Weighted Sobolev spaces on
metric measure spaces
By Luigi Ambrosio at Pisa, Andrea Pinamonti at Bologna and Gareth Speight at Cincinnati
Abstract. We investigate weighted Sobolev spaces on metric measure spaces .X; d; m/. Denoting by the weight function, we compare the space W1;p.X; d; m/ (which always coincides with the closure H1;p.X; d; m/ of Lipschitz functions) with the weighted Sobolev spaces W1;p.X; d; m/ and H1;p.X; d; m/ defined as in the Euclidean theory of weighted
Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1;p.X; d; m/D H1;p.X; d; m/. We also adapt the results in [23] and in the
recent paper [27] to the metric measure setting, considering appropriate conditions on that ensure the equality W1;p.X; d; m/D H1;p.X; d; m/.
1. Introduction
The theory of Sobolev spaces W1;p.X; d; m/ with p2 .1; 1/ on metric measure spaces .X; d; m/ has by now reached a mature stage, after the seminal papers [10, 24], the more recent developments in [5] and the monographs [8, 19]. In this context, it is natural to investigate to what extent the Sobolev space is sensitive to the reference measure m. It is clear that the measure m is involved, since we impose Lp.m/ summability of the weak gradient, but things are more subtle. Indeed, the measure m is also involved in the definition of .p; m/-modulus Modp;m(Definition 2.2) which, in turn, plays a role in the axiomatization in [24]: by definition,
f 2 W1;p.X; d; m/ if there exist a representative ef 2 Lp.m/ of f and g2 Lp.m/ such that jef . .1// ef . .0//j
Z 1
0
g. .t //j P .t/j dt
along Modp;m-a.e. absolutely continuous curve W Œ0; 1 ! X. If such a function g exists, then
there is one with minimal Lp.m/ norm which is called the minimal gradient.
The authors acknowledge the support of the grant ERC ADG GeMeThNES. The second author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The definition adopted in [10], instead, is equivalent but based on the approximation in Lp.m/ with functions having an upper gradient in Lp.m/. More recently, in [4, 5] it has been proved that W1;p.X; d; m/D H1;p.X; d; m/, where the latter space is defined as the collection of all Lp.m/ functions for which there exist fn2 Lip.X/\Lp.m/ withRXjfn fjpdm! 0
and sup n2N Z Xjrf njpdm <1
(here and in the sequel jrgj denotes the local Lipschitz constant of g). In this case, fol-lowing [10], the minimal gradient is defined by considering all functions larger than weak Lp.m/ limits of local Lipschitz constants of sequences of Lipschitz functions converging to f in Lp.m/, and considering the function with smallest Lp.m/ norm. This general “H D W ” result does not depend on structural assumptions on the metric measure structure: .X; d/ com-plete and separable and m a locally finite Borel measure are sufficient for the validity of this identification theorem. In this introduction we shall denote by jrf jw the minimal gradient
arising from both unweighted W and H definitions, not emphasizing its potential dependence on p (see [13]).
Given a Borel weight function W X ! Œ0; 1, in this paper we compare spaces H D W relative to the metric measure structure .X; d; m/ with the weighted spaces built as in the Euclidean theory (namely X D Rn, dDEuclidean distance, m D Ln, the Lebesgue measure in Rn). The first weighted space is
(1.1) W1;p WD ¹f 2 W1;1.X; d; m/W jf j C jrf jw 2 Lp.m/º
endowed with the norm kf k WD Z Xjf j p dm C Z Xjrf j p w dm 1=p ;
where jrf jw is the minimal 1-weak gradient of f with respect to the unweighted space
.X; d; m/. Minimal regularity requirements (which provide respectively local finiteness of m and a basic embedding in W1;1) are that 2 L1loc.m/ and
1
2 L1=.p 1/.m/. The
sec-ond space we will consider is the subspace H1;p of W1;pdefined by
(1.2) H1;p WD Lip.X/ \ W1;p kk
:
Even when the metric measure structure is Euclidean, it is well known that H1;p can be
strictly included in W1;p, see Section 5 for a more detailed discussion and examples. This gap
suggests a discrepancy between the weighted spaces H D W of the metric theory, obtained by considering m as reference measure, and the spaces W1;p, H1;p.
Our first main result states that if 2 L1loc.m/, 1
2 L1=.p 1/.m/, .X; d; m/ is
doub-ling and supports a 1-Poincaré inequality for Lipschitz functions, then (1.3) W1;p.X; d; m/D H1;p
and the two spaces are isometric. Hence, the smaller of the two weighted Sobolev spaces can be naturally identified with the Sobolev space of the metric theory, with weighted reference measure m.
In light of the equality H D W and the Euclidean counterexamples to H1;p D W1;p, it
is natural to expect that stronger integrability properties of are needed to establish the equality W1;p.X; d; m/D W1;p;
namely
(1.4) W1;p.X; d; m/D ¹f 2 W1;1.X; d; m/W jf j C jrf jw 2 Lp.m/º:
Notice that the inclusion readily follows by (1.3). Our second main result shows that (1.4) holds provided .X; d; m/ is doubling, supports a 1-Poincaré inequality for Lipschitz functions and satisfies the asymptotic condition
(1.5) lim inf n!1 1 np Z X ndm 1=nZ X ndm 1=n <1:
This condition appeared first in the Euclidean context in [27], dealing with H1;2D W1;2,
see also the recent extension [26] to any power p > 1 and even to variable exponents. As we illustrate below, the proof in [27] is sufficiently robust to be adapted, with minor variants, to a nonsmooth context.
In view of the characterization in [24], we believe that (1.4) is conceptually interesting. Indeed, functions on the left hand side of (1.4) are absolutely continuous (modulo the choice of an appropriate representative) along Modp;m-a.e. curve, while functions on the right hand
side are absolutely continuous along Mod1;m-a.e. curve. On the other hand, even with pD 1, it
seems very difficult to connect the two notions of negligibility if and 1are unbounded. As a matter of fact our proof is very indirect and it would be nice to find a more direct explanation of the validity of (1.4).
We conclude the introduction by describing the structure of the paper. In Section 2 we recall aspects of the theory of Sobolev spaces on metric measure spaces; we detail approxima-tion results and the noapproxima-tion of measurable differentiable structure from [10].
In Section 3 we introduce the weighted Sobolev spaces W1;p and H1;p, showing first
completeness of W1;p under the assumption 1 2 L1=.p 1/.m/ and then reflexivity, under
the additional assumption that .X; d; m/ is doubling and satisfies a 1-Poincaré inequality. The proof of reflexivity is particularly tricky and it passes, as in [10] and [2], through the construc-tion of an equivalent uniformly convex norm. This involves a Lusin-type approximaconstruc-tion by Lipschitz functions. Notice this is not necessarily an approximation in the norm of W1;p, since
we know that additional assumptions on are needed to get density of Lipschitz functions, namely the equality W1;p D H1;p. Then, using reflexivity and the H D W theorem of the
metric theory, we prove (1.3) in Theorem 3.6.
Section 4 is devoted to the proof of (1.4), obtained in Theorem 4.1 under the assump-tion (1.5). Here we follow closely [27], with some minor adaptaassump-tions due to the lack of differ-entiability of f 7!R jrf jpwdm (potentially even for pD 2).
In Section 5 we recall an example from [12], showing that H1;p can be strictly
in-cluded in W1;p, and we explore some variants of our results. In particular, we relax the
1-Poincaré assumption to a p-Poincaré assumption, modifying consequently the definitions of W1;p and H1;p. Our main results still work, under the p-Poincaré assumption, for these
spaces and we prove that the new definitions coincide with (1.1) and (1.2) assuming the vali-dity of the 1-Poincaré inequality. Finally, we discuss the notion of Muckenhoupt weight and the invariance of our assumptions on .X; d; m/ under the replacement of m by m, with a Muckenhoupt weight.
