The weak Cartan property for the p-fine topology on metric spaces

The weak Cartan property for the p-fine topology on metric spaces Anders Björn arXiv:1310.8101v1 [math.AP] 30 Oct 2013 Department of Mathematics, Linköpings universitet, SE-581 83 Linköping, Sweden; [email protected] Jana Björn Department of Mathematics, Linköpings universitet, SE-581 83 Linköping, Sweden; [email protected] Visa Latvala Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland ; [email protected] Abstract. We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all psuperharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients and do not rely on a vector-valued differentiable structure. Key words and phrases: capacity, coarsest topology, doubling, fine topology, finely continuous, metric space, p-harmonic, Poincaré inequality, quasicontinuous, superharmonic, thick, thin, weak Cartan property, Wiener criterion. Mathematics Subject Classification (2010): Primary: 31E05; Secondary: 30L99, 31C40, 31C45, 35J92, 49Q20. 1. Introduction The aim of this paper is to study the p-fine topology and the fine potential theory associated with p-harmonic functions on a complete metric space X equipped with a doubling measure µ supporting a p-Poincaré inequality, 1 < p < ∞. Nonlinear potential theory associated with p-harmonic functions has been studied since the 1960s. For extensive treatises and notes on the history, see the monographs Adams–Hedberg [1] and Heinonen–Kilpeläinen–Martio [32], the latter developing the theory on weighted Rn (with respect to p-admissible weights). Starting in the 1990s a lot of attention has been given to analysis on metric spaces, see e.g. Hajlasz [24], [25], Hajlasz–Koskela [28], Heinonen [29], [30], and Heinonen–Koskela [33]. Around 2000 this initiated studies of p-harmonic and psuperharmonic functions on metric spaces without a differentiable structure, by e.g. Shanmugalingam [56], Kinnunen–Martio [39], Kinnunen–Shanmugalingam [41], Björn–MacManus–Shanmugalingam [20] and Björn–Björn–Shanmugalingam [12], 1 2 Anders Björn, Jana Björn and Visa Latvala [13]. The theory has later been further developed by these and other authors, see the monograph Björn–Björn [7] and the references therein. While p-harmonic functions are known to be locally Hölder continuous (even on metric spaces, see [41]), p-superharmonic functions are in general only lower semicontinuous. However, at points of discontinuity they still exhibit more regularity than just lower semicontinuity, namely, the limit lim u(x), as x → x0 , exists along a substantial (in a capacitary sense) part of x0 ’s neighbourhood and equals u(x0 ). The topology giving rise to such neighbourhoods and limits is called the p-fine topology. Together with the associated fine potential theory it goes back to Cartan in the 1940s in the linear case p = 2, which has been later systematically studied, see e.g. Fuglede [22], [23] and Lukeš–Malý–Zajı́ček [48]. The nonlinear fine potential theory started in the 1970s, with papers by e.g. Maz′ ya [50], Maz′ ya–Havin [51], [52], Hedberg [26], Adams–Meyers [3], Meyers [53], Hedberg–Wolff [27], Adams–Lewis [2] and Lindqvist–Martio [47]. See also the notes to Chapter 12 in Heinonen–Kilpeläinen–Martio [32] and Section 2.6 in Malý– Ziemer [49]. In the 1990s the fine potential theory associated with p-harmonic functions was developed further in Heinonen–Kilpeläinen–Martio [31], Kilpeläinen– Malý [36], [37], Latvala [44], [45], [46], and the monograph Malý–Ziemer [49] for unweighted Rn . The monograph [32] is the main source for fine potential theory on weighted Rn (note that Chapter 21, which is only in the second addition, contains some more recent results). See also Mikkonen [54] for related results (in weighted Rn ) on the Wolff potential. In fact, the Wolff potential appeared already in Maz′ ya–Havin [52]. The fine potential theory in metric spaces is more recent, starting with Kinnunen– Latvala [38], J. Björn [18] and Korte [42], where it was shown that p-superharmonic functions on open subsets of metric spaces are p-finely continuous. There are also some related more recent results in Björn–Björn [8] and [9]. As in the classical situation, the p-fine topology on metric spaces is defined by means of p-capacity and p-thin sets, see Section 4. From now on we drop the p from the notation and just write e.g. fine and superharmonic even though the notions depend on p. Our first main result complements the results in [18], [31], [38] and [42] as follows. Theorem 1.1. The fine topology is the coarsest topology making all superharmonic functions on open subsets of X continuous. The superharmonic functions considered in this and most of the earlier papers on metric spaces are defined through upper gradients (see later sections for precise definitions), which in particular means that we have no equation, only variational inequalities, to work with. In this way the results do not depend on any differentiable structure of the metric space. The proofs of our main results are based on pointwise estimates of capacitary potentials. These estimates lead in a natural way to a central property which we call the weak Cartan property, see Theorem 5.1. The following consequence is a slight reformulation and extension of the weak Cartan property. Theorem 1.2. Let E ⊂ X be an arbitrary set, and let x0 ∈ E \ E. Then the following are equivalent : (a) E is thin at x0 ; p p (b) x0 ∈ / E , where E is the fine closure of E; (c) X \ E is a fine neighbourhood of x0 ; (d) there are k ≥ 2 superharmonic functions u1 , ... , uk in an open neighbourhood of x0 such that the function v = max{u1 , ... , uk } satisfies v(x0 ) < lim inf v(x); E∋x→x0 (1.1) The weak Cartan property for the p-fine topology on metric spaces 3 (e) condition (d) holds with k = 2 nonnegative bounded superharmonic functions. Here and elsewhere, a set U is a fine neighbourhood of a point x0 if it contains a finely open set V ∋ x0 ; it is not required that U itself is finely open. Note also that if x0 ∈ E, then E is thin at x0 if and only if Cp ({x0 }) = 0 and E \ {x0 } is thin at x0 . This is a consequence of the following generalization of Theorem 6.33 in Heinonen–Kilpeläinen–Martio [32]. Proposition 1.3. If Cp ({x0 }) > 0, then {x0 } is thick at x0 . Note that the converse statement is trivially true. At points with positive capacity we further improve Theorem 1.2 and obtain the usual Cartan property (with k = 1), see Proposition 6.3. (Note that in weighted Rn and in metric spaces it can happen that some points have positive capacity while others do not. A sharp condition for when Cp ({x0 }) > 0 is given in Proposition 8.3 in Björn–Björn– Lehrbäck [10].) Proposition 6.3 also shows that E is thin at x0 ∈ E \ E with Cp ({x0 }) > 0 if and only if the seemingly weaker condition lim Cp (E ∩ B(x0 , ρ)) = 0 ρ→0 holds. This characterization fails for points with zero capacity. The classical Cartan property says that if E ⊂ Rn is thin at x0 ∈ E \ E, then for every r > 0 there is a nonnegative bounded superharmonic function u on B(x0 , r) such that u(x0 ) < lim inf u(x), E∋x→x0 see Theorem 1.3 in Kilpeläinen–Malý [37] or Theorem 2.130 in Malý–Ziemer [49] for the nonlinear case on unweighted Rn , and Theorem 21.26 in Heinonen–Kilpeläinen– Martio [32] (only in the second edition) for weighted Rn . In the generality of this