aa r X i v : . [ m a t h . M G ] M a r NEWTONIAN LORENTZ METRIC SPACES
S¸ERBAN COSTEA AND MICHELE MIRANDA JR.
Abstract.
This paper studies Newtonian Sobolev-Lorentz spaces. We prove thatthese spaces are Banach. We also study the global p, q -capacity and the p, q -modulusof families of rectifiable curves. Under some additional assumptions (that is, X carriesa doubling measure and a weak Poincar´e inequality), we show that when 1 ≤ q
. We also provide acounterexample to the density result in the Euclidean setting when 1 < p ≤ n and q = ∞ . Introduction
In this paper (
X, d ) is a complete metric space endowed with a nontrivial Borelregular measure µ. We assume that µ is finite and nonzero on nonempty bounded opensets. In particular, this implies that the measure µ is σ -finite. Further restrictions onthe space X and on the measure µ will be imposed later.The Sobolev-Lorentz relative p, q -capacity was studied in the Euclidean setting byCostea [6] and Costea-Maz’ya [8]. The Sobolev p -capacity was studied by Maz’ya [24]and Heinonen-Kilpel¨ainen-Martio [16] in R n and by Costea [7] and Kinnunen-Martio[21] and [22] in metric spaces. The relative Sobolev p -capacity in metric spaces wasintroduced by J. Bj¨orn in [2] when studying the boundary continuity properties ofquasiminimizers.After recalling the definition of p, q -Lorentz spaces in metric spaces, we study someuseful property of the p, q -modulus of families of curves needed to give the notion of p, q -weak upper gradients. Then, following the approach of Shanmugalingam in [27] and[28], we generalize the notion of Newtonian Sobolev spaces to the Lorentz setting. Thereare several other definitions of Sobolev-type spaces in the metric setting when p = q ;see Haj lasz [12], Heinonen-Koskela [17], Cheeger [4], and Franchi-Haj lasz-Koskela [11].It has been shown that under reasonable hypotheses, the majority of these definitionsyields the same space; see Franchi-Haj lasz-Koskela [11] and Shanmugalingam [27].We prove that these spaces are Banach. In order to this, we develop a theory ofSobolev p, q -capacity. Some of the ideas used here when proving the properties of the p, q -capacity follow Kinnunen-Martio [21] and [22] and Costea [7]. We also use thistheory to prove that, in the case 1 ≤ q < p, Lipschitz functions are dense in theNewtonian Sobolev-Lorentz space if the space X carries a doubling measure µ and aweak (1 , L p,q )-Poincar´e inequality. Newtonian Banach-valued Sobolev-Lorentz spaceswere studied by Podbrdsky in [26].We prove that under certain restrictions (when 1 < q ≤ p and the space ( X, d )carries a doubling measure µ and a certain weak Poincar´e inequality) this capacity isa Choquet set function. Mathematics Subject Classification.
Primary: 31C15, 46E35.
Key words and phrases.
Newtonian spaces, Lorentz spaces, capacity. e recall the standard notation and definitions to be used throughout this paper. Wedenote by B ( x, r ) = { y ∈ X : d ( x, y ) < r } the open ball with center x ∈ X and radius r > , while B ( x, r ) = { y ∈ X : d ( x, y ) ≤ r } is the closed ball with center x ∈ X andradius r > . For a positive number λ, λB ( a, r ) = B ( a, λr ) and λB ( a, r ) = B ( a, λr ) . Throughout this paper, C will denote a positive constant whose value is not nec-essarily the same at each occurrence; it may vary even within a line. C ( a, b, . . . ) is aconstant that depends only on the parameters a, b, . . . . For E ⊂ X, the boundary, theclosure, and the complement of E with respect to X will be denoted by ∂E, E, and X \ E, respectively; diam E is the diameter of E with respect to the metric d .2. Lorentz spaces
Let f : X → [ −∞ , ∞ ] be a µ -measurable function. We define µ [ f ] , the distributionfunction of f as follows (see Bennett-Sharpley [1, Definition II.1.1]): µ [ f ] ( t ) = µ ( { x ∈ X : | f ( x ) | > t } ) , t ≥ . We define f ∗ , the nonincreasing rearrangement of f by f ∗ ( t ) = inf { v : µ [ f ] ( v ) ≤ t } , t ≥ . (See Bennett-Sharpley [1, Definition II.1.5].) We note that f and f ∗ have the samedistribution function. For every positive α we have( | f | α ) ∗ = ( | f | ∗ ) α and if | g | ≤ | f | µ -almost everywhere on X, then g ∗ ≤ f ∗ . (See [1, Proposition II.1.7].)We also define f ∗∗ , the maximal function of f ∗ by f ∗∗ ( t ) = m f ∗ ( t ) = 1 t Z t f ∗ ( s ) ds, t > . (See [1, Definition II.3.1].)Throughout the paper, we denote by p ′ the H¨older conjugate of p ∈ [1 , ∞ ].The Lorentz space L p,q ( X, µ ) , < p < ∞ , ≤ q ≤ ∞ , is defined as follows: L p,q ( X, µ ) = { f : X → [ −∞ , ∞ ] : f is µ -measurable, || f || L p,q ( X,µ ) < ∞} , where || f || L p,q ( X,µ ) = || f || p,q = Z ∞ ( t /p f ∗ ( t )) q dtt ! /q , ≤ q < ∞ , sup t> tµ [ f ] ( t ) /p = sup s> s /p f ∗ ( s ) , q = ∞ . (See Bennett-Sharpley [1, Definition IV.4.1] and Stein-Weiss [29, p. 191].)If 1 ≤ q ≤ p, then || · || L p,q ( X,µ ) represents a norm, but for p < q ≤ ∞ it represents aquasinorm, equivalent to the norm || · || L ( p,q ) ( X,µ ) , where || f || L ( p,q ) ( X,µ ) = || f || ( p,q ) = Z ∞ ( t /p f ∗∗ ( t )) q dtt ! /q , ≤ q < ∞ , sup t> t /p f ∗∗ ( t ) , q = ∞ . See [1, Definition IV.4.4].) Namely, from [1, Lemma IV.4.5] we have that || f || L p,q ( X,µ ) ≤ || f || L ( p,q ) ( X,µ ) ≤ p ′ || f || L p,q ( X,µ ) for every q ∈ [1 , ∞ ] and every µ -measurable function f : X → [ −∞ , ∞ ] . It is known that ( L p,q ( X, µ ) , || · || L p,q ( X,µ ) ) is a Banach space for 1 ≤ q ≤ p, while( L p,q ( X, µ ) , || · || L ( p,q ) ( X,µ ) ) is a Banach space for 1 < p < ∞ , ≤ q ≤ ∞ . In addition,if the measure µ is nonatomic, the aforementioned Banach spaces are reflexive when1 < q < ∞ . (See Hunt [18, p. 259-262] and Bennett-Sharpley [1, Theorem IV.4.7and Corollaries I.4.3 and IV.4.8].) (A measure µ is called nonatomic if for everymeasurable set A of positive measure there exists a measurable set B ⊂ A such that0 < µ ( B ) < µ ( A ).) Definition 2.1. (See [1, Definition I.3.1].) Let 1 < p < ∞ and 1 ≤ q ≤ ∞ . Let Y = L p,q ( X, µ ) . A function f in Y is said to have absolutely continuous norm in Y ifand only if || f χ E k || Y → E k of µ -measurable sets satisfying E k → ∅ µ -almost everywhere.Let Y a be the subspace of Y consisting of functions of absolutely continuous normand let Y b be the closure in Y of the set of simple functions. It is known that Y a = Y b whenever 1 ≤ q ≤ ∞ . (See Bennett-Sharpley [1, Theorem I.3.13].) Moreover, since( X, µ ) is a σ -finite measure space, we have Y b = Y whenever 1 ≤ q < ∞ . (See Hunt[18, p. 258-259].)We recall (see Costea [6]) that in the Euclidean setting (that is, when µ = m n isthe n -dimensional Lebesgue measure and d is the Euclidean distance on R n ) we have Y a = Y for Y = L p, ∞ ( X, m n ) whenever X is an open subset of R n . Let X = B (0 , \{ } . As in Costea [6] we define u : X → R ,u ( x ) = ( | x | − np if 0 < | x | <
10 if 1 ≤ | x | ≤ . (1)It is easy to see that u ∈ L p, ∞ ( X, m n ) and moreover, || uχ B (0 ,α ) || L p, ∞ ( X,m n ) = || u || L p, ∞ ( X,m n ) = m n ( B (0 , /p for every α > . This shows that u does not have absolutely continuous weak L p -normand therefore L p, ∞ ( X, m n ) does not have absolutely continuous norm. Remark . It is also known (see [1, Proposition IV.4.2]) that for every p ∈ (1 , ∞ )and 1 ≤ r < s ≤ ∞ there exists a constant C ( p, r, s ) such that || f || L p,s ( X,µ ) ≤ C ( p, r, s ) || f || L p,r ( X,µ ) (2)for all measurable functions f ∈ L p,r ( X, µ ) . In particular, the embedding L p,r ( X, µ ) ֒ → L p,s ( X, µ ) holds.
