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CHARACTERIZATIONS OF SOBOLEV INEQUALITIES ON METRIC SPACES JUHA KINNUNEN AND RIIKKA KORTE Abstract. WepresentisocapacitarycharacterizationsofSobolev 8 inequalities in very general metric measure spaces. 0 0 2 n a 1. Introduction J 4 1 ThereisawellknownconnectionbetweentheisoperimetricandSobolev inequalities. By the isoperimetric inequality, we have ] P |E|(n−1)/n ≤ c(n)Hn−1(∂E), (1.1) A h. where E is a smooth enough subset of Rn, |E| is the Lebesgue measure at and Hn−1 is the (n−1)-dimensional Hausdorff measure. The constant m c(n) is chosen so that (1.1) becomes an equality when E is a ball. The [ Sobolev inequality states that 2 (n−1)/n v |u|n/(n−1)dx ≤ c(n) |∇u|dx (1.2) 7 9 (cid:18)ZRn (cid:19) ZRn 0 for every u ∈ C∞(Rn). The smallest constant in (1.2) is the same as 1 0 . the constant in (1.1). The Sobolev inequality follows from the isoperi- 9 0 metric inequality through the co-area formula. On the other hand, the 7 isoperimetric inequality can be deduced from the Sobolev inequality, 0 see for example [5]. This shows that the isoperimetric and Sobolev : v inequalities are different aspects of the same phenomenon. Originally i X this observation is due to Federer and Fleming [6] and Maz′ya [11]. r a When the gradient is integrable to a power which is greater than one, the isoperimetric inequality has to be replaced with an isocapacitary inequality. When the exponent is one, capacity and Hausdorff content are equivalent and hence it does not matter which one we choose. In this case due to the boxing inequality, it is enough to have the isocapacitary inequality for balls instead of all sets. This elegant approach to Sobolev inequalities is due to Maz′ya, see [12] and [13]. Usually this characterization leads to descriptions of the best possible constants in Sobolev inequalities. However, the aim of the present work is not so much to study best possible constants but rather study necessary and 2000 Mathematics Subject Classification. 46E35, 31C45. 1 2 JUHA KINNUNENAND RIIKKAKORTE sufficient conditions for Sobolev–Poincaré inequalities in a general met- ricspacecontext. InweightedEuclideanspaces, thesecharacterizations have been studied in [20]. Rather standard assumptions in analysis on metric measure spaces in- clude a doubling condition for the measure and validity of some kind of Sobolev–Poincaré inequality. Despite the fact that plenty of analysis has been done in this general context, very little is known about the basic assumptions. Several necessary conditions are known, but unfortunately only few sufficient conditions are available so far. On Riemannian manifolds, Grigor’yan and Saloff-Coste observed that the doubling condition and the Poincaré inequality are not only sufficient but also necessary conditions for a scale invariant parabolic Harnack principle for the heat equation, see [15], [16] and [7]. It also known that Maz′ya type characterizations of Sobolev inequalities are available on Riemannian manifolds. The purpose of this work is to show that this is also the case on a very general metric measure spaces. 