paper, for superharmonic functions defined through upper gradients on metric spaces, it is not known whether the classical Cartan property (with k = 1) holds, since its proof is based on the equation rather than on the minimization problem. Using variational methods, we have only been able to prove it for points with positive capacity in Proposition 6.3. However, the weak Cartan property provides us with two superharmonic functions whose maximum in many situations can be used instead of the usual Cartan property (but not always, since the maximum need not be superharmonic). In particular Theorem 1.1 follows quite easily. The (strong) Cartan property is closely related to the necessity part of the Wiener criterion, as it provides a superharmonic function which is not continuous at x0 , and can thus be used to obtain a p-harmonic function which does not attain its continuous boundary values at x0 . The weak Cartan property only leads to the necessity part of the Wiener criterion for certain domains, see Remark 5.6. Due to the lack of equation, the necessity part of the Wiener criterion for general domains in metric spaces is not known for p-harmonic functions defined by means of upper gradients, while for Cheeger p-harmonic functions based on a vector-valued differentiable structure it was proved in J. Björn [17]. The sufficiency part of the Wiener criterion in metric spaces was proved in Björn–MacManus–Shanmugalingam [20] and J. Björn [18]. In Euclidean spaces, the Wiener criterion was obtained in Maz′ ya [50], Lindqvist–Martio [47], Heinonen–Kilpeläinen–Martio [32], Kilpeläinen–Malý [36] and Mikkonen [54]. The outline of the paper is as follows: In Sections 2 and 3 we introduce the necessary background on metric spaces, upper gradients, Newtonian spaces, capacity and superharmonic functions. In Section 4 we introduce the fine topology, cite the necessary background results, and establish a number of auxiliary results not requiring the weak Cartan property nor the capacitary estimates used to establish 4 Anders Björn, Jana Björn and Visa Latvala it. We also conclude the following generalization of a result by J. Björn [18] and Korte [42], who (independently) established the result corresponding to (b) for open sets U , see Theorem 4.3. Theorem 1.4. (a) Any quasiopen set U ⊂ X can be written as U = V ∪ E, where V is finely open and Cp (E) = 0. (b) Let u be a quasicontinuous function on a quasiopen or finely open set U . Then u is finely continuous q.e. in U . A fundamental step in the proof is the fact that the capacity of a set coincides with the capacity of its fine closure, see Lemma 4.8 which generalizes Corollary 4.5 in J. Björn [18]. Section 5 is devoted to the proof of the weak Cartan property (Theorem 5.1). Also Theorem 1.2 is established. In the last section, Section 6, we draw a number of consequences of the weak Cartan property, including Theorem 1.1 and Proposition 1.3, and end the paper by proving the following characterization of fine continuity, which as pointed out in Malý–Ziemer [49] is by no means trivial. Theorem 1.5. Let u be a function on a fine neighbourhood U of x0 . Then the following conditions are equivalent : (a) u is finely continuous at x0 ; (b) the set {x ∈ U : |u(x) − u(x0 )| ≥ ε} is thin at x for each ε > 0; (c) there exists a set E which is thin at x0 such that u(x0 ) = lim U\E∋x→x0 u(x), where the limit is taken with respect to the metric topology. Many of the results in this paper are known on weighted Rn , but as far as we know, Theorem 1.4 and Proposition 6.3 are new on weighted Rn and Lemma 4.8 is new even on unweighted Rn . Note also that many of our proofs in Sections 5 and 6 differ from the proofs on weighted Rn , since our approach is purely based on variational inequalities, not on an equation. The proofs of the auxiliary results in Section 4 are analogous to the Euclidean ones, but we have given proofs whenever some technical modifications are required. Acknowledgement. The first two authors were supported by the Swedish Research Council. Part of this research was done during several visits of the third author to Linköpings universitet in 2009, 2012 and 2013. The first of these visits was supported by the Scandinavian Research Network Analysis and Application, and the others by Linköpings universitet. The paper was completed while all three authors visited Institut Mittag-Leffler in the autumn of 2013. They want to thank the institute for the hospitality, and the third author also wishes to thank the Department of Mathematics at Linköpings universitet for its hospitality. 2. Notation and preliminaries We assume throughout the paper that 1 < p < ∞ and that X = (X, d, µ) is a metric space equipped with a metric d and a positive complete Borel measure µ such that 0 < µ(B) < ∞ for all (open) balls B ⊂ X. The σ-algebra on which µ is defined is obtained by the completion of the Borel σ-algebra. It follows that X is separable. Towards the end of the section we further assume that X is complete and supports a p-Poincaré inequality, and that µ is doubling, which are then assumed throughout the rest of the paper. We also always assume that Ω ⊂ X is a nonempty open set. The weak Cartan property for the p-fine topology on metric spaces 5 We say that µ is doubling if there exists a doubling constant C > 0 such that for all balls B = B(x0 , r) := {x ∈ X : d(x, x0 ) < r} in X, 0 < µ(2B) ≤ Cµ(B) < ∞. Here and elsewhere we let δB = B(x0 , δr). A metric space with a doubling measure is proper (i.e. closed and bounded subsets are compact) if and only if it is complete. See Heinonen [29] for more on doubling measures. A curve is a continuous mapping from an interval, and a rectifiable curve is a curve with finite length. We will only consider curves which are nonconstant, compact and rectifiable. A curve can thus be parameterized by its arc length ds. We follow Heinonen and Koskela [33] in introducing upper gradients as follows (they called them very weak gradients). Definition 2.1. A nonnegative Borel function g on X is an upper gradient of an extended real-valued function f on X if for all nonconstant, compact and rectifiable curves γ : [0, lγ ] → X, Z g ds, |f (γ(0)) − f (γ(lγ ))| ≤ (2.1) γ where we follow the convention that the left-hand side is ∞ whenever at least one of the terms therein is infinite. If g is a nonnegative measurable function on X and if (2.1) holds for p-almost every curve (see below), then g is a p-weak upper gradient of f . Here we say that a property holds for p-almost every curve if it fails only for a curve family Γ with zero p-modulus, i.e. there exists 0 ≤ ρ ∈ Lp (X) such that R γ ρ ds = ∞ for every curve γ ∈ Γ. Note that a p-weak upper gradient need not be a Borel function, it is only required to be measurable. On the other hand, every measurable function g can be modified R on a set of measure zero to obtain a Borel function, from which it follows that γ g ds is defined (with a value in [0, ∞]) for p-almost every curve γ. For proofs of these and all other facts in this section we refer to Björn–Björn [7] and Heinonen–Koskela–Shanmugalingam–Tyson [34]. The p-weak upper gradients were introduced in Koskela–MacManus [43]. It was also shown there that if g ∈ Lploc (X) is a p-weak upper gradient of f , then one can p find a sequence {gj }∞ j=1 of upper gradients of f such