Self-improvement of uniform fatness revisited
aa r X i v : . [ m a t h . C A ] D ec SELF-IMPROVEMENT OF UNIFORM FATNESS REVISITED
JUHA LEHRB ¨ACK, HELI TUOMINEN, AND ANTTI V. V ¨AH ¨AKANGAS
Abstract.
We give a new proof for the self-improvement of uniform p -fatness in the settingof general metric spaces. Our proof is based on rather standard methods of geometricanalysis, and in particular the proof avoids the use of deep results from potential theoryand analysis on metric spaces that have been indispensable in the previous proofs of theself-improvement. A key ingredient in the proof is a self-improvement property for localHardy inequalities. Introduction
Self-improvement is among the most profound and beautiful phenomena in mathematicalanalysis, and a source of important tools in the proofs of several deep and perhaps evensurprising results. Important examples of concepts enjoying self-improvement include re-verse H¨older inequalities, Muckenhoupt’s A p classes of weights, Poincar´e inequalities, andthe main topics of this paper: Hardy inequalities and uniform p -fatness related to the vari-ational p -capacity.That a uniformly p -fat set E , for 1 < p < ∞ , is actually uniformly q -fat for some1 ≤ q < p as well, was first proven by Lewis [14] in the Euclidean case E ⊂ R n . In fact,Lewis studied more general ( α, p )-fatness conditions related to Riesz capacities, but when α = 1 his setting is equivalent to that of the variational p -capacity. Another proof for the self-improvement of uniform p -fatness in (weighted) R n was given by Mikkonen [18], and in [2]Bj¨orn, MacManus and Shanmugalingam generalized the self-improvement to more generalmetric spaces, essentially proving the following theorem (although in [2] the assumptions onthe space X were slightly stronger). Theorem 1.1.
Let < p < ∞ and let X be a complete metric measure space equipped witha doubling measure µ and supporting a (1 , p ) -Poincar´e inequality. Assume that E ⊂ X is auniformly p -fat closed set. Then there exists < q < p such that E is also uniformly q -fat(quantitatively). The proofs of the versions of Theorem 1.1 in [2, 14, 18] utilize deep results from linearand non-linear potential theory, and moreover the proof in [2] is based on the impressivetheory of differential structures on metric spaces, established by Cheeger in [4].In this paper, we use a different approach and establish a new proof for Theorem 1.1with the help of local Hardy inequalities and their self-improvement properties. Our proofis completely new also in R n , where all previously known proofs have been based on theideas either in [14] or in [18]. In addition, it turns out that with our approach it is possibleto obtain the following generalization of Theorem 1.1 to a non-complete space X , whereCheeger’s theory is not available. Theorem 1.2.
Let < p < p < ∞ and let X be a metric measure space equipped witha doubling measure µ and supporting a ( p, p ) -Poincar´e inequality. Assume that E ⊂ X isa uniformly p -fat closed set and that E ∩ B ( w, r ) is compact for all w ∈ E and all r > .Then there exists p < q < p such that E is also uniformly q -fat (quantitatively). Mathematics Subject Classification.
Primary 31C15, Secondary 31E05, 35A23.
Key words and phrases.
Self-improvement, uniform fatness, local Hardy inequality, metric space.
We will also formulate a slightly stronger version of Theorem 1.2 later in Theorem 4.1.Recall that if X is as in Theorem 1.1 (i.e., complete, equipped with a doubling measureand supporting a (1 , p )-Poincar´e inequality), then a ( p, p )-Poincar´e inequality as in The-orem 1.2 follows from the well-known self-improvement properties of Poincar´e inequalities;see Section 2.3 for more discussion.It should perhaps be noted here that we define the variational p -capacity using Lipschitztest functions (the precise definition is given in Section 2.4). If X is complete, this definitionagrees with the definition using Newtonian (or Sobolev) test functions, but in a non-completespace the resulting capacities can be different.Let us turn to an outline of the ideas behind the proofs of Theorems 1.1 and 1.2. In [11](see also [10, Theorem 3.3]) it was shown (essentially) that if X is as in Theorem 1.1, thena closed set E ⊂ X is uniformly p -fat if and only if there is C > boundary p -Poincar´e inequality Z B ( w,r ) | u | p dµ ≤ Cr p Z B ( w,τr ) g p dµ (1)holds for all w ∈ E and all r >
0, whenever u is a Lipschitz function in X such that u = 0 in E and g is a ( p -weak) upper gradient of u (in R n one can always take g = |∇ u | ).Hence to obtain the self-improvement of uniform p -fatness, it would suffice to prove theself-improvement directly to inequality (1); this was actually mentioned in [11, p. 718] as apossible and interesting approach to self-improvement.We will not give a direct proof for the self-improvement of (1), but we show in Theorem 3.1that if X is as in Theorem 1.2 (in particular not necessarily complete) and E ⊂ X isuniformly p -fat, then there exist ε > C > p − ε )-versionof (1) holds for all w ∈ E and all r >
0, whenever u is a Lipschitz function in X such that u = 0 in E : Z B ( w,r ) | u | p − ε dµ ≤ Cr p − ε Z B ( w,τr ) Lip( u, · ) p − ε dµ . (2)Here Lip( u, x ) is the upper pointwise Lipschitz constant of u at x ∈ X . We remark that in R n inequality (2) can be obtained directly with |∇ u | instead of Lip( u, · ) on the right-handside. More generally, if the space X is complete, then Lip( u, · ) is actually known to bea minimal weak upper gradient of u by the results of Cheeger [4]. Hence we can connectfrom inequality (2) back to uniform fatness, and now indeed to the better ( p − ε )-uniformfatness, thus proving Theorem 1.1. In [11] this connection was established with the helpof the so-called pointwise Hardy inequalities, but, for the sake of completeness, we showin Section 3 how Theorem 1.1 follows directly from the validity of (2). In particular, thisway we avoid the use of pointwise Hardy inequalities in our proofs of Theorems 1.1 and 1.2,although it should be noted that our general approach has been partially suggested andmotivated by these pointwise inequalities.In a non-complete space X the validity of (2) does not immediately yield the uniform( p − ε )-fatness of E . Nevertheless, using as an additional tool the connection betweenuniform fatness and density conditions for suitable Hausdorff contents, we show in Section 4how the improved boundary Poincar´e inequality (2) can be used to prove also Theorem 1.2.It is the proof of (2) (assuming uniform p -fatness) that constitutes the main challenge inour proofs of Theorems 1.1 and 1.2. In fact, we will establish (2) via a self-improvementproperty of suitable local Hardy inequalities. Recall that one of the consequences of theself-improvement of uniform fatness, noted in each of [2, 14, 18], is the validity of a p -Hardy inequality in the complement of a uniformly p -fat set E ⊂ X . However, usinga method originating from Wannebo [19] (see also [11, Section 5]), it is also possible toprove such a p -Hardy inequality without using the self-improvement of uniform fatness. Weuse an adaptation of this latter method together with a novel ‘local absorbtion argument’ ELF-IMPROVEMENT OF UNIFORM FATNESS 3 (Lemma 5.3), and prove in the end of Section 5.2 the following local p -Hardy inequalitywhen E ⊂ X is uniformly p -fat. Theorem 1.3.
Let < p < ∞ and let X be a metric measure space equipped with a doublingmeasure µ and supporting a (1 , p ) -Poincar´e inequality. Assume that E ⊂ X is a uniformly p -fat closed set. Then there exists a constant C > such that the local p -Hardy inequality Z B \ E (cid:18) | u | d E (cid:19) p dµ ≤ C Z τ B g p dµ (3) holds whenever u is a Lipschitz function in X such that u = 0 in E , g is a p -weak uppergradient of u , and B = B ( w, r ) is a ball with w ∈ E and < r < (1 /
32) diam( X ) . Above we have abbreviated d E ( x ) = dist( x, E ). Notice in particular that we do not needto assume in Theorem 1.3 that the space X is complete.The next step towards (2) is a self-improvement property for local p -Hardy inequalities (3).Here we need the assumption that X supports a ( p, p )-Poincar´e inequality for some 1
p − ε , but now with the p -weak upper gradient g onthe right-hand side of (3) replaced with the upper pointwise Lipschitz constant Lip( u, · ); seeProposition 5.7. The proof of this self-improvement for local Hardy inequalities is based onideas used by Koskela and Zhong [13] in connection with the self-improvement of usual p -Hardy inequalities; the ideas in [13] were, in turn, inspired by the work of Lewis [15]. Againthe absorbtion Lemma 5.3 is needed to obtain the local inequalities. The ( p − ε )-version ofthe local Hardy inequality now easily yields the ( p − ε )-version of the boundary Poincar´einequality (2), see Section 3, concluding the proofs of Theorems 1.1 and 1.2.Admittedly, the proofs that were outlined above are somewhat lengthy and in many placesstill quite technical and delicate in the level of details, but one could argue that our generalapproach is nevertheless based on rather ‘elementary’ (or ‘standard’) tools. In particular,we do not need any sophisticated prerequisites concerning potential theory and we canalso avoid completely the use of Cheeger’s deep theory—or, if this theory is used to givea more direct proof to Theorem 1.1, the use is very explicit and localized; cf. the proof ofTheorem 1.1 at the end of Section 3. In this sense we believe that our proof of Theorem 1.1is more transparent (also in R n ) than its predecessors and thus hopefully easier to adaptto further problems, for instance in connection with weighted capacities or capacities offractional order smoothness. 2. Preliminaries
Metric spaces.