Remark . Via Bennett-Sharpley [1, Proposition II.1.7 and Definition IV.4.1] it iseasy to see that for every p ∈ (1 , ∞ ) , q ∈ [1 , ∞ ] and 0 < α ≤ min( p, q ) , we have || f || αL p,q ( X,µ ) = || f α || L pα , qα ( X,µ ) for every nonnegative function f ∈ L p,q ( X, µ ) . .1. The subadditivity and superadditivity of the Lorentz quasinorms.
Werecall the known results and present new results concerning the superadditivity andthe subadditivity of the Lorentz p, q -quasinorm. For the convenience of the reader, wewill provide proofs for the new results and for some of the known results.The superadditivity of the Lorentz p, q -norm in the case 1 ≤ q ≤ p was stated inChung-Hunt-Kurtz [5, Lemma 2.5]. Proposition 2.4. (See [5, Lemma 2.5] .) Let ( X, µ ) be a measure space. Suppose that ≤ q ≤ p. Let { E i } i ≥ be a collection of pairwise disjoint µ -measurable subsets of X with E = ∪ i ≥ E i and let f ∈ L p,q ( X, µ ) . Then X i ≥ || χ E i f || pL p,q ( X,µ ) ≤ || χ E f || pL p,q ( X,µ ) . A similar result concerning the superadditivity was obtained in Costea-Maz’ya [8,Proposition 2.4] for the case 1 < p < q < ∞ when X = Ω was an open set in R n and µ was an arbitrary measure. That result is valid for a general measure space ( X, µ ) . Proposition 2.5.
Let ( X, µ ) be a measure space. Suppose that < p < q < ∞ . Let { E i } i ≥ be a collection of pairwise disjoint µ -measurable subsets of X with E = ∪ i ≥ E i and let f ∈ L p,q ( X, µ ) . Then X i ≥ || χ E i f || qL p,q ( X,µ ) ≤ || χ E f || qL p,q ( X,µ ) . Proof.
We mimic the proof of Proposition 2.4 from Costea-Maz’ya [8]. We replace Ωwith X. (cid:3) We have a similar result for the subadditivity of the Lorentz p, q -quasinorm. When1 < p < q ≤ ∞ we obtain a result that generalizes Theorem 2.5 from Costea [6].
Proposition 2.6.
Let ( X, µ ) be a measure space. Suppose that < p < q ≤ ∞ . Suppose f i , i = 1 , , . . . is a sequence of functions in L p,q ( X, µ ) and let f = sup i ≥ | f i | . Then || f || pL p,q ( X,µ ) ≤ ∞ X i =1 || f i || pL p,q ( X,µ ) . Proof.
Without loss of generality we can assume that all the functions f i , i = 1 , , . . . are nonnegative. We have to consider two cases, depending on whether p < q < ∞ or q = ∞ . Let µ [ f i ] be the distribution function of f i for i = 0 , , , . . . . It is easy to see that µ [ f ] ( s ) ≤ ∞ X i =1 µ [ f i ] ( s ) for every s ≥ . (3)Suppose that p < q < ∞ . We have (see Kauhanen-Koskela-Mal´y [20, Proposition2.1]) || f i || pL p,q ( X,µ ) = (cid:18) p Z ∞ s q − µ [ f i ] ( s ) qp ds (cid:19) pq (4) or i = 0 , , , . . . . From this and (3) we obtain || f || pL p,q (Ω ,µ ) = (cid:18) p Z ∞ s q − µ [ f ] ( s ) qp ds (cid:19) pq ≤ X i ≥ (cid:18) p Z ∞ s q − µ [ f i ] ( s ) qp ds (cid:19) pq = X i ≥ || f i || pL p,q (Ω ,µ ) . Now, suppose that q = ∞ . From (3) we obtain s p µ [ f ] ( s ) ≤ X i ≥ ( s p µ [ f i ] ( s )) for every s > , which implies s p µ [ f ] ( s ) ≤ X i ≥ || f i || pL p, ∞ ( X,µ ) for every s > . (5)By taking the supremum over all s > (cid:3) We recall a few results concerning Lorentz spaces.
Theorem 2.7. (See [6, Theorem 2.6] .) Suppose < p < q ≤ ∞ and ε ∈ (0 , . Let f , f ∈ L p,q ( X, µ ) . We denote f = f + f . Then f ∈ L p,q ( X, µ ) and || f || pL p,q ( X,µ ) ≤ (1 − ε ) − p || f || pL p,q ( X,µ ) + ε − p || f || pL p,q ( X,µ ) . Proof.
The proof of Theorem 2.6 from Costea [6] carries verbatim. We replace Ω with X. (cid:3) Theorem 2.7 has an useful corollary.
Corollary 2.8. (See [6, Corollary 2.7] .) Suppose < p < ∞ and ≤ q ≤ ∞ . Let f k bea sequence of functions in L p,q ( X, µ ) converging to f with respect to the p, q -quasinormand pointwise µ -almost everywhere in X. Then lim k →∞ || f k || L p,q ( X,µ ) = || f || L p,q ( X,µ ) . Proof.
The proof of Corollary 2.7 from Costea [6] carries verbatim. We replace Ω with X. (cid:3) p,q-modulus of the path family In this section, we establish some results about p, q -modulus of families of curves.Here (
X, d, µ ) is a metric measure space. We say that a curve γ in X is rectifiable ifit has finite length. Whenever γ is rectifiable, we use the arc length parametrization γ : [0 , ℓ ( γ )] → X, where ℓ ( γ ) is the length of the curve γ. Let Γ rect denote the family of all nonconstant rectifiable curves in X. It may well bethat Γ rect = ∅ , but we will be interested in metric spaces for which Γ rect is sufficientlylarge. efinition 3.1. For Γ ⊂ Γ rect , let F (Γ) be the family of all Borel measurable functions ρ : X → [0 , ∞ ] such that Z γ ρ ≥ γ ∈ Γ . Now for each 1 < p < ∞ and 1 ≤ q ≤ ∞ we defineMod p,q (Γ) = inf ρ ∈ F (Γ) || ρ || pL p,q ( X,µ ) . The number Mod p,q (Γ) is called the p,q-modulus of the family Γ . Basic properties of the p,q -modulus.