that gj − g → 0 in L (X). If p f has an upper gradient in Lloc (X), then it has a minimal p-weak upper gradient gf ∈ Lploc (X) in the sense that for every p-weak upper gradient g ∈ Lploc (X) of f we have gf ≤ g a.e., see Shanmugalingam [56] and Hajlasz [25]. The minimal p-weak upper gradient is well defined up to a set of measure zero in the cone of nonnegative functions in Lploc (X). Following Shanmugalingam [55], we define a version of Sobolev spaces on the metric measure space X. Definition 2.2. Let kf kN 1,p (X) = Z X p |f | dµ + inf g Z X p g dµ 1/p , where the infimum is taken over all upper gradients of f . The Newtonian space on X is N 1,p (X) = {f : kf kN 1,p (X) < ∞}. The space N 1,p (X)/∼, where f ∼ h if and only if kf − hkN 1,p (X) = 0, is a Banach space and a lattice, see Shanmugalingam [55]. In this paper we assume that functions in N 1,p (X) are defined everywhere, not just up to an equivalence class in the corresponding function space. For a measurable set E ⊂ X, the Newtonian 6 Anders Björn, Jana Björn and Visa Latvala space N 1,p (E) is defined by considering (E, d|E , µ|E ) as a metric space on its own. 1,p We say that f ∈ Nloc (Ω) if for every x ∈ Ω there exists a ball Bx ∋ x such that 1,p 1,p Bx ⊂ Ω and f ∈ N (Bx ). If f, h ∈ Nloc (X), then gf = gh a.e. in {x ∈ X : f (x) = h(x)}, in particular gmin{f,c} = gf χ{f <c} for c ∈ R. Definition 2.3. The Sobolev capacity of an arbitrary set E ⊂ X is Cp (E) = inf kukpN 1,p (X) , u where the infimum is taken over all u ∈ N 1,p (X) such that u ≥ 1 on E. The capacity is countably subadditive. We say that a property holds quasieverywhere (q.e.) if the set of points for which the property does not hold has capacity zero. The capacity is the correct gauge for distinguishing between two Newtonian functions. If u ∈ N 1,p (X), then u ∼ v if and only if u = v q.e. Moreover, Corollary 3.3 in Shanmugalingam [55] shows that if u, v ∈ N 1,p (X) and u = v a.e., then u = v q.e. A set U ⊂ X is quasiopen if for every ε > 0 there is an open set G ⊂ X such that Cp (G) < ε and G ∪ U is open. A function u on a quasiopen set U ⊂ X is quasicontinuous if for every ε > 0 there is an open set G ⊂ X such that Cp (G) < ε and u|U\G is finite and continuous. Definition 2.4. We say that X supports a p-Poincaré inequality if there exist constants C > 0 and λ ≥ 1 such that for all balls B ⊂ X, all integrable functions f on X and all upper gradients g of f , Z 1/p Z |f − fB | dµ ≤ C(diam B) g p dµ , (2.2) B where fB := R B f dµ := λB R B f dµ/µ(B). In the definition of Poincaré inequality we can equivalently assume that g is a p-weak upper gradient—see the comments above. If X is complete and supports a p-Poincaré inequality and µ is doubling, then Lipschitz functions are dense in N 1,p (X), see Shanmugalingam [55]. Moreover, all functions in N 1,p (X) and those in N 1,p (Ω) are quasicontinuous, see Björn–Björn–Shanmugalingam [14]. This means that in the Euclidean setting, N 1,p (Rn ) is the refined Sobolev space as defined in Heinonen–Kilpeläinen–Martio [32, p. 96], see Björn–Björn [7] for a proof of this fact valid in weighted Rn . This is the main reason why, unlike in the classical Euclidean setting, we do not need to require the functions admissible in the definition of capacity to be 1 in a neighbourhood of E. In Section 4 the fine topology is defined by means of thin sets, which in turn use the variational capacity capp . To be able to define the variational capacity we first need a Newtonian space with zero boundary values. We let, for an arbitrary set A ⊂ X, N01,p (A) = {f |A : f ∈ N 1,p (X) and f = 0 on X \ A}. One can replace the assumption “f = 0 on X \A” with “f = 0 q.e. on X \A” without changing the obtained space N01,p (A). Functions from N01,p (A) can be extended by zero in X \ A and we will regard them in that sense if needed. Definition 2.5. Let A ⊂ X be arbitrary. The variational capacity of E ⊂ A with respect to A is Z gup dµ, capp (E, A) = inf where the infimum is taken over all u ∈ u X 1,p N0 (A) such that u ≥ 1 on E. The weak Cartan property for the p-fine topology on metric spaces 7 Remark 2.6. The infimum above can equivalently be taken over u ∈ N 1,p (X) such that u ≥ 1 q.e. on E and u = 0 q.e. outside A. We will call such functions admissible for the capacity capp (E, A). Similarly, one can test the capacity Cp (E) by any function u ∈ N 1,p (X) such that u ≥ 1 q.e. on E, and we will call such a function admissible for Cp (E). We will mainly be interested in the variational capacity with respect to open sets A, but in Lemma 4.8 we will generalize an earlier result for the variational capacity to arbitrary sets. The variational capacity with respect to nonopen sets was recently studied and used in Björn–Björn [8] and [9]. (Note that the roles of A and E are reversed in [8] and [9] compared with this paper.) Throughout the rest of the paper, we assume that X is complete and supports a p-Poincaré inequality, and that µ is doubling. The following lemma from J. Björn [16] compares the capacities Cp and capp , and the measure µ. Here and elsewhere, the letter C denotes various positive constants whose values may vary even within a line. Lemma 2.7. Let E ⊂ B = B(x0 , r) with 0 < r < 1 6 diam(X). Then Cµ(B) µ(E) ≤ capp (E, 2B) ≤ p Cr rp and   1 Cp (E) p 1 + Cp (E). ≤ cap (E, 2B) ≤ 2 p C(1 + rp ) rp In particular, µ(B) Cµ(B) ≤ capp (B, 2B) ≤ . Crp rp We will also need the following result from Björn–Björn–Shanmugalingam [14]. (It was recently extended to arbitrary bounded sets Ω in Björn–Björn [9], but we will not need that generality here.) Recall that E ⋐ Ω if E is a compact subset of Ω. Theorem 2.8. Let Ω ⊂ X be a bounded open set. The variational capacity capp is an outer capacity for sets E ⋐ Ω, i.e. capp (E, Ω) = 3. inf G open E⊂G⊂Ω capp (G, Ω). (2.3) Superminimizers and superharmonic functions In this section we introduce superminimizers and superharmonic functions, as well as obstacle problems, which all will be needed in later sections. For further discussion and references on these topics see Kinnunen–Martio [39] and [40], and also Björn–Björn [7] (which also contains proofs of the facts mentioned in this section, but for Lemma 3.7). 1,p Definition 3.1. A function u ∈ Nloc (Ω) is a (super )minimizer in Ω if Z Z p gu+ϕ dµ for all (nonnegative) ϕ ∈ N01,p (Ω). gup dµ ≤ {ϕ6=0} {ϕ6=0} A function u is a subminimizer if −u is a superminimizer. A p-harmonic function is a continuous minimizer. 