(B(x, 2p)) _< Cr-'(2p)'-q#(B(x, 2p)) <_ Cr- 'p '-q#(B(y, p)),

i.e., (27) holds for all x E 28 and 0 < p < 1-!6r. Therefore, for x E 2B,

f ( D(B(x,p)) 1/(q-l) f,,o jo <_ Cjo ,7 , -y <

Hence by Theorem 4�9 we can conclude that 1) E N~'q(2B) *. Then by

Proposition 3.9, there exists a unique q-supersolution v E N~'q(2B) such that

the identity (7) holds with ;, on the right-hand side�9 Lemma 3.4 yields v > 0. From

Corollary 4.11, we have for x E B

v ( x ) < C ~ ine v + W ~ ( x , ~ r ) ) < C ( inf v + l ) ; - - \ B ( : z , r / 2 0 ) - - \ B ( ~ , r / 2 0 )

and it is shown as in the proof o f Theorem 4.12 that infB(~,~/20)v _< M, for some

M < oo independent o f x and r. Hence v _< M ' in B and, by the strong maximum

principle applied to 2B \ B, also v _< M' in 2B. Let vl = v/M'. Then 0 _< vl <_ 1,

and (7) holds with Ul = CP on the right-hand side for some C > 0 independent

of r. Therefore Lemmas 2.6, 3.10 and 5.5 together with the uniform p-fatness of

F yield

Capq(E n B;2B) ___ Capq(F; 28) _> ul(F) ,~ rP-qu(F) = rP-qCapp(F;5B)

>_ CrP-qCapp( B; 28) ~, r-q #( B ) ,~ Cape(S; 28).

Thus E is unift, rmly q-fat, and the proof is complete. []

As a corollary to Theorem 1.2 we have the following Hardy inequality.

368 J. BJORN, E MACMANUS AND N. SHANMUGALINGAM

C o r o l l a r y 6.1. Let X be a proper LLC metric space endowed with a doubling

regular Borel measure supporting a (1, qo)-Poincarb inequality for some qo < p,

and suppose that f~ is a bounded domain in X such that X \ f~ is uniformly p-fat.

Then there is a constant C(fL p) > 0 such that a function u E N I'p ( X ) is in N~'P ( f~ )

i f and only i f the following inequality holds."

f~ I~(x)l~ f~ dist(x, X \ f~)p d#(x) < C(f~,p) gu(x) p d#(x).

The proof of this corollary uses Theorem 1.2 and follows the same lines of

reasoning as in [KM2]; it will therefore be omitted here. See [L], [Ku], and [Wa]

for more on Hardy inequalities.

[Bj]

[C]

[FHK]

[HaK]

[HW]

[HKM]

[HeK]

[KaSh]

[KL]

[KMI]

[KM2]

[KiSh]

[KMc]

[Ku]

[L]

[LM]

[Li]

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dana Bj6rn DEPART M ENT OF M ATHEM ATICS

LUND INSTITUTE OF TECHNOLOGY P.O. BOX 118

SE-22100 LUND, SWEDEN email: [email protected]

Paul MacManas DEPARTMENT OF MATHEMATICS

PHILLIPS EXETER ACADEMY 20 MAIN STREEX

EXETER, NH 03833, USA email: pmacmanusOexeter.ed u

Nageswari Shanmugalingam DEPARTMENT OF MATHEMATICS

UNIVERSITY OF TEXAS AT AUSTIN AUSTIN, TX 78712, USA

email: nageswari~math.utexas.ed u

(Received November 5, 2000)

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Fat sets and pointwise boundary estimates for p -harmonic functions in metric spaces