Usually, a modulus is a monotone andsubadditive set function. The following theorem will show, among other things, thatthis is true in the case of the p, q -modulus.
Theorem 3.2.
Suppose < p < ∞ and ≤ q ≤ ∞ . The set function Γ → Mod p,q (Γ) , Γ ⊂ Γ rect , enjoys the following properties: (i) Mod p,q ( ∅ ) = 0 . (ii) If Γ ⊂ Γ , then Mod p,q (Γ ) ≤ Mod p,q (Γ ) . (iii) Suppose ≤ q ≤ p. Then
Mod p,q ( ∞ [ i =1 Γ i ) q/p ≤ ∞ X i =1 Mod p,q (Γ i ) q/p . (iv) Suppose p < q ≤ ∞ . Then
Mod p,q ( ∞ [ i =1 Γ i ) ≤ ∞ X i =1 Mod p,q (Γ i ) . Proof. (i) Mod p,q ( ∅ ) = 0 because ρ ≡ ∈ F ( ∅ ) . (ii) If Γ ⊂ Γ , then F (Γ ) ⊂ F (Γ ) and hence Mod p,q (Γ ) ≤ Mod p,q (Γ ) . (iii) Suppose that 1 ≤ q ≤ p. The case p = q corresponds to the p -modulus and theclaim certainly holds in that case. (See for instance Haj lasz [13, Theorem 5.2 (3)].) Sowe can look at the case 1 ≤ q < p. We can assume without loss of generality that ∞ X i =1 Mod p,q (Γ i ) q/p < ∞ . Let ε > ρ i ∈ F (Γ i ) such that || ρ i || qL p,q ( X,µ ) < Mod p,q (Γ i ) q/p + ε − i . Let ρ := ( P ∞ i =1 ρ qi ) /q . We notice via Bennett-Sharpley [1, Proposition II.1.7 and Defi-nition IV.4.1] and Remark 2.3 applied with α = q that ρ qi ∈ L pq , ( X, µ ) and || ρ qi || L pq , ( X,µ ) = || ρ i || qL p,q ( X,µ ) . (6)for every i = 1 , , . . . . By using (6) and Remark 2.3 together with the definition of ρ and the fact that || · || L pq , ( X,µ ) is a norm when 1 ≤ q ≤ p, it follows that ρ ∈ F (Γ) andMod p,q (Γ i ) q/p ≤ || ρ || qL p,q ( X,µ ) ≤ ∞ X i =1 || ρ i || qL p,q ( X,µ ) < ∞ X i =1 Mod p,q (Γ i ) q/p + 2 ε. Letting ε → , we complete the proof when 1 ≤ q ≤ p. iv) Suppose now that p < q ≤ ∞ . We can assume without loss of generality that ∞ X i =1 Mod p,q (Γ i ) < ∞ . Let ε > ρ i ∈ F (Γ i ) such that || ρ i || pL p,q ( X,µ ) < Mod p,q (Γ i ) + ε − i . Let ρ := sup i ≥ ρ i . Then ρ ∈ F (Γ) . Moreover, from Proposition 2.6 it follows that ρ ∈ L p,q ( X, µ ) andMod p,q (Γ) ≤ || ρ || pL p,q ( X,µ ) ≤ ∞ X i =1 || ρ i || pL p,q ( X,µ ) < ∞ X i =1 Mod p,q (Γ i ) + 2 ε. Letting ε → , we complete the proof when p < q ≤ ∞ . (cid:3) So we proved that the modulus is a monotone function. Also, the shorter the curves,the larger the modulus. More precisely, we have:
Lemma 3.3.
Let Γ , Γ ⊂ Γ rect . If each curve γ ∈ Γ contains a subcurve that belongsto Γ , then Mod p,q (Γ ) ≤ Mod p,q (Γ ) . Proof. F (Γ ) ≤ F (Γ ) . (cid:3) The following theorem provides an useful characterization of path families that have p, q -modulus zero.
Theorem 3.4.
Let Γ ⊂ Γ rect . Then
Mod p,q (Γ) = 0 if and only if there exists a Borelmeasurable function ≤ ρ ∈ L p,q ( X, µ ) such that R γ ρ = ∞ for every γ ∈ Γ . Proof.
Sufficiency. We notice that ρ/n ∈ F (Γ) for every n and henceMod p,q (Γ) ≤ lim n →∞ || ρ/n || pL p,q ( X,µ ) = 0 . Necessity. There exists ρ i ∈ F (Γ) such that || ρ i || L ( p,q ) ( X,µ ) < − i and R γ ρ i ≥ γ ∈ Γ . Then ρ := P ∞ i =1 ρ i has the required properties. (cid:3) Corollary 3.5.
Suppose < p < ∞ and ≤ q ≤ ∞ are given. If ≤ g ∈ L p,q ( X, µ ) is Borel measurable, then R γ g < ∞ for p, q -almost every γ ∈ Γ rect . The following theorem is also important.
Theorem 3.6.
Let u k : X → R = [ −∞ , ∞ ] be a sequence of Borel functions whichconverge to a Borel function u : X → R in L p,q ( X, µ ) . Then there is a subsequence ( u k j ) j such that Z γ | u k j − u | → as j → ∞ , for p, q -almost every curve γ ∈ Γ rect . Proof.
We follow Haj lasz [13]. We take a subsequence ( u k j ) j such that || u k j − u || L p,q ( X,µ ) < − j . (7) et g j = | u k j − u | , and let Γ ⊂ Γ rect be the family of curves such thatlim sup j →∞ Z γ g j > . We want to show that Mod p,q (Γ) = 0 . Denote by Γ j the family of curves in Γ rect forwhich R γ g j > − j . Then 2 j g j ∈ F (Γ j ) and hence Mod p,q (Γ j ) < − pj as a consequenceof (7). We notice that Γ ⊂ ∞ \ i =1 ∞ [ j = i Γ j . Thus Mod p,q (Γ) /p ≤ ∞ X j = i Mod p,q (Γ j ) /p ≤ ∞ X j = i − j = 2 − i for every integer i ≥ , which implies Mod p,q (Γ) = 0 . (cid:3) Upper gradient.Definition 3.7.
Let u : X → [ −∞ , ∞ ] be a Borel function. We say that a Borelfunction g : X → [0 , ∞ ] is an upper gradient of u if for every rectifiable curve γ parametrized by arc length parametrization we have | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g (8)whenever both u ( γ (0)) and u ( γ ( ℓ ( γ ))) are finite and R γ g = ∞ otherwise. We say that g is a p, q -weak upper gradient of u if (8) holds on p, q -almost every curve γ ∈ Γ rect . The weak upper gradients were introduced in the case p = q by Heinonen-Koskelain [17]. See also Heinonen [15] and Shanmugalingam [27] and [28].If g is an upper gradient of u and e g = g, µ -almost everywhere, is another nonnegativeBorel function, then it might happen that e g is not an upper gradient of u. However,we have the following result.
Lemma 3.8. If g is a p, q -weak upper gradient of u and e g is another nonnegative Borelfunction such that e g = g µ -almost everywhere, then e g is a p, q -weak upper gradient of u. Proof.