8 Anders Björn, Jana Björn and Visa Latvala For characterizations of minimizers and superminimizers see A. Björn [5]. Minimizers were first studied for functions in N 1,p (X) in Shanmugalingam [56]. For a superminimizer u, it was shown by Kinnunen–Martio [39] that its lower semicontinuous regularization u∗ (x) := ess lim inf u(y) = lim ess inf u y→x r→0 B(x,r) (3.1) is also a superminimizer and u∗ = u q.e. For an alternative proof of this fact see Björn–Björn–Parviainen [11]. If u is a minimizer, then u∗ is continuous, and thus p-harmonic, see Kinnunen–Shanmugalingam [41]. We will need the following weak Harnack inequalities. Theorem 3.2. (Weak Harnack inequality for subminimizers) Let q > 0. Then there is C > 0 such that for all subminimizers u in Ω and all balls B ⊂ 2B ⊂ Ω, ess sup u ≤ C B Z q u+ dµ 2B 1/q . Here u+ := max{u, 0}. Theorem 3.3. (Weak Harnack inequality for superminimizers) There are q > 0 and C > 0, such that for all nonnegative superminimizers u in Ω, Z q u dµ 2B 1/q ≤ C ess inf u B (3.2) for every ball B ⊂ 50λB ⊂ Ω. These Harnack inequalities were in metric spaces first obtained for minimizers by Kinnunen–Shanmugalingam [41], using De Giorgi’s method, whereas Kinnunen– Martio [39] soon afterwards modified the arguments for sub- and superminimizers. See Björn–Marola [15], p. 363, for some necessary modifications of the statements in [41] and [39], and for alternative proofs using Moser iteration. For a nonempty bounded open set G ⊂ X with Cp (X \ G) > 0 we consider the following obstacle problem. (If X is unbounded then the condition Cp (X \ G) > 0 is of course immediately fulfilled.) Definition 3.4. For f ∈ N 1,p (G) and ψ : G → R let Kψ,f (G) = {v ∈ N 1,p (G) : v − f ∈ N01,p (G) and v ≥ ψ q.e. in G}. A function u ∈ Kψ,f (G) is a solution of the Kψ,f (G)-obstacle problem if Z Z p gu dµ ≤ gvp dµ for all v ∈ Kψ,f (G). G G A solution to the Kψ,f (G)-obstacle problem is easily seen to be a superminimizer in G. Conversely, a superminimizer u in Ω is a solution of the Ku,u (G)-obstacle problem for all open G ⋐ Ω with Cp (X \ G) > 0. If Kψ,f (G) 6= ∅, then there is a solution of the Kψ,f (G)-obstacle problem, and this solution is unique up to equivalence in N 1,p (G). Moreover, u∗ is the unique lower semicontinuously regularized solution. If the obstacle ψ is continuous, then u∗ is also continuous. The obstacle ψ, as a continuous function, is even allowed to take the value −∞. For f ∈ N 1,p (G), we let HG f denote the continuous solution of the K−∞,f (G)-obstacle problem; this function is p-harmonic in G and has the same boundary values (in the Sobolev sense) as f on ∂G, and hence is also called the solution of the Dirichlet problem with Sobolev boundary values. The weak Cartan property for the p-fine topology on metric spaces 9 Definition 3.5. A function u : Ω → (−∞, ∞] is superharmonic in Ω if (i) u is lower semicontinuous; (ii) u is not identically ∞ in any component of Ω; (iii) for every nonempty open set G ⋐ Ω with Cp (X \ G) > 0 and all functions v ∈ Lip(X), we have HG v ≤ u in G whenever v ≤ u on ∂G. This definition of superharmonicity is equivalent to the ones in Heinonen–Kilpeläinen–Martio [32] and Kinnunen–Martio [39], see A. Björn [4]. A locally bounded superharmonic function is a superminimizer, and all superharmonic functions are lower semicontinuously regularized. Conversely, any lower semicontinuously regularized superminimizer is superharmonic. We will need the following comparison lemma for solutions to obstacle problems from Björn–Björn [6]. Lemma 3.6. (Comparison principle) Assume that Ω is bounded and such that Cp (X \ Ω) > 0. Let ψj : Ω → R and fj ∈ N 1,p (Ω) be such that Kψj ,fj 6= ∅, and let uj be the lower semicontinuously regularized solution of the Kψj ,fj -obstacle problem, j = 1, 2. If ψ1 ≤ ψ2 q.e. in Ω and (f1 − f2 )+ ∈ N01,p (Ω), then u1 ≤ u2 in Ω. The following simple localization lemma will be useful in the coming proofs. For a proof in the metric space setting see Farnana [21], Lemma 3.6. Lemma 3.7. Let u be the lower semicontinuously regularized solution of the Kψ,f (Ω)obstacle problem and let Ω′ ⊂ Ω be open. Then u is the lower semicontinuously regularized solution of the Kψ,u (Ω′ )-obstacle problem. 4. Fine topology In this section we introduce the main concepts of this paper and present the necessary auxiliary results. At the end of the section, we prove Theorem 1.4. A set E ⊂ X is thin at x ∈ X if  Z 1 capp (E ∩ B(x, r), B(x, 2r)) 1/(p−1) dr < ∞. (4.1) capp (B(x, r), B(x, 2r)) r 0 A set U ⊂ X is finely open if X \ U is thin at each point x ∈ U . It is easy to see that the finely open sets give rise to a topology, which is called the fine topology, see Proposition 11.36 in Björn–Björn [7]. Every open set is finely open, but the converse is not true in general. For any E ⊂ X, the base bp (E) is the set of all points x ∈ X so that E is thick, p i.e. not thin, at x. We also let E be the fine closure of E and fine-int E be the fine interior of E, both taken with respect to the fine topology. In the definition of thinness, and in the sum (4.2) below, we make the convention that the integrand is 1 whenever capp (B(x, r), B(x, 2r)) = 0. This happens e.g. if X = B(x, 2r) is bounded, but never e.g. if r < 12 diam X. Note that thinness is a local property, i.e. E is thin at x if and only if E ∩ B(x, δ) is thin at x, where δ > 0 is arbitrary. Definition 4.1. A function u : U → R, defined on a finely open set U , is finely continuous if it is continuous when U is equipped with the fine topology and R with the usual topology. Note that u is finely continuous in U if and only if it is finely continuous at every x ∈ U in the sense that for all ε > 0 there exists a finely open set V ∋ x such 10 Anders Björn, Jana Björn and Visa Latvala that |u(y) − u(x)| < ε for all y ∈ V , if u(x) ∈ R, and such that ±u(y) > 1/ε for all y ∈ V , if u(x) = ±∞, or equivalently if and only if the sets {x ∈ U : u(x) > k} and {x ∈ U : u(x) < k} are finely open for all k ∈ R. Since every open set is finely open, the fine topology generated by the finely open sets is finer than the metric topology. In fact, it is so fine that all superharmonic functions become finely continuous. This is the content of the following theorem. Theorem 4.2. Let u be a superharmonic function in an open set Ω. Then u is finely continuous in Ω. By Theorem 1.1, which we prove in Section 6, the fine topology is the coarsest topology with this property. Together with its consequence Theorem 4.3 below, Theorem 4.2 was obtained by J. Björn [18], Theorems 4.4 and 4.6, and independently by Korte [42], Theorem 4.3 and Corollary 4.4, (they can also be found in Björn– Björn [7] as Theorems 11.38 and 11.40). Theorem 4.3. Let Ω be open. Then every quasicontinuous function u : Ω → R is 1,p finely continuous at q.e. x ∈ Ω. In particular, this is true for all u ∈ Nloc (Ω). At the end of this section (when proving Theorem 1.4), we extend the first part of the above result to finely open and quasiopen sets. We next give some auxiliary lemmas. The following characterization was essentially obtained in Björn–Björn [8]. Lemma 4.4. Let E ⊂ X and x ∈ X. Then x ∈ fine-int E if and only if x ∈ E and X \ E is thin at x. Moreover, we have p E = E ∪ bp (E). Proof. For the characterization of fine interior points, see Proposition 7.8 in [8]. p Accordingly, x ∈ / E if and only if x is a fine interior point of X \ E, i.e. x ∈ / E and p E is thin at x. Thus x ∈ E if and only if x ∈ E or E is thick at x. In Sections 5 and 6 we will use the fact that the integral (4.1) can be replaced by a sum and the factor 2 in (4.1) can be replaced by an arbitrary factor greater than 1. To prove this (see Lemma 4.6), we need the following simple lemma, whose proof can be found e.g. in Björn–Björn [7], Lemma 11.22. Lemma 4.5. Let B = B(x0 , r) and E ⊂ B. Then for every 1 < τ < t <   tp capp (E, tB). capp (E, tB) ≤ capp (E, τ B) ≤ C 1 + (τ − 1)p 1 4 diam X, Lemma 4.6. Let E ⊂ X, x ∈ X, r0 > 0 and σ > 1. Then E is thin at x if and only if  ∞  X capp (E ∩ B(x, σ −j r0 ), B(x, σ 1−j r0 )) 1/(p−1) < ∞. (4.2) capp (B(x, σ −j r0 ), B(x, σ 1−j r0 )) j=1 Proof. Let Bs = B(x, s) for s > 0. Let ρ < 81 diam X and ρ/σ ≤ r ≤ ρ. Then, by Lemma 4.5 and the monotonicity of the capacity, 1 capp (E ∩ Bρ/σ , Bρ ) ≤ capp (E ∩ Br , B2r ) ≤ C capp (E ∩ Bρ , Bσρ ), C which together with the doubling property of µ and Lemmas 2.7 and 4.5 shows that  1/(p−1) Z ρ   capp (E ∩ Br , B2r ) 1/(p−1) dr 1 capp (E ∩ Bρ/σ , Bρ ) ≤ C capp (Bρ/σ , Bρ ) capp (Br , B2r ) r ρ/σ  1/(p−1) capp (E ∩ Bρ , Bσρ ) ≤C . capp (Bρ , Bσρ ) Hence (4.1) converges if and only if (4.2) converges. The weak Cartan property for the p-fine topology on metric spaces 11 Lemma 4.7. Let E ⊂ X be thin at x ∈ E \ E. Then there is an open neighbourhood G of E such that G is thin at x and x ∈ / G. Proof. Let Bj = B(x, 2−j ), j = 1, 2, ... . By Lemma 4.5, capp (E ∩ B j , 2Bj ) ≤ C capp (E ∩ B j , 4Bj ) ≤ C capp (E ∩ 2Bj , 4Bj ). Since the variational capacity is an outer capacity, by Theorem 2.8, we can find open sets Gj ⊃ E ∩ B j such that  capp (Gj , 2Bj ) capp (Bj , 2Bj ) 1/(p−1) ≤  capp (E ∩ B j , 2Bj ) capp (Bj , 2Bj ) 1/(p−1) + 2−j . Let G = (X \ B 1 ) ∪ (G1 \ B 2 ) ∪ ((G1 ∩ G2 ) \ B 3 ) ∪ ((G1 ∩ G2 ∩ G3 ) \ B 4 ) ∪ ... . Then G is open and contains E, and x ∈ / G. Moreover G ∩ Bj ⊂ Gj and thus, by combining the estimates and using Lemmas 2.7 and 4.6,  ∞  X capp (G ∩ Bj , 2Bj ) 1/(p−1) j=1 capp (Bj , 2Bj ) ≤C  ∞  X capp (E ∩ 2Bj , 4Bj ) 1/(p−1) capp (2Bj , 4Bj ) j=1 + 1 < ∞. Hence the claim follows from Lemma 4.6. Theorem 4.3 can be used to prove the following generalization of Corollary 4.5 in J. Björn [18] (which can also be found as Corollary 11.39 in [7]), where (4.3) was obtained for bounded open A with Cp (X \ A) > 0 and E ⋐ A. There is also an intermediate version in Björn–Björn [9], Corollary 4.7. In [18], Corollary 4.5 was used to obtain Theorem 4.3. Here we instead use Theorem 4.3 to obtain Lemma 4.8, i.e. to improve Corollary 4.5 from [18]. p Lemma 4.8. If E ⊂ X, then Cp (E ) = Cp (E). Moreover, if E ⊂ A, then p capp (E, A) = capp (E ∩ A, A). (4.3) p If furthermore capp (E, A) < ∞, then Cp (E \ fine-int A) = 0 and p p capp (E, A) = capp (E ∩ A, A) = capp (E ∩ fine-int A, fine-int A). (4.4) p Proof. The inequality Cp (E ) ≤ Cp (E) follows since any v ∈ N 1,p (X) admissible p p for the capacity Cp (E) is also admissible for the capacity Cp (E ). Indeed, if x ∈ E is a fine continuity point of v, then v(x) = fine lim v(y) ≥ 1. y→x Since q.e. point in X is a fine continuity point for v ∈ N 1,p (X), by Theorem 4.3, p we conclude that v ≥ 1 q.e. in E . The converse inequality is trivial. p 1,p Similarly, if u ∈ N0 (A) is admissible for capp (E, A) then u ≥ 1 q.e. in E and u = 0 q.e. in X \ fine-int A. This proves the nontrivial inequality in (4.4), and p also that Cp (E \ fine-int A) = 0 if there exists such a u. Finally, (4.3) is trivial if capp (E, A) = ∞. As a main consequence of Lemma 4.8, we end this section by proving Theorem 1.4. 12 Anders Björn, Jana Björn and Visa Latvala Proof of Theorem 1.4. (a) For each j = 1, 2, ..., find an open set Gj with Cp (Gj ) < p 2−j so that U ∪ Gj is open. By Lemma 4.8, we have Cp (Gj ) = Cp (Gj ) < 2−j . Let T p E := U ∩ ∞ j=1 Gj . Then Cp (E) = 0. Moreover, p p Vj := U \ Gj = (U ∪ Gj ) \ Gj S∞ is finely open, and thus V := j=1 Vj = U \ E is finely open. (b) By (a) we may assume that U = V ∪E, where V is finely open and Cp (E) = 0. As u is quasicontinuous, we can for each j = 1, 2, ... find an open set Gj with p Cp (Gj ) < 2−j so that u|V \Gj is continuous. By Lemma 4.8, we have Cp (Gj ) = Cp (Gj ) < 2−j . Hence the set   ∞ \ p A := E ∪ V ∩ Gj j=1 p is of capacity zero. If x ∈ U \ A, then x belongs to the finely open set V \ Gk for some k, and the fine continuity of u at x follows from the continuity of u|V \Gk since the fine topology is finer than the metric topology. 5. The weak Cartan property Our aim in this section is to obtain the following weak Cartan property. Theorem 5.1. (Weak Cartan property) Assume that E is thin at x0 ∈ / E. Then there exist a ball B centred at x0 and superharmonic functions u, u′ ∈ N 1,p (B) such that 0 ≤ u ≤ 1, 0 ≤ u′ ≤ 1, u(x0 ) < 1, u′ (x0 ) < 1 and E ∩ B ⊂ F ∪ F ′, where F = {x ∈ B : u(x) = 1} and F ′ = {x ∈ B : u′ (x) = 1}. In particular, with v = max{u, u′ } we have v(x0 ) < 1 and v = 1 in E ∩ B. Note that u, u′ and v above are lower semicontinuous, quasicontinuous and finely continuous in B. In the proof we will use two lemmas which are also of independent interest (see e.g. the proof of Proposition 1.3). We shall frequently use the following notion. Definition 5.2. We say that a function u is the capacitary potential of a set E in B ⊃ E if it is the lower semicontinuously regularized solution of the KχE ,0 (B)obstacle problem. Lemma 5.3. Let B = B(x0 , r) and B0 be balls such that 50λB ⊂ B0 and Cp (X \
B0 ) > 0. Also let E ⊂ 12 B0 be such that E ∩ 2B \ 12 B = ∅ and let u be the capacitary potential of E in B0 . Then sup u ≤ C ′ ∂B  capp (E, B0 ) capp (B, B0 ) 1/(p−1) . (5.1) Proof. Let m = inf B u. If m = 0, then the left-hand sideR in (5.1) is 0, by Theorem 3.3, and (5.1) follows. If m = 1, then capp (B, B0 ) ≤ B0 gup dµ = capp (E, B0 ) and (5.1) holds for any C ′ ≥ 1. Assume therefore that 0 < m < 1. Thus the functions nu o u − mu1 ,1 and u2 = u1 = min m 1−m The weak Cartan property for the p-fine topology on metric spaces 13 are admissible in the definition of capp (B, B0 ) and capp (E, B0 ), respectively, (in view of Remark 2.6). Note that for a.e. x ∈ B0 , at least one of gu1 (x) and gu2 (x) vanishes. As u is the capacitary potential of E in B0 , we therefore obtain that Z Z gup dµ gup dµ + capp (E, B0 ) = {u>m} {u≤m} Z Z p p gup2 dµ gu1 dµ + (1 − m)p =m B0 B0 ≥ mp capp (B, B0 ) + (1 − m)p capp (E, B0 ). It follows that capp (B, B0 ) ≤ 1 − (1 − m)p capp (E, B0 ) ≤ pm1−p capp (E, B0 ) mp and equivalently, m≤  p capp (E, B0 ) capp (B, B0 ) 1/(p−1) . (5.2) Now, let B ′ = B(x′ , r′ ) be such that x′ ∈ ∂B, r′ = 15 r, and sup∂B u ≤ supB ′ u. Then 2B ′ ⋐ 2B \ 21 B and as u is a nonnegative lower semicontinuously regularized minimizer in 2B\ 21 B, the weak Harnack inequality for subminimizers (Theorem 3.2) implies that for every q > 0, there exists a constant Cq , independent of u and B ′ , such that Z  1/q uq dµ sup u ≤ Cq B′ . (5.3) 2B ′ Finally, as u is a superminimizer in B0 , the weak Harnack inequality for superminimizers (Theorem 3.3) and the doubling property of µ imply that for some q > 0 e > 0, independent of u, B and B ′ , and C e m = inf u ≥ C B Z uq dµ 2B 1/q ≥C Z uq dµ 2B ′ 1/q . (5.4) Combining (5.2)–(5.4) gives (5.1). Remark 5.4. Lemma 3.9 in J. Björn [18] (Lemma 11.20 in [7] or Lemma 5.6 in Björn–MacManus–Shanmugalingam [20] in linearly locally connected spaces) provides us with the converse inequality to (5.1), viz. inf u ≥ C ′′ ∂B  capp (E, B0 ) capp (B, B0 ) 1/(p−1) . (5.5) Proposition 5.5. For a ball B = B(x0 , r) with Cp (X \ B) > 0 let Bj = σ −j B, j =
0, 1, ..., where σ ≥ 50λ is fixed. Assume that E ⊂ 12 B is such that E ∩ 2Bj \ 1 2 Bj = ∅ for all j = 0, 1, ..., and let u be the capacitary potential of E in B. Then 1− ∞ Y j=0 where (1 − caj ) ≤ u(x0 ) ≤ 1 − ∞ Y (1 − aj ), j=0