Acknowledgement. We would like to thank M. D. Surnachev for kindly explaining the change of weight in [27] (which we adapted in Proposition 4.4) and for informing us of his generalization of Zhikov’s result to more general exponents [26].
2. Sobolev spaces
Throughout this paper we will denote by .X; d/ a complete separable metric space and by m a locally finite (i.e. finite on bounded sets) Borel regular measure on X . In metric spaces Lipschitz functions play the role of smooth functions. We recall that a function f W X ! R is called Lipschitz if there exists L 0 such that jf .x/ f .y/j Ld.x; y/ for all x; y 2 X; we denote the smallest such constant L by Lip.f / and denote the set of Lipschitz functions on X by Lip.X /.
For a Lipschitz function f , a natural candidate for the modulus of gradient is given by the slopejrf j W X ! R, defined by
jrf j.x/ WD lim sup
y!x
jf .y/ f .x/j d.y; x/ :
Definition 2.1 (Absolute continuity). Let J R be a closed interval. Consider a curve W J ! X. We say that is absolutely continuous if
d. .t /; .s// Z t s g.r/ dr for all s; t2 J; s < t; (2.1) for some g2 L1.J /.
It is well known (see [3, Proposition 4.4] for the proof) that every absolutely continuous curve admits a minimal g satisfying (2.1), called metric speed, denoted byj P .t/j and given for a.e. t 2 J by
j P .t/j D lim
s!t
d. .s/; .t // js tj :
We will denote by C.Œ0; 1I X/ the space of continuous curves from Œ0; 1 to .X; d/ endowed with the sup norm and by AC.Œ0; 1; X / the subset of absolutely continuous curves. Using the metric derivative we can easily define curvilinear integrals, namely
Z gWD Z 1 0 g. .t //j P .t/j dt for all gW X ! Œ0; 1 Borel and 2 AC.Œ0; 1I X/.
Definition 2.2 (Modulus). Given p 1 and a subset AC.Œ0; 1; X/, the p-modulus Modp;m./ is defined by Modp;m./WD inf ²Z X hpdmW Z h 1 for all 2 ³ ;
where the infimum is taken over all nonnegative Borel functions hW X ! Œ0; 1. We say that is Modp;m-negligible if Modp;m./D 0.
We can now give the definition of weak gradient and Sobolev space which we will use, see [24].
Definition 2.3 (p-upper gradient). For p 1, a Borel function g W X ! Œ0; 1 with Z
X
gpdm <1
is said to be a p-weak upper gradient of f if there exist a function ef and a Modp;m-negligible
set such that ef D f m-a.e. in X and jef . 0/ f . e 1/j
Z
g ds for all 2 AC.Œ0; 1; X/ n .
The following theorem is classical, see [19, 24] for a proof.
Theorem 2.4. For everyp 1 the collection of all p-weak upper gradients of a map f W X ! R is a closed convex lattice in Lp.m/. Moreover, if the collection of all p-weak upper gradients off is nonempty, then it contains a unique element of smallest Lp.m/ norm. We shall denote it byjrf jp;m.
From now on we denote the 1-weak gradient of f with respect to m simply byjrf jw,
sojrf jw D jrf j1;m. Following [24] we can now define the Sobolev space from which we
will define weighted Sobolev spaces on metric measure spaces.
Definition 2.5 (Sobolev space W1;p.X; d; m/). For p 1 we define W1;p.X; d; m/ to be the Banach space of (m-a.e. equivalence classes of) functions f 2 Lp.m/ having a p-weak upper gradient, endowed with the norm
kf kpW1;p.m/WD Z Xjf j p dmC Z Xjrf j p p;mdm:
It can be proved (see [19, Proposition 5.3.25]) thatjrf jp;m is local, namely
jrf jp;m D jrgjp;m m-a.e. on¹f D gº
for all f; g2 W1;p.X; d; m/.
Definition 2.5 is by now classical and it goes back to the pioneering work [24], where the author also proved that if p > 1, then the space W1;p.m/ coincides with the Sobolev space defined by Cheeger [10] in terms of approximation by pairs .fn; gn/, with fn! f in Lp.m/,
gnan upper gradient of fnand¹gnºn2N bounded in Lp.m/. More recently, the first named
author, Gigli and Savaré (see [5] for pD 2 and [4] for p > 1) improved this equivalence result proving existence of an approximation by Lipschitz functions, with slopes (or even asymptotic Lipschitz constants, see [1]) as upper gradients. More precisely, defining
H1;p.X; d; m/D ²
f 2 Lp.m/W there exists fn2 Lip.X/ \ Lp.m/
with fn! f in Lp.m/ and sup n Z Xjrf njpdm <1 ³ ; the following result holds.
Theorem 2.6. W1;p.X; d; m/D H1;p.X; d; m/ for all p > 1. In addition, for all func-tionsf 2 W1;p.X; d; m/ the following holds:
(2.2) there exist fn2 Lip.X/\Lp.m/ with fn! f and jrfnj ! jrf jp;minLp.m/.
On the contrary, the picture for pD 1 is far from being complete since at least three definitions are available (see also [2] and the forthcoming paper [6] for a discussion on this subject).
For our analysis of weighted Sobolev spaces we will require that the measure m is doub-ling and that a p-Poincaré inequality holds; we recall these properties now. Doubdoub-ling metric measure spaces which satisfy a p-Poincaré inequality are often calledPIpspaces and there are
many known examples which differ from ordinary Euclidean spaces, see for instance [15–17]. Definition 2.7 (Doubling). A locally finite Borel measure m on .X; d/ is doubling if it gives finite positive measure to balls and there exists a constant C > 0 such that
m.B.x; 2r// C m.B.x; r// for all x 2 X; r > 0: (2.3)
In this case we also say that the metric measure space .X; d; m/ is doubling.
A metric measure space gains additional structure if a Poincaré inequality is imposed; this type of inequality is a control on the local variation of a Lipschitz function using infinitesimal behavior encoded by the slope.
Definition 2.8 (p-Poincaré). For p2 Œ1; 1/, we say that a p-Poincaré inequality holds for Lipschitz functions if there exist constants ; ƒ > 0 such that for every f 2 Lip.X/ and for every x2 supp.m/, r > 0, the following inequality holds:
« B.x;r/jf fB.x;r/j dm r « B.x;ƒr/jrf j pdm 1=p ; (2.4)
where, here and in the sequel, fAD « A f dmWD 1 m.A/ Z A f dm:
We will say that a constant is structural if it depends only on the doubling constant in (2.3) and the constants ; ƒ in (2.4).
Notice that, by the Hölder inequality, thePIpcondition becomes weaker as p increases, so
PI1 is the strongest possible assumption. A remarkable result (see [21]) is thatPIp is an open
ended condition in .1;1/, namelyPIp for some p2 .1; 1/ impliesPIq for some exponent
q2 .1; p/.
Thanks to (2.2), for all p > 1 and f 2 W1;p.X; d; m/, under thePIpassumption it holds
« B.x;r/jf fB.x;r/j dm r « B.x;ƒr/jrf j p p;mdm 1=p (2.5)
for all x2 supp.m/ and r > 0, where ; ƒ are as in (2.4) (see also [20, Theorem 2]). The inequality is still valid with pD 1 under thePI1assumption. Indeed, the space W1;1.X; d; m/
con-sidered in [2]. The equivalence result with Miranda’s definition of BV (which parallels the ideas provided in [5] for the Sobolev spaces) provided in [2] ensures the existence of sequence .fn/ Lip.X/ \ L1.m/ with fn! f in L1.m/ andjrfnjm ! jDf j weakly as measures.
Taking the limit, one obtains (2.5) withjDf j.B.x; ƒr// on the right hand side. In the case when f 2 W1;1.X; d; m/ BV.X; d; m/, one can use the inequality jDf j jrf j1;mm to
conclude, see also [6] for a more detailed discussion.
We now recall the relevant properties of the maximal operator.