2. Preliminaries We assume that X = (X,d,µ) is a metric measure space equipped with ametricdandaBorelregularoutermeasureµsuchthat0 < µ(B) < ∞ for all balls B = B(x,r) = {y ∈ X : d(x,y) < r}. In what follows, Ω stands for an open bounded subset of X unless otherwise stated. The measure µ is said to be doubling if there exists a constant c ≥ 1, D called the doubling constant, such that µ(B(x,2r)) ≤ c µ(B(x,r)) D for all x ∈ X and r > 0. In this paper, a path in X is a rectifiable non-constant continuous map- ping from a compact interval to X. A path can thus be parameterized by arc length. By saying that a condition holds for p-almost every path with 1 ≤ p < ∞, we mean that it fails only for a path family with zero p-modulus. A family Γ of curves is of zero p-modulus if there is a non-negative Borel measurable function ρ ∈ Lp(X) such that for all curves γ ∈ Γ, the path integral ρds is infinite. γ A nonnegative Borel function g on X is an upper gradient of an ex- R tended real valued function u on X if for all paths γ joining points x and y in X we have |u(x)−u(y)| ≤ gds, (2.1) Zγ CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 3 whenever both u(x) and u(y) are finite, and gds = ∞ otherwise. γ If g is a nonnegative measurable function on X and if (2.1) holds for R p-almost every path, then g is a p-weak upper gradient of u. Let 1 ≤ p < ∞. If u is a function that is integrable to power p in X, let 1/p kuk = |u|pdµ+inf gpdµ , N1,p(X) g (cid:16)ZX ZX (cid:17) where the infimum is taken over all p-weak upper gradients of u. The Newtonian space on X is the quotient space N1,p(X) = {u : kuk < ∞}/∼, N1,p(X) where u ∼ v if and only if ku − vk = 0. For properties of N1,p(X) Newtonian spaces, we refer to [18]. Let E beasubset of Ω. Wewriteu ∈ A(E,Ω) ifu = 1andu = 0. |E |X\Ω Definition 2.1. Let E ⊂ Ω. The p–capacity of E with respect to Ω is cap (E,Ω) = inf gpdµ, p u ZΩ where the infimum is taken over all continuous functions u ∈ A(E,Ω) with p–weak upper gradients g . If there are no such functions, then u cap (E,Ω) = ∞. p We say that a property regarding points in X holds p–quasieverywhere (p–q.e.) if the set of points for which the property does not hold has capacity zero. To be able to compare the boundary values of Newtonian functions, we need a Newtonian space with zero boundary values. Let E be a measurable subset of X. The Newtonian space with zero boundary values is the space N1,p(E) = {u| : u ∈ N1,p(X) and u = 0 p–q.e. on X \E}. 0 E Note that if cap (X \E,X) = 0, then N1,p(E) = N1,p(X). The space p 0 N1,p(E) equipped with the norm inherited from N1,p(X) is a Banach 0 space, see Theorem 4.4 in [19]. Definition 2.2. We say that X supports a weak (1,p)–Poincaré inequality if there exist constants c > 0 and τ ≥ 1 such that for all balls p B(x,r) of X, all integrable functions u on X and all upper gradients g of u, we have u 1/p |u−u |dµ ≤ c r gpdµ , (2.2) B(x,r) p u ZB(x,r) (cid:18)ZB(x,τr) (cid:19) where 1 u = udµ = udµ. B µ(B) ZB ZB 4 JUHA KINNUNENAND RIIKKAKORTE 3. Functions with zero boundary