Let Γ ⊂ Γ rect be the family of all nonconstant rectifiable curves γ : [0 , ℓ ( γ )] → X for which R γ | g − e g | > . The constant sequence g n = | g − e g | converges to 0 in L p,q ( X, µ ) , so from Theorem 3.6 it follows that Mod p,q (Γ ) = 0 and R γ | g − e g | = 0 for everynonconstant rectifiable curve γ : [0 , ℓ ( γ )] → X that is not in Γ . Let Γ ⊂ Γ rect be the family of all nonconstant rectifiable curves γ : [0 , ℓ ( γ )] → X for which the inequality | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then Mod p,q (Γ ) = 0 . Thus Mod p,q (Γ ∪ Γ ) = 0 . For every γ ∈ Γ rect not in Γ ∪ Γ we have | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g = Z γ e g. This finishes the proof. (cid:3) he next result shows that p, q -weak upper gradients can be nicely approximated byupper gradients. The case p = q was proved by Koskela-MacManus [23]. Lemma 3.9. If g is a p, q -weak upper gradient of u which is finite µ -almost everywhere,then for every ε > there exists an upper gradient g ε of u such that g ε ≥ g everywhere on X and || g ε − g || L p,q ( X,µ ) ≤ ε. Proof.
Let Γ ⊂ Γ rect be the family of all nonconstant rectifiable curves γ : [0 , ℓ ( γ )] → X for which the inequality | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then Mod p,q (Γ) = 0 and hence, from Theorem 3.4 it follows thatthere exists 0 ≤ ρ ∈ L p,q ( X, µ ) such that R γ ρ = ∞ for every γ ∈ Γ . Take g ε = g + ερ/ || ρ || L p,q ( X,µ ) . Then g ε is a nonnegative Borel function and | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g ε for every curve γ ∈ Γ rect . This finishes the proof. (cid:3) If A is a subset of X let Γ A be the family of all curves in Γ rect that intersect A and letΓ + A be the family of all curves in Γ rect such that the Hausdorff one-dimensional measure H ( | γ | ∩ A ) is positive. Here and throughout the paper | γ | is the image of the curve γ. The following lemma will be useful later in this paper.
Lemma 3.10.
Let u i : X → R , i ≥ , be a sequence of Borel functions such that g ∈ L p,q ( X ) is a p, q -weak upper gradient for every u i , i ≥ . We define u ( x ) = lim i →∞ u i ( x ) and E = { x ∈ X : | u ( x ) | = ∞} . Suppose that µ ( E ) = 0 and that u is well-definedoutside E. Then g is a p, q -weak upper gradient for u. Proof.
For every i ≥ ,i to be the set of all curves γ ∈ Γ rect for which | u i ( γ (0)) − u i ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then Mod p,q (Γ ,i ) = 0 and hence Mod p,q (Γ ) = 0 , where Γ = ∪ ∞ i =1 Γ ,i . Let Γ be the collection of all paths γ ∈ Γ rect such that R γ g = ∞ . Then we have viaTheorem 3.4 that Mod p,q (Γ ) = 0 since g ∈ L p,q ( X, µ ) . Since µ ( E ) = 0 , it follows that Mod p,q (Γ + E ) = 0 . Indeed, ∞ · χ E ∈ F (Γ + E ) and ||∞ · χ E || L p,q ( X,µ ) = 0 . Therefore Mod p,q (Γ ∪ Γ + E ∪ Γ ) = 0 . For any path γ in the family Γ rect \ (Γ ∪ Γ + E ∪ Γ ) , by the fact that the path is not inΓ + E , there exists a point y in | γ | such that y is not in E, that is y ∈ | γ | and | u ( y ) | < ∞ . For any point x ∈ | γ | , we have (since γ is not in Γ ,i ) | u i ( x ) | − | u i ( y ) | ≤ | u i ( x ) − u i ( y ) | ≤ Z γ g < ∞ . Therefore | u i ( x ) | ≤ | u i ( y ) | + Z γ g. Taking limits on both sides and using the facts that | u ( y ) | < ∞ and that γ is not inΓ ∪ Γ , we see thatlim i →∞ | u i ( x ) | ≤ lim i →∞ | u i ( y ) | + Z γ g = | u ( y ) | + Z γ g < ∞ nd therefore x is not in E. Thus Γ E ⊂ Γ ∪ Γ + E ∪ Γ and Mod p,q (Γ E ) = 0 . Next, let γ be a path in Γ rect \ (Γ ∪ Γ + E ∪ Γ ) . The above argument showed that | γ | does not intersect E. If we denote by x and y the endpoints of γ, we have | u ( x ) − u ( y ) | = | lim i →∞ u i ( x ) − lim i →∞ u i ( y ) | = lim i →∞ | u i ( x ) − u i ( y ) | ≤ Z γ g. Therefore g is a p, q -weak upper gradient for u as well. (cid:3) The following proposition shows how the upper gradients behave under a change ofvariable.
Proposition 3.11.
Let F : R → R be C and let u : X → R be a Borel function. If g ∈ L p,q ( X, µ ) is a p, q -weak upper gradient for u, then | F ′ ( u ) | g is a p, q -weak uppergradient for F ◦ u. Proof.
Let Γ to be the set of all curves γ ∈ Γ rect for which | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then Mod p,q (Γ ) = 0 . Let Γ ⊂ Γ rect be the collection of all curveshaving a subcurve in Γ . Then F (Γ ) ⊂ F (Γ ) and hence Mod p,q (Γ ) ≤ Mod p,q (Γ ) = 0 . Let Γ be the set of curves γ ∈ Γ rect for which R γ g = ∞ . Then we have via Theorem3.4 that Mod p,q (Γ ) = 0 since g ∈ L p,q ( X, µ ) . Thus Mod p,q (Γ ∪ Γ ) = 0 . The claim will follow immediately after we show that | ( F ◦ u )( γ (0)) − ( F ◦ u )( γ ( ℓ ( γ ))) | ≤ Z ℓ ( γ )0 ( | F ′ ( u ( γ ( s ))) | + ε ) g ( γ ( s )) ds. (9)for all curves γ ∈ Γ rect \ (Γ ∪ Γ ) and for every ε > . So fix ε > γ ∈ Γ rect \ (Γ ∪ Γ ) . Let ℓ = ℓ ( γ ) . We noticeimmediately that u ◦ γ is uniformly continuous on [0 , ℓ ] and F ′ is uniformly continuouson the compact interval I := ( u ◦ γ )([0 , ℓ ]) . Let δ, δ > | ( F ′ ◦ u ◦ γ )( t ) − ( F ′ ◦ u ◦ γ )( s ) | + | ( u ◦ γ )( t ) − ( u ◦ γ )( s ) | < δ for all t, s ∈ [0 , ℓ ] with | t − s | < δ and such that | F ′ ( u ) − F ′ ( v ) | < ε for all u, v ∈ I with | u − v | < δ . Fix an integer n > /δ and put ℓ i = ( iℓ ) /n, i = 0 , . . . , n − . For every i = 0 , . . . , n − | ( F ◦ u ◦ γ )( ℓ i +1 ) − ( F ◦ u ◦ γ )( ℓ i ) | = | F ′ ( t i,i +1 )) | | ( u ◦ γ )( ℓ i +1 ) − ( u ◦ γ )( ℓ i ) |≤ | F ′ ( t i,i +1 )) | Z ℓ i +1 ℓ i g ( γ ( s )) ds, where t i,i +1 ∈ I i,i +1 := ( u ◦ γ )(( ℓ i , ℓ i +1 )) . From the choice of δ it follows that | ( F ◦ u ◦ γ )( ℓ i +1 ) − ( F ◦ u ◦ γ )( ℓ i ) | ≤ Z ℓ i +1 ℓ i ( | F ′ ( u ( γ ( s ))) | + ε ) g ( γ ( s )) ds, for every i = 0 , . . . , n − . If we sum over i we obtain easily (9). This finishes theproof. (cid:3) As a direct consequence of Proposition 3.11, we have the following corollaries. orollary 3.12. Let r ∈ (1 , ∞ ) be fixed. Suppose u : X → R is a bounded nonnegativeBorel function. If g ∈ L p,q ( X, µ ) is a p, q -weak upper gradient of u, then ru r − g is a p, q -weak upper gradient for u r . Proof.