Definition 2.9 (Maximal operator). Given f 2 L1loc.m/ we define the maximal function
Mf W X ! Œ0; 1 associated to f by
(2.6) Mf .x/WD sup
r>0
«
B.x;r/jf j dm:
Since m is doubling, we know [16, Theorem 2.2] that for q > 1 the maximal operator is a bounded linear map from Lq.m/ to Lq.m/; more precisely, there exists a constant C > 0 depending only on the doubling constant such that
kM.f /kLq.m/
C
.q 1/1=qkf kLq.m/
for all f 2 Lq.m/. For qD 1 the maximal operator is also weakly bounded, namely
(2.7) sup
>0
m.¹M.g/ > º/ Z
Xjgj dm:
We will also need the asymptotic estimate
(2.8) lim
!1m.¹M.g/ > º/ D 0:
This asymptotic version follows by (2.7), taking the inclusion ¹M.g/ > 2º ¹M..jgj /C/ > /º into account.
Recall that x is a Lebesgue point of a locally integrable function u if lim
r#0
«
B.x;r/ju.y/
u.x/j dm.y/ D 0:
This notion is sensitive to modification of u in m-negligible sets. With a slight abuse of notation we shall also apply this notion to Sobolev functions, meaning that we have chosen a representative in the equivalence class.
We now state a key approximation property for functions in W1;p.X; d; m/, valid under the doubling and p-Poincaré assumptions. We give a sketch of proof for the reader’s conve-nience, but these facts are well known, see for instance [10, 24] or the more recent paper [1] where some proofs are revisited.
Proposition 2.10. Assume thatp 2 Œ1; 1/ and that .X; d; m/ is aPIp metric measure
setsEnwith:
(i) En EnC1andsupnnpm.Xn En/ <1, so that m.X n [nEn/D 0,
(ii) jfnj n, Lip.fn/ C n, f D fnm-a.e. inEn,
(iii) jfn fj ! 0 and jr.fn f /jp;m! 0 in Lp.m/.
Furthermore, there is a structural constantc such that for all f 2 Lip.X/ \ W1;p.X; d; m/, (2.9) jrf jp;m jrf j cjrf jp;m m-a.e. inX :
Proof. Recall the definition of the maximal operator M with respect to m from (2.6). By iterating the estimate (2.5) on concentric balls (see for instance [10] or [1, Lemma 8.2]) one can obtain the estimate
(2.10) jf .x/ f .y/j C d.x; y/.M1=p.jrf jp;mp /.x/C M1=p.jrf j p
p;m/.y//
whenever f 2 W1;p.X; d; m/ and x; y 2 X are Lebesgue points of (a representative of) f . Set now
gWD max¹jf j; M1=p.jrf jp;mp /º:
We also define
EnWD ¹x 2 X W x is a Lebesgue point of f and g.x/ nº:
Notice that since¹g > nº is contained in ¹jf j > nº [ ¹M.jrf jp;mp / > npº, the set X n En
has finite m-measure, and more precisely Markov inequality and the weak maximal estimate give that supnnpm.X n En/ <1.
Using (2.10) and the definition of g we obtain
jf .x/ f .y/j C nd.x; y/; jf .x/j n
for all x; y2 En. By the McShane lemma (see for instance [16] for a simple proof) we can
extend fjEn to a Lipschitz function fn on X preserving the Lipschitz constant and the sup
estimate, namely Lip.fn/ C n and jfnj n. We claim that jrfnjp;m C n m-a.e. in X.
Indeed, sincejrfnjp;m jrfnj m-a.e in X, we get
jrfnjp;m jrfnj Lip.fn/ C n m-a.e. in X.
Furthermore, by the locality of the weak gradient we obtainjr.fn f /jp;m D 0 m-a.e. on
the set En.
Using these facts and (2.8) it is straightforward to check, by dominated convergence, that fn! f in Lp.m/ and thatjr.fn f /jp;m ! 0 in Lp.m/.
The proof of (2.9) relies on a localized version of (2.10), namely
(2.11) jf .x/ f .y/j C d.x; y/ M2ƒr1=p.jrf jp;mp /.x/C M2ƒr1=p.jrf j p p;m/.y/
for all Lebesgue points x; y 2 X of f with d.x; y/ < r, where Msis the maximal operator on
scale s (i.e. the supremum in (2.6) is restricted to balls with radius smaller than s). The idea of the proof is to differentiate at Lebesgue points x ofjrf jp;mp , letting eventually r # 0 and
using the fact that Mr.g/# jgj at Lebesgue points of g as r # 0, see [10] or [1, Proposition 47]
With a similar proof, using the boundedness of the maximal operator, one can prove the following proposition (see [5, 13] for counterexamples showing that thePIqassumption cannot
be removed).
Proposition 2.11. Let.X; d; m/ be a PIq metric measure space andp > q. If it holds
thatf 2 W1;q.X; d; m/ and both f andjrf jq;mbelong toLp.m/, then f 2 W1;p.X; d; m/.
In the sequel we will use the fact thatPIp spaces for some p 1 admit a differentiable
structure; we conclude this section by recalling some aspects of Cheeger’s remarkable theory [10, 22], which provides a differentiable structure that will play a role in the reflexivity of the weighted Sobolev spaces.
Definition 2.12. A measurable differentiable structure on a metric measure space .X; d; m/ is a countable collection of pairs¹.U˛; '˛/º, called local charts, that satisfy the
fol-lowing conditions:
(i) Each U˛is a measurable subset of X with positive measure, and m.X nS˛U˛/D 0.
(ii) Each '˛is a Lipschitz map from X to RN.˛/for some integer N.˛/ 1, and moreover
N WD sup˛N.˛/ <1.
(iii) For every f 2 Lip.X/ and for every ˛ there exists an m-measurable function d˛f W U˛! RN.˛/ such that lim sup y!x jf .y/ f .x/ d˛f .x/ .'˛.y/ '˛.x//j d.x; y/ D 0 for m-a.e. x 2 X
and d˛f is unique up to m-negligible sets.
The following theorem is proved in [10], here we state it in the form needed in this paper. Theorem 2.13 (Existence of a measurable differentiable structure). If .X; d; m/ is aPIp
metric measure space for somep 1, then X admits a measurable differentiable structure and the integerN in Definition 2.12 (ii) depends only on the structural constants. Moreover, for all ˛ and m-a.e. x2 U˛, there is a Hilbertian normk kx onRN.˛/ such thatx7! kd˛f .x/kx
ism-measurable inU˛and
kd˛f .x/kx jrf j.x/ M kd˛f .x/kx form-a.e.x2 U˛, for allf 2 Lip.X/,
whereM > 0 is a constant independent of ˛.
3. Weighted Sobolev spaces
In this section we will define weighted Sobolev spaces and prove that, under natural integrability assumptions on the weight, we obtain a reflexive Banach space. Recall the 1-weak upper gradient of f with respect to m is denoted byjrf jw, sojrf jw D jrf j1;m.
Definition 3.1 (Weighted space W1;p.X; d; m/). Let p > 1 and W X ! Œ0; 1 a Borel
function satisfying 12 L1=.p 1/.m/; we define the weighted Sobolev space W1;p.X; d; m/
by (3.1) W1;p.m/WD ² f 2 W1;1.X; d; m/W Z Xjf j p dm C Z Xjrf j p w dm <1 ³ : We endow W1;p.X; d; m/, shortened to W1;p, with the norm
kf kp WD Z Xjf j p dm C Z Xjrf j p w dm:
Remark 3.2. The fact thatk k is a norm is a consequence of the observation > 0
m-a.e. in X and of the following elementary properties:
jr.f C g/jw jrf jwC jrgjw m-a.e. in X;
jr.f /jw D jjjrf jw m-a.e. in X; for all 2 R:
Note that, using Hölder’s inequality, it follows that (3.2) kf kW1;1 2
Z
p11 dm
.p 1/=p
kf k:
That is, W1;p.m/ embeds continuously into W1;1.X; d; m/ provided 12 L1=.p 1/.m/;
a similar calculation shows that the definition of W1;p.m/ would be unchanged if we replace
W1;1.X; d; m/ by Wloc1;1.X; d; m/ (namely the space of functions whose 1-weak upper gradient is integrable on bounded sets) in (3.1).