values In this section, we give necessary and sufficient conditions for Sobolev inequalities of type 1/q 1/p |u|qdν ≤ c gpdµ , S u (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19) where the constant c is independent of u ∈ N1,p(Ω)∩C(Ω), and µ and S 0 ν are Borel regular outer measures. We consider two ranges of indices separately. We have chosen to study continuous Newtonian functions, but our ar- guments do not depend on the choice of the function space. For example, similar results also hold for all Newtonian functions and Lipschitz functions if the definition of p–capacity is adjusted accordingly. Remark 3.1. In the standard versions of Poincaré inequality, the inequality depends on the diameter of the set. Therefore the constant c S also depends strongly on the diameter of the set in many cases. 3.1. The case 1 ≤ p ≤ q < ∞. Let u : Ω → [−∞,∞] be a µ– measurable function. By the well-known Cavalieri principle ∞ |u|pdµ = p λp−1µ(E )dλ, λ ZΩ Z0 where E = {x ∈ Ω : |u(x)| > λ}. λ The following simple integral inequality will be useful for us. Notice that the equality occurs when p = q. Lemma 3.2. If u : Ω → [−∞,∞] is µ–measurable and 0 < p ≤ q < ∞, then 1/q ∞ 1/p |u|qdµ ≤ p λp−1µ(E )p/qdλ . (3.1) λ (cid:18)ZΩ (cid:19) (cid:18) Z0 (cid:19) Proof. We have p/q (|u|p)q/p dµ = sup |u|pf dµ, (cid:18)ZX (cid:19) kfks≤1ZX where s = q/(q−p) is the Hölder conjugate of q/p. By kfk we denote s the Ls(µ)–norm of f. Define a measure µ as µ(A) = |f|edµ ZA e CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 5 for every µ–measurable set A ⊂ X. If µ(E) > 0 and kfk ≤ 1, we s conclude that 1/s µ(E) = |f|dµ ≤ µ(E)1−1/s |f|sdµ ZE (cid:18)ZE (cid:19) ≤ µ(E)1−1/s = µ(E)p/q. e Here we used the Hölder inequality. Hence ∞ |u|pf dµ ≤ |u|pdµ = p λp−1µ(E )dλ λ ZX ZX Z0 ∞ ≤ p λp−e1µ(E )p/qdλ. e λ Z0 Taking supremum over all functions f with kfk ≤ 1 completes the s (cid:3) proof. Next we prove a strong type inequality for the capacity. When p = 1 the obtained estimate reduces to the co-area formula. The proof is based on a general truncation argument, see page 110 in [12]. A similar argument has been used for example in [3], [15] and [8]. Lemma 3.3. Let u ∈ N1,p(Ω)∩C(Ω) and 1 ≤ p < ∞. Then 0 ∞ λp−1cap (E ,Ω)dλ ≤ 22p−1 gpdµ, p λ u Z0 ZΩ where g is a p-weak upper gradient of u. u Proof. A straightforward calculation shows that ∞ λp−1cap (E ,Ω)dλ p λ Z0 ∞ 2j = λp−1cap (E ,Ω)dλ p λ j=−∞Z2j−1 X ∞ ≤ (2j −2j−1)2j(p−1)capp(E2j−1,Ω) j=−∞ X ∞ 1 = 2 2jpcapp(E2j−1,Ω) j=−∞ X ∞ = 2p−1 2jpcap (E ,Ω). p 2j j=−∞ X Let 1, if u ≥ 2j, u = 21−j|u|−1, if 2j−1 < u < 2j, j  0, if u ≤ 2j−1.   6 JUHA KINNUNENAND RIIKKAKORTE Then u ∈ A(E ,Ω). This implies that j 2j cap (E ,Ω) ≤ 2p(1−j) gpdµ p 2j u ZE2j−1\E2j and consequently ∞ ∞ 2jpcap (E ,Ω) ≤ 2jp+p(1−j) gpdµ p 2j u j=−∞ j=−∞ ZE2j−1\E2j X X ≤ 2p gpdµ. u ZΩ (cid:3) The claim follows from this. The following result gives a necessary and sufficient condition for a Sobolev inequality in terms of an isocapacitary inequality. This is a metric space version of a corollary on page 113 of [12]. Remark 3.4. We do not need the doubling condition