    capp E ∩ 21 Bj , Bj 1/(p−1) ′ aj = min 1, C , capp (Bj+1 , Bj ) c = C ′′ /C ′ > 0 and C ′ and C ′′ are as in (5.1) and (5.5). 14 Anders Björn, Jana Björn and Visa Latvala Remark 5.6. (a) The case c ≥ 1 is not excluded in Proposition 5.5. However, by (5.1) and (5.5), the case c > 1 holds true only if aj = 0 for all j = 0, 1, ... . By (5.7), the case c = 1 holds true only if inf uj = sup uj ∂Bj+1 ∂Bj+1 for all j = 0, 1, ... . See the proof below for the notation here. (b) The first inequality in Proposition 5.5 can be obtained from Lemma 5.7 in Björn–MacManus–Shanmugalingam [20] (in linearly locally connected spaces) or from Proposition 3.10 in J. Björn [18] (alternatively Theorem 11.21 in [7]). In this paper we will not need it, but we have chosen to include it here as the proof below shows that both inequalities can be obtained simultaneously. In fact, by taking logarithms, the left estimate in Proposition 5.5 implies   X ∞ aj , 1 − u(x0 ) ≤ exp −c j=0 which in particular shows that if E is thick at x0 then u(x0 ) = 1. As Qnfor the right estimate in Proposition 5.5, it is easily shown by induction that 1 − j=0 (1 − aj ) ≤ Pn j=0 aj and hence we obtain the qualitative estimate
 ∞  X capp E ∩ 21 Bj , Bj 1/(p−1) u(x0 ) ≤ C , capp (Bj+1 , Bj ) j=0 ′ (5.6) which in Rn , with p < n, reduces to a special case of the estimate (6.1) in Maz′ ya–Havin [52]. It corresponds to the Wolff potential estimates for superharmonic functions in e.g. Kilpeläinen–Malý [37], Mikkonen [54] and Björn–MacManus– Shanmugalingam [20] and partly generalizes Theorem 3.6 in J. Björn [19]. More precisely, the Wolff potential for the capacitary measure of E is easily seen to be functions comparable to the sum in (5.6). The estimates for general superharmonic R in [37], [54] and [20] contain an additional term, such as ( B0 up dµ)1/p , but since the potential u has boundary values 0 on ∂B0 , this term can be avoided in this case, cf. [19, Theorem 3.6]. In particular, (5.6) implies the necessity part of the Wiener criterion in certain domains (such that (2Bj \ 21 Bj ) \ Ω) = ∅ for all sufficiently large j and some σ > 0), since for a sufficiently small ball B = B(x0 , r), the capacitary potential of 21 B \ Ω in B will not attain its boundary value 1 at x0 . Note that the necessity part of the Wiener criterion is still open for p-harmonic functions (based on upper gradients) in metric spaces. Proof. For j = 0, 1, ..., let uj be the capacitary potential of Ej = E ∩ 12 Bj in Bj . Then u = u0 . Lemma 5.3 and Remark 5.4 imply that for all j = 0, 1, ..., caj ≤ inf uj ≤ sup uj ≤ aj . ∂Bj+1 (5.7) ∂Bj+1 We shall show by induction that for all k = 1, 2, ..., 1 − sup u ≥ ∂Bk k−1 Y j=0 (1 − aj ) =: bk and 1 − inf u ≤ ∂Bk k−1 Y (1 − caj ) =: b′k . (5.8) j=0 By (5.7), this clearly holds for k = 1. Assume that (5.8) holds for some k ≥ 1 and let Gk = {x ∈ Bk : u(x) > 1 − bk }. Then Gk is open by the lower semicontinuity of u, and since sup∂Bk u ≤ 1 − bk , we have vk := (u − (1 − bk ))+ ∈ N01,p (Gk ). The weak Cartan property for the p-fine topology on metric spaces 15 Lemma 3.7 shows that vk is the lower semicontinuously regularized solution of the Kψk ,0 (Gk )-obstacle problem, where ψk = (χE0 − (1 − bk ))+ = bk χEk in Bk . On the other hand, by the minimum principle for superharmonic functions, we have u ≥ 1 − b′k in Bk and Lemma 3.7 again shows that vk′ := u − (1 − b′k ) ≥ 0 is the lower semicontinuously regularized solution of the Kψk′ ,vk′ (Bk )-obstacle problem, where ψk′ = (χE0 − (1 − b′k ))+ = b′k χEk in Bk . Since 0 ≤ uk ∈ N01,p (Bk ) is the lower semicontinuously regularized solution of the KχEk ,0 (Bk )-obstacle problem, the comparison principle (Lemma 3.6) yields that vk′ ≥ b′k uk in Bk and that vk ≤ bk uk in Gk , and hence in Bk . In particular, by (5.7), sup vk ≤ sup bk uk ≤ ak bk ∂Bk+1 and ∂Bk+1 inf vk′ ≥ inf b′k uk ≥ cak b′k . ∂Bk+1 ∂Bk+1 Hence sup u ≤ sup vk + 1 − bk ≤ ak bk + 1 − bk = 1 − bk (1 − ak ) = 1 − bk+1 ∂Bk+1 ∂Bk+1 and inf u = inf vk′ + 1 − b′k ≥ cak b′k + 1 − b′k = 1 − b′k (1 − cak ) = 1 − b′k+1 , ∂Bk+1 ∂Bk+1 which proves (5.8) for k + 1. By induction, (5.8) holds for all k = 1, 2 ... . Since u is lower semicontinuously regularized, letting k → ∞ gives u(x0 ) = lim inf u(x) ≤ 1 − lim bk = 1 − x→x0 k→∞ ∞ Y (1 − aj ) j=0 and, by the minimum principle, u(x0 ) ≥ 1 − lim k→∞ b′k =1− ∞ Y (1 − caj ). j=0 We are now ready to prove the weak Cartan property. The proof uses a separation argument which has been inspired by Theorem 3.2 in Heinonen–Kilpeläinen– Martio [31], and whose idea goes back to Lindqvist–Martio [47]. Proof of Theorem 5.1. By Lemma 4.7, we can assume that E is open. For r > 0 let Bj = σ −j
B(x0 , r) with S σ = 50λ be as in Proposition 5.5. Also let
Dj = ∞ 1 1 B \ 2B ∩ E and E = j j+1 i=j Di , j = 0, 1, ... . Note that E0 ∩ 2Bj \ 2 Bj = ∅ 2 j for all j = 0, 1, ... . Proposition 5.5 then implies that the capacitary potential u of E0 in B0 = B(x0 , r) satisfies u(x0 ) ≤ 1 − ∞ Y (1 − aj ), j=0 where     capp (Ej , Bj ) 1/(p−1) aj = min 1, C ′ capp (Bj+1 , Bj ) r > 0 so that all and C ′ is as P in Lemma 5.3. Since E is thin at x0 , we can find P ∞ ∞ 4.6). Hence the series aj ≤ 21 and a < ∞ (by Lemma j j=0 log(1 − aj ) j=0 Q∞ converges as well, which implies that j=0 (1 − aj ) > 0, i.e. that u(x0 ) < 1. On the other hand, we have u = 1 in E0 , as E0 is open.
Similarly, since 2B j \ 21 Bj ⊂ 15 21 Bj−1 \2B j , replacing r by r′ = 51 r in the above ′ ′ argument provides us with the capacitary

potential u in B(x0 , r ) which
satisfies 1 ′ ′ ′ u (x0 ) < 1 and u = 1 in E ∩ B x0 , 2 r \ E0 . Letting B = B x0 , 12 r′ concludes the proof. 16 Anders Björn, Jana Björn and Visa Latvala We end this section by proving Theorem 1.2. Proof of Theorem 1.2. (a) ⇔ (b) ⇔ (c) This follows directly from Lemma 4.4. (a) ⇒ (e) This follows from the weak Cartan property (Theorem 5.1). (e) ⇒ (d) This is trivial. (d) ⇒ (b) We can find δ and a ball B ∋ x0 such that v(x0 ) < δ < v(x) for all x ∈ B ∩ E. As v is finely continuous, by Theorem 4.2, V := {x ∈ B : v(x) < δ} is a p finely open fine neighbourhood of x0 . Since E ∩ V = ∅, we see that x0 ∈ /E . 6. Consequences of the weak Cartan property In this section we establish several consequences of the weak Cartan property. First, we prove Theorem 1.1, i.e. that the fine topology is the coarsest topology making all superharmonic functions continuous, and that the base of its neighbourhoods is given by finite intersections of level sets of superharmonic functions. The coarsest topology related to Theorem 1.1 is traditionally formulated using global superharmonic functions on Rn . This definition relies on the following extension result: If u is superharmonic in Ω ⊂ Rn and G ⋐ Ω, then there is a superharmonic function v on Rn such that v = u in G, see Theorem 3.1 in Kilpeläinen [35] (for unweighted Rn ) and Theorem 7.30 in Heinonen–Kilpeläinen–Martio [32] (for weighted Rn ). Such an extension result is not known for unbounded metric spaces, while it is false for bounded metric spaces as there are only constant superharmonic functions on X if X is bounded. Therefore we directly prove the following local formulation. Theorem 6.1. A set U ⊂ X is a fine neighbourhood of x0 if and only if there exist constants cj and bounded superharmonic functions uj in some ball B ∋ x0 , j = 1, 2, ... , k, such that x0 ∈ k \ {x ∈ B : uj (x) < cj } ⊂ U. (6.1) j=1 The proof shows that the neighbourhood base condition always holds with k = 2. Recall that a set U is a fine neighbourhood of a point x0 if it contains a finely open set V ∋ x0 ; it is not required that U itself is finely open. Proof. Let U ⊂ X. First, we assume that there exist constants cj and bounded superharmonic functions uj in a ball B ∋ x0 , j = 1, 2, ... , k, such