We use the embedding of W1;p.m/ to prove completeness of the weighted Sobolev
space, building on the completeness of W1;1.X; d; m/.
Proposition 3.3 (Completeness of W1;p). For everyp > 1, the weighted Sobolev space
.W1;p;k k/ is a Banach space whenever 12 L1=.p 1/.m/.
Proof. Suppose that .fi/1i D1is a Cauchy sequence in W1;p, and let !i # 0 be such that
kfn fmkp !i whenever n; m i. From (3.2) and the completeness of W1;1.X; d; m/ we
obtain that there exists f 2 W1;1.X; d; m/ such that fn! f in W1;1.X; d; m/ and hence,
up to a subsequence, we can also assume that fn! f pointwise m-a.e. Since Lp.m/ is
a Banach space and Lp.m/ convergence implies the existence of subsequences convergent m-a.e. in X , we deduce f is also the limit of fn in Lp.m/. As jr.fn f /jw ! 0 in
L1.m/ impliesjr.fn fm/jw ! jr.f fm/jw in L1.m/, we can pass to the limit in
kjr.fn fm/jwkLpp.m/ !i
to obtain Z
Xjr.f
fm/jpw dm !i for all m i:
This proves that
Z
Xjrf j p
w dm <1;
Theorem 3.4 (Reflexivity of W1;p). Suppose.X; d; m/ is aPI1metric measure space,
2 L1loc.m/ and 1
2 L1=.p 1/.m/. Then .W1;p;k k/ is reflexive for all p > 1.
Proof. By assumption, .X; d; m/ is a P I1 space; therefore, by Theorem 2.13, it
ad-mits a differentiable structure consisting of charts .U˛; '˛/, where U˛ X is measurable and
'˛W X ! RN.˛/are Lipschitz with N.˛/ N , with respect to which Lipschitz functions are
differentiable. Further, for all f 2 Lip.X/,
(3.3) kd˛f .x/kx jrf j.x/ M kd˛f .x/kx for m-a.e. x 2 U˛;
where k kx is an inner product norm on RN.˛/ for x 2 U˛ and M is a positive constant
depending only on N .
Without loss of generality we assume the sets U˛ are disjoint and denote derivatives by
df .x/ instead of d˛f .x/ when x 2 U˛. We now observe that we can also assume N.˛/D N
for all ˛ by replacing:
the inner product normk kxon RN.˛/by the semi-inner product normkp. /kx on RN,
where pW RN ! RN.˛/ is the projection onto the first N.˛/ coordinates for x2 U˛.
Here a semi-inner producth ; i satisfies the usual properties of an inner product except for positive definiteness – the corresponding semi-inner product norm is then given by kvk2D hv; vi.
the derivative df .x/2 RN.˛/by .df .x/; 0/2 RN for x2 U˛.
After this replacement the map d still satisfies the equivalence (3.3). Clearly, d is a linear map from Lip.X / to the space of (m-a.e. defined) RN valued measurable functions on X . We split the proof of reflexivity into three steps. Note that the first step is known and follows from results of [14], but we present it explicitly for the sake of readability.
Step 1. We construct an equivalent norm onW1;p.
We first define a (nonlinear) map D from W1;p to nonnegative (m-a.e. defined)
measur-able functions on X , which we denote by Dx.g/ instead of D.g/.x/, satisfying
Dx.g/D jjDx.g/ m-a.e. x2 X; for all g 2 W1;p; 2 R;
Dx.gC h/ Dx.g/C Dx.h/ m-a.e. x2 X; for all g; h 2 W1;p;
(3.4)
Dx.f /D kdf .x/kx m-a.e. x2 X; for all f 2 Lip.X/ \ W1;p;
Dx.g/ jrgjw.x/ MDx.g/ m-a.e. x2 X; for all g 2 W1;p:
(3.5)
Fix g2 W1;p W1;1. By Proposition 2.10, there exists a sequence of Lipschitz functions gn
such thatjr.gn g/jw ! 0 in L1.m/ and 1
X
nD1
m¹g ¤ gnº < 1:
Let AnWD ¹g D gnº and GT WDTnT An; from the Borel–Cantelli lemma it follows that
S
T GT has full m-measure. For n; m > T we have, by Proposition 2.10 (i),
Hence, by inequality (2.9), n; m > T impliesjr.gn gm/j D 0 m-a.e. in GT. We now claim
that, for each fixed x2 GT,kdgn.x/kxis constant as a function of n > T . Indeed, for n; m > T ,
using (3.3),
jkdgn.x/kx kdgm.x/kxj kd.gn gm/.x/kx jr.gn gm/j.x/ D 0:
We define Dx.g/WD kdgn.x/kx for x 2 GT and n > T . It is easy to show that if we took
a different sequence of Lipschitz functionsegnwithP1nD1m¹g ¤egnº < 1, then we obtain the same definition of Dx.g/ up to m-a.e. equivalence.
Using the measurability of the differential map we easily obtain the measurability of x7! Dx.g/. Clearly, for every f 2 W1;p we have Dx.f / 0. For every f; g 2 Lip.X/ and
2 R we know
d.f C g/ D d.f / C d.g/ and d.f / D d.f / m-a.e. in X.
This implies (3.4), since approximations for f and g give rise to approximations for f C g and f . In order to prove (3.5), we first remark that, by Proposition 2.10 (iii) and (3.5), we get
(3.6) kdgn.x/kx jrgnjw M kdgn.x/kx:
Sincejrgnjw ! jrgjw in L1.m/, it follows, up to a subsequence,jrgnjw.x/! jrgjw.x/
for m-a.e. x 2 X. Therefore, the conclusion follows by letting n ! 1 in (3.6) and recalling thatkdgn.x/kx is constant for large n.
Using (3.4) and (3.5) it is easy to see that the following expression defines an equivalent norm on W1;p: kf kCh;WD Z X .jf .x/jpC .Dx.f //p/.x/ dm.x/ 1=p :
Step 2. SupposeQW X ! Œ0; 1/ is an m-measurable function; then the seminorm
f 7! kf kCh;Q D
Z
X
.jf .x/jpC .Dx.f //p/Q.x/ dm.x/
1=p
is uniformly convex on the intersectionLip.X /\ W1;p, with modulus of convexity independent
ofQ.
Suppose .Y;F / is a measurable space and, for each y 2 Y , Rnis equipped with a semi-inner product normk kysuch that y 7! kvkyis measurable for any v 2 Rn. By polarization,
the map y7! hv; wiy is measurable for any v; w2 Rn, where h ; iy denotes the induced
semi-inner product. Representing the semi-inner product by a symmetric positive semidefinite matrix y 7! Ay in the canonical basis of Rn, it is clear that the entries of Ayare also
measur-able.
It is well known (see for instance [7]) that for any symmetric positive semidefinite matrix A there exists a unique symmetric matrixpA such thatpApAD A; in addition, the map A7!pA is continuous. As the composition of a continuous and a measurable map, the entries of pAy are measurable. Further, we can write hv; wiy D .pAyv/t.
p
Ayw/, which
Using the discussion above, for each x2 X, we choose an N N matrix Bx such that
kvkx D jBxvj for all v 2 RN. Let X1 and X2be two disjoint copies of X supporting copies
m1and m2of m respectively. We define the following function bf W X1[ X2 ! RN,
b f .x/WD
´
.Q.x/1=pf .x/; 0; : : : ; 0/; x 2 X1;
Q.x/1=pBxdf .x/; x 2 X2:
Clearly, bf is measurable. By using the equality Dx.f /D kdf .x/kx m-a.e. in X it is simple
to verify
kbfkLp.m
1Cm2/ D kf kCh;Q:
Since the transformation f 7! bf is linear, the usual uniform convexity of Lp spaces implies uniform convexity of the normk kCh;Qon Lip.X /\ W
1;p
(with modulus independent of Q).