Let
M > ≤ u ( x ) < M for all x ∈ X. We apply Proposition3.11 to any C function F : R → R satisfying F ( t ) = t r , ≤ t ≤ M. (cid:3) Corollary 3.13.
Let r ∈ (0 , be fixed. Suppose that u : X → R is a nonnegativefunction that has a p, q -weak upper gradient g ∈ L p,q ( X, µ ) . Then r ( u + ε ) r − g is a p, q -weak upper gradient for ( u + ε ) r for all ε > . Proof.
Fix ε > . We apply Proposition 3.11 to any C function F : R → R satisfying F ( t ) = t r , ε ≤ t < ∞ . (cid:3) Corollary 3.14.
Suppose ≤ q ≤ p < ∞ . Let u , u be two nonnegative boundedreal-valued functions defined on X. Suppose g i ∈ L p,q ( X, µ ) , i = 1 , are p, q -weak uppergradients for u i , i = 1 , . Then L p,q ( X, µ ) ∋ g := ( g q + g q ) /q is a p, q -weak uppergradient for u := ( u q + u q ) /q . Proof.
The claim is obvious when q = 1 , so we assume without loss of generality that1 < q ≤ p. We prove first that g ∈ L p,q ( X, µ ) . Indeed, via Remark 2.3 it is enough toshow that g q ∈ L pq , ( X, µ ) . But g q = g q + g q and g qi ∈ L pq , ( X, µ ) since g i ∈ L p,q ( X, µ ) . (See Remark 2.3.) This, the fact that || · || L pq , ( X,µ ) is a norm whenever 1 < q ≤ p, andanother appeal to Remark 2.3 yield g ∈ L p,q ( X, µ ) with || g || qL p,q ( X,µ ) = || g q || L pq , ( X,µ ) ≤ || g q || L pq , ( X,µ ) + || g q || L pq , ( X,µ ) = || g || qL p,q ( X,µ ) + || g || qL p,q ( X,µ ) . For i = 1 , i, be the family of nonconstant rectifiable curves in Γ rect for which | u i ( x ) − u i ( x ) | ≤ Z γ g i is not satisfied. Then Mod p,q (Γ i, ) = 0 since g i is a p, q -weak upper gradient for u i , i = 1 , . Let Γ i, be the family of nonconstant rectifiable curves in Γ rect for which R γ g i = ∞ . Then for i = 1 , p,q (Γ i, ) = 0 via Theorem 3.4 because by hypothesis g i ∈ L p,q ( X, µ ) , i = 1 , . Let Γ = Γ , ∪ Γ , ∪ Γ , ∪ Γ , . Then Mod p,q (Γ ) = 0 . Fix ε > . By applying Corollary 3.12 with r = q to the functions u i for i = 1 , , we see that L p,q ( X, µ ) ∋ q ( u i + ε ) q − g i is a p, q -weak upper gradient of ( u i + ε ) q for i = 1 , . Thus via H¨older’s inequality it follows that G ε is a p, q -weak upper gradientfor U ε , where G ε := q (( u + ε ) q + ( u + ε ) q ) ( q − /q ( g q + g q ) /q and U ε := ( u + ε ) q + ( u + ε ) q . We notice that G ε ∈ L p,q ( X, µ ) . Indeed, G ε = qU ( q − /qε g, with U ε nonnegative abounded and g ∈ L p,q ( X, µ ) , so G ε ∈ L p,q ( X, µ ) . Now we apply Corollary 3.13 with r = 1 /q, u = U ε and g = G ε to obtain that u ε := U /qε has 1 /qU (1 − q ) /qε G ε = g as a p, q -weak upper gradient that belongs to L p,q ( X, µ ) . In fact, by looking at the proof of Proposition 3.11, we see that | u ε ( x ) − u ε ( y ) | ≤ Z γ g or every curve γ ∈ Γ rect that is not in Γ . Letting ε → , we obtain the desiredconclusion. This finishes the proof of the corollary. (cid:3) Lemma 3.15. If u i , i = 1 , are nonnegative real-valued Borel functions in L p,q ( X, µ ) with corresponding p, q -weak upper gradients g i ∈ L p,q ( X, µ ) , then g := max( g , g ) ∈ L p,q ( X, µ ) is a p, q -weak upper gradient for u := max( u , u ) ∈ L p,q ( X, µ ) . Proof.
It is easy to see that u, g ∈ L p,q ( X, µ ) . For i = 1 , ,i ⊂ Γ rect be thefamily of nonconstant rectifiable curves γ for which R γ g i = ∞ . Then we have viaTheorem 3.4 that Mod p,q (Γ ,i ) = 0 because g i ∈ L p,q ( X, µ ) . Thus Mod p,q (Γ ) = 0 , where Γ = Γ , ∪ Γ , . For i = 1 , i, ⊂ Γ rect be the family of curves γ ∈ Γ rect \ Γ for which | u i ( γ (0)) − u i ( γ ( ℓ ( γ ))) | ≤ Z γ g i is not satisfied. Then Mod p,q (Γ ,i ) = 0 since g i is a p, q -weak upper gradient for u i ,i = 1 , . Thus Mod p,q (Γ ) = 0 , where Γ = Γ , ∪ Γ , . It is easy to see that | u ( x ) − u ( y ) | ≤ max( | u ( x ) − u ( y ) | , | u ( x ) − u ( y ) | ) . (10)On every curve γ ∈ Γ rect \ (Γ ∪ Γ ) we have | u i ( x ) − u i ( y ) | ≤ Z γ g i ≤ Z γ g. This and (10) show that | u ( x ) − u ( y ) | ≤ Z γ g on every curve γ ∈ Γ rect \ (Γ ∪ Γ ) . This finishes the proof. (cid:3)
Lemma 3.16.
Suppose g ∈ L p,q ( X, µ ) is a p, q -weak upper gradient for ≤ u ∈ L p,q ( X, µ ) . Let λ ≥ be fixed. Then u λ := min( u, λ ) ∈ L p,q ( X, µ ) and g is a p, q -weakupper gradient for u λ . Proof.