in this theorem. Theorem 3.5. Suppose that 1 ≤ p ≤ q < ∞. (i) If there is a constant γ such that ν(E)p/q ≤ γcap (E,Ω) (3.2) p for every E ⊂ Ω, then 1/q 1/p |u|qdν ≤ c gpdµ (3.3) S u (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19) for every u ∈ N1,p(Ω)∩C(Ω) with c depending only on γ and 0 S p. (ii) If (3.3) holds for every u ∈ N1,p(Ω)∩C(Ω) and if the constant 0 c is independent of u, then (3.2) holds for every E ⊂ Ω with S γ = c . S Proof. (i) By Lemma 3.2, (3.2) and Lemma 3.3, we obtain 1/q ∞ 1/p |u|qdν ≤ p λp−1ν(E )p/qdλ λ (cid:18)ZΩ (cid:19) (cid:18) Z0 (cid:19) ∞ 1/p ≤ γp λp−1cap (E ,Ω)dλ p λ (cid:18) Z0 (cid:19) 1/p ≤ γp22p−1 1/p gpdµ . u (cid:18)ZΩ (cid:19) (cid:0) (cid:1) (ii) If u ∈ A(E,Ω) is continuous, then by (3.3), we have 1/q 1/p ν(E)1/q ≤ |u|qdν ≤ c gpdµ . S u (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19) (cid:3) The claim follows by taking the infimum on the right-hand side. CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 7 Remark 3.6. The previous theorem gives a necessary and sufficient condition for the Hardy inequality p |u(x)| dµ ≤ c gpdµ, (3.4) dist(x,X \Ω) H u ZΩ(cid:18) (cid:19) ZΩ where the constant c is independent of u ∈ N1,p(Ω)∩C(Ω). Indeed, H 0 (3.4) holds if and only if 1 dµ ≤ γcap (E,Ω) dist(x,X \Ω)p p ZE for every E ⊂ Ω. Thus Theorem 3.5 is a generalization of Theorem 4.1 in [10]. In the metric space context, the Hardy inequality has also been studied in [4]. 3.2. The case p = 1. When p = 1 and Ω = X, the isocapacitary inequalities reduce to isoperimetric inequalities. Moreover, in this case we can improve Theorem 3.5 under the additional assumptions that the measure is doubling and the space supports a Poincaré inequality. Indeed, it is enough that condition (3.2) is satisfied for all balls. To prove that, we will need equivalence of the capacity of order one and the Hausdorff content of co-dimension one ∞ ∞ µ(B(x ,r )) Hh (K) = inf i i : K ⊂ B(x ,r ) . ∞ r i i ( i ) i=1 i=1 X [ Theorem 3.7. Let X be a complete metric space with a doubling measure µ. Suppose that X supports a weak (1,1)–Poincaré inequality. Let K be a compact subset of X. Then 1 cap (K) ≤ Hh (K) ≤ ccap (K), c 1 ∞ 1 where c depends only on the doubling constant and the constants in the weak (1,1)–Poincaré inequality. The proof is based on co-area formula and a metric space version of so–called boxing inequality. For more details, see [9]. A similar result has been studied in Mäkäläinen [14]. Theorem 3.8. Let X be a complete metric space with a doubling measure µ. Suppose that X supports a weak (1,1)–Poincaré inequality. Suppose that 1 ≤ q < ∞. If there is a constant γ such that ν(B)1/q ≤ γcap (B,X) (3.5) 1 for every ball B ⊂ X, then 1/q |u|qdν ≤ c g dµ, (3.6) S u (cid:18)ZX (cid:19) ZX where c is independent of u ∈ N1,1(X)∩C (X). S 0 8 JUHA KINNUNENAND RIIKKAKORTE Proof. First we prove that if the space satisfies (3.5) for all balls in X, then it satisfies the same condition for all compact sets with a different constant. Let K ⊂ X be compact, ε > 0 and {B(x ,r )}∞ be a covering of K i i i=1 such that ∞ µ(B(x ,r )) Hh (K) ≥ i i −ε. ∞ r i i=1 X Since q ≥ 1, we have ∞ ν(K)1/q ≤ ν(B(x ,r ))1/q. i i i=1 X Because u (x) = (1−dist(x,B(x ,r ))/r ) i i i i + belongs to A(B(x ,r ),X), and g = χ /r is an upper gradient i i i B(xi,2ri) i of u , we have i µ(B(x ,r )) i i cap (B(x ,r )) ≤ g dµ ≤ c . 