Step 3. The normk kCh;is uniformly convex onW 1;p
.
It is an easy consequence of (3.4), (3.5) and locality for the weak gradient that if g; h2 W1;p and gD h on a measurable set E, then DxgD Dxh for m-a.e. x 2 E. We use
this locality property throughout the sequel.
Given "2 .0; 1/, let ı D ı."/ 2 .0; 1/ be given by the uniform convexity proved in the previous step. Suppose f; g2 W1;p satisfy kf kCh;D kgkCh; D 1 and kf gkCh; ".
Using Proposition 2.10, we can find an increasing family of bounded sets Ensuch that
m X n[ n En D 0
on which fjEnand gjEnare Lipschitz. Set n.x/WD .x/ En.x/, where Enis the
character-istic function of En. We first extend fjEnand gjEnto Lipschitz functions fnand gnon X with
bounded support. An easy argument using locality of the weak gradient and local integrability of shows that fn; gn2 Lip.X/ \ W1;p.
Next, let efnWD fn=znandegnWD gn=wnfor some scalars znand wnsuch that kefnkpCh;nD Z X .jefn.x/jpC .Dx.efn//p/n.x/ dm.x/D 1 and kegnkpCh;n D Z X .jegn.x/jpC .Dx.egn// p/ n.x/ dm.x/D 1: Since znp D Z X .jfn.x/jpC .Dx.fn//p/n.x/ dm.x/ D Z En .jf .x/jp C .Dx.f //p/.x/ dm.x/
andkf kCh; D 1, the monotone convergence theorem yields zn" 1; similarly we obtain that
wn" 1. In our choice of f and g we assumed that
kf gkpCh;D Z
X
By Fatou’s lemma, using locality of Dx to justify pointwise convergence of the integrand, it
follows that for n sufficiently large, kefn egnkCh;n D Z X .jefn.x/ egn.x/j p C .Dx.efn egn// p/ n.x/ dm.x/ > "p:
Hence, since efn;egn2 Lip.X/ \ W
1;p
, the uniform convexity of the norm k kCh;n on the
space Lip.X /\ W1;p, proved in Step 2, gives
e fnCegn 2 p Ch;n D Z X ˇ ˇ ˇ ˇ e fn.x/Cegn.x/ 2 ˇ ˇ ˇ ˇ p C Dx e fnCegn 2 p! n.x/ dm.x/ .1 ı/p:
By using the locality of D and the definitions of f and g, we obtain Z En ˇ ˇ ˇ ˇ f .x/=znC g.x/=wn 2 ˇ ˇ ˇ ˇ p C Dx f =znC g=wn 2 p! .x/ dm.x/ .1 ı/p: By letting n! 1 and using the dominated convergence theorem, we obtain the inequality k.f C g/=2kCh; .1 ı/. Hence the normk kCh;is uniformly convex on W
1;p
and so the
Banach space W1;pis reflexive.
As previously stated (see (5.3)) another natural definition of weighted Sobolev space is provided by the closure of Lipschitz functions with respect to the weighted norm.
Definition 3.5 (Weighted space H1;p.X; d; m/). Let p > 1 and let W X ! Œ0; 1 be
a Borel function satisfying 12 L1=.p 1/.m/. We define H1;p.X; d; m/ as the closure of
Lipschitz functions in W1;p, namely
H1;p.X; d; m/WD Lip.X/ \ W1;p kk
: As for the W space, we will adopt the shorter notation H1;p.
We can now use the general identification Theorem 2.6 and the reflexivity of W1;p
to prove our first main result, namely that H1;p coincides with the metric Sobolev space
W1;p.X; d; m/ under thePI1assumption on .X; d; m/ and no extra assumption on the weight,
besides local integrability of and global integrability of 1=.1 p/.
Theorem 3.6. Suppose.X; d; m/ is aPI1metric measure space. Letp > 1 and suppose
W X ! Œ0; 1 is a Borel function satisfying 2 L1loc.m/ and 12 L1=.p 1/.m/. Then W1;p.X; d; m/D H1;p:
Proof. Let f 2 W1;p.X; d; m/. By Theorem 2.6 and the inequality jrgjw jrgj
for g Lipschitz, we can approximate f in Lp.m/ by functions fn2 Lip.X/ \ Lp.m/ with
kfnk uniformly bounded and evenjrfnj ! jrf jp;m in Lp.m/. By reflexivity we have
that fnweakly converge to the function f in W1;p, and since H 1;p
by definition is a closed
subspace, it follows that f 2 H1;p as well. In addition, the weak lower semicontinuity of the
norm gives (3.7) Z X .jf jp C jrf jpw/ dm Z X .jf jpC jrf jp;mp / dm:
Conversely, let f 2 H1;p and let fn2 Lip.X/ \ W1;p be convergent to f in W1;p
norm. Using (2.9) of Proposition 2.10 we obtain that lim
n;m!1
Z
Xjr.f
n fm/jp dmD 0:
It follows thatR jrfnjpdm is uniformly bounded, therefore one obtains f 2 H1;p.X; d; m/
and therefore, by Theorem 2.6, f 2 W1;p.X; d; m/.
Remark 3.7 (W1;p.X; d; m/ and H1;pare isometric). The second part of the proof of
Theorem 3.6 can be improved if we use the finer information (see [19, Theorem 12.5.1], while [10] covered only the case p > 1) that
(3.8) jrf j D jrf jw m-a.e. in X
for all f 2 Lip.X/ \ W1;1.X; d; m/, under thePI1assumption (recall thatjrf jw stands for
jrf j1;m). Indeed, using (3.8) one can get
Z Xjrf j p p;m dm Z Xjrf j p w dm
which, combined with (3.7), gives that the spaces are isometric.
4. Identification of weighted Sobolev spaces
In this section we prove that for p > 1, under certain assumptions on the space .X; d; m/ and on the weight , the weighted Sobolev spaces W1;p and H1;pcoincide. Unless otherwise
stated, all integrals are with respect to m. Our second main result is the following.
Theorem 4.1 (Identification of weighted and metric Sobolev spaces). Let .X; d; m/ be a PI1 metric measure space. Let p > 1 and let W X ! Œ0; 1 be such that 2 L1loc.m/,
12 L1=.p 1/.m/ and L WD lim inf n!1 1 np Z X ndm 1=nZ X ndm 1=n <1: ThenW1;p.X; d; m/D W1;pand, in particular,H1;p D W1;p.
Suppose that the hypotheses of Theorem 4.1 hold. Since L<1, there exists N 2 N
such thatkkLN.m/k 1kLN.m/<1. The assumptions 2 L1loc.m/ and 12 L1=.p 1/.m/
imply that is not identically 0 or1. Hence
kkLN.m/<1 and k 1kLN.m/<1:
These imply m¹ 1º and m¹ 1º respectively; hence m.X/ < 1. Further, the integrabil-ity 2 LN.m/ (for some N ) and m.X / <1 imply 2 L1.m/. We use these facts freely in what follows.
We already know that we can identify H1;p with W1;p.X; d; m/, by Theorem 3.6.
Since H1;p is a closed subspace of the reflexive space W1;p (Theorem 3.4), there exists an
element u2 W1;pn H1;psuch that
ku C vk kuk for any v2 H1;p
(it suffices, given z2 W1;pn H1;p, to minimizekz hk as h runs in H1;p and then define
uD z h, where h is a minimizer). Now, suppose v is of the form vD tw for t 2 .0; 1/, with w2 H1;p and wD u on a Borel set E X. Then we obtain,
Z E .jujpC jrujpw/ dmC Z X nE .jujpC jrujpw/ dm .1 t /p Z E .jujpC jrujpw/ dmC Z X nE .ju t wjpC jr.u t w/jpw/ dm and hence .1 .1 t /p/ t Z E .jujpC jrujpw/ dm Z X nE
.juj C tjwj/p jujpC .jrujwC tjrwjw/p jrujpw
t dm:
By letting t ! 0 and using the dominated convergence theorem we obtain (4.1) Z E .jujpC jrujpw/ dm Z X nE .jujp 1jwj C jrujp 1w jrwjw/ dm:
To apply (4.1) we need to use u2 W1;pto construct an appropriate test function in H1;p.