that (6.1) holds. By Theorem 4.2, each uj is finely continuous and hence Vj := {x ∈ B : uj (x) < cj } T is finely open. It follows that kj=1 Vj is finely open and hence U is a fine neighbourhood of x0 . To prove the converse, let E = X \ U . Then x ∈ / E and E is thin at x. Let B, F , F ′ , u, and u′ be as given by the weak Cartan property (Theorem 5.1). Then B ∩ U = B \ E ⊃ B \ (F ∪ F ′ ) = {x ∈ B : u(x) < 1} ∩ {x ∈ B : u′ (x) < 1}, i.e. the fine neighbourhood base condition holds with k = 2. Proof of Theorem 1.1. By Theorem 4.2, the fine topology makes all superharmonic functions on all open subsets of X continuous. To show that it is the coarsest topology with this property, let T be such a topology on X, and let U ⊂ X be finely open. We shall show that for every x0 ∈ U there exists V ∈ T such that The weak Cartan property for the p-fine topology on metric spaces 17 x0 ∈ V ⊂ U . Indeed, let u1 and u2 be the superharmonic functions provided by Theorem 6.1 and so that (6.1) holds. Since T makes all superharmonic functions continuous, we get that the level sets {x ∈ B : uj (x) < cj } belong to T , and so does their intersection. In view of (6.1) this concludes the proof. Note that here it is not enough to only consider all superharmonic functions on X, as these may be just the constants (if X is bounded). Therefore, superharmonic functions on all open sets (or balls) in X have to be considered in Theorem 1.1. As a consequence of Proposition 5.5 we can also deduce Proposition 1.3. Proof of Proposition 1.3. Let σ = 50λ, E = {x0 }, B = B(x0 , r), Bj and u be as in Proposition 5.5. Since Cp ({x0 }) > 0, we have u(x0 ) = 1. Proposition 5.5 yields u(x0 ) ≤ 1 − ∞ Y (1 − aj ), j=0 where     capp ({x0 }, Bj ) 1/(p−1) aj = min 1, C ′ capp (Bj+1 , Bj ) and C ′ is asPin Lemma 5.3. If E were thin at x0 , we couldPfind r > 0 so that all ∞ ∞ aj ≤ 21 and j=0 aj < ∞ (by Lemma 4.6). Hence the series j=0 log(1 − aj ) would Q∞ converge as well, implying that j=0 (1 − aj ) > 0, i.e. that u(x0 ) < 1, which is a contradiction. Thus {x0 } is thick at x0 . The proof of the following lemma has been inspired by the proof of Lemma 12.24 in Heinonen–Kilpeläinen–Martio [32], but here we make use of the weak Cartan property to simplify the argument. Lemma 6.2. If a set E is thin at x0 then for every ball B ∋ x0 lim capp (E ∩ B(x0 , ρ), B) = 0. ρ→0 Proof. Without loss of generality we may assume that diam B < 61 diam X. Since the variational capacity is an outer capacity, by Theorem 2.8, we see that capp (E ∩ B(x0 , ρ), B) ≤ capp (B(x0 , ρ), B) → capp ({x0 }, B), as ρ → 0, and thus the result is trivial if capp ({x0 }, B) = 0. If x0 ∈ E and capp ({x0 }, B) > 0, then Cp ({x0 }) > 0, by Lemma 2.7. Proposition 1.3 then implies that E is thick at x0 , a contradiction. We can therefore assume that x0 ∈ / E and capp ({x0 }, B) > 0. Let 0 < ε < capp ({x0 }, B) be arbitrary. By the weak Cartan property (Theorem 5.1), there exist a ball B ′ ⊂ 2B ′ ⊂ B, containing x0 , and v ∈ N 1,p (B ′ ) such that v(x0 ) < 1 and v = 1 in E ∩ B ′ . Since v ∈ N 1,p (B ′ ) it is quasicontinuous in B ′ , see the discussion after Definition 2.4. Thus Lemma 2.7 shows that there is an open set G ⊂ B ′ such that capp (G, B) < ε and v|B ′ \G is continuous. As ε < capp ({x0 }, B), we see that x0 ∈ / G and v|B ′ \G is continuous at x0 . Thus, there exists ρ > 0 such that B(x0 , ρ) ⊂ B ′ and v < 1 in B(x0 , ρ) \ G. Since v = 1 in E ∩ B ′ , we must have E ∩ B(x0 , ρ) ⊂ G, and hence capp (E ∩ B(x0 , ρ), B) ≤ capp (G, B) < ε. As a corollary of Lemma 6.2 we obtain the following strong Cartan property at points of positive capacity, which also gives a new characterization of thin sets at such points. 18 Anders Björn, Jana Björn and Visa Latvala Proposition 6.3. Assume that Cp ({x0 }) > 0 and that x0 ∈ E \ E. Then the following are equivalent. (a) E is thin at x0 ; (b) for every (some) ball B ∋ x0 with Cp (X \ B) > 0, lim capp (E ∩ B(x0 , ρ), B) = 0; ρ→0 (c) for every (some) ball B ∋ x0 with Cp (X \ B) > 0 there exists a nonnegative superharmonic function u in B such that lim E∋x→x0 u(x) = ∞ > u(x0 ). Remark 6.4. By letting v := min{u, u(x0 ) + 1}, we obtain a bounded superharmonic function satisfying (1.1). Proof. (a) ⇒ (b) This is a special case of Lemma 6.2. (b) ⇒ (c) For j = 1, 2, ..., find rj > 0 such that capp (E ∩ B(x0 , rj ), B) < 2−jp . Since capp is an outer capacity, by Theorem 2.8, there exist open sets Gj 6∋ x0 such that Gj ⊃ E ∩B(x0 , rj ) and capp (Gj , B) < 2−jp . Let vj be the capacitary potential of Gj in B. The Poincaré inequality for N01,p (also known as Friedrichs’ inequality), see Corollary 5.54 in Björn–Björn [7], shows that Z Z p vj dµ ≤ CB gvpj dµ < CB 2−jp , B B eB 2−j . It follows that v := P∞ vj ∈ N 1,p (B). and hence kvj kN 1,p (X) ≤ C 0 j=1 Let u be the lower semicontinuously regularized solution of the Kv,0 (B)-obstacle problem. Then u ∈ N01,p (B) is a nonnegative superharmonic function in B and (as Gj are open) u ≥ k in G1 ∩...∩Gk , k = 1, 2, ... . It follows that limE∋x→x0 u(x) = ∞. On the other hand, as u ∈ N01,p (B) and Cp ({x0 }) > 0, we have u(x0 ) < ∞ by Definition 2.3. (c) ⇒ (a) Since superharmonic functions are finely continuous, by Theorem 4.2, the set U = {x ∈ B : u(x) < u(x0 ) + 1} is finely open. As x0 ∈ U , we get that B \ U is thin at x0 , and hence E is also thin at x0 . Another consequence of Lemma 6.2 is the following result, which is proved in the same way as the first part of Lemma 2.138 in Malý–Ziemer [49], although we use the variational capacity instead of the Sobolev capacity. We include a short proof for the reader’s convenience. Lemma 6.5. If E is thin at x0 and ε > 0, then there exists ρ > 0 such that Z 1 0 capp (E ∩ B(x0 , ρ) ∩ B(x0 , r), B(x0 , 2r)) capp (B(x0 , r), B(x0 , 2r)) 1/(p−1) dr < ε. r Proof. Lemma 6.2 implies that the functions fj (r) :=  capp (E ∩ B(x0 , 1/j) ∩ B(x0 , r), B(x0 , 2r)) capp (B(x0 , r), B(x0 , 2r)) 1/(p−1) 1 r decrease pointwise to zero on (0, 1). As E is thin at x0 , we see that f1 is integrable on R1 (0, 1), and hence by dominated convergence, 0 fj (r) dr → 0, as j → ∞. Choosing ρ = 1/j for some sufficiently large j concludes the proof. The weak Cartan property for the p-fine topology on metric spaces 19 Now we can deduce the following result which we will need when proving Theorem 1.5. Lemma 6.6. Assume that the sets Ej , j = 1, 2, ..., are thin at x0 . Then there exist radii rj > 0 such that the set E= ∞ [ (Ej ∩ B(x0 , rj )) j=1 is thin at x0 . S Note that in general the union ∞ j=1 Ej need not be thin at x0 . This happens e.g. if Ej = ∂B(x0 , 1/j). To obtain a similar example where x0 ∈ E j , j = 1, 2, ..., let Ej = ∂B(x0 , 1/j) ∪ E0 , where E0 is an arbitrary set thin at x0 and such that x0 ∈ E 0 . Proof. The proof of the corresponding result for weighted Rn in Heinonen–Kilpeläinen– Martio [32], Lemma 12.25, carries over verbatim to metric spaces. However, instead of appealing to their Lemma 12.24 (i.e. our Lemma 6.2), it is more straightforward to appeal to our Lemma 6.5. We end this paper with the proof of Theorem 1.5. Proof of Theorem 1.5. (a) ⇒ (c) For each j = 1, 2, ... there is a finely open set Uj ∋ x0 such that |u(x) − u(x0 )| < 1/j for every x ∈ Uj . Since the sets Ej := X \ Uj are thin at x0 , Lemma 6.6 implies that there are radii rj > 0 such that the set E= ∞ [ (Ej ∩ B(x0 , rj )) j=1 is thin at x0 . It follows that |u(x) − u(x0 )| < 1/j for every x ∈ U ∩ B(x0 , rj ) \ E, and we conclude that (c) holds. The implication (c) ⇒ (b) is immediate and (b) ⇒ (a) follows from Lemma 4.4. References 1. Adams, D. R. and Hedberg, L. I., Function Spaces and Potential Theory, Springer, Berlin–Heidelberg, 1996. 