1 i i i D r ZX i By combining the above estimates and (3.5), we conclude ∞ ν(K)1/q ≤ ν(B(x ,r ))1/q i i i=1 X ∞ ≤ γ cap (B(x ,r )) 1 i i i=1 X ∞ µ(B(x ,r )) i i ≤ γc D r i i=1 X ≤ γc (Hh (K)+ε). D ∞ The claim follows by Theorem 3.7 as ε → 0. Now the theorem follows as in the proof of Theorem 3.5. Note that since u has compact support and is continuous, we can as well consider compact level sets {|u| ≥ t} instead of open sets. (cid:3) 3.3. The case 1 ≤ q < p < ∞. In the case 1 ≤ q < p < ∞, the isocapacitary inequality takes a different form. Let E , j = −N,−N + j 1,...,N,N+1 be such that E ⊂ Ω and E ⊂ E for j = −N,−N+ j j j+1 1,...,N. We define (p−q)/q N ν(E )p/q q/(p−q) j γ = sup , (3.7) cap (E ,E ) "j=−N(cid:18) p j j+1 (cid:19) # X where the supremum is taken over all sequences of sets as above. The following result is a metric space version of a theorem on page 120 of [12]. CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 9 Theorem 3.9. Suppose that 1 ≤ q < p < ∞. (i) If γ < ∞, then 1/q 1/p |u|qdν ≤ c gpdµ , (3.8) S u (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19) where c is independent of u ∈ N1,p(Ω)∩C(Ω). S 0 (ii) If (3.8) holds for every u ∈ N1,p(Ω)∩C(Ω) and if the constant 0 c is independent of u, then γ < ∞. S Proof. (i) We have ∞ 2j+1 |u|qdν = q λq−1ν(E )dλ λ ZΩ j=−∞ Z2j X ∞ ≤q 2(j+1)(q−1)2jν(E ) 2j j=−∞ X ∞ =q2q−1 2jqν(E ). 2j j=−∞ X By the Hölder inequality, ∞ 2jqν(E ) 2j j=−∞ X ∞ ν(E )p/q q/p =j=−∞(cid:18)capp(E22jj,E2j−1)(cid:19) 2jpcapp(E2j,E2j−1) q/p X (cid:0) (cid:1) ∞ ν(E )p/q q/(p−q) (p−q)/p ≤ 2j j=−∞(cid:18)capp(E2j,E2j−1)(cid:19) ! X ∞ q/p × 2jpcapp(E2j,E2j−1) . ! j=−∞ X Let 1, if |u| > 2j, |u|−2j−1 uj =  , if 2j−1 < |u| ≤ 2j,  2j−1  0, if |u| ≤ 2j−1.  Then   capp(E2j,E2j−1) ≤ gupj dµ ≤ 2−(j−1)p gupdµ ZΩ ZE2j−1\E2j 10 JUHA KINNUNENAND RIIKKAKORTE It follows that ∞ ∞ 2jpcapp(E2j,E2j−1) ≤ 2p gupdµ j=−∞ j=−∞ZE2j−1\E2j X X = 2p gpdµ u ZΩ and consequently q/p |u|qdν ≤ c gpdµ . u ZΩ (cid:18)ZΩ (cid:19) (ii) Let E be as as in the statement of the theorem, and define j N 1/(p−q) ν(E ) i λ = , j = −N,−N +1,...,N, j cap (E ,E ) i=j (cid:18) p i i+1 (cid:19) X and λ = 0. Let u ∈ A(E ,E ) be continuous, and define N+1 j j j+1 (λ −λ )u +λ in E \E , j j+1 j j+1 j+1 j u = λ in E ,  −N −N 0 in Ω\E . N+1 Then u ∈ N1,p(Ω)∩C(Ω). By the Cavalieri principle 0 ∞ N λj |u|qdν =q λq−1ν(E )dλ = q λq−1ν(E )dλ λ λ ZΩ Z0 j=−N Zλj+1 X N ≥ ν(E )(λq −λq ). j j j+1 j=−N X From this we conclude that p/q N p/q N ν(E )(λ −λ )q ≤ ν(E )(λq −λq ) j j j+1 j j j+1 ! (cid:18)j=−N (cid:19) j=−N X X p/q N ≤ |u|qdν ≤ c gpdν = c gpdµ S u S u (cid:18)ZΩ (cid:19) ZΩ j=−NZEj+1\Ej X N ≤ c (λ −λ )p gp dµ. S j j+1 uj j=−N ZEj+1\Ej X Taking the infimum on the right-hand side, we arrive at p/q N N ν(E )(λ −λ )q ≤ c (λ −λ )pcap (E ,E ). j j j+1 S j j+1 p j j+1 ! j=−N j=−N X X