To do this, as in the proof of Proposition 2.10, we use a maximal operator estimate to obtain Lipschitz bounds on the restriction of u to a smaller subset and then extend this restriction to a Lipschitz map on X .
Now let u2 W1;pn H1;p such thatku C vk kukfor any v 2 H1;p and set
gWD max¹juj; M.jrujw/º;
where M is the maximal operator with respect to m, defined in (2.6). We also define the set FWD ¹x 2 X W x is a Lebesgue point of u and g.x/ º:
Arguing as in the proof of Proposition 2.10 we obtain a Lipschitz function u withjuj ,
Lip.u/ C , and equal to u m-a.e. in F. Since m.X / <1 and 2 L1.m/, we deduce
u 2 H1;p.
Now we apply (4.1) with wD uand ED Fto obtain
(4.2) Z F .jujpC jrujpw/ dm C Z X nF .jujp 1C jrujp 1w / dm:
Next, we prove the following estimate:
Proposition 4.2. Letf W .0; 1/ ! .0; 1/ be a continuously differentiable and strictly decreasing function such thatf ./! 0 as ! 1 and Rt
0f0./ d <1 for all t > 0.
Then, under assumption(4.2), there exists a constant C > 0 such that Z X f .g/.jujpC jrujpw/ dm C Z X ˚.g/.jujp 1C jrujp 1w / dm with˚.t /D Rt 0f0./ d.
Proof. First of all we notice that the statement makes sense, since g > 0 m-a.e. on the set wherejujpC jrujpwis positive (therefore, understanding the integrand as being null on
¹g D 0º). As in [27, Lemma 1], we can apply Cavalieri’s formulaR d D R01.¹ > tº/ dt
with Borel nonnegative and a finite Borel measure. For k2 L1.m/, choosing D km and D f .g/, D ˚.g/ and using the change of variable D f 1.t / yields
Z X f .g/k D Z 1 0 f0./ Z F k d; (4.3) Z X ˚.g/k D Z 1 0 ˚0./ Z X nF k d (4.4)
for all k2 L1.m/ nonnegative, and eventually for any nonnegative Borel k. Now observe that multiplying inequality (4.2) by f0./ (recall that f0./ 0 in .0; 1/) and integrating from 0 to1 we get Z 1 0 f0./ Z F .jujp C jrujpw/ d (4.5) C Z 1 0 f0./ Z X nF .jujp 1C jrujp 1w / d D C Z 1 0 ˚0./ Z X nF .jujp 1C jrujp 1w / d: By applying (4.3) to (4.5) with kD .jujp C jrujpw/ we get
Z X f .g/.jujpC jrujpw/ C Z 1 0 ˚0./ Z X nF .jujp 1C jrujp 1w / d: (4.6)
Further, we choose kD .jujp 1C jrujp 1w / and apply (4.4)–(4.6) to obtain the thesis.
We modify the estimate from Proposition 4.2 by using Hölder’s inequality for the mea-sure m; for any nonnegative Borel function G on X we have
Z X ˚.g/Gp 1 Z X Gp 1=p0Z X ˚.g/p 1=p ;
where 1=pC 1=p0D 1. By applying Proposition 4.2 then using the above inequality with G D juj and G D jrujw we deduce
Z X f .g/.jujpC jrujpw/ (4.7) C Z Xjuj p 1=p0 C Z Xjruj p w 1=p0!Z X ˚.g/p 1=p C Z X .jujp C jrujpw/ 1=p0Z X ˚.g/p 1=p D C kukp=p 0 Z X ˚.g/p 1=p : Now we fix p2 .1; p/ and choose " > 0 such that
" < min ² 1 2.p 1/; 1 p 1 1 p p ³ :
Therefore, .p 1/" < 1=2 and p"WD p.1 C .1 p/"/ > p. Our next goal is to prove the
inequality (4.8)
Z
X
g.1 p/".jujp C jrujpw/ C "kukp=p 0 Z
X
gp"
1=p : In order to prove (4.8), let f ./WD .1 p/", so that
˚./WD .p 1/" 1 .p 1/"
1 .p 1/":
Now we observe that
k˚ ı gkLpp.m/D .p 1/p"p .1 .p 1/"/p Z X gp.1 .p 1/"/ 2p.p 1/p"p Z X gp.1 .p 1/"/;
by our choice of ". Hence, by applying (4.7) with our choice of f and ˚ we obtain (4.8). Now, recalling that gD max¹juj; M.jrujw/º and using the triangle inequality, we
esti-mate
(4.9) kgkp"=p
Lp".m/ .kukLp".m/C kM.jrujw/kLp".m//p"=p:
We will use Hölder’s inequality and boundedness of the maximal operator to bound the right hand side in terms ofkukand . Notice that the constants C appearing in the estimates below
are independent of ", since we are going to apply the maximal estimates with exponent p"r and
p"r > p, by our choice of ". We handlekukLp".m/andkM.jrujw/kLp".m/separately but
with a similar argument; we apply Hölder’s inequality twice with the following exponents: r D r"WD
2C .1 p/"
2C 2.1 p/"; sD s"WD
2 2C .1 p/":
It is easy to see that r; s > 1 and that the conjugate Hölder exponents r0; s0are respectively given by
r0D 2C .1 p/" .p 1/" ; s
0D 2
.p 1/":
Furthermore, these exponents satisfy the equations p"rsD p and s0D r0s. Now, let us derive
the inequalities for M.jrujw/; the case ofjuj is similar but easier, since we do not need to use
boundedness of the maximal operator: Z X M.jrujw/p" Z X M.jrujw/p"r 1=rZ X r0 1=r0 (4.10) C Z Xjruj p"r w 1=s 1=s 1=rZ X r0 1=r0 C Z Xjruj p"rs w 1=.rs/Z X r0 1=r0Z X s0=s 1=.rs0/ C kukp=.rs/ Z X r0 1=r0Z X r0 1=.rs0/ :
Similarly, we obtain Z jujp" C kukp=.rs/ Z X r0 1=r0Z X r0 1=rs0 : Hence we have .kukLp".m/C kM.jrujw/kLp".m//p"=p (4.11) C kuk1=.rs/ Z X r0 1=.pr0/Z X r0 1=.prs0/ : By combining our estimates (4.8), (4.9) and (4.11) we obtain
Z
g.1 p/".jujpC jrujpw/ C "kuk
p p0C 1 rs Z X r0 1=pr0Z X r0 1=prs0 : (4.12)
As "! 0, Fatou’s lemma gives Z
X
.jujp C jrujpw/ lim inf
"#0
Z
X
g.1 p/".jujpC jrujpw/: (4.13)
Therefore in order to estimateRX.jujp C jrujpw/ from above we can estimate the right hand
side of (4.12) as "# 0 (notice that r"; s" # 1, while p" " p) in the following lemma.
Lemma 4.3. The following inequality holds:
(4.14) lim inf "#0 " pZ X r0 1=r0Z X r0 1=.rs0/ 2 p .p 1/pL;
whereLis defined as in Theorem4.1.
Proof. Setting "D 2=Œ.n C 1/.p 1/ gives n D r0and r s0D n.n C 1/=.n 1/. Hence the left hand side of (4.14) is bounded from above by
lim inf n!1 2p .p 1/p.nC 1/p Z X n n1Z X n n.nnC1/1 :
Since m.X / <1 and 12 L1=.p 1/.m/, it follows that k 1kLn.m/! k 1kL1.m/ as
n! 1. Hence if 0 < k 1kL1.m/<1, then lim inf n!1 2p .p 1/p.nC 1/pkkLn.m/k 1 k n 1 nC1 Ln.m/ D lim infn!1 2 p .p 1/p.nC 1/pkkLn.m/k 1 kLn.m/ D 2 p .p 1/pL
and the thesis follows. Ifk 1kL1.m/D 1, then k 1kLn.m/> 1 for sufficiently large n and
we can use the trivial inequality k 1k n 1 nC1 Ln.m/<k 1 kLn.m/
for every n to obtain the same bound.