2. Adams, D. R. and Lewis, J. L., Fine and quasiconnectedness in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 35:1 (1985), 57–73. 3. Adams, D. R. and Meyers, N. G., Thinness and Wiener criteria for nonlinear potentials, Indiana Univ. Math. J. 22 (1972/73), 169–197. 4. Björn, A., Characterizations of p-superharmonic functions on metric spaces, Studia Math. 169 (2005), 45–62. 5. Björn, A., A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809–825. 6. Björn, A. and Björn, J., Boundary regularity for p-harmonic functions and solutions of the obstacle problem, J. Math. Soc. Japan 58 (2006), 1211–1232. 7. Björn, A. and Björn, J., Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics 17, European Math. Soc., Zurich, 2011. 8. Björn, A. and Björn, J., Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, to appear in Rev. Mat. Iberoam. 20 Anders Björn, Jana Björn and Visa Latvala 9. Björn, A. and Björn, J., The variational capacity with respect to nonopen sets in metric spaces, to appear in Potential Anal. DOI:10.1007/s11118-013-9341-1 10. Björn, A., Björn, J. and Lehrbäck, J., Sharp capacity estimates for annuli in weighted Rn and metric spaces, In preparation. 11. Björn, A., Björn, J. and Parviainen, M., Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces, Rev. Mat. Iberoam. 26 (2010), 147–174. 12. Björn, A., Björn, J. and Shanmugalingam, N., The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173–203. 13. Björn, A., Björn, J. and Shanmugalingam, N., The Perron method for p-harmonic functions, J. Differential Equations 195 (2003), 398–429. 14. Björn, A., Björn, J. and Shanmugalingam, N., Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), 1197–1211. 15. Björn, A. and Marola, N., Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (2006), 339–366. 16. Björn, J., Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383–403. 17. Björn, J., Wiener criterion for Cheeger p-harmonic functions on metric spaces, in Potential Theory in Matsue, Advanced Studies in Pure Mathematics 44, pp. 103–115, Mathematical Society of Japan, Tokyo, 2006. 18. Björn, J., Fine continuity on metric spaces, Manuscripta Math. 125 (2008), 369–381. 19. Björn, J., Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations, Calc. Var. Partial Differential Equations 35 (2009), 481–496. 20. Björn, J., MacManus, P. and Shanmugalingam, N., Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces, J. Anal. Math. 85 (2001), 339–369. 21. Farnana, Z., The double obstacle problem on metric spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 261–277. 22. Fuglede, B., The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier (Grenoble) 21:1 (1971), 123–169. 23. Fuglede, B., Finely Harmonic Functions, Springer, Berlin–New York, 1972. 24. Hajlasz, P., Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415. 25. Hajlasz, P., Sobolev spaces on metric-measure spaces, in Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces (Paris, 2002 ), Contemp. Math. 338, pp. 173–218, Amer. Math. Soc., Providence, RI, 2003. 26. Hedberg, L. I., Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319. 27. Hedberg, L. I. and Wolff, T. H., Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33:4 (1983), 161–187. 28. Hajlasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145:688 (2000). 29. Heinonen, J., Lectures on Analysis on Metric Spaces, Springer, New York, 2001. 30. Heinonen, J., Nonsmooth calculus, Bull. Amer. Math. Soc. 44 (2007), 163–232. 31. Heinonen, J., Kilpeläinen, T. and Martio, O., Fine topology and quasilinear elliptic equations, Ann. Inst. Fourier (Grenoble) 39:2 (1989), 293–318. The weak Cartan property for the p-fine topology on metric spaces 21 32. Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006. 33. Heinonen, J. and Koskela, P., Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61. 34. Heinonen, J., Koskela, P., Shanmugalingam, N. and Tyson, J. T., Sobolev Spaces on Metric Measure Spaces: an Approach Based on Upper Gradients, In preparation. 35. Kilpeläinen, T., Potential theory for supersolutions of degenerate elliptic equations, Indiana Univ. Math. J. 38 (1989), 253–275. 36. Kilpeläinen, T. and Malý, J., Supersolutions to degenerate elliptic equation on quasi open sets, Comm. Partial Differential Equations 17 (1992), 371–405. 37. Kilpeläinen, T. and Malý, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161. 38. Kinnunen, J. and Latvala, V., Fine regularity of superharmonic functions on metric spaces, in Future Trends in Geometric Function Theory RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dep. Math. Stat. 92, pp. 157–167, University of Jyväskylä, Jyväskylä, 2003. 39. Kinnunen, J. and Martio, O., Nonlinear potential theory on metric spaces, Illinois Math. J. 46 (2002), 857–883. 40. Kinnunen, J. and Martio, O., Sobolev space properties of superharmonic functions on metric spaces, Results Math. 44 (2003), 114–129. 41. Kinnunen, J. and Shanmugalingam, N., Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401–423. 42. Korte, R., A Caccioppoli estimate and fine continuity for superminimizers on metric spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 597–604. 43. Koskela, P. and MacManus, P., Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1–17. 44. Latvala, V., Finely Superharmonic Functions of Degenerate Elliptic Equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 96 (1994). 45. Latvala, V., Fine topology and Ap -harmonic morphisms, Ann. Acad. Sci. Fenn. Math. 22 (1997), 237–244. 46. Latvala, V., A theorem on fine connectedness, Potential Anal. 12 (2000), 221–232. 47. Lindqvist, P. and Martio, O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153–171. 48. Lukeš, J., Malý, J. and Zajı́ček, L., Fine Topology Methods in Real Analysis and Potential Theory, Springer, Berlin–New York, 1986. 49. Malý, J. and Ziemer, W. P., Fine Regularity of Solutions of Elliptic Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997. 50. Maz′ ya, V. G., On the continuity at a boundary point of solutions of quasilinear elliptic equations. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25:13 (1970), 42–55 (Russian). English transl.: Vestnik Leningrad Univ. Math. 3 (1976), 225–242. 51. Maz′ ya, V. G. and Havin, V. P., A nonlinear analogue of the Newtonian potential, and metric properties of (p, l)-capacity. Dokl. Akad. Nauk SSSR 194 (1970), 770–773 (Russian). English transl.: Soviet Math. Dokl. 11 (1970), 1294– 1298. 52. Maz′ ya, V. G. and Havin, V. P., A nonlinear potential theory. Uspekhi Mat. Nauk 27:6 (1972), 67–138 (Russian). English transl.: Russian Math. Surveys 27:6 (1974), 71–148. 53. Meyers, N. G., Continuity properties of potentials, Duke Math. J 42 (1975), 157–166. 54. Mikkonen, P., On the Wolff Potential and Quasilinear Elliptic Equations Involving Measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996). 22 Anders Björn, Jana Björn and Visa Latvala 55. Shanmugalingam, N., Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279. 56. Shanmugalingam, N., Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.