Obviously 0 ≤ u λ ≤ u on X, so it follows via Bennett-Sharpley [1, Propo-sition I.1.7] and Kauhanen-Koskela-Mal´y [20, Proposition 2.1] that u λ ∈ L p,q ( X, µ )with || u λ || L p,q ( X,µ ) ≤ || u || L p,q ( X,µ ) . The second claim follows immediately since | u λ ( x ) − u λ ( y ) | ≤ | u ( x ) − u ( y ) | for every x, y ∈ X. (cid:3) Newtonian L p,q spaces We denote by f N ,L p,q ( X, µ ) the space of all Borel functions u ∈ L p,q ( X, µ ) that havea p, q -weak upper gradient g ∈ L p,q ( X, µ ) . We note that f N ,L p,q ( X, µ ) is a vector space,since if α, β ∈ R and u , u ∈ f N ,L p,q ( X, µ ) with respective p, q -weak upper gradients g , g ∈ L p,q ( X, µ ) , then | α | g + | β | g is a p, q -weak upper gradient of αu + βu . Definition 4.1. If u is a function in f N ,L p,q ( X, µ ) , let || u || e N ,Lp,q := (cid:16) || u || qL p,q ( X,µ ) + inf g || g || qL p,q ( X,µ ) (cid:17) /q , ≤ q ≤ p, (cid:16) || u || pL p,q ( X,µ ) + inf g || g || pL p,q ( X,µ ) (cid:17) /p , p < q ≤ ∞ , here the infimum is taken over all p, q -integrable p, q -weak upper gradients of u. Similarly, let || u || e N ,L ( p,q ) := (cid:16) || u || qL ( p,q ) ( X,µ ) + inf g || g || qL ( p,q ) ( X,µ ) (cid:17) /q , ≤ q ≤ p, (cid:16) || u || pL ( p,q ) ( X,µ ) + inf g || g || pL ( p,q ) ( X,µ ) (cid:17) /p , p < q ≤ ∞ , where the infimum is taken over all p, q -integrable p, q -weak upper gradients of u. If u, v are functions in f N ,L p,q ( X, µ ) , let u ∼ v if || u − v || e N ,Lp,q = 0 . It is easy to seethat ∼ is an equivalence relation that partitions f N ,L p,q ( X, µ ) into equivalence classes.We define the space N ,L p,q ( X, µ ) as the quotient f N ,L p,q ( X, µ ) / ∼ and || u || N ,Lp,q = || u || e N ,Lp,q and || u || N ,L ( p,q ) = || u || e N ,L ( p,q ) Remark . Via Lemma 3.9 and Corollary 2.8, it is easy to see that the infima inDefinition 4.1 could as well be taken over all p, q -integrable upper gradients of u. Wealso notice (see the discussion before Definition 2.1) that ||·|| N ,L ( p,q ) is a norm whenever1 < p < ∞ and 1 ≤ q ≤ ∞ , while || · || N ,Lp,q is a norm when 1 ≤ q ≤ p < ∞ . Moreover(see the discussion before Definition 2.1) || u || N ,Lp,q ≤ || u || N ,L ( p,q ) ≤ p ′ || u || N ,Lp,q for every 1 < p < ∞ , ≤ q ≤ ∞ and u ∈ N ,L p,q ( X, µ ) . Definition 4.3.
Let u : X → [ −∞ , ∞ ] be a given function. We say that(i) u is absolutely continuous along a rectifiable curve γ if u ◦ γ is absolutely contin-uous on [0 , ℓ ( γ )] . (ii) u is absolutely continuous on p, q -almost every curve (has ACC p,q property) iffor p, q -almost every γ ∈ Γ rect , u ◦ γ is absolutely continuous. Proposition 4.4. If u is a function in f N ,L p,q ( X, µ ) , then u is ACC p,q . Proof.
We follow Shanmugalingam [27]. By the definition of f N ,L p,q ( X, µ ) , u has a p, q -weak upper gradient g ∈ L p,q ( X, µ ) . Let Γ be the collection of all curves in Γ rect for which | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then by the definition of p, q -weak upper gradients, Mod p,q (Γ ) =0 . Let Γ be the collection of all curves in Γ rect that have a subcurve in Γ . ThenMod p,q (Γ ) ≤ Mod p,q (Γ ) = 0 . Let Γ be the collection of all curves in Γ rect such that R γ g = ∞ . Then Mod p,q (Γ ) = 0because g ∈ L p,q ( X, µ ) . Hence Mod p,q (Γ ∪ Γ ) = 0 . If γ is a curve in Γ rect \ (Γ ∪ Γ ) , then γ has no subcurves in Γ , and hence | u ( γ ( β )) − u ( γ ( α )) | ≤ Z βα g ( γ ( t )) dt, provided [ α, β ] ⊂ [0 , ℓ ( γ )] . This implies the absolute continuity of u ◦ γ as a consequence of the absolute continuityof the integral. Therefore u is absolutely continuous on every curve γ in Γ rect \ (Γ ∪ Γ ) . (cid:3) Lemma 4.5.
Suppose u is a function in f N ,L p,q ( X, µ ) such that || u || L p,q ( X,µ ) = 0 . Thenthe family
Γ = { γ ∈ Γ rect : u ( x ) = 0 for some x ∈ | γ |} has zero p, q -modulus. roof. We follow Shanmugalingam [27]. Since || u || L p,q ( X,µ ) = 0 , the set E = { x ∈ X : u ( x ) = 0 } has measure zero. With the notation introduced earlier, we haveΓ = Γ E = Γ + E ∪ (Γ E \ Γ + E ) . We can disregard the family Γ + E , sinceMod p,q (Γ + E ) ≤ ||∞ · χ E || pL p,q ( X,µ ) = 0 , where χ E is the characteristic function of the set E. The curves γ in Γ E \ Γ + E intersect E only on a set of linear measure zero, and hence with respect to the linear measure almosteverywhere on γ the function u is equal to zero. Since γ also intersects E, it follows that u is not absolutely continuous on γ. By Proposition 4.4, we have Mod p,q (Γ E \ Γ + E ) = 0 , yielding Mod p,q (Γ) = 0 . This finishes the proof. (cid:3)
Lemma 4.6.
Let F be a closed subset of X. Suppose that u : X → [ −∞ , ∞ ] is aBorel ACC p,q function that is constant µ -almost everywhere on F. If g ∈ L p,q ( X, µ ) isa p, q -weak upper gradient of u, then gχ X \ F is a p, q -weak upper gradient of u. Proof.
We can assume without loss of generality that u = 0 µ -almost everywhere on F. Let E = { x ∈ F : u ( x ) = 0 } . Then by assumption µ ( E ) = 0 . Hence Mod p,q (Γ + E ) = 0because ∞ · χ E ∈ F (Γ + E ) . Let Γ ⊂ Γ rect be the family of curves on which u is not absolutely continuous or onwhich | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ g is not satisfied. Then Mod p,q (Γ ) = 0 . Let Γ ⊂ Γ rect be the family of curves that havea subcurve in Γ . Then F (Γ ) ⊂ F (Γ ) and thus Mod p,q (Γ ) ≤ Mod p,q (Γ ) = 0 . Let Γ ⊂ Γ rect be the family of curves on which R γ g = ∞ . Then via Theorem 3.4 wehave Mod p,q (Γ ) = 0 because g ∈ L p,q ( X, µ ) . Let γ : [0 , ℓ ( γ )] → X be a curve in Γ rect \ (Γ ∪ Γ ∪ Γ + E ) connecting x and y. Weshow that | u ( x ) − u ( y ) | ≤ Z γ gχ X \ F for every such curve γ. The cases | γ | ⊂ F \ E and | γ | ⊂ ( X \ F ) ∪ E are trivial. So is the case when both x and y are in F \ E. Let K := ( u ◦ γ ) − ( { } ) . Then K is a compact subset of [0 , ℓ ( γ )]because u ◦ γ is continuous on [0 , ℓ ( γ )] . Hence K contains its lower bound c and itsupper bound d. Let x = γ ( c ) and y = γ ( d ) . Suppose that both x and y are in ( X \ F ) ∪ E. Then we see that [ c, d ] ⊂ (0 , ℓ ( γ ))and γ ([0 , c ) ∪ ( d, ℓ ( γ )]) ⊂ ( X \ F ) ∪ E. Moreover, since γ is not in Γ and u ( x ) = u ( y ) , we have | u ( x ) − u ( y ) | ≤ | u ( x ) − u ( x ) | + | u ( y ) − u ( y ) | ≤ Z γ ([0 ,c ]) g + Z γ ([ d,ℓ ( γ )]) g ≤ Z γ gχ X \ F because the subcurves γ | [0 ,c ] and γ | [ d.ℓ ( γ )] intersect E on a set of Hausdorff 1-measurezero.Suppose now by symmetry that x ∈ ( X \ F ) ∪ E and y ∈ F \ E. This means in termsof our notation that c > d = ℓ ( γ ) . We notice that γ ([0 , c )) ⊂ ( X \ F ) ∪ E and ( x ) = u ( y ) and thus | u ( x ) − u ( y ) | = | u ( x ) − u ( x ) | ≤ Z γ ([0 ,c ]) g ≤ Z γ gχ X \ F because the subcurve γ | [0 ,c ] intersect E on a set of Hausdorff 1-measure zero.This finishes the proof of the lemma. (cid:3) Lemma 4.7.
Assume that u ∈ N ,L p,q ( X, µ ) , and that g, h ∈ L p,q ( X, µ ) are p, q -weakupper gradients of u. If F ⊂ X is a closed set, then ρ = gχ F + hχ X \ F is a p, q -weak upper gradient of u as well.Proof. We follow Haj lasz [13]. Let Γ ⊂ Γ rect be the family of curves on which R γ ( g + h ) = ∞ . Then via Theorem 3.4 it follows that Mod p,q (Γ ) = 0 because g + h ∈ L p,q ( X, µ ) . Let Γ ⊂ Γ rect be the family of curves on which u is not absolutely continuous. Thenvia Proposition 4.4 we see that Mod p,q (Γ ) = 0 . Let Γ ′ ⊂ Γ rect be the family of curves on which | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ min (cid:18)Z γ g, Z γ h (cid:19) is not satisfied. Let Γ ⊂ Γ rect be the family of curves which contain subcurves be-longing to Γ ′ . Since F (Γ ′ ) ⊂ F (Γ ) , we have Mod p,q (Γ ) ≤ Mod p,q (Γ ′ ) = 0 . Now itremains to show that | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ ρ for all γ ∈ Γ rect \ (Γ ∪ Γ ∪ Γ ) . If | γ | ⊂ F or | γ | ⊂ X \ F, then the inequality is obvious.Thus we can assume that the image | γ | has a nonempty intersection both with F andwith X \ F. The set γ − ( X \ F ) is open and hence it consists of a countable (or finite) number ofopen and disjoint intervals. Assume without loss of generality that there are countablymany such intervals. Denote these intervals by (( t i , s i )) ∞ i =1 . Let γ i = γ | [ t i ,s i ] . We have | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ | u ( γ (0)) − u ( γ ( t )) | + | u ( γ ( t )) − u ( γ ( s )) | + | u ( γ ( s )) − u ( γ ( ℓ ( γ ))) | ≤ Z γ \ γ g + Z γ h, where γ \ γ denotes the two curves obtained from γ by removing the interior part γ , that is the curves γ | [0 ,t ] and γ | [ s ,b ] . Similarly we can remove a larger number ofsubcurves of γ. This yields | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ \∪ ni =1 γ i g + Z ∪ ni =1 γ i h for each positive integer n. By applying Lebesgue dominated convergence theorem tothe curve integral on γ, we obtain | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ gχ F + Z γ hχ X \ F = Z γ ρ. (cid:3) ext we show that when 1 < p < ∞ and 1 ≤ q < ∞ , every function u ∈ N ,L p,q ( X, µ ) has a ‘smallest’ p, q -weak upper gradient. For the case p = q see Kallunki-Shanmugalingam [19] and Shanmugalingam [28]. Theorem 4.8.
Suppose that < p < ∞ and ≤ q < ∞ . For every u ∈ N ,L p,q ( X, µ ) , there exists the least p, q -weak upper gradient g u ∈ L p,q ( X, µ ) of u. It is smallest inthe sense that if g ∈ L p,q ( X, µ ) is another p, q -weak upper gradient of u, then g ≥ g u µ -almost everywhere.Proof. We follow Haj lasz [13]. Let m = inf g || g || L p,q ( X,µ ) , where the infimum is takenover the set of all p, q -weak upper gradients of u. It suffices to show that there exists a p, q -weak upper gradient g u of u such that || g u || L p,q ( X,µ ) = m. Indeed, if we suppose that g ∈ L p,q ( X, µ ) is another p, q -weak upper gradient of u such that the set { g < g u } haspositive measure, then by the inner regularity of the measure µ there exists a closedset F ⊂ { g < g u } such that µ ( F ) > . Via Lemma 4.7 it follows that the function ρ := gχ F + g u χ X \ F is a p, q -weak upper gradient. Via Kauhanen-Koskela-Mal´y [20,Proposition 2.1] that would give || ρ || L p,q ( X,µ ) < || g u || L p,q ( X,µ ) = m, in contradiction withthe minimality of || g u || L p,q ( X,µ ) . Thus it remains to prove the existence of a p, q -weak upper gradient g u such that || g u || L p,q ( X,µ ) = m. Let ( g i ) ∞ i =1 be a sequence of p, q -weak upper gradients of u such that || g i || L p,q ( X,µ ) < m + 2 − i . We will show that it is possible to modify the sequence ( g i ) insuch a way that we will obtain a new sequence of p, q -weak upper gradients ( ρ i ) of u satisfying || ρ i || L p,q ( X,µ ) < m + 2 − i , ρ ≥ ρ ≥ ρ ≥ . . . µ -almost everywhere . The sequence ( ρ i ) ∞ i =1 will be defined by induction. We set ρ = g . Suppose the p, q -weak upper gradients ρ , ρ , . . . , ρ i have already been chosen. We will now define ρ i +1 . Since ρ i ∈ L p,q ( X, µ ) , the measure µ is inner regular and the ( p, q )-norm has theabsolute continuity property whenever 1 < p < ∞ and 1 ≤ q < ∞ (see the discussionafter Definition 2.1), there exists a closed set F ⊂ { g i +1 < ρ i } such that || ρ i χ { g i +1 <ρ i }\ F || L p,q ( X,µ ) < − i − . Now we set ρ i +1 = g i +1 χ F + ρ i χ X \ F . Then ρ i +1 ≤ ρ i and ρ i +1 ≤ g i +1 χ F ∪{ g i +1 ≥ ρ i } + ρ i χ { g i +1 <ρ i }\ F . Suppose first that 1 ≤ q ≤ p. Since || · || L p,q ( X,µ ) is a norm, we see that || ρ i +1 || L p,q ( X,µ ) ≤ || g i +1 χ F ∪{ g i +1 ≥ ρ i } || L p,q ( X,µ ) + || ρ i χ { g i +1 <ρ i }\ F || L p,q ( X,µ ) < m + 2 − i − + 2 − i − = m + 2 − i . Suppose now that p < q < ∞ . Then we have via Proposition 2.6 || ρ i +1 || pL p,q ( X,µ ) ≤ || g i +1 χ F ∪{ g i +1 ≥ ρ i } || pL p,q ( X,µ ) + || ρ i χ { g i +1 <ρ i }\ F || pL p,q ( X,µ ) < ( m + 2 − i − ) p + 2 − p ( i +1) < ( m + 2 − i ) p . Thus, no matter what q ∈ [1 , ∞ ) is, we showed that m ≤ || ρ i +1 || L p,q ( X,µ ) < m + 2 − i . The sequence of p, q -weak upper gradients ( ρ i ) ∞ i =1 converges pointwise to a function ρ. The absolute continuity of the ( p, q )-norm (see Bennett-Sharpley [1, Proposition I.3.6]and the discussion after Definition 2.1) yieldslim i →∞ || ρ i − ρ || L p,q ( X,µ ) = 0 . bviously || ρ || L p,q ( X,µ ) = m. The proof will be finished as soon as we show that ρ is a p, q -weak upper gradient for u. By taking a subsequence if necessary, we can assume that || ρ i − ρ || L p,q ( X,µ ) ≤ − i forevery i ≥ . Let Γ ⊂ Γ rect be the family of curves on which R γ ( ρ + ρ i ) = ∞ for some i ≥ . Thenvia Theorem 3.4 and the subadditivity of Mod p,q ( · ) /p we see that Mod p,q (Γ ) = 0 since ρ + ρ i ∈ L p,q ( X, µ ) for every i ≥ . For any integer i ≥ ,i ⊂ Γ rect be the family of curves for which | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ ρ i is not satisfied. Then Mod p,q (Γ ,i ) = 0 because ρ i is a p, q -weak upper gradient for u. Let Γ = ∪ ∞ i =1 Γ ,i . Let Γ ⊂ Γ rect be the family of curves for which lim sup i →∞ R γ | ρ i − ρ | > . Then itfollows via Theorem 3.6 that Mod p,q (Γ ) = 0 . Let γ be a curve in Γ rect \ (Γ ∪ Γ ∪ Γ ) . On any such curve we have (since γ is notin Γ ,i ) | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ Z γ ρ i for every i ≥ . By letting i → ∞ , we obtain (since γ is not in Γ ∪ Γ ) | u ( γ (0)) − u ( γ ( ℓ ( γ ))) | ≤ lim i →∞ Z γ ρ i = Z γ ρ < ∞ . This finishes the proof of the theorem. (cid:3) Sobolev p, q -capacity
In this section, we establish a general theory of the Sobolev-Lorentz p, q -capacityin metric measure spaces. If (
X, d, µ ) is a metric measure space, then the Sobolev p, q -capacity of a set E ⊂ X isCap p,q ( E ) = inf {|| u || pN ,Lp,q : u ∈ A ( E ) } , where A ( E ) = { u ∈ N ,L p,q ( X, µ ) : u ≥ } . We call A ( E ) the set of admissible functions for E. If A ( E ) = ∅ , then Cap p,q ( E ) = ∞ . Remark . It is easy to see that we can consider only admissible functions u forwhich 0 ≤ u ≤ . Indeed, for u ∈ A ( E ) , let v := min( u + , , where u + = max( u, . Wenotice that | v ( x ) − v ( y ) | ≤ | u ( x ) − u ( y ) | for every x, y in X, which implies that every p, q -weak upper gradient for u is also a p, q -weak upper gradient for v. This impliesthat v ∈ A ( E ) and || v || N ,Lp,q ≤ || u || N ,Lp,q . Basic properties of the Sobolev p, q -capacity.
A capacity is a monotone,subadditive set function. The following theorem expresses, among other things, thatthis is true for the Sobolev p, q -capacity.
Theorem 5.2.
Suppose that < p < ∞ and ≤ q ≤ ∞ . Suppose that ( X, d, µ ) is acomplete metric measure space. The set function E Cap p,q ( E ) , E ⊂ X, enjoys thefollowing properties: (i) If E ⊂ E , then Cap p,q ( E ) ≤ Cap p,q ( E ) . ii) Suppose that µ is nonatomic. Suppose that < q ≤ p. If E ⊂ E ⊂ . . . ⊂ E = S ∞ i =1 E i ⊂ X, then Cap p,q ( E ) = lim i →∞ Cap p,q ( E i ) . (iii) Suppose that p < q ≤ ∞ . If E = S ∞ i =1 E i ⊂ X, then Cap p,q ( E ) ≤ ∞ X i =1 Cap p,q ( E i ) . (iv) Suppose that ≤ q ≤ p. If E = S ∞ i =1 E i ⊂ X, then Cap p,q ( E ) q/p ≤ ∞ X i =1 Cap p,q ( E i ) q/p . Proof.
Property (i) is an immediate consequence of the definition.(ii) Monotonicity yields L := lim i →∞ Cap p,q ( E i ) ≤ Cap p,q ( E ) . To prove the opposite inequality, we may assume without loss of generality that
L < ∞ . The reflexivity of L p,q ( X, µ ) (guaranteed by the nonatomicity of µ whenever 1 < q ≤ p < ∞ ) will be used here in order to prove the opposite inequality.Let ε > i = 1 , , . . . we choose u i ∈ A ( E i ) , ≤ u i ≤ g i such that || u i || qN ,Lp,q < Cap p,q ( E i ) q/p + ε ≤ L q/p + ε. (11)We notice that u i is a bounded sequence in N ,L p,q ( X, µ ) . Hence there exists a subse-quence, which we denote again by u i and functions u, g ∈ L p,q ( X, µ ) such that u i → u weakly in L p,q ( X, µ ) and g i → g weakly in L p,q ( X, µ ) . It is easy to see that u ≥ µ -almost everywhere and g ≥ µ -almost everywhere . Indeed, since u i converges weakly to u in L p,q ( X, µ ) which is the dual of L p ′ ,q ′ ( X, µ )(see Hunt [18, p. 262]), we havelim i →∞ Z X u i ( x ) ϕ ( x ) dµ ( x ) = Z X u ( x ) ϕ ( x ) dµ ( x )for all ϕ ∈ L p ′ ,q ′ ( X, µ ) . For nonnegative functions ϕ ∈ L p ′ ,q ′ ( X, µ ) this yields0 ≤ lim i →∞ Z X u i ( x ) ϕ ( x ) dµ ( x ) = Z X u ( x ) ϕ ( x ) dµ ( x ) , which easily implies u ≥ µ -almost everywhere on X. Similarly we have g ≥ µ -almosteverywhere on X. From the weak- ∗ lower semicontinuity of the p, q -norm (see Bennett-Sharpley [1,Proposition II.4.2, Definition IV.4.1 and Theorem IV.4.3] and Hunt [18, p. 262]), itfollows that || u || L p,q ( X,µ ) ≤ lim inf i →∞ || u i || L p,q ( X,µ ) and || g || L p,q ( X,µ ) ≤ lim inf i →∞ || g i || L p,q ( X,µ ) . (12)Using Mazur’s lemma simultaneously for u i and g i , we obtain sequences v i withcorrespondent upper gradients e g i such that v i ∈ A ( E i ) , v i → u in L p,q ( X, µ ) and µ -almost everywhere and e g i → g in L p,q ( X, µ ) and µ -almost everywhere. These sequencescan be found in the following way. Let i be fixed. Since every subsequence of ( u i , g i )converges to ( u, g ) weakly in the reflexive space L p,q ( X, µ ) × L p,q ( X, µ ) , we may usethe Mazur lemma (see Yosida [30, p. 120]) for the subsequence ( u i , g i ) , i ≥ i . e obtain finite convex combinations v i and e g i of the functions u i and g i , i ≥ i asclose as we want in L p,q ( X, µ ) to u and g respectively. For every i = i , i + 1 , . . . wesee that u i = 1 in E i ⊃ E i . The intersection of finