A new Cartan-type property and strict quasicoverings when p=1 in metric spaces
aa r X i v : . [ m a t h . M G ] J a n A new Cartan-type property and strictquasicoverings when p = 1 in metric spaces ∗ Panu LahtiJune 11, 2018
Abstract
In a complete metric space that is equipped with a doubling mea-sure and supports a Poincar´e inequality, we prove a new Cartan-typeproperty for the fine topology in the case p = 1. Then we use thisproperty to prove the existence of 1-finely open strict subsets and strictquasicoverings of 1-finely open sets. As an application, we study fineNewton-Sobolev spaces in the case p = 1, that is, Newton-Sobolevspaces defined on 1-finely open sets. Nonlinear fine potential theory in metric spaces has been studied in severalpapers in recent years, see [6, 7, 8]. Much of nonlinear potential theory, for1 < p < ∞ , deals with p -harmonic functions, which are local minimizers ofthe L p -norm of |∇ u | . Such minimizers can be defined also in metric measurespaces by using upper gradients , and the notion can be extended to the case p = 1 by considering functions of least gradient, which are BV functions thatminimize the total variation locally; see Section 2 for definitions.Nonlinear fine potential theory is concerned with studying p -harmonicfunctions and related superminimizers by means of the p-fine topology . For ∗ : 30L99, 31E05, 26B30. Keywords : metric measure space, function of bounded variation, fine topology, Cartanproperty, strict quasicovering, fine Newton-Sobolev space < p < ∞ , see especially the monographs [1, 15, 23], as well as the mono-graph [3] in the metric setting. The typical assumptions of a metric space,which we make also in this paper, are that the space is complete, equippedwith a doubling measure, and supports a Poincar´e inequality.A central result in fine potential theory is the (weak) Cartan property forsuperminimizer functions. In [21] we proved the following formulation of thisproperty in the case p = 1. Theorem 1.1 ([21, Theorem 1.1]) . Let A ⊂ X and let x ∈ X \ A suchthat A is -thin at x . Then there exist R > and u , u ∈ BV( X ) that are -superminimizers in B ( x, R ) such that max { u ∧ , u ∧ } = 1 in A ∩ B ( x, R ) and u ∨ ( x ) = 0 = u ∨ ( x ) . In [22] we used this property to prove the so-called Choquet propertyconcerning finely open and quasiopen sets in the case p = 1, similarly ascan be done when 1 < p < ∞ (see [7]). On the other hand, it is naturalto consider an alternative version of the weak Cartan property. In the case p >
1, superminimizers are Newton-Sobolev functions, but in the case p = 1they are only BV functions and so the question arises whether the functions u , u above can be replaced by a Newton-Sobolev function (even though itwould no longer be a superminimizer). In Theorem 3.11 we show that sucha new Cartan-type property indeed holds.It is said that a set A is a p - strict subset of a set D if there exists aNewton-Sobolev function u ∈ N ,p ( X ) such that u = 1 on A and u = 0 on X \ D . In [6] it was shown that if U is a p -finely open set (1 < p < ∞ )and x ∈ U , then there exists a p -finely open strict subset V ⋐ U such that x ∈ V . The proof was based on the weak Cartan property. In Theorem 4.3we show that the analogous result is true in the case p = 1. Here we needthe Cartan-type property involving a Newton-Sobolev function (instead ofthe BV superminimizer functions).This result on the existence of 1-strict subsets can be combined with the quasi-Lindel¨of principle to prove the existence of strict quasicoverings of 1-finely open sets, that is, countable coverings by 1-finely open strict subsets.We do this in Proposition 5.4, and it is again analogous to the case 1 < p < ∞ , see [6]. Such coverings will be useful in future research when consideringpartition of unity arguments in finely open sets. In this paper, we apply strictquasicoverings in defining and studying fine Newton-Sobolev spaces , that is,Newton-Sobolev spaces defined on finely open or quasiopen sets. In the case2 < p < ∞ , these were studied in [6]. In Section 5 we show that the theorywe have developed allows us to prove directly analogous results in the case p = 1. In this section we introduce the notation, definitions, and assumptions usedin the paper.Throughout this paper, (
X, d, µ ) is a complete metric space that is equip-ped with a metric d and a Borel regular outer measure µ that satisfies adoubling property, meaning that there exists a constant C d ≥ < µ ( B ( x, r )) ≤ C d µ ( B ( x, r )) < ∞ for every ball B ( x, r ) := { y ∈ X : d ( y, x ) < r } . We also assume that X supports a (1 , X contains atleast 2 points. For a ball B = B ( x, r ) and a >