We assume throughout the paper that X = ( X, d, µ ) is a metricmeasure space equipped with a metric d and a positive complete Borel measure µ such that0 < µ ( B ( x, r )) < ∞ for all balls B = B ( x, r ) = { y ∈ X : d ( y, x ) < r } . As in [1, p. 2], weextend µ as a Borel regular (outer) measure on X . In particular, the space X is separable.Let us emphasize that we do not, in general, require X to be complete. If completeness isneeded somewhere in the paper, we will mention this explicitly.We also assume that the measure µ is doubling , meaning that there is a constant C D ≥ doubling constant of µ , such that µ (2 B ) ≤ C D µ ( B ) (4)for all balls B = B ( x, r ) of X . Here we use for 0 < t < ∞ the notation tB = B ( x, tr ).When A ⊂ X , we let A denote the closure of A , and hence B always refers to the closure ofthe ball B , not to the corresponding closed ball.Let A ⊂ X . A function u : A → R is said to be ( L -)Lipschitz , for 0 ≤ L < ∞ , if | u ( x ) − u ( y ) | ≤ Ld ( x, y ) for all x, y ∈ A .
J. LEHRB ¨ACK, H. TUOMINEN, AND A.V. V ¨AH ¨AKANGAS If u : A → R is an L -Lipschitz function, then the classical McShane extension˜ u ( x ) = inf y ∈ A { u ( y ) + Ld ( x, y ) } , x ∈ X , (5)defines an L -Lipschitz function ˜ u : X → R which satisfies ˜ u | A = u . The set of all Lipschitzfunctions u : A → R is denoted by Lip( A ), andLip ( A ) = { u ∈ Lip( X ) : u = 0 in X \ A } . (Weak) upper gradients. By a curve we mean a nonconstant, rectifiable, continuousmapping from a compact interval to X . We say that a Borel function g ≥ X is an uppergradient of an extended real-valued function u on X , if for all curves γ joining arbitrarypoints x and y in X we have | u ( x ) − u ( y ) | ≤ Z γ g ds , (6)whenever both u ( x ) and u ( y ) are finite, and R γ g ds = ∞ otherwise. In addition, when1 ≤ p < ∞ , a measurable function g ≥ X is a p -weak upper gradient of an extendedreal-valued function u on X if inequality (6) holds for p -almost every curve γ joining arbitrarypoints x and y in X ; that is, there exists a non-negative Borel function ρ ∈ L p ( X ) such that R γ ρ ds = ∞ whenever (6) does not hold for the curve γ . We refer to [1] for more informationon p -weak upper gradients.When u is a (locally) Lipschitz function on X , the upper pointwise Lipschitz constant of u at x ∈ X is defined as Lip( u, x ) = lim sup r → sup y ∈ B ( x,r ) | u ( y ) − u ( x ) | r . (7)The Borel function Lip( u, · ) is an upper gradient of u ; cf. [1, Proposition 1.14]. Moreover, if X is complete and 1 < p < ∞ , then Lip( u, · ) is actually a so-called minimal p -weak uppergradient of u (in particular, this implies that Lip( u, · ) ≤ g a.e. whenever g ∈ L p ( X ) is a p -weak upper gradient of u ). This is a deep result of Cheeger, we refer to [4, Theorem 6.1]and [1, p. 342].2.3. Poincar´e inequalities.
We say that the space X supports a ( q, p ) -Poincar´e inequality ,for 1 ≤ q, p < ∞ , if there exist constants C > λ ≥ B ⊂ X ,all measurable functions u on X , and for all p -weak upper gradients g of u , (cid:18)Z B | u − u B | q dµ (cid:19) p/q ≤ C diam( B ) p Z λB g p dµ . (8)Here u B = Z B u dµ = 1 µ ( B ) Z B u dµ is the integral average of u over the ball B , and the left-hand side of (8) is interpreted as ∞ whenever u B is not defined. We remark that X supports a ( q, p )-Poincar´e inequality withconstants C > λ ≥ B ⊂ X , allfunctions u ∈ L ( X ), and all upper gradients g of u ; see [4, Proposition 4.13].If 1 < p < ∞ and X supports a (1 , p )-Poincar´e inequality (and the measure µ is doubling,as we assume throughout the paper), then X supports also a ( p, p )-Poincar´e inequality; see[1, Corollary 4.24]. If in addition X is complete, then there is an exponent 1 < p < p such that X supports a ( p, p )-Poincar´e inequality and, consequently, also ( q, q )-Poincar´einequalities with uniform constants whenever p ≤ q ≤ p ; for details we refer to [8] (see also[1, Theorem 4.30]) and to [1, Theorem 4.21]. Therefore the following (PI) condition, for acomplete space X supporting a (1 , p )-Poincar´e inequality, is valid with the above exponents1 < p < p . ELF-IMPROVEMENT OF UNIFORM FATNESS 5
However, since we do not in general assume that X is complete, we use in many of ourresults the following a priori assumption concerning the validity of (improved) Poincar´einequalities with uniform constants:(PI) Let 1 < p < ∞ be given. We assume that there are 1 < p < p , C P > τ ≥ X supports the ( q, q )-Poincar´e inequality Z B | u − u B | q dµ ≤ C P diam( B ) q Z τB g q dµ (9)for every p ≤ q ≤ p .For simplicity, we will in the sequel use Poincar´e inequalities with the constants C P > τ ≥
1. Indeed, if only a (1 , p )-Poincar´e inequality is assumed, this is just a matter ofnotation (in this case we may use both (1 , p )-Poincar´e and ( p, p )-Poincar´e inequality withthe above constants). And if (PI) is assumed, the (1 , q )-Poincar´e inequalities (with C P > τ ≥
1) for p ≤ q ≤ p are all trivial consequences of (9) and H¨older’s inequality.2.4. Capacity and fatness.
Let Ω ⊂ X be a bounded open set and let K ⊂ Ω be a closedset. We define the (Lipschitz) variational p -capacity of K with respect to Ω to becap p ( K, Ω) = inf Z Ω g p dµ , (10)where the infimum is taken over all functions u ∈ Lip (Ω), such that u ≥ K , and all p -weak upper gradients g of u . If there are no such functions u , we set cap p ( K, Ω) = ∞ . Remark 2.1.