Using (4.12)–(4.14) and the fact that p=p0D p 1 we get kukp CL 1=p kuk p
for some structural constant C . Since u¤ 0, we obtain
(4.15) 1 CL:
As in [27], the strategy is now to use the fact that C is independent of to derive a contradiction. Following the notation of Theorem 4.1 we write
L D lim inf n!1 1 npkkLn.m/k 1 kLn.m/
for any Borel W X ! Œ0; 1.
Proposition 4.4. For allı > 0 there exists a weighteW X ! Œ0; 1/ satisfying the hypo-theses of Theorem4.1 such that L
ı, W 1;p D W1;p andH 1;p D H1;p .
Proof. Throughout this proof we simply write bounded or unbounded instead of essenti-ally bounded (i.e. in L1.m/) or essentially unbounded (not in L1.m/). We start by observing that if both and 1are bounded, then L D 0, so there is nothing to prove.
If and 1are both unbounded, then for every t > 0, we define t.x/WD
´1
t.x/ if .x/ 1;
t.x/ if .x/ > 1:
Clearly, t satisfies the assumptions of Theorem 4.1 since it is bounded by constant multiples
of . For the same reason, the W and H weighted Sobolev spaces induced by and t are the
same. Observe that (4.16) Z X nt t nm.X /C tn Z X n; Z X tn t nm.X /C tn Z X n: Since is unbounded, we get
lim n!1 1 m.X / nZ X nD 1 for all > 1: (4.17)
Using (4.17) with D t 2we get t nm.X / tn
Z
X
n for n sufficiently large. Therefore the first inequality in (4.16) gives
ktkLn.m/ 2 1
ntkkLn.m/ for n sufficiently large.
(4.18)
Arguing in the same way using the unboundedness of 1 and the second inequality in (4.16) we obtain
kt 1kLn.m/ 2 1
ntk 1kLn.m/ for n sufficiently large.
(4.19)
Let us now assume that is bounded but 1is unbounded. For every t > 0 we define t.x/WD
´1
t.x/ if 0 .x/ t;
.x/ if .x/ > t: As before, t satisfies the hypotheses of Theorem 4.1. We observe,
Z X tnD tn Z ¹tº nC Z ¹>tº n tn Z X nC t nm.X /: Since 1is unbounded proceeding as in (4.18), we obtain
kt1kLn.m/ 2 1 ntk 1kLn.m/; (4.20) while ktkLn.m/ kkL1.m/C 1 (4.21)
for every n. Because is bounded, we have LD lim inf
n!1
kkL1.m/k 1kLn.m/
np :
(4.22)
Putting together (4.20), (4.21) and (4.22) we get lim inf n!1 1 npktkLn.m/k 1 t kLn.m/ 2tL.1C kk 1 L1.m//
and we conclude, again, choosing t > 0 sufficiently small. The case where 1is bounded and is unbounded is analogous.
We can now conclude the proof of Theorem 4.1. Choose ı2 .0; C 1/, where C > 0 is the constant in (4.15) and apply Proposition 4.4 to find a weighte satisfying the hypotheses of Theorem 4.1 such that L
ı, W 1;p D W1;p and H 1;p D H1;p
. Then the assumption that
W1;pn H1;p ¤ ¿ implies W1;p n H 1;p ¤ ¿
and we may repeat all of our arguments to obtain an analogue of (4.15) withe in place of ; hence,
1 CL C ı < 1
which gives a contradiction.
5. Examples and extensions
In this section we discuss some examples and generalize our results by considering Muckenhoupt weights or by requiring a weaker Poincaré inequality.
5.1. An example where W1;p © H1;p. Let us consider the standard Euclidean struc-ture, X is the closed unit ball of R2, d is the Euclidean distance, mD L2. In [12] examples of weights 2 L1.m/ with 1 2 L1=.p 1/.m/ and H1;p ¨W1;p are given for any p > 1.
Here we report only the example with pD 2, with a weight in all Lq spaces having also the inverse in all Lq spaces, 1 q < 1.
Let D B.0; 1/ R2and "2 .0; =2/. Set S"WD ² .x1; x2/2 W tan."/ < x2 x1 < tan 2 " ³ ;
S"CWD S"\ ¹x2 > 0º and S" WD S"\ ¹x2< 0º. Let us consider W R2n ¹0º ! Œ0; 1/
de-fined by .x/WD 8 < : ln 2 jxje k arccos x1 jx1j if 0 <jxj 1; 1 ifjxj > 1;
where kW R ! Œ 1; 0 is a -periodic smooth function such that k0.0/D 0 and k 1 in "; 2 " ; k 0 in 2; : It follows that 2 C0.R2n ¹0º/ and that
; 12 \
q2Œ1;1/
Lqloc.R2/:
It is proved by a direct calculation in [12] that the function
u.x/WD 8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ : 1 if x1> 0, x2 > 0; 0 if x1< 0, x2 < 0; x2 jxj if x1< 0, x2 > 0; x1 jxj if x1> 0, x2 < 0; belongs to W1;2n H1;2.
5.2. Muckenhoupt weights and thePIp condition. An important class of weights is
the one introduced by Muckenhoupt [23] to study the boundedness of the maximal operator in Lp spaces. In Euclidean spaces, for this p-dependent class of weights it is known that W1;p D H1;p, see for instance [11, 18] and Theorem 5.8 below. Let us recall the definition of
Muckenhoupt weight in the context of metric measure spaces.
Definition 5.1 (Muckenhoupt weight). Let .X; d; m/ be a metric measure space and let W X ! Œ0; 1 be locally integrable. For p > 1, we say that is an Ap-weight if
sup B « B dm « B 1=.p 1/dm p 1 <1;
where the supremum runs among all balls B. We say that is an A1-weight if there exists
a constant C such that «
B
dm C ess inf
B u
It is immediate to see, using the Hölder inequality, that .ª
B dm/.
ª
B
1=.p 1/dm/p 1
is always larger than 1. This easily yields that .X; d; m/ is doubling whenever .X; d; m/ is doubling; indeed, for p > 1,
« 2B dm C « 2B 1=.p 1/dm 1 p C « B 1=.p 1/dm 1 p C « B dm
and a similar argument works for pD 1. It follows that the maximal operator with respect to m is bounded in Lp.m/ for all p > 1. A remarkable fact, proved in the Euclidean case by Muckenhoupt [23], with a proof that extends readily also to doubling metric measure spaces .X; d; m/ (see [25, Theorem 9]), is the fact that even the maximal operator M in (2.6), namely the maximal operator respect to m, is bounded in Lp.m/ for all p > 1, and weakly bounded if pD 1.
For p > 1 it is well known that an Apweight on a Euclidean space is p-admissible [18];
this means that the weighted space .Rn;j j; Ln/ satisfiesPIp. A converse holds in dimension
one but it is an open problem for higher dimensions [9]. We generalize the result to metric measure spaces.