0, we sometimes abbrevi-ate aB := B ( x, ar ); note that in metric spaces, a ball (as a set) does notnecessarily have a unique center and radius, but we will always understandthese to be predetermined for the balls that we consider. By iterating thedoubling condition, we obtain for any x ∈ X and any y ∈ B ( x, R ) with0 < r ≤ R < ∞ that µ ( B ( y, r )) µ ( B ( x, R )) ≥ C d (cid:16) rR (cid:17) Q , (2.1)where Q > C d . When we wantto state that a constant C depends on the parameters a, b, . . . , we write C = C ( a, b, . . . ). When a property holds outside a set of µ -measure zero, wesay that it holds almost everywhere, abbreviated a.e.As a complete metric space equipped with a doubling measure, X isproper, that is, closed and bounded sets are compact. For any µ -measurableset D ⊂ X , we define Lip loc ( D ) to be the space of functions u on D such thatfor every x ∈ D there exists r > u ∈ Lip( D ∩ B ( x, r )). For anopen set Ω ⊂ X , a function u ∈ Lip loc (Ω) is then in Lip(Ω ′ ) for every openΩ ′ ⋐ Ω; this notation means that Ω ′ is a compact subset of Ω. Other localspaces of functions are defined analogously.For any A ⊂ X and 0 < R < ∞ , the restricted Hausdorff content of3odimension one is defined to be H R ( A ) := inf ( ∞ X i =1 µ ( B ( x i , r i )) r i : A ⊂ ∞ [ i =1 B ( x i , r i ) , r i ≤ R ) . The codimension one Hausdorff measure of A ⊂ X is then defined to be H ( A ) := lim R → H R ( A ) . All functions defined on X or its subsets will take values in [ −∞ , ∞ ].By a curve we mean a nonconstant rectifiable continuous mapping from acompact interval of the real line into X . A nonnegative Borel function g on X is an upper gradient of a function u on X if for all curves γ , we have | u ( x ) − u ( y ) | ≤ Z γ g ds, (2.2)where x and y are the end points of γ and the curve integral is defined byusing an arc-length parametrization, see [16, Section 2] where upper gradientswere originally introduced. We interpret | u ( x ) − u ( y ) | = ∞ whenever at leastone of | u ( x ) | , | u ( y ) | is infinite.Let 1 ≤ p < ∞ ; we are going to work solely with p = 1, but we givedefinitions that cover all values of p where it takes no extra work. We saythat a family of curves Γ is of zero p -modulus if there is a nonnegative Borelfunction ρ ∈ L p ( X ) such that for all curves γ ∈ Γ, the curve integral R γ ρ ds is infinite. A property is said to hold for p -almost every curve if it fails onlyfor a curve family with zero p -modulus. If g is a nonnegative µ -measurablefunction on X and (2.2) holds for p -almost every curve, we say that g is a p -weak upper gradient of u . By only considering curves γ in a set D ⊂ X ,we can talk about a function g being a ( p -weak) upper gradient of u in D .Let D ⊂ X be a µ -measurable set. We define the norm k u k N ,p ( D ) := k u k L p ( D ) + inf k g k L p ( D ) , where the infimum is taken over all p -weak upper gradients g of u in D . Theusual Sobolev space W ,p is replaced in the metric setting by the Newton-Sobolev space N ,p ( D ) := { u : k u k N ,p ( D ) < ∞} , which was first introduced in [25]. We understand every Newton-Sobolevfunction to be defined at every x ∈ D (even though k · k N ,p ( D ) is then only4 seminorm). It is known that for any u ∈ N ,p loc ( D ), there exists a minimal p -weak upper gradient of u in D , always denoted by g u , satisfying g u ≤ g a.e.on D for any p -weak upper gradient g ∈ L p loc ( D ) of u in D , see [3, Theorem2.25].For any D ⊂ X , the space of Newton-Sobolev functions with zero bound-ary values is defined to be N ,p ( D ) := { u | D : u ∈ N ,p ( X ) and u = 0 on X \ D } . This is a subspace of N ,p ( D ) when D is µ -measurable, and it can always beunderstood to be a subspace of N ,p ( X ).The p -capacity of a set A ⊂ X isCap p ( A ) := inf k u k N ,p ( X ) , where the infimum is taken over all functions u ∈ N ,p ( X ) such that u ≥ A . If a property holds outside a set A ⊂ X with Cap p ( A ) = 0, we saythat it holds p -quasieverywhere, or p -q.e. If D ⊂ X is µ -measurable, then k u k N ,p ( D ) = 0 if u = 0 p -q.e. on D, (2.3)see [3, Proposition 1.61].We know that Cap p is an outer capacity, meaning thatCap p ( A ) = inf W open A ⊂ W Cap p ( W )for any A ⊂ X , see e.g. [3, Theorem 5.31]. By [14, Theorem 4.3, Theorem5.1], for any A ⊂ X it holds thatCap ( A ) = 0 if and only if H ( A ) = 0 . (2.4)We say that a set U ⊂ X is p -quasiopen if for every ε > G ⊂ X such that Cap p ( G ) < ε and U ∪ G is open. We say that a function u defined on a set D ⊂ X is p -quasicontinuous on D if for every ε > G ⊂ X such that Cap p ( G ) < ε and u | D \ G is continuous (as areal-valued function). It is a well-known fact that Newton-Sobolev functionsare quasicontinuous; for a proof of the following theorem, see [10, Theorem1.1] or [3, Theorem 5.29]. Theorem 2.5.
Let Ω ⊂ X be open and let u ∈ N ,p (Ω) (with ≤ p < ∞ ).Then u is p -quasicontinuous on Ω . p -capacity of a set A ⊂ D with respect to D ⊂ X is givenby cap p ( A, D ) := inf Z X g pu dµ, where the infimum is taken over functions u ∈ N ,p ( D ) such that u ≥ A ,and g u is the minimal p -weak upper gradient of u (in X ). By truncation, wesee that we can also assume that 0 ≤ u ≤ X (and the same applies to the p -capacity). For basic properties satisfied by capacities, such as monotonicityand countable subadditivity, see [3, 5].Next we recall the definition and basic properties of functions of boundedvariation on metric spaces, following [24]. See also the monographs [2, 11,12, 13, 26] for the classical theory in the Euclidean setting. Let Ω ⊂ X bean open set. Given u ∈ L (Ω), the total variation of u in Ω is defined to be k Du k (Ω) := inf (cid:26) lim inf i →∞ Z Ω g u i dµ : u i ∈ Lip loc (Ω) , u i → u in L (Ω) (cid:27) , where each g u i is the minimal 1-weak upper gradient of u i in Ω. (In [24], localLipschitz constants were used instead of upper gradients, but the propertiesof the total variation can be proved similarly with either definition.) We saythat a function u ∈ L (Ω) is of bounded variation, and denote u ∈ BV(Ω),if k Du k (Ω) < ∞ . For an arbitrary set A ⊂ X , we define k Du k ( A ) := inf W open A ⊂ W k Du k ( W ) . If u ∈ L (Ω) and k Du k (Ω) < ∞ , then k Du k ( · ) is a Radon measure onΩ by [24, Theorem 3.4]. A µ -measurable set E ⊂ X is said to be of finiteperimeter if k Dχ E k ( X ) < ∞ , where χ E is the characteristic function of E .The perimeter of E in Ω is also denoted by P ( E, Ω) := k Dχ E k (Ω).The lower and upper approximate limits of a function u on X are definedrespectively by u ∧ ( x ) := sup (cid:26) t ∈ R : lim r → µ ( B ( x, r ) ∩ { u < t } ) µ ( B ( x, r )) = 0 (cid:27) and u ∨ ( x ) := inf (cid:26) t ∈ R : lim r → µ ( B ( x, r ) ∩ { u > t } ) µ ( B ( x, r )) = 0 (cid:27) . µ -equivalence classes. To consider fine properties, we need to consider thepointwise representatives u ∧ and u ∨ .We will assume throughout the paper that X supports a (1 , C P > λ ≥ B ( x, r ), every u ∈ L ( X ), and every upper gradient g of u ,we have Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C P r Z B ( x,λr ) g dµ, (2.6)where u B ( x,r ) := Z B ( x,r ) u dµ := 1 µ ( B ( x, r )) Z B ( x,r ) u dµ. The (1 , x ∈ X , 0 < r < diam X , and u ∈ N , ( B ( x, r )), then Z B ( x,r ) | u | dµ ≤ C S r Z B ( x,r ) g u dµ (2.7)for some constant C S = C S ( C d , C P ) ≥
1, see [3, Theorem 5.51]. By applyingthis to approximating functions in the definition of the total variation, weobtain for any x ∈ X , 0 < r < diam X , and any µ -measurable set E ⊂ B ( x, r ) µ ( E ) ≤ C S rP ( E, X ) . (2.8)Next we define the fine topology in the case p = 1. Definition 2.9.
We say that A ⊂ X is 1-thin at the point x ∈ X iflim r → r cap ( A ∩ B ( x, r ) , B ( x, r )) µ ( B ( x, r )) = 0 . We say that a set U ⊂ X is 1-finely open if X \ U is 1-thin at every x ∈ U .Then we define the 1-fine topology as the collection of 1-finely open sets on X (see [20, Lemma 4.2] for a proof of the fact that this is indeed a topology).We denote the 1-fine interior of a set H ⊂ X , i.e. the largest 1-finely openset contained in H , by fine-int H . We denote the 1-fine closure of H ⊂ X , i.e.the smallest 1-finely closed set containing H , by H . We define the b H of H ⊂ X to be the set of points in X where H is not u defined on a set U ⊂ X is 1-finely continuousat x ∈ U if it is continuous at x when U is equipped with the induced 1-finetopology on U and [ −∞ , ∞ ] is equipped with the usual topology.7y [3, Proposition 6.16], for all x ∈ X and 0 < r < diam X (in fact, thesecond inequality holds for all r > µ ( B ( x, r ))2 C S r ≤ cap ( B ( x, r ) , B ( x, r )) ≤ C d µ ( B ( x, r )) r , (2.10)and so obviously W ⊂ b W for any open set W ⊂ X .The following fact is given in [19, Proposition 3.3]:Cap ( A ) = Cap ( A ) for any A ⊂ X. (2.11)The following result describes the close relationship between finely openand quasiopen sets. Theorem 2.12 ([22, Corollary 6.12]) . A set U ⊂ X is -quasiopen if andonly if it is the union of a -finely open set and a H -negligible set. For an open set Ω ⊂ X , we denote by BV c (Ω) the class of functions ϕ ∈ BV(Ω) with compact support in Ω, that is, spt ϕ ⋐ Ω. Definition 2.13.
We say that u ∈ BV loc (Ω) is a 1-minimizer in Ω if for all ϕ ∈ BV c (Ω), k Du k (spt ϕ ) ≤ k D ( u + ϕ ) k (spt ϕ ) . (2.14)We say that u ∈ BV loc (Ω) is a 1-superminimizer in Ω if (2.14) holds for allnonnegative ϕ ∈ BV c (Ω).More precisely, we should talk about spt | ϕ | ∨ , since ϕ is only a.e. defined.In the literature, 1-minimizers are usually called functions of least gradient. In this section we prove the new Cartan-type property, given in Theorem3.11. First we take note of a few results that we will need in the proofs; thefollowing is given in [3, Lemma 11.22].
Lemma 3.1.
Let x ∈ X , r > , and A ⊂ B ( x, r ) . Then for every < s < t with tr < diam X , we have cap ( A, B ( x, tr )) ≤ cap ( A, B ( x, sr )) ≤ C S (cid:18) ts − (cid:19) cap ( A, B ( x, tr )) , where C S is the constant from the Sobolev inequality (2.7) . heorem 3.2 ([21, Theorem 3.16]) . Let u be a -superminimizer in an openset Ω ⊂ X . Then u ∧ : Ω → ( −∞ , ∞ ] is lower semicontinuous. As mentioned in the introduction, in [21] we proved a weak Cartan prop-erty for p = 1, more precisely in the following form. Theorem 3.3 ([21, Theorem 5.3]) . Let A ⊂ X and let x ∈ X \ A be such that A is -thin at x . Then there exist R > and E , E ⊂ X such that χ E , χ E ∈ BV( X ) , χ E and χ E are -superminimizers in B ( x, R ) , max { χ ∧ E , χ ∧ E } = 1 in A ∩ B ( x, R ) , χ ∨ E ( x ) = 0 = χ ∨ E ( x ) , { max { χ ∨ E , χ ∨ E } > } is -thin at x ,and lim r → r P ( E , B ( x, r )) µ ( B ( x, r )) = 0 , lim r → r P ( E , B ( x, r )) µ ( B ( x, r )) = 0 . Now we collect a few facts that are not included in the above statement,but follow from the proof given in [21]. Defining B j := B ( x, − j R ) and H j := B j \ B j +1 for j = 0 , , . . . , there exists an open set W ⊃ A that is1-thin at x , W ∩ [ j =0 , ,... H j ⊂ E and W ∩ [ j =1 , ,... H j ⊂ E , (3.4)and E ⊂ (cid:0) B \ B (cid:1) ∪ ∞ [ j =2 , ,... B j \ B j +1 and E ⊂ (cid:0) B \ B (cid:1) ∪ ∞ [ j =3 , ,... B j \ B j +1 . (3.5)Moreover, by [21, Eq (5.6)], for all i = 2 , , , . . . we have P ( E ∩ B i , X ) ≤ C S cap ( W ∩ B i , B i ) , (3.6)and similarly for all i = 3 , , , . . . , P ( E ∩ B i , X ) ≤ C S cap ( W ∩ B i , B i ) . (3.7)From the proof it can also be seen that if R > B j and H j be defined as above.9 emma 3.8. Let A ⊂ X and let x ∈ X \ A be such that A is -thin at x .Then there exists a number R > , an open set W ⊃ A that is -thin at x ,and open sets F j ⊃ W ∩ H j such that F j ⊂ B j \ B j +1 for all j = 0 , , . . . ,and ∞ X j = i P ( F j , X ) ≤ C S cap ( W ∩ B i , B i ) (3.9) for all i = 0 , , . . . .Proof. By using the weak Cartan property (Theorem 3.3), choose
R > E , E ⊂ X such that χ E , χ E ∈ BV( X ) and χ E and χ E are 1-super-minimizers in B ( x, R ). We can assume that R < diam X . Also let W ⊃ A be an open set that is 1-thin at x , as described above. Define F j := { χ ∧ E > } ∩ B j \ B j +1 for j = 2 , , . . . , and F j := { χ ∧ E > } ∩ B j \ B j +1 for j = 3 , , . . . . By (3.4), we have F j ⊃ W ∩ H j for all j = 2 , , . . . as desired. The sets F j are open by Theorem 3.2. By Lebesgue’s differentiation theorem, the sets { χ ∧ E > } and { χ ∧ E > } differ from E and E , respectively, only by a set of µ -measure zero. Thus by (3.5) and the fact that the sets F j are at a positivedistance from each other, we find that for all i = 2 , , . . . , P ( E ∩ B i , X ) = P [ j = i,i +2 ,... F j , X ! = X j = i,i +2 ,... P ( F j , X ) , and similarly for all i = 3 , , . . . , P ( E ∩ B i , X ) = X j = i,i +2 ,... P ( F j , X ) . Combining these with (3.6) and (3.7), and using Lemma 3.1, we have for all i = 2 , , . . . ∞ X j = i P ( F j , X ) ≤ C S (cap ( W ∩ B i , B i ) + cap ( W ∩ B i +1 , B i +1 )) ≤ C S (cap ( W ∩ B i , B i ) + 5 C S cap ( W ∩ B i +1 , B i +1 )) ≤ C S (cap ( W ∩ B i , B i ) + cap ( W ∩ B i , B i +1 ))= 50 C S cap ( W ∩ B i , B i ) . Then by replacing R with R/
4, we have the result.10ecall the constant λ ≥ µ ( X ) = ∞ , but the proof reveals that we can alternativelyassume µ ( F ) < µ ( X ) / Theorem 3.10.
Let F ⊂ X be an open set of finite perimeter with µ ( F ) <µ ( X ) / (in particular, µ ( F ) is finite). Then there exists a collection of balls { B k = B ( x k , r k ) } k ∈ N such that the balls λB k are disjoint, F ⊂ S ∞ k =1 λB k , C d ≤ µ ( B k ∩ F ) µ ( B k ) ≤ for all k ∈ N , and ∞ X k =1 µ (5 λB k )5 λr k ≤ C B P ( F, X ) for some constant C B = C B ( C d , C P , λ ) . Now we can show the following Cartan-type property.
Theorem 3.11.
Let A ⊂ X and let x ∈ X \ A be such that A is -thin at x . Then there exists a number R > , open sets G ⊂ V ⊂ X , and a function η ∈ N , ( V ) such that A ∩ B ( x, R ) ⊂ G , V is -thin at x , ≤ η ≤ on X , η = 1 on G , and lim r → rµ ( B ( x, r )) k η k N , ( B ( x,r )) = 0 . (3.12) Proof.