If cap p ( K, Ω) < ∞ , then the infimum in (10) can be restricted to u ∈ Lip (Ω)satisfying χ K ≤ u ≤ p -weak upper gradients g of u such that g = gχ Ω ∈ L p ( X ).Indeed, if u is an admissible test function for cap p ( K, Ω) and g is a p -weak upper gradient of u such that g ∈ L p (Ω), then ˜ u = max { , min { , u }} belongs to Lip (Ω) and χ K ≤ ˜ u ≤ X . Moreover, the function g is clearly a p -weak upper gradient of ˜ u . By the glueing lemma[1, Lemma 2.19], we may further assume that g = 0 outside Ω. (Actually, since g need notbelong to L p ( X ) but this is needed in the glueing lemma, we first define a function˜ g = gχ Ω + Lip(˜ u, · ) χ X \ Ω ∈ L p ( X )that is a p -weak upper gradient of ˜ u , cf. the proof of [1, Theorem 2.6]. Now the glueinglemma applies, with ˜ g , showing that gχ Ω is a p -weak upper gradient of ˜ u .)Let us remark here that if the metric space X is complete and supports a (1 , p )-Poincar´einequality, then the above definition of cap p ( K, Ω) is equivalent to the definition where thefunction u is assumed to belong to the Newtonian space N ,p (Ω). However, we will not usethe theory of Newtonian spaces in this paper, but rather refer to [1] for an introductionand basic properties of Newtonian functions. In particular, see [1, Theorem 6.19(x)] for theabove-mentioned equivalence of capacities in the complete case.On the other hand, if X = R n , equipped with the Euclidean metric and the Lebesguemeasure (or more generally a p -admissible weight, see [6, 18]), then by standard approxi-mation cap p ( K, Ω) = inf (cid:26) Z Ω |∇ u | p dx : u ∈ C ∞ (Ω) , u ≥ K (cid:27) (11)for all closed (compact) K ⊂ Ω, and therefore cap p ( K, Ω) is the usual variational p -capacityof K . In this case all our results (and computations) concerning Lipschitz functions andtheir p -weak upper gradients (or upper pointwise Lipschitz constants) can be restated usingfunctions in C ∞ (Ω) and the norms of their gradients. We recall that our approach is neweven in this special case. J. LEHRB ¨ACK, H. TUOMINEN, AND A.V. V ¨AH ¨AKANGAS
We say that a closed set E ⊂ X is uniformly p -fat , for 1 ≤ p < ∞ , if there exists aconstant 0 < c ≤ p ( E ∩ B ( x, r ) , B ( x, r )) ≥ c cap p ( B ( x, r ) , B ( x, r )) (12)for all x ∈ E and all 0 < r < (1 /
8) diam( X ). If there exists a constant r > x ∈ E and all 0 < r < r , the closed set E is said to be locallyuniformly p -fat . Remark 2.2.
Both Theorem 1.1 and Theorem 1.2 are formulated in terms of uniformfatness. However, the corresponding results are valid also when ‘uniform fatness’ is replacedby ‘local uniform fatness’ (in the assumptions with exponent p and in the conclusions withexponent q ). In the sequel, we will exclusively focus on the case of uniformly fat sets. Theminor modifications (required throughout the paper) in the local case are straightforward.The self-improvement of uniform p -fatness (that is formulated, e.g., in Theorem 1.1) iscritical in various applications; examples beyond the scope of Hardy inequalities includeglobal higher integrability of both the gradients of solutions to PDE’s [9, 18] and the uppergradients of certain quasiminimizers in metric measure spaces [16]. In [12] a quite simpleproof for the self-improvement of uniform Q -fatness is provided in the setting of Ahlfors Q -regular metric measure spaces.In the Euclidean space R n , the self-improvement property is known to hold also for moregeneral ( α, p )-fatness conditions related to Riesz capacities by the results of Lewis [14]. For α = 1 these conditions are equivalent to the uniform p -fatness; cf. [9, p. 902].In the rest of this paper (and hence in particular in our proof of the self-improvementof uniform fatness), we only need the following two basic facts concerning the variational p -capacity, which hold under the assumption that the space X supports a (1 , p )-Poincar´einequality (and hence also a ( p, p )-Poincar´e inequality). First, there is a constant C > u on X , all p -weak upper gradients g of u , and for allballs B ⊂ X , we have Z B | u | p dµ ≤ C cap p (2 − B ∩ { u = 0 } , B } Z τB g p dµ . (13)Here { u = 0 } = { x ∈ X : u ( x ) = 0 } and τ is the dilatation from the ( p, p )-Poincar´einequality (9). This ‘capacitary Poincar´e inequality’ is in the classical Euclidean case dueto Maz’ya [17, Ch. 10]. For the metric space version, cf. [1, Proposition 6.21].The second fact is a comparison between p -capacity and measure. Namely, there is aconstant C > B = B ( x, r ) with 0 < r < (1 /
8) diam( X ) and for eachclosed set E ⊂ B , µ ( E ) C r p ≤ cap p ( E, B ) ≤ C D µ ( B ) r p ; (14)see, for instance [1, Proposition 6.16]. The (1 , p )-Poincar´e inequality is needed to ensure thevalidity of the lower bound in inequality (14).2.5. Tracking constants.
Our results are based on quantitative estimates and absorptionarguments, where it is often crucial to track the dependencies of constants quantitatively.For this purpose, we will use the following notational convention: • C X, ∗ , ··· , ∗ denotes a positive constant which quantitatively depends on the quantitiesindicated by the ∗ ’s and (possibly) on: the doubling constant C D of the measure µ in (4), the constants C P and τ appearing in the ( q, q )-Poincar´e inequalities (9)and the constants appearing in the capacitary Poincar´e inequality (13) and thecomparison inequality (14).Observe that C X, ∗ , ··· , ∗ can implicitly depend on p via the estimates in inequalities (9), (13)and (14). However, any further dependencies on the exponent p will be explicitly indicated. ELF-IMPROVEMENT OF UNIFORM FATNESS 7 Improved boundary Poincar´e inequalities
Recall, for the rest of the paper, that we assume X to be a metric space (not necessarilycomplete) equipped with a doubling measure µ . Further assumptions, concerning e.g. thevalidity of Poincar´e inequalities, will be stated separately in each of the following results. Our proof of the self-improvement of uniform fatness is based on the following improvedboundary Poincar´e inequalities.
Theorem 3.1.
Let < p < ∞ and suppose that X supports the improved ( q, q ) -Poincar´einequalities (PI) for p ≤ q ≤ p . Assume that E ⊂ X is a uniformly p -fat closed set. Thenthere exists constants < ε < p − p and C > , quantitatively, such that inequality Z B ( w,ρ ) | u | p − ε dµ ≤ Cρ p − ε Z B ( w,τρ ) Lip( u, · ) p − ε dµ holds whenever w ∈ E , ρ > , and u ∈ Lip ( X \ E ) .Proof. Fix w ∈ E , a radius ρ >
0, and a function u ∈ Lip ( X \ E ). Clearly, we may assumethat ρ < (3 /
2) diam( X ). It is convenient to write r = ρ/ (12 τ ) and B = B ( w, r ). Let usassume, for the time being, that 0 < ε < p − p is given and E B ⊂ E ∩ B is any closed setsuch that w ∈ E B . Since Z B ( w,ρ ) | u ( x ) | p − ε ρ p − ε dµ ( x ) ≤ Z B ( w,ρ ) \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) , it suffices to find quantitative constants 0 < ε < p − p and C > E B ⊂ E as above) such that Z B ( w,ρ ) \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C Z B ( w,τρ ) Lip( u, x ) p − ε dµ ( x ) . (15)We establish this improved local Hardy inequality below in Proposition 5.7, and this provesthe theorem. Let us remark here that the proof of Proposition 5.7 is rather involved anddivided in Section 5 to the following three stages: ‘Truncation’ in § § § (cid:3) From Theorem 3.1 (that is based on postponed Proposition 5.7) we obtain the followingestimate for the capacity test-functions related to cap p ( E ∩ B, B ). This estimate will beused in various settings to prove the self-improvement of uniform fatness. Proposition 3.2.
Let < p < ∞ and suppose that X supports the improved ( q, q ) -Poincar´einequalities (PI) for p ≤ q ≤ p . Assume that E ⊂ X is a uniformly p -fat closed set. Thenthere exist constants C > and < ε < p − p , quantitatively, such that for all balls B = B ( w, R ) , with w ∈ E and < R < (1 /
8) diam( X ) , and for all functions v ∈ Lip (2 B ) ,with ≤ v ≤ and v = 1 in E ∩ B , it holds that µ ( B ) R − ( p − ε ) ≤ C Z B Lip( v, · ) p − ε dµ . (16) Proof.
This proof is based on a similar idea as the proof of Lemma 2 in [11]. Let 0 < ε
2, where τ isthe dilatation constant from the ( q, q )-Poincar´e inequality (9). Fix w , R , and v as in thestatement of the proposition. The doubling inequality (4) implies that there is a constant C = C C D ,τ > µ ( ℓB ) ≥ C µ ( B ). If v B > C /
4, we obtain from condition (PI)and the Sobolev inequality [1, Theorem 5.51] for v ∈ Lip (2 B ) that C / ≤ Z B | v | dµ ≤ C D Z B | v | dµ ≤ CR (cid:18) Z B Lip( v, · ) q dµ (cid:19) /q , and from this (16) follows easily. J. LEHRB ¨ACK, H. TUOMINEN, AND A.V. V ¨AH ¨AKANGAS
We may hence assume that v B ≤ C /
4. Let ψ ∈ Lip ( B ) be a cut-off function, defined as ψ ( x ) = max n , − R dist (cid:0) x, B (cid:1)o , and take u = min { ψ, − v } . Since 1 − v = 0 in E ∩ B and ψ = 0 in X \ B , we have that u ∈ Lip ( X \ E ). Observe that u coincides with 1 − v on (1 / B , and therefore Lip( u, · ) | (1 / B = Lip( v, · ) | (1 / B .Let F = { x ∈ ℓB : u ( x ) > / } . We claim that µ ( F ) ≥ ( C / µ ( B ). To prove this claimwe assume the contrary, namely, that µ ( F ) < ( C / µ ( B ). Since v ≥ v = 1 − u ≥ / ℓB \ F , we obtain from the assumptions µ ( ℓB ) ≥ C µ ( B ) and µ ( F ) < ( C / µ ( B ) that Z B v dµ ≥ Z ℓB \ F v dµ ≥ (cid:0) µ ( ℓB ) − µ ( F ) (cid:1) > (cid:0) C µ ( B ) − ( C / µ ( B ) (cid:1) = C µ ( B ) . This contradicts the assumption v B ≤ C /
4, and thus indeed µ ( F ) ≥ ( C / µ ( B ).Theorem 3.1, with ρ = ℓR , now implies that( C / µ ( B ) ≤ µ ( F ) ≤ q Z ℓB | u | q dµ ≤ CR q Z τℓB Lip( u, · ) q dµ ≤ CR q Z B Lip( v, · ) q dµ . This proves estimate (16) and concludes the proof. (cid:3)
In a Euclidean space R n , which supports the (1 , p )-Poincar´e inequalities for all 1 ≤ p < ∞ ,Proposition 3.2 yields immediately the self-improvement of uniform p -fatness. Indeed, wecan replace in our argument the Lipschitz function v ∈ Lip (2 B ) with a function ˜ v ∈ C ∞ (2 B )and the pointwise Lipschitz constant Lip( v, · ) with |∇ ˜ v | , whence the uniform ( p − ε )-fatnessof E follows from estimates (14) and (16).More generally, in a complete metric space X supporting a (1 , p )-Poincar´e inequality, wecan deduce the self-improvement of uniform fatness from Proposition 3.2 with the help ofsome deep facts concerning analysis on metric spaces (see the proof below). Nevertheless,with an additional argument using the interplay between uniform fatness and density condi-tions for suitable Hausdorff contents, it is possible to obtain a version of the self-improvementin a non-complete setting as well (Theorem 1.2), and hence in particular without the useof Cheeger’s differentiation theory, but then the ( p, p )-Poincar´e inequality, or at least thevalidity of improved Poincar´e inequalities (PI) for p ≤ q ≤ p , has to be explicitly assumedfor some exponent 1 < p < p ; see Section 4 for details. Proof of Theorem 1.1.
Since the space X is assumed to be complete, the validity of theimproved Poincar´e inequalities (PI) follows from the (1 , p )-Poincar´e inequality, as discussedin Section 2.3. Hence we can apply Proposition 3.2. Moreover, by the deep result of Cheeger,[4, Theorem 6.1] (see also [1, Theorem A.7]), the upper pointwise Lipschitz constant Lip( v, · )is a minimal ( p − ε )-weak gradient of the Lipschitz function v , and so we obtain fromestimates (14) and (16) (and Remark 2.1) that the set E is indeed uniformly ( p − ε )-fat. (cid:3) Self-improvement of uniform fatness in non-complete spaces
In this section we provide the additional argument that is needed for the proof of theself-improvement result in the setting of non-complete metric spaces, Theorem 1.2. In fact,we prove the following slightly stronger result (by H¨older’s inequality, the ( p, p )-Poincar´einequality that was assumed in Theorem 1.2 implies the improved Poincar´e inequalities (PI)for p ≤ q ≤ p .) Theorem 4.1.
Let < p < ∞ and suppose that X supports the improved ( q, q ) -Poincar´einequalities (PI) for p ≤ q ≤ p . Assume that E ⊂ X is a uniformly p -fat closed set andthat E ∩ B ( w, r ) is compact for all w ∈ E and all r > . Then there exists < ε < p − p ELF-IMPROVEMENT OF UNIFORM FATNESS 9 such that E is uniformly ( p − ε ) -fat; here both ε and the constant of uniform ( p − ε ) -fatnessare quantitative. The proof of Theorem 4.1 is based on Proposition 3.2, but we also need some auxil-iary results related to Hausdorff contents. Note that these auxiliary results are essentiallyestablished in [11], but there the space X is assumed to be complete.The Hausdorff content of codimension q of a set K ⊂ X is defined by e H qρ ( K ) = inf (cid:26)X k µ ( B ( x k , r k )) r − qk : K ⊂ [ k B ( x k , r k ) , x k ∈ K, < r k ≤ ρ (cid:27) . Density conditions for these Hausdorff contents are known to be closely related to uniformfatness. Indeed, from Proposition 3.2 we obtain the following result.
Lemma 4.2.
Let < p < ∞ and suppose that X supports the improved ( q, q ) -Poincar´einequalities (PI) for p ≤ q ≤ p . Assume that E ⊂ X is a uniformly p -fat closed set. Thenthere exist constants C > and p < q < p , quantitatively, such that e H qR/ (cid:0) E ∩ B ( w, R ) (cid:1) ≥ Cµ (cid:0) B ( w, R ) (cid:1) R − q (17) whenever w ∈ E and < R < (1 /
8) diam( X ) are such that E ∩ B ( w, R ) is compact.Proof. Fix w ∈ E and 0 < R < (1 /
8) diam( X ), write B = B ( w, R ), and assume that E ∩ B is compact. Let { B k } , where B k = B ( x k , r k ) with x k ∈ E ∩ B and 0 < r k ≤ R/
2, bea cover of E ∩ B . Since E ∩ B is compact, we may assume that this cover is finite, i.e. E ∩ B ⊂ S Nk =1 B k . Also let q = p − ε , where ε is as in Proposition 3.2.Define v ( x ) = max ≤ k ≤ N (cid:8) , − r k − dist( x, B k ) (cid:9) . Then v is a Lipschitz function, v = 1 in E ∩ B , v = 0 outside 2 B , and 0 ≤ v ≤
1. Moreover,the upper pointwise Lipschitz constant of v satisfies Lip( v, x ) ≤ max ≤ k ≤ N r k − χ B k ( x ) forall x ∈ X , and hence Lip( v, x ) q ≤ N X k =1 r k − q χ B k ( x )for all x ∈ B . Thus we obtain from Proposition 3.2 (and the doubling condition) that µ ( B ) R − q ≤ C Z B Lip( v, x ) q dµ ( x ) ≤ C N X k =1 µ (cid:0) B k (cid:1) r k − q ≤ C N X k =1 µ ( B k ) r k − q . Taking the infimum over all such covers of E ∩ B yields the claim. (cid:3) On the other hand, from (17) we get back to t -uniform fatness, for any t > q . Lemma 4.3.
Let < q < ∞ and suppose that X supports a (1 , t ) -Poincar´e inequality forall t > q . Let E ⊂ X be a closed set. If there exists C > such that the density condition e H qR/ (cid:0) E ∩ B ( w, R ) (cid:1) ≥ Cµ (cid:0) B ( w, R ) (cid:1) R − q (18) holds for all w ∈ E and all < R < (1 /
8) diam( X ) , then E is uniformly t -fat for all t > q .Proof. Fix t > q . Let w ∈ E and 0 < R < (1 /
8) diam( X ), write B = B ( w, R ), and let u ∈ Lip (2 B ) be such that 0 ≤ u ≤ u = 1 in E ∩ B . By the capacity comparisonestimate (14) and Remark 2.1, it suffices to show that there exists a constant C > w , R and u , such that µ ( B ) R − t ≤ C Z B g t dµ (19)for all t -weak upper gradients g of u such that g = gχ B ∈ L t ( X ). If u B ≥ /
2, then it follows from the Sobolev inequality [1, Theorem 5.51] that1 / ≤ Z B u dµ ≤ C X,p R (cid:18) Z B g t dµ (cid:19) /t , and from this (19) follows easily.On the other hand, if u B < /
2, we can use similar reasoning as in [11, p. 729] (whichis based on the proof of [7, Theorem 5.9]), but let us recall the main steps for convenience.Since t > q and 1 / < u ( x ) − u B = | u ( x ) − u B | for each x ∈ E ∩ B , we can apply a well-known chaining argument (using also the continuity of u and the (1 , t )-Poincar´e inequality)to find for each x ∈ E ∩ B a ball B x = B ( x, r x ) with 0 < r x ≤ R such that µ ( B x ) r − qx ≤ C X,t,q R t − q Z τB x g t dµ . (20)The 5 r -covering lemma then yields us a countable collection of points x , x , . . . ∈ E ∩ B such that the corresponding balls B k = τ B x k are pairwise disjoint, but the balls 5 B k cover E ∩ B . Using the assumption (18) for this particular cover and the doubling property of µ ,we find that µ ( B ) R − q ≤ C X k µ ( B x k ) r − qx k , (21)whence estimate (20) and the pairwise disjointness of the balls B k yield the claim (19). (Inparticular, here we may assume that the radii of the balls 5 B k are all less than R/
2, sinceotherwise the claim readily follows from the doubling property of µ and inequality (20)applied to a ball B x k with 5 τ r x k > R/ (cid:3) The proof of Theorem 4.1.
Since we assumed that E is uniformly p -fat and E ∩ B ( w, R ) iscompact for all w ∈ E and all R >
0, we have by Lemma 4.2 that e H qR/ (cid:0) E ∩ B ( w, R ) (cid:1) ≥ Cµ (cid:0) B ( w, R ) (cid:1) R − q for all w ∈ E and all 0 < R < (1 /
8) diam( X ), where p < q < p . But now we can fix q < t < p , and Lemma 4.3 yields that E is uniformly t -fat. Notice, in particular, that the(1 , t )-Poincar´e inequality that is needed in the proof of Lemma 4.3 is valid by the assumedimproved Poincar´e inequalities (PI) since p < t < p . (cid:3) Improved local Hardy inequalities
This section is devoted to the proof of the improved local Hardy inequality (15) that isreformulated as Proposition 5.7. The proof of this proposition is divided in the followingthree parts. In § E and proving a local absorption lemma. In § p , and in § Truncation.
We begin with some technical tools that will be needed in the proofs ofthe local Hardy inequalities. The following truncation procedure provides us with the closedset E B ⊂ B that was required in the proof of Theorem 3.1. A similar procedure was usedin [14, p. 180] when proving the self-improvement of uniform ( α, p )-fatness conditions in R n , and later also in [18], for weighted R n , and in [2], for general metric spaces.We write N = { , , , . . . } and N = N ∪ { } . Lemma 5.1.
Assume that E ⊂ X is a closed set and that B = B ( w, r ) for w ∈ E and r > . Let E B = E ∩ B , define inductively, for every j ∈ N , that E jB = [ x ∈ E j − B E ∩ B ( x, − j − r ) , and set E B = [ j ∈ N E jB . Then the following statements hold:
ELF-IMPROVEMENT OF UNIFORM FATNESS 11 (a) w ∈ E B (b) E B ⊂ E (c) E B ⊂ B (d) E j − B ⊂ E jB ⊂ E B for every j ∈ N .Proof. Part (a) is is true since w ∈ E B . Part (b) follows from the facts that E is closed and ∪ j E jB ⊂ E by definition. To verify (c), we fix x ∈ E jB . If j = 0, then x ∈ B . If j >
0, thenby induction we find a sequence x j , . . . , x such x j = x and, for each k = 0 , . . . , j , x k ∈ E kB and x k ∈ E ∩ B ( x k − , − k − r ) if k >
0. It follows that d ( x, w ) ≤ j X k =1 d ( x k , x k − ) + d ( x , w ) ≤ j X k =1 − k − r + 2 − r < r . Hence, x ∈ B ( w, r ) ⊂ B . We have shown that E jB ⊂ B whenever j ∈ N , whence it followsthat also E B ⊂ B . To prove (d) we fix j ∈ N and x ∈ E j − B . By definition we have x ∈ E and, hence, x ∈ E ∩ B ( x, − j − r ) ⊂ E jB . (cid:3) Next we show that Lemma 5.1, in fact, truncates the set E to B in such a way that thereare always certain balls b B whose intersection with the truncated set E B contain big piecesof the original set E (by these balls we later employ the uniform fatness of E ). Lemma 5.2.
Let E , B , and E B be as in Lemma 5.1. Suppose that m ∈ N and x ∈ X is such that d E B ( x ) < − m +1 r . Then there exists a ball b B = B ( y x,m , − m − r ) such that y x,m ∈ E , − b B ∩ E = 2 − b B ∩ E B , (22) and σ b B ⊂ B ( x, σ − m +2 r ) for every σ ≥ .Proof. In this proof we will apply Lemma 5.1 several times without further notice. Since d E B ( x ) < − m +1 r there exists y ∈ ∪ j ∈ N E jB ⊂ E such that d ( y, x ) < − m +1 r . Let us fix j ∈ N such that y ∈ E jB . There are two cases to be treated.First, let us consider the case when j > m ≥
0. By induction, there are points y k ∈ E kB with k = m, . . . , j such that y j = y and y k ∈ E ∩ B ( y k − , − k − r ) for every k = m + 1 , . . . , j .It follows that d ( y m , y ) = d ( y j , y m ) ≤ j X k = m +1 d ( y k , y k − ) ≤ j X k = m +1 − k − r < − m − r . Take y x,m = y m ∈ E mB ⊂ E and b B = B ( y m , − m − r ). If σ ≥ z ∈ σ b B , then d ( z, x ) ≤ d ( z, y m ) + d ( y m , y ) + d ( y, x ) ≤ σ − m − r + 2 − m − r + 2 − m +1 r < σ − m +2 r , and thus σ b B ⊂ B ( x, σ − m +2 r ). Moreover, since y m ∈ E mB , we have2 − b B ∩ E = E ∩ B ( y m , − m − r ) ⊂ [ z ∈ E mB E ∩ B ( z, − m − r ) = E m +1 B ⊂ E B . On the other hand E B ⊂ E , and thus 2 − b B ∩ E = 2 − b B ∩ E B .Let us then consider the case m ≥ j ≥
0. We take y x,m = y ∈ E and b B = B ( y, − m − r ).Then, for every σ ≥ z ∈ σ b B , d ( z, x ) ≤ d ( z, y ) + d ( y, x ) < σ − m − r + 2 − m +1 r < σ − m +2 r , and so σ b B ⊂ B ( x, σ − m +2 r ). Since y ∈ E jB ⊂ E mB ⊂ E B we can repeat the argument above,with y m replaced by y , and it follows as above that 2 − b B ∩ E = 2 − b B ∩ E B . (cid:3) One of the reasons for truncating the set E , in the first place , is to obtain the absorptionLemma 5.3. This lemma is needed twice during the rest of the paper, with slightly differentcontexts, and hence there are two different assumptions concerning the validity of Poincar´einequalities. The dependencies of the constants below are rather delicate, and it is importantto track them carefully; to this end, recall our notational convention from § Lemma 5.3.
Suppose that either(i) ≤ q = p < ∞ and X supports a (1 , p ) -Poincar´e inequality; or(ii) < p < p < ∞ and X supports the improved Poincar´e inequalities (PI) for expo-nents p ≤ q ≤ p .In addition, let E , B , and E B be as in Lemma 5.1, let σ ≥ and ς ≥ , and write B ∗ = ςB .Assume that u ∈ Lip( X ) is such that u = 0 on E B , and that g is a q -weak upper gradientof u such that inequality Z B ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) ≤ C Z σB ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) + C Z σB ∗ g ( x ) q dµ ( x ) holds with some constants C , C > . Then C Z σB ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) ≤ C Z τσB ∗ g ( x ) q dµ ( x ) , where C = 1 − C (1 + C C D ,σ,ς,p ) and C = (1 + C ) C X,σ,ς,q .Proof.
Since E B ⊂ B and ς ≥
2, we obtain the estimate Z ( σB ∗ \ E B ) \ B ∗ | u ( x ) | q d E B ( x ) q dµ ( x ) ≤ r − q Z σB ∗ | u ( x ) | q dµ ( x ) ≤ q r − q (cid:18) Z σB ∗ | u ( x ) − u σB ∗ | q dµ ( x ) + µ ( σB ∗ ) | u σB ∗ − u B ∗ | q + µ ( σB ∗ ) | u B ∗ | q (cid:19) . By the doubling inequality (4) and the ( q, q )-Poincar´e inequality (9) (recall that in case (i),i.e. for q = p , this is a consequence of the (1 , p )-Poincar´e inequality, cf. [1, Corollary 4.24])we obtain3 q r − q (cid:18) Z σB ∗ | u ( x ) − u σB ∗ | q dµ ( x ) + µ ( σB ∗ ) | u σB ∗ − u B ∗ | q (cid:19) ≤ C σ,q,C D r − q Z σB ∗ | u ( x ) − u σB ∗ | q dµ ( x ) ≤ C X,σ,ς,q Z τσB ∗ g ( x ) q dµ ( x ) . On the other hand, by the assumption,3 q r − q µ ( σB ∗ ) | u B ∗ | q ≤ q C σ,C D r − q Z B ∗ \ E B | u ( x ) | q dµ ( x ) ≤ q ς q C σ,C D Z B ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) ≤ p ς p C σ,C D C Z σB ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) + 3 q ς q C σ,C D C Z σB ∗ g ( x ) q dµ ( x ) . Combining the estimates above, we find that Z σB ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) = Z B ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) + Z ( σB ∗ \ E B ) \ B ∗ | u ( x ) | q d E B ( x ) q dµ ( x ) ≤ C (1 + C C D ,σ,ς,p ) Z σB ∗ \ E B | u ( x ) | q d E B ( x ) q dµ ( x ) + C Z τσB ∗ g ( x ) q dµ ( x ) , where C = (1 + C ) C X,σ,ς,q . This concludes the proof. (cid:3)
ELF-IMPROVEMENT OF UNIFORM FATNESS 13
Local Hardy.
In this section we prove Proposition 5.4 that gives a local p -Hardyinequality with respect to the truncated set E B (Theorem 1.3 is also proved at the end ofthis section). This is done by adapting the proof of [11, Theorem 3] to the present setting.The proof in [11] is, in turn, based on the ideas of Wannebo [19]; see also [3].Throughout this section, we will assume that X supports a (1 , p )-Poincar´e inequality,whence X supports also a ( p, p )-Poincar´e inequality. Both of these inequalities are assumedto be valid with constants C P > τ ≥ Proposition 5.4.
Let < p < ∞ and suppose that X supports a (1 , p ) -Poincar´e inequality.Assume that E ⊂ X is a uniformly p -fat closed set, let w ∈ E and < r < (1 /
8) diam( X ) ,and let E B be as in Lemma 5.1 for B = B ( w, r ) . Let u ∈ Lip( X ) and let g be a p -weakupper gradient of u such that u = 0 = g in an open set U ⊂ X satisfying the condition dist( E B , X \ U ) > (or the condition X = U ). Then Z τB \ E B | u ( x ) | p d E B ( x ) p dµ ( x ) ≤ C H Z τ B g ( x ) p dµ ( x ) . (23) Here C H = C X,p,c and the number < c ≤ is the constant from the uniform p -fatnesscondition (12) for E . Recall that we do not assume X to be complete, and hence it is not necessarily true thatdist( K, X \ U ) > K is a closed subset of a bounded open set U . For this reasonwe make in Proposition 5.4 the explicit assumption that dist( E B , X \ U ) > p , X , E , B = B ( w, r ), and E B are as in Proposition 5.4(these are considered arbitrary but fixed).For each m ∈ Z , let us write G m = { x ∈ τ B : 2 − m r ≤ d E B ( x ) < − m +1 r } and e G m = ∞ [ k = m G k = { x ∈ τ B : 0 < d E B ( x ) < − m +1 r } . For every m ∈ N we let G m be a (countable) cover of G m with open balls e B that arecentered at G m and of radius 2 − m +2 r . Moreover, we require that { − e B : e B ∈ G m } is adisjoint family, whence there exits C = C C D ,τ > X e B ∈G m χ τ e B ≤ C . (24)The existence of such a cover follows using a maximal packing argument and the doublingproperty of µ . Lemma 5.5.
Let us define ℓ = ⌈ log ( τ ) ⌉ +2 . Then, for each m ∈ N and every ball e B ∈ G m ,we have τ e B \ E B ⊂ e G m − ℓ . (25) Proof.
Fix m ∈ N and e B ∈ G m . By definition, we have e B = B ( x e B , − m +2 r ) with x e B ∈ G m .Let x ∈ τ e B \ E B . Then d E B ( x ) >
0. Moreover, d E B ( x ) = dist( x, E B ) ≤ d ( x, x e B ) + dist( x e B , E B ) < τ − m +2 r + 2 − m +1 r < τ − m +3 r ≤ − ( m − ℓ )+1 r . Since m ≥
0, a modification of the previous estimate also yields d ( x, w ) ≤ dist( x, B ) + r ≤ dist( x, E B ) + r < τ r + 2 r + r < τ r , and it follows that x ∈ B ( w, τ r ) = 8 τ B . We can now conclude that x ∈ e G m − ℓ . (cid:3) The uniform p -fatness of E is exclusively used in the following lemma. Lemma 5.6.
Let v be a Lipschitz function on X such that v = 0 on E B and let g be a p -weak upper gradient of v . Then, for every m ∈ N and each e B ∈ G m , Z e B | v ( x ) | p dµ ( x ) ≤ C X,p c − mp r p Z τ e B g ( x ) p dµ ( x ) . (26) Proof.
Fix m ∈ N and e B = B ( x e B , − m +2 r ) ∈ G m . Then, by definition, x e B ∈ G m . We applyLemma 5.2 and thereby associate to e B a smaller open ball b B ⊂ e B , centered at E and ofradius 2 − m − r < (1 /
8) diam( X ). Note first that Z e B | v ( x ) | p dµ ( x ) ≤ C p (cid:18) Z e B | v ( x ) − v e B | p dµ ( x ) + | v e B − v b B | p + | v b B | p (cid:19) . Here, by H¨older’s inequality and the doubling condition (4), | v e B − v b B | p ≤ (cid:18) Z b B | v ( x ) − v e B | dµ ( x ) (cid:19) p ≤ C C D Z e B | v ( x ) − v e B | p dµ ( x ) , and therefore, by the ( p, p )-Poincar´e inequality, we have that C p (cid:18) Z e B | v ( x ) − v e B | p dµ ( x ) + | v e B − v b B | p (cid:19) ≤ C X,p − mp r p Z τ e B g ( x ) p dµ ( x ) . On the other hand, by the capacitary Poincar´e inequality (13), | v b B | p ≤ Z b B | v ( x ) | p dµ ( x ) ≤ C X cap p (2 − b B ∩ { v = 0 } , b B ) Z τ b B g ( x ) p dµ ( x ) . Recall that v ( x ) = 0 whenever x ∈ E B (by assumption) and 2 − b B ∩ E B = 2 − b B ∩ E byLemma 5.2. Therefore, using monotonicity, the uniform p -fatness condition (12), and thecomparison inequality (14), we obtaincap p (2 − b B ∩ { v = 0 } , b B ) ≥ cap p (2 − b B ∩ E B , b B ) = cap p (2 − b B ∩ E, b B ) ≥ c cap p (2 − b B, b B ) ≥ c µ (2 − b B ) C X,p − mp r p . Finally, since τ b B ⊂ τ e B , it follows that C p | v b B | p ≤ C p Z b B | v ( x ) | p dµ ( x ) ≤ C X,p c − mp r p µ (2 − b B ) Z τ e B g ( x ) p dµ ( x ) . Inequality (26) follows from the above estimates and the doubling condition (4). (cid:3)
Proof of Proposition 5.4.
Let us first assume that v ∈ Lip ( X \ E B ) and that g v is a p -weakupper gradient of v that also vanishes in the set E B . Then, by summing the inequalities (26)and using (24) and (25) we obtain, for every m ∈ N , Z G m | v ( x ) | p dµ ( x ) ≤ X e B ∈G m Z e B | v ( x ) | p dµ ( x ) ≤ C X,p c − mp r p X e B ∈G m Z τ e B \ E B g v ( x ) p dµ ( x ) ≤ C X,p c − mp r p Z e G m − ℓ g v ( x ) p dµ ( x ) . (27) ELF-IMPROVEMENT OF UNIFORM FATNESS 15
Let 0 < β < m ( p + β ) r − p − β and sum the inequalities to obtain the estimate Z B \ E B | v ( x ) | p d E B ( x ) p + β dµ ( x ) ≤ Z e G | v ( x ) | p d E B ( x ) p + β dµ ( x ) = ∞ X m =0 Z G m | v ( x ) | p d E B ( x ) p + β dµ ( x ) ≤ ∞ X m =0 m ( p + β ) r − p − β Z G m | v ( x ) | p dµ ( x ) ≤ C X,p c r − β ∞ X m =0 mβ Z e G m − ℓ g v ( x ) p dµ ( x )= C X,p c r − β ∞ X k = − ℓ k + ℓ X m =0 mβ Z G k g v ( x ) p dµ ( x ) ≤ C X,p c β ∞ X k = − ℓ kβ r − β Z G k g v ( x ) p dµ ( x ) ≤ C X,p c β Z τB \ E B g v ( x ) p d E B ( x ) β dµ ( x ) . (28)Now we come to the main line of the argument. Let u be a Lipschitz function on X andlet g be a p -weak upper gradient of u , both of which vanish in an open set U ⊂ X satisfyingthe condition dist( E B , X \ U ) >
0. We aim to show that inequality (23) holds, and so wemay assume that g ∈ L p (8 τ B ) (recall that τ ≥ A = 8 τ B ∪ ( X \ τ B ) that coincides with u in 8 τ B and vanishes outside 10 τ B , and let ˜ u be the McShane extension (5) of this function to allof X . Then the function ˜ g = gχ τB + Lip(˜ u, · ) χ X \ τB ∈ L p ( X )is a p -weak upper gradient of ˜ u , cf. the proof of [1, Theorem 2.6]. Define v ( x ) = ˜ u ( x ) d E B ( x ) β/p for every x ∈ X . Then v ( x ) = u ( x ) d E B ( x ) β/p for every x ∈ τ B and in particular v vanishesin E B . Moreover, by applying the assumptions on u and g in combination with the Leibnizand chain rules of Theorems 2.15 and 2.16 in [1], we find that v has a p -weak upper gradient g v such that g v ( x ) ≤ g ( x ) d E B ( x ) β/p + βp | u ( x ) | d E B ( x ) β/p − for every x ∈ τ B ; in particular, also g v vanishes on the set E B . Using estimate (28) forthe pair v and g v , we obtain Z B \ E B | u ( x ) | p d E B ( x ) p dµ ( x ) = Z B \ E B | v ( x ) | p d E B ( x ) p + β dµ ( x ) ≤ C X,p c β Z τB \ E B g v ( x ) p d E B ( x ) β dµ ( x ) ≤ C X,p c β Z τB g ( x ) p dµ ( x ) + C X,p c β p − Z τB \ E B | u ( x ) | p d E B ( x ) p dµ ( x ) . We can now apply Lemma 5.3 with parameters ς = 2 , σ = 4 τ , q = p , C = C X,p c β p − , C = C X,p c β . Recall our convention in § < β <
1, depending on C X , p , and c , such that C = 1 − C (1 + C C D ,σ,ς,p ) ≥ . Then, Lemma 5.3 yields that Z τB \ E B | u ( x ) | p d E B ( x ) p dµ ( x ) ≤ C X,p,c Z τ B g ( x ) p dµ ( x ) , and this concludes the proof. (cid:3) Before entering the final stage in our proof of the self-improvement of uniform p -fatness, wetake a small side step and give a proof for Theorem 1.3 that was stated in the introduction. Proof of Theorem 1.3.
Fix w ∈ E and 0 < r < (1 /
8) diam( X ), and let E B be as inLemma 5.1 for the ball B = B ( w, r ). Fix u ∈ Lip ( X \ E ) and a p -weak upper gradient g of u . Since we first aim to prove estimate (29) below, we can assume that g ∈ L p (8 τ B ).For every δ >
0, we define a Lipschitz function u δ = max { , | u | − δ } . Since g is clearly a p -weak upper gradient of u δ , it is straightforward to show that the function h = gχ τ B + Lip( u δ , · ) χ X \ τ B is a p -weak upper gradient of u δ , cf. the proof of [1, Theorem 2.6]. Since u δ vanishes in the set U δ = {| u | < δ } we can apply the local version of the glueing lemma [1, Lemma 2.19] with u δ and h . From this we can deduce that g δ = hχ X \ U δ is a p -weak upper gradient of u δ . Observethat both u δ and g δ vanish in the open neighbourhood U δ of E and dist( E B , X \ U δ ) > U δ = X . Since E ∩ (1 / B ⊂ E B and w ∈ E B , we see that d E = d E B in (1 / B . Hence, bymonotone convergence and Proposition 5.4, we conclude that Z (1 / B \ E | u ( x ) | p d E ( x ) p dµ ( x ) = lim δ → Z (1 / B \ E | u δ ( x ) | p d E B ( x ) p dµ ( x ) ≤ C H lim inf δ → Z τ B g δ ( x ) p dµ ( x ) ≤ C H Z τ B g ( x ) p dµ ( x ) . (29)The desired inequality (3) now follows by a simple change of variables. (cid:3) Improvement.
In this section we improve the ‘local integral Hardy inequality’, thatwas established in Proposition 5.4, by adapting ideas from Koskela–Zhong [13] to the presentsetting and applying again the absorption Lemma 5.3. This improvement argument consti-tutes the final step in the proof of Theorem 3.1.
Proposition 5.7.
Let < p < ∞ and suppose that X supports the improved ( q, q ) -Poincar´einequalities (PI) for p ≤ q ≤ p . Assume that E ⊂ X is a uniformly p -fat closed set, let w ∈ E and < r < (1 /
8) diam( X ) , and let E B be as in Lemma 5.1 for B = B ( w, r ) . Thenthere exists constants < ε = ε X,p ,p,C H < p − p and C > such that the inequality Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C Z τ B Lip( u, x ) p − ε dµ ( x ) (30) holds whenever u ∈ Lip ( X \ E ) . Here C H = C X,p,c is the constant from Proposition 5.4. In the proof of Proposition 5.7, we use the restricted maximal function M R u at x ∈ X , for R : X → [0 , ∞ ) and a locally integrable function u on X , that is defined by M R u ( x ) = | u ( x ) | if R ( x ) = 0, and otherwise by M R u ( x ) = sup r Z B ( x,r ) | u ( y ) | dµ ( y ) , where the supremum is taken over all radii 0 < r < R ( x ). Proof of Proposition 5.7.
Without loss of generality, we may assume that C H ≥ u ∈ Lip( X ) is suchthat u = 0 in an open set U ( X for which dist( E, X \ U ) >
0. Throughout this proof, wewrite g = Lip( u, · ); in particular, also g = 0 in U .Fix a number λ >
0, and define F λ = H λ ∩ G λ , where H λ = { x ∈ τ B : | u ( x ) | ≤ λd E B ( x ) } ,G λ = (cid:8) x ∈ τ B : (cid:0) M d EB ( x ) / g p ( x ) (cid:1) /p ≤ λ (cid:9) . ELF-IMPROVEMENT OF UNIFORM FATNESS 17
We claim that the restriction of u to F λ is ( C X,p λ )-Lipschitz. Indeed, let x, y ∈ F λ , x = y ,be such that d E B ( y ) ≤ d E B ( x ). If d E B ( x ) ≥ τ d ( x, y ), then d E B ( y ) ≥ d E B ( x ) − d ( x, y ) ≥ τ d ( x, y ) . Thus a standard chaining argument [5, Theorem 3.2], which is based on the facts that µ is doubling and that the (1 , p )-Poincar´e inequality holds for the pair u and g = Lip( u, · ),yields that | u ( x ) − u ( y ) | ≤ C X,p d ( x, y ) (cid:16)(cid:0) M τd ( x,y ) g p ( x ) (cid:1) /p + (cid:0) M τd ( x,y ) g p ( y ) (cid:1) /p (cid:17) ≤ C X,p d ( x, y ) (cid:16)(cid:0) M d EB ( x ) / g p ( x ) (cid:1) /p + (cid:0) M d EB ( y ) / g p ( y ) (cid:1) /p (cid:17) ≤ C X,p λd ( x, y ) . On the other hand, if d E B ( y ) ≤ d E B ( x ) ≤ τ d ( x, y ), then | u ( x ) − u ( y ) | ≤ | u ( x ) | + | u ( y ) | ≤ λ ( d E B ( x ) + d E B ( y )) ≤ τ λd ( x, y ) . These two estimates show that u is Lipschitz on F λ .Next we use the McShane extension (5) and extend the restriction of u on A = F λ to a( C X,p λ )-Lipschitz function ˜ u on X . Then also ˜ u vanishes on an open set e U ⊂ U such thatdist( E B , X \ e U ) >
0; indeed, if x ∈ e U = { x ∈ τ B : d E B ( x ) < dist( E B , X \ U ) / } then u ( x ) = 0 and x ∈ F λ (here we use the fact that g = Lip( u, · ) = 0 in U ), whence ˜ u ( x ) = 0.By [1, Lemma 2.19], the bounded function˜ g ( x ) = χ F λ ( x ) g ( x ) + C X,p λχ X \ F λ ( x )is a p -weak upper gradient of ˜ u that vanishes on e U . Hence, applying Proposition 5.4 to thepair ˜ u and ˜ g , we find that Z (8 τB \ E B ) ∩ F λ | u ( x ) | p d E B ( x ) p dµ ( x ) ≤ Z τB \ E B | ˜ u ( x ) | p d E B ( x ) p dµ ( x ) ≤ C H Z F λ g ( x ) p dµ ( x ) + C H C pX,p λ p µ (8 τ B \ F λ )and, since C H ≥
1, that Z (8 τB \ E B ) ∩ H λ | u ( x ) | p d E B ( x ) p dµ ( x ) ≤ C H Z F λ g ( x ) p dµ ( x ) + C H C pX,p λ p µ (8 τ B \ F λ ) + Z ( H λ \ E B ) \ G λ | u ( x ) | p d E B ( x ) p dµ ( x ) ≤ C H Z G λ g ( x ) p dµ ( x ) + C H C X,p ,p λ p (cid:0) µ (8 τ B \ F λ ) + µ ( H λ \ G λ ) (cid:1) ≤ C H Z G λ g ( x ) p dµ ( x ) + C H C X,p ,p λ p (cid:0) µ (8 τ B \ H λ ) + µ (8 τ B \ G λ ) (cid:1) . The above estimate holds for all λ >
0. We multiply it by λ − − ε (here 0 < ε < ( p − p ) / , ∞ ). This gives ε − Z τB \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C H Z ∞ λ − − ε Z G λ g ( x ) p dµ ( x ) dλ + C H C X,p ,p Z ∞ λ p − − ε (cid:0) µ (8 τ B \ H λ ) + µ (8 τ B \ G λ ) (cid:1) dλ . By the definition of G λ , and the observation that g ( x ) ≤ (cid:0) M d EB ( x ) / g p ( x ) (cid:1) /p for a.e. x ∈ τ B , we find that the first term on the right-hand side is dominated by C H ε − Z τ B g ( x ) p − ε dµ ( x ) . Using the definitions of H λ and G λ , the second term on the right-hand side can be estimatedfrom above by C H C X,p ,p p − ε (cid:18) Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) + Z τ B (cid:0) M d EB ( x ) / g p ( x ) (cid:1) p − εp dµ ( x ) (cid:19) . Since d E B ( x ) / ≤ τ r for all x ∈ τ B , we have by the Hardy–Littlewood maximal theorem,see e.g. [1, Theorem 3.13], that Z τ B (cid:0) M d EB ( x ) / g p ( x ) (cid:1) p − εp dµ ( x ) ≤ Z X (cid:0) M ( χ τ B g p )( x ) (cid:1) p − εp dµ ( x ) ≤ C C D ,p ,p,ε Z τ B g ( x ) p − ε dµ ( x ) ;here M denotes the usual unrestricted maximal operator.By combining the estimates above, we obtain Z τB \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) + C Z τ B g ( x ) p − ε dµ ( x ) , where C = C H C X,p ,p ε ( p − ε ) − and C = C H (1 + C X,p ,p ε ( p − ε ) − C C D ,p ,p,ε ) . In order to apply Lemma 5.3, we write ς = 8 τ , σ = 3 τ / , q = p − ε . Recall our convention in § < ε < ( p − p ) /
2, depending on X , p , p , and C H , in such a way that C = 1 − C (1 + C C D ,σ,ς,p ) ≥ / . Then, Lemma 5.3 yields that Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C Z τ B g ( x ) p − ε dµ ( x ) , where C = (1 + C ) C X,σ,ς,p − ε . This proves the claim in the case where u = 0 in an open set U ⊂ X with dist( E, X \ U ) > u ∈ Lip ( X \ E ). Clearly, we may assume that u does notvanish everywhere in X . For every δ >
0, we define a Lipschitz function u δ = max { , | u |− δ } .Now Lip( u δ , · ) ≤ Lip( u, · ) and u δ vanishes in the open neighbourhood U δ = {| u | < δ } of E .Thus, by monotone convergence and the special case of inequality (30) that was establishedabove we conclude that Z τ B \ E B | u ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) = lim δ → Z τ B \ E B | u δ ( x ) | p − ε d E B ( x ) p − ε dµ ( x ) ≤ C lim inf δ → Z τ B Lip( u δ , x ) p − ε dµ ( x ) ≤ C Z τ B Lip( u, x ) p − ε dµ ( x ) . This proofs the claim in the general case u ∈ Lip ( X \ E ). (cid:3) ELF-IMPROVEMENT OF UNIFORM FATNESS 19
References [1] A. Bj¨orn and J. Bj¨orn.
Nonlinear potential theory on metric spaces , volume 17 of
EMS Tracts inMathematics . European Mathematical Society (EMS), Z¨urich, 2011.[2] J. Bj¨orn, P. MacManus, and N. Shanmugalingam. Fat sets and pointwise boundary estimates for p -harmonic functions in metric spaces. J. Anal. Math. , 85:339–369, 2001.[3] S. M. Buckley and P. Koskela. Orlicz-Hardy inequalities.
Illinois J. Math. , 48(3):787–802, 2004.[4] J. Cheeger. Differentiability of Lipschitz functions on metric measure spaces.
Geom. Funct. Anal. ,9(3):428–517, 1999.[5] P. Haj lasz and P. Koskela. Sobolev met Poincar´e.
Mem. Amer. Math. Soc. , 145(688):x+101, 2000.[6] J. Heinonen, T. Kilpel¨ainen, and O. Martio.
Nonlinear potential theory of degenerate elliptic equations .Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993.[7] J. Heinonen and P. Koskela. Quasiconformal maps in metric spaces with controlled geometry.
ActaMath. , 181(1):1–61, 1998.[8] S. Keith and X. Zhong. The Poincar´e inequality is an open ended condition.
Ann. of Math. (2) ,167(2):575–599, 2008.[9] T. Kilpel¨ainen and P. Koskela. Global integrability of the gradients of solutions to partial differentialequations.
Nonlinear Anal. , 23(7):899–909, 1994.[10] J. Kinnunen and R. Korte. Characterizations for the Hardy inequality. In
Around the research ofVladimir Maz’ya. I , volume 11 of
Int. Math. Ser. (N. Y.) , pages 239–254. Springer, New York, 2010.[11] R. Korte, J. Lehrb¨ack, and H. Tuominen. The equivalence between pointwise Hardy inequalities anduniform fatness.
Math. Ann. , 351(3):711–731, 2011.[12] R. Korte and N. Shanmugalingam. Equivalence and self-improvement of p -fatness and Hardy’s inequal-ity, and association with uniform perfectness. Math. Z. , 264(1):99–110, 2010.[13] P. Koskela and X. Zhong. Hardy’s inequality and the boundary size.
Proc. Amer. Math. Soc. ,131(4):1151–1158 (electronic), 2003.[14] J. L. Lewis. Uniformly fat sets.
Trans. Amer. Math. Soc. , 308(1):177–196, 1988.[15] J. L. Lewis. On very weak solutions of certain elliptic systems.
Comm. Partial Differential Equations ,18(9-10):1515–1537, 1993.[16] O. E. Maasalo and A. Zatorska-Goldstein. Stability of quasiminimizers of the p -Dirichlet integral withvarying p on metric spaces. J. Lond. Math. Soc. (2) , 77(3):771–788, 2008.[17] V. G. Maz’ya.
Sobolev spaces . Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.Translated from the Russian by T. O. Shaposhnikova.[18] P. Mikkonen. On the Wolff potential and quasilinear elliptic equations involving measures.
Ann. Acad.Sci. Fenn. Math. Diss. , (104):71, 1996.[19] A. Wannebo. Hardy inequalities.
Proc. Amer. Math. Soc. , 109(1):85–95, 1990.(J.L.)
University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35,FI-40014 University of Jyvaskyla, Finland
E-mail address : [email protected] (H.T.) University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35,FI-40014 University of Jyvaskyla, Finland
E-mail address : [email protected] (A.V.V.) University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box35, FI-40014 University of Jyvaskyla, Finland
E-mail address ::