Proposition 5.2 (Invariance of PIp). If.X; d; m/ is a PIp space for some p > 1 and
2 Ap.m/, then .X; d; m/ is also aPIp space.
Proof. Letme D m. For u locally Lipschitz, we start from (5.1) ju.x/ u.y/j C d.x; y/ M2ƒr1=p.jruj
p/.x/
CM2ƒr1=p.jruj p/.y/;
d.x; y/ < r; obtained from (2.11) by replacing weak gradients with slope. Recall that Msdenotes the
max-imal operator with respect to m on scale s; using multiplication by a cut-off function it is easy to check that boundedness of M in Lp.m/ yields the localized LQ p boundedness
Z B.z;s/ .Msg/pdmQ C Z B.z;2s/ gpdm:Q
We multiply both sides of (5.1) by .x/.y/, integrate with respect to m m, divide by .m.B.x; r///e 2and eventually use the boundedness of M in Lp.m/ for p > 1 to gete
« B.x;r/ ˇ ˇ ˇ ˇ u.x/ « B.x;r/ u dme ˇ ˇ ˇ ˇ dm.x/e C r « B.x;2r/jruj p dme 1=p ; where now the integrals are averaged with respect tom. This proves the result.e
Let us now compare the Muckenhoupt condition with the Zhikov one, introduced in [27] and used also in the present paper.
Definition 5.3. Let W X ! Œ0; 1 be Borel and let p > 1. We say that belongs to the class Zp.m/ if lim inf n!1 1 npkkLn.m/k 1 kLn.m/<1:
Even though both the Muckenhoupt and Zhikov conditions lead to the identification of the weighted Sobolev spaces, the following simple examples show that they are not compa-rable, even in the Euclidean case. One of the reasons is that the class Ap involves a more
local condition; for instance there is no reason for .X; d; m/ to be doubling when .X; d; m/ is doubling and 2 Zp.m/. As a matter of fact, the Zhikov condition is easier to check. For
instance, if both exp.t/ and exp.t 1/ belong to L1.m/ for some t > 0, then 2 Z2.m/ (see
the simple proof in [27], still valid in the metric measure setting).
Example 5.4. Let X be the unit ball of R2and let W X ! R be given by
.x/WD 8 < : log jxj1 if x1x2> 0; 1 log.jxj1/ if x1x2< 0:
It is easy to check that exp./; exp. 1/2 L1.L2/ (see also [27]), which implies 2 Z2.L2/.
Further, one can easily prove that … Ap.L2/ for any p 1; indeed, the average of both
and 1on balls centered at the origin tends to infinity as the radius of the ball tends to zero. Example 5.5. Let XD .0; 1/. Then, by a direct computation, jxj˛ 2 Ap.L1/ provided
1 < ˛ < p 1. Hence 1=px2 Ap.L1/ for any p 1; however, 1=px… Zp.L1/ for any
p 1 since 1=px… Ln.L1/ for any n 2.
5.3. Relaxation of thePI1assumption toPIp. Suppose 12 L˛.m/ for some
expo-nent ˛2 .1=.p 1/;1 and set q D p˛=.˛ C 1/ (q D p if ˛ D 1). Then the definition (5.2) W;q1;p WD ² f 2 W1;q.X; d; m/W Z Xjf j p dm C Z Xjrf j p q;m dm <1 ³ ; with the corresponding norm, and the corresponding definition of H;p1;p(namely the closure of
the space Lip.X /\ W;q1;p inside W;q1;p) are much more natural. Indeed, there is already a
nat-ural embedding in the space W1;q.X; d; m/ that is missing in the general case, so there is no necessity to invoke the space W1;1.X; d; m/ and thePI1 structure (notice that qD 1
corre-sponds precisely to ˛D 1=.p 1/). The embedding provides completeness of W;q1;p, via the
completeness of W1;q.X; d; m/, and also the proof of reflexivity can be immediately adapted to the space W;q1;p.
Assume now that 12 L˛.m/ for any ˛2 .1=.p 1/; 1/, as it happens when L<1;
in this case we can choose the power q in (5.2) as close to p as we wish, and use the fact that
PIp is an open ended condition to choose q in such a way thatPIq still holds. This leads to
the following result (which also shows that the space W;q1;p is essentially independent of the
exponent q).
Theorem 5.6. Let.X; d; m/ be a PIp metric measure space, and let 2 L1loc.m/ be
a nonnegative Borel function satisfying 12 L˛.m/ for all ˛2 .1=.p 1/;1/. Then the spaceW;q1;p in(5.2) and its norm do not depend on the choice of q 2 Œ1; p/, as soon asPIq
holds, and
(5.3) H;q1;p D W1;p.X; d; m/ wheneverPIqholds.
Proof. Recall that (see [19, Theorem 12.5.1], while [10] covered only the case q > 1) jrf j D jrf jq;m m-a.e. in X , for all f 2 Lip.X/ \ W1;q.X; d; m/
for all q2 Œ1; 1/, under thePIqcondition. This, combined with the locality of weak gradients
and the Lusin approximation with Lipschitz functions gives jrf jq;mD jrf jq0;m m-a.e. in X
whenever f 2 W1;q.X; d; m/\ W1;q0.X; d; m/ and bothPIq andPIq0 hold. Then, the
inde-pendence of W;q1;pwith respect to q follows by Proposition 2.11. The identity (5.3) and the last
statement can be obtained repeating respectively the proofs of Theorem 3.6 and Theorem 4.1 with this new class of weighted spaces.
5.4. Combination of Zhikov and Muckenhoupt weights. Zhikov [27] proves identi-fication of weighted Sobolev spaces for weights expressable as a product D MZ, where
M 2 Ap.Ln/ and Z 2 Zp.MLn/. The minor adaptations needed to include Muckenhoupt
weights work also in the metric setting.
Theorem 5.7. Suppose .X; d; m/ is a PI1 metric measure space. Let p > 1 and let
D MZ, whereM 2 Ap.m/ and Z2 Zp.Mm/. If 2 L1loc.m/ and 12 L1=.p 1/.m/,
thenW1;p.m/D H 1;p
.m/.
Proof. As remarked in the discussion before Theorem 5.8 we know that the maximal operator with respect to m is bounded in Lp.Mm/ if p > 1. To obtain the identification
W1;p.m/D H1;p.m/ we apply exactly the same argument as in the proof of Theorem 4.1
apart from the fact that in (4.10) we apply Hölder’s inequality and boundedness of the maximal operator with respect to the measure Mm; hence (4.10) changes to
Z X M.jrujw/p"ZM Z X M.jrujw/p"rM 1=rZ X rZ0M 1=r0 C Z Xjruj p"r w 1=s Z 1=s Z M 1=rZ X Zr0M 1=r0 C Z Xjruj p"rs w ZM 1=.rs/Z X rZ0M 1=r0 Z X Zs0=sM 1=.rs0/ C kukp=.rs/ Z X rZ0M 1=r0Z X Zr0M 1=.rs0/ :
With this estimate we are able to use the assumption Z 2 Zp.Mm/ to obtain again the
iden-tification W1;p.m/D H1;p.m/.
The following result, which is well known in the Euclidean setting, easily follows from Theorem 5.7 by taking Z D 1.
Corollary 5.8. Suppose.X; d; m/ is aPI1space. Let2 Ap.m/ with 2 L1loc.m/ and
A more direct proof of Corollary 5.8 can be obtained using the same approach described in Proposition 2.10. More precisely,
« Bjf j dm 1 m.B/ Z Bjf j p dm p1Z B p11dm pp1 « Bjf j p dm p1« B dm « B p11dm p 1p1
and since 2 Ap.m/, we get
Mmf .x/ c Mmfp.x/
p1 ; (5.4)
where c > 0 and depends only on . Given f 2 W1;p W1;1.X; d; m/, by (2.10) and (5.4)
we have jf .x/ f .y/j C d.x; y/ Mmjrf jp1;m.x/ 1 p C Mmjrf jp1;m.y/ 1 p
and, proceeding exactly as in Proposition 2.10, we obtain the inclusion W1;p H1;p and
therefore W1;p D H1;p.
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Luigi Ambrosio, Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa 56126, Italy e-mail: luigi.ambrosio@sns.it
Andrea Pinamonti, Dipartimento di Matematica, Universita di Bologna, Piazza di Porta San Donato 5, Bologna 40126, Italy
e-mail: andrea.pinamonti@gmail.com
Gareth Speight, Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221, USA
e-mail: gareth.speight@uc.edu