Take
R >
0, an open set W ⊃ A , and open sets F j ⊂ B j \ B j +1 asgiven by Lemma 3.8. Let δ := 12 (680 λ ) Q C d C S , where Q > R ≤ min (cid:8) , diam X (cid:9) .Since µ ( { x } ) = 0 (see [3, Corollary 3.9]), we can also assume R to be so smallthat µ ( B ) < µ ( X ), and so also µ ( F j ) < µ ( X ) for all j = 0 , , . . . . Since W is 1-thin at x , we can further assume that R is so small that2 − j R cap ( W ∩ B j , B j ) µ ( B j ) < δ (3.13)11or all j = 0 , , . . . . Fix j . By the boxing inequality (Theorem 3.10) we finda collection of balls { B jk = B ( x jk , r jk ) } ∞ k =1 such that the balls λB jk are disjoint, F j ⊂ S ∞ k =1 λB jk , 12 C d ≤ µ ( B jk ∩ F j ) µ ( B jk ) ≤
12 (3.14)for all k ∈ N , and ∞ X k =1 µ (5 λB jk )5 λr jk ≤ C B P ( F j , X ) . (3.15)Thus we have µ ( B jk ) ≤ C d µ ( B jk ∩ F j ) ≤ C d µ ( F j ) ≤ − j RC d C S P ( F j , X ) by (2.8) ≤ − j RC d C S cap ( W ∩ B j , B j ) by (3.9) . Thus for all k ∈ N , µ ( B jk ) µ ( B j ) ≤ C d C S − j R cap ( W ∩ B j , B j ) µ ( B j ) ≤ C d C S δ (3.16)by (3.13). By (3.14) we necessarily have F j ∩ B jk = ∅ for all k ∈ N , and so B j ∩ B jk = ∅ . Now if r jk ≥ − j R for some k ∈ N , then B j ⊂ B jk and so µ ( B jk ) µ ( B j ) ≥ C d , contradicting (3.16) by our choice of δ . Thus r jk ≤ − j R for all k ∈ N , sothat x jk ∈ B j , and thus by (2.1), r jk − j +2 R ! Q ≤ C d µ ( B jk ) µ (4 B j ) ≤ C d µ ( B jk ) µ ( B j ) ≤ C d C S δ by (3.16), so that by our choice of δ , r jk ≤ (2 C d C S δ ) /Q − j +2 R = 2 − j R λ . (3.17)12hus recalling that F j ∩ B jk = ∅ , so that ( B j \ B j +1 ) ∩ B jk = ∅ , we concludethat 20 λB jk ⊂ B j − \ B j +2 (let B − := B ( x, R )). Now, define Lipschitzfunctions ξ jk := max ( , − dist( · , λB jk )10 λr jk ) , k ∈ N , so that ξ jk = 1 on 10 λB jk and ξ jk = 0 on X \ λB jk . Using the basic propertiesof 1-weak upper gradients, see [3, Corollary 2.21], we obtain Z X g ξ jk dµ ≤ µ (20 λB jk )10 λr jk . Define V := S ∞ j =0 S ∞ k =1 λB jk . Now for every i = 1 , , . . . ,cap ( V ∩ B i , B i ) ≤ cap ∞ [ j = i − ∞ [ k =1 λB jk , B i ! ≤ ∞ X j = i − ∞ X k =1 cap (cid:0) λB jk , B i (cid:1) ≤ ∞ X j = i − ∞ X k =1 Z X g ξ jk dµ ≤ ∞ X j = i − ∞ X k =1 µ (20 λB jk )10 λr jk ≤ C d C B ∞ X j = i − P ( F j , X ) by (3.15) ≤ C d C B C S cap ( W ∩ B i − , B i − ) by (3.9) . (3.18)Thus2 − i R cap ( V ∩ B i , B i ) µ ( B i ) ≤ C d C B C S − i +1 R cap ( W ∩ B i − , B i − ) µ ( B i − ) → i → ∞ , since W is 1-thin at x . By Lemma 3.1 it is then straightforwardto show that V is also 1-thin at x . Let us also define the Lipschitz functions η jk := max ( , − dist( · , λB jk )5 λr jk ) , j = 0 , , . . . , k = 1 , , . . . ,
13o that η jk = 1 on 5 λB jk and η jk = 0 on X \ λB jk , and then let η := sup j =0 , ,..., k =1 , ,... η jk . Recall from Lemma 3.8 that S ∞ j =0 F j ⊃ W ∩ B ( x, R ); thus η ≥ G := ∞ [ j =0 ∞ [ k =1 λB jk ⊃ ∞ [ j =0 F j ⊃ W ∩ B ( x, R ) ⊃ A ∩ B ( x, R ) . By [3, Lemma 1.52] we know that g η ≤ P ∞ j =0 P ∞ k =1 g η jk . Thus for any i =1 , , . . . , Z B i g η dµ ≤ ∞ X j =0 ∞ X k =1 Z B i g η jk dµ ≤ ∞ X j = i − ∞ X k =1 Z X g η jk dµ ≤ C d C B C S cap ( W ∩ B i − , B i − ) , where the last inequality follows just as in the last four lines of (3.18). Sincewe assumed R ≤ λr jk ≤ k η k L ( B i ) ≤ C d C B C S cap ( W ∩ B i − , B i − ) . Using the fact that W is 1-thin at x and the doubling property of µ , we get(3.12). Estimating just as in the last four lines of (3.18), now with i = 1, weget Z X g η dµ ≤ ∞ X j =0 ∞ X k =1 Z X g η jk dµ ≤ C d C B C S cap ( W ∩ B , B ) < ∞ . Thus η ∈ N , ( X ). Clearly η = 0 on X \ V , and so η ∈ N , ( V ). -strict subsets In this section we study 1-strict subsets which are defined as follows.
Definition 4.1.
A set A ⊂ D is a of D ⊂ X if there is afunction u ∈ N , ( D ) such that u = 1 on A .14quivalently, A is a 1-strict subset of D if cap ( A, D ) < ∞ . In [22,Proposition 6.7] we proved the following result by using the weak Cartanproperty (Theorem 3.3). Proposition 4.2.
Let U ⊂ X be -finely open and let x ∈ U . Then thereexists a -finely open set W such that x ∈ W ⊂ U , and a function w ∈ BV( X ) such that ≤ w ≤ on X , w ∧ = 1 on W , and spt w ⋐ U . This kind of formulation is sufficient for some purposes, but now we areable to improve it by replacing w ∈ BV( X ) with w ∈ N , ( X ). The followingis our main result on the existence of 1-strict subsets. Theorem 4.3.
Let U ⊂ X be -finely open and let x ∈ U . Then there existsa -finely open set W such that x ∈ W ⊂ U , and a function w ∈ N , ( U ) such that ≤ w ≤ on X , w = 1 on W , and spt w ⋐ U .Moreover, if Cap ( { x } ) = 0 , then k w k N , ( X ) can be made arbitrarilysmall.Proof. Applying Theorem 3.11 with the choice A = X \ U , we find a number R >
0, open sets G ⊂ V ⊂ X , and a function η ∈ N , ( V ) such that B ( x, R ) ⊂ G ∪ U , V is 1-thin at x , 0 ≤ η ≤ X , η = 1 on G , andlim r → rµ ( B ( x, r )) k η k N , ( B ( x,r )) = 0 . Choose 0 < r ≤ R such that r k η k N , ( B ( x,r )) /µ ( B ( x, r )) ≤ ρ := max (cid:26) , − · , B ( x, r/ r (cid:27) ∈ Lip( X ) , so that 0 ≤ ρ ≤ X , ρ = 1 on B ( x, r/ ρ ⋐ B ( x, r ). Thenlet w := (1 − η ) ρ . By the Leibniz rule (see [3, Theorem 2.15]), we have w ∈ N , ( X ) and Z X g w dµ = Z B ( x,r ) g w dµ ≤ Z B ( x,r ) g η dµ + Z B ( x,r ) g ρ dµ ≤ µ ( B ( x, r )) r + 4 µ ( B ( x, r )) r . Thus k w k N , ( X ) ≤ (5 /r +1) µ ( B ( x, r )). If Cap ( { x } ) = 0, then also H ( { x } ) =0 by (2.4), and so we can make µ ( B ( x, r )) /r as small as we like by choosingsuitable r . Then we can also make k w k N , ( X ) arbitrarily small.15egardless of the value of Cap ( { x } ), the set V is 1-thin at x , that is, x / ∈ b V . Since V is open we have V ⊂ b V ; recall (2.10) and the commentafter it. We know that V = V ∪ b V by [19, Corollary 3.5], so in conclusion x / ∈ V . Thus W := B ( x, r/ \ V ⊂ { w = 1 } is a 1-finely open neighborhood of x . Finally, spt w is compact andspt w ⊂ spt ρ \ G ⊂ ( U ∪ G ) \ G ⊂ U, so that spt w ⋐ U . Clearly now w ∈ N , ( U ).Let us make a few more observations concerning 1-strict subsets. Ingeneral it is not clear which subsets A of a set D are 1-strict subsets. If A isa compact subset of an open set D , we obviously have cap ( A, D ) < ∞ , andthe test function can even be chosen to be Lipschitz. When A is a compactsubset of a 1- quasiopen set D , we cannot necessarily choose a Lipschitz testfunction but one might nonetheless suspect that cap ( A, D ) < ∞ . However,this is not always the case. Example 4.4.
Let X = R (unweighted), denote the origin by 0, and let A := ∞ [ j =1 A j ∪ { } with A j := { − j } × [ − / (2 j ) , / (2 j )] . Denoting A εj := { x ∈ X : dist( x, A j ) < ε } , with ε >
0, let D := ∞ [ j =1 D j ∪ { } with D j := A − j j . Since all the sets D j are disjoint, it is straightforward to check thatcap ( A, D ) = ∞ X j =1 cap ( A j , D j ) = ∞ X j =1 j = ∞ . Now A is clearly a compact set, and D is 1-quasiopen since D ∪ B (0 , r ) is anopen set for every r > A, D connected by adding the line (0 , / ×{ } to A , and by adding e.g. the sets (2 − j − , − j ) × ( − − j − , − j − ) to D ; then westill have cap ( A, D ) = ∞ but the calculation is somewhat more complicated.The variational 1-capacity is an outer capacity in the following weak sense.16 roposition 4.5. Let A ⊂ D ⊂ X . Then cap ( A, D ) = inf V -quasiopen A ⊂ V ⊂ D cap ( V, D ) . Proof.
We can assume that cap ( A, D ) < ∞ . Fix 0 < ε <
1. Take u ∈ N , ( D ) such that u = 1 on A and R X g u dµ < cap ( A, D ) + ε . The set V := { u > − ε } is 1-quasiopen by Theorem 2.5, andcap ( V, D ) ≤ Z X g u/ (1 − ε ) dµ = R X g u dµ − ε ≤ cap ( A, D ) + ε − ε . Since 0 < ε <
Open Problem. If D ⊂ X and A ⊂ fine-int D , do we havecap ( A, D ) = inf V A ⊂ V ⊂ D cap ( V, D )?Note that according to Theorem 4.3, the above property does hold in thevery special case when A is a point with 1-capacity zero.Let us say that a set K ⊂ X is if X \ K is 1-quasiopen.Now we can show that 1-strict subsets have the following continuity. Proposition 4.6.
Let D ⊂ X and let K ⊃ K ⊃ . . . be bounded -quasiclosed subsets of D such that cap ( K , D ) < ∞ . Then for K := T ∞ i =1 K i we have cap ( K, D ) = lim i →∞ cap ( K i , D ) . We will show in Example 4.7 below that the assumption cap ( K , D ) < ∞ is needed. Proof.
Fix ε >
0. By Proposition 4.5 we find a 1-quasiopen set V such that K ⊂ V ⊂ D and cap ( V, D ) < cap ( K, D ) + ε . For each j ∈ N we find anopen set e G j ⊂ X such that V ∪ e G j is open and Cap ( e G j ) → j → ∞ . Foreach i, j ∈ N , we find an open set G i,j ⊂ X such that K i \ G i,j is compactand Cap ( G i,j ) < − i − j . Letting G j := e G j ∪ S ∞ i =1 G i,j for each j ∈ N , wehave that each V ∪ G j is open, each K i \ G j is compact, and Cap ( G j ) → j → ∞ . Then for each j ∈ N we find a function w j ∈ N , ( X ) such that0 ≤ w j ≤ X , w j = 1 on G j , and k w j k N , ( X ) → j → ∞ . Passing toa subsequence (not relabeled), we can assume that w j → ( K , D ) < ∞ , we find v ∈ N , ( D ) such that 0 ≤ v ≤ X and v = 1 on K . For each j ∈ N , let ρ j := vw j . Then k ρ j k L ( X ) → j → ∞ , and by the Leibniz rule (see [3, Theorem 2.15]), Z X g ρ j dµ ≤ Z X g w j dµ + Z X w j g v dµ → j → ∞ ; for the second term this follows from Lebesgue’s dominatedconvergence theorem. Thus cap ( G j ∩ K , D ) →
0. Fix j ∈ N such thatcap ( G j ∩ K , D ) < ε . Since every K i \ G j is compact and V ∪ G j is open,for some i ∈ N we have K i \ G j ⊂ V ∪ G j . Thus K i ⊂ V ∪ G j . Thencap ( K i , D ) ≤ cap ( V ∪ ( G j ∩ K ) , D ) ≤ cap ( V, D ) + cap ( G j ∩ K , D ) ≤ cap ( K, D ) + ε + cap ( G j ∩ K , D ) ≤ cap ( K, D ) + 2 ε. Since ε >
Example 4.7.
In the notation of Example 4.4, let K i := S ∞ j = i A j ∪ { } foreach i ∈ N . These are compact sets and similarly as in Example 4.4 wefind that cap ( K i , D ) = ∞ for every i ∈ N . However, cap ( K, D ) = 0 for K := T ∞ i =1 K i = { } , by the fact that a point has 1-capacity zero and byusing (2.3). Bj¨orn–Bj¨orn–Latvala [6] have studied different definitions of Newton-Sobolevspaces on quasiopen sets in metric spaces in the case 1 < p < ∞ . As anapplication of the theory we have developed, we show that the analogousresults hold for p = 1.First we prove the following fact in a very similar way as it is proved inthe case 1 < p < ∞ , see [7, Theorem 1.4(b)] and [8, Theorem 4.9(b)]. Recallthat a function u defined on a set U ⊂ X is 1-quasicontinuous on U if forevery ε > G ⊂ X such that Cap ( G ) < ε and u | U \ G iscontinuous (as a real-valued function).18 heorem 5.1. A function u on a -quasiopen set U is -quasicontinuous on U if and only if it is finite -q.e. and -finely continuous -q.e. on U .Proof. To prove one direction, suppose there is a set N ⊂ U such thatCap ( N ) = 0 and u is finite and 1-finely continuous at every point in V := U \ N . By Theorem 2.12, we can assume that V is 1-finely open. Let { ( a j , b j ) } ∞ j =1 be an enumeration of all intervals in R with rational endpointsand let V j := { x ∈ V : a j < u ( x ) < b j } . By the 1-fine continuity of u , the sets V j are 1-finely open. Hence by Theorem2.12, they are also 1-quasiopen. Fix ε >
0. There are open sets G j ⊂ X suchthat Cap ( G j ) < − j − ε and each V j ∪ G j is open. Since Cap is an outercapacity, there is also an open set G N ⊃ N such that Cap ( G N ) < ε/
2. Now G := G N ∪ ∞ [ j =1 G j is an open set such that Cap ( G ) < ε , and u | U \ G is continuous since V j ∪ G are open sets.To prove the converse direction, by Theorem 2.12 we know that U = V ∪ N , where V is 1-finely open and H ( N ) = 0, and then also Cap ( N ) = 0by (2.4). By the quasicontinuity of u , for each j ∈ N we find an open set G j ⊂ X such that Cap ( G j ) < /j and u | V \ G j is continuous. By (2.11), wehave Cap ( G j ) = Cap ( G j ) for each j ∈ N , and so the set A := N ∪ ∞ \ j =1 G j satisfies Cap ( A ) = 0. If x ∈ U \ A , then x ∈ V \ G j for some j ∈ N . Since V \ G j is a 1-finely open set and u | V \ G j is continuous, it follows that u isfinite and 1-finely continuous at x .We will need the following quasi-Lindel¨of principle from [22]. Theorem 5.2 ([22, Theorem 5.2]) . For every family V of -finely open setsthere is a countable subfamily V ′ such that Cap [ V ∈V V \ [ V ′ ∈V ′ V ′ ! = 0 . U ⊂ X is always a 1-quasiopen set. Note that 1-quasiopensets are µ -measurable by [4, Lemma 9.3]. Definition 5.3.
A family B of 1-quasiopen sets is a of U ifit is countable, S V ∈B V ⊂ U , and Cap (cid:0) U \ S V ∈B V (cid:1) = 0. If every V ∈ B is a 1-finely open 1-strict subset of U with V ⋐ U , then B is a of U . Moreover, we say that1. u ∈ N , − loc ( U ) if u ∈ N , ( V ) for every 1-finely open 1-strict subset V ⋐ U ;2. u ∈ N , − loc ( U ) (respectively L − loc ( U )) if there is a 1-quasicovering B of U such that u ∈ N , ( V ) (respectively L ( V )) for every V ∈ B . Proposition 5.4.
There exists a -strict quasicovering B of U . Moreover,the associated Newton-Sobolev functions can be chosen compactly supportedin U .Proof. By Theorem 2.12, we have U = V ∪ N , where V is 1-finely openand H ( N ) = 0, and then also Cap ( N ) = 0 by (2.4). For every x ∈ V , byTheorem 4.3 we find a 1-finely open set V x ∋ x such that V x ⋐ V and anassociated function v x ∈ N , ( V ) such that 0 ≤ v x ≤ X , v x = 1 on V x ,and spt v x ⋐ V . The collection B ′ := { V x } x ∈ V covers V , and by the quasi-Lindel¨of principle (Theorem 5.2) and the fact that Cap ( U \ V ) = 0, thereexists a countable subcollection B ⊂ B ′ such that Cap (cid:0) U \ S V x ∈B V x (cid:1) =0. It follows that N , − loc ( U ) ⊂ N , − loc ( U ). From now on, since the proofsgiven in [6] in the case 1 < p < ∞ apply almost verbatim also in our setting,we will only point out the differences with [6]. Theorem 5.5.
Let u ∈ N , − loc ( U ) . Then u if finite -q.e. and -finelycontinuous -q.e. on U . Thus u is also -quasicontinuous on U .Proof. Follow verbatim the proof of [6, Theorem 4.4], except that replace thereference to [6, Proposition 4.2] by Proposition 5.4, and the references to [8,Theorem 4.9(b)] and [7, Theorem 1.4(b)] by Theorem 5.1.
Definition 5.6.
A nonnegative function e g u on U is a of u ∈ N , − loc ( U ) if there is a quasicovering B of U such that for every V ∈ B , u ∈ N , ( V ) and e g u = g u,V a.e. on V , where g u,V is the minimal1-weak upper gradient of u in V . 20 emma 5.7. If u ∈ N , − loc ( U ) , then it has a unique (in the a.e. sense) -fine upper gradient e g u .Proof. Follow verbatim the proof of [6, Lemma 5.2].
Theorem 5.8. If u ∈ N , − loc ( U ) and e g u is a -fine upper gradient of u ,then e g u is a -weak upper gradient of u in U .Proof. Follow verbatim the proof of [6, Theorem 5.3].
Proposition 5.9. If u ∈ N , − loc ( U ) , then there is a -strict quasicovering B of U such that for every V ∈ B , there exists u V ∈ N , ( X ) with u = u V on V .Proof. Follow verbatim the proof of [6, Proposition 5.5], except that replacethe reference to [6, Theorem 4.4] by Theorem 5.5, and [6, Proposition 4.2]by Proposition 5.4.The following definition is originally from Kilpel¨ainen–Mal´y [17].
Definition 5.10.
Let U ⊂ R n . A function u ∈ L ( U ) is in W , ( U ) if1. there is a quasicovering B of U such that for every V ∈ B there is anopen set G V ⊃ V and u V ∈ W , ( G V ) such that u = u V on V , and2. the fine gradient ∇ u , defined by ∇ u = ∇ u V a.e. on each V ∈ B , alsobelongs to L ( U ).Moreover, let k u k W , ( U ) := Z U ( | u | + |∇ u | ) dx. Recall that we constantly assume U to be a 1-quasiopen set. Theorem 5.11.
Let U ⊂ R n . Then u ∈ W , ( U ) if and only if there exists v ∈ N , ( U ) such that v = u a.e. on U . Moreover, g v = |∇ u | a.e. on U and k v k N , ( U ) = k u k W , ( U ) . Here g v is the minimal 1-weak upper gradient of v in U . Proof.
Follow verbatim the proof of [6, Theorem 5.7], except that replace thereference to [6, Proposition 5.5] by Proposition 5.9, [3, Proposition A.12] by[3, Corollary A.4], and [6, Theorem 5.4] by Theorem 5.8.21eturning momentarily to the metric space setting, define the space b N , ( U ) := { u : u = v a.e. for some v ∈ N , ( U ) } . Theorem 5.12.
Let u ∈ b N , ( U ) . Then u ∈ N , ( U ) if and only if u is -quasicontinuous on U .Proof. Assume that u is 1-quasicontinuous on U . There is v ∈ N , ( U ) suchthat u = v a.e. on U . By Theorem 5.5, v is 1-quasicontinuous on U . By[3, Proposition 5.23] and [9, Proposition 4.2], u = v U , and thus u ∈ N , ( U ) by (2.3).The converse follows from Theorem 5.5. Theorem 5.13.
Let U ⊂ R n , and let u be an everywhere defined function on U . Then u ∈ N , ( U ) if and only if u ∈ W , ( U ) and u is -quasicontinuous.Moreover, then k u k N , ( U ) = k u k W , ( U ) .Proof. This follows from Theorems 5.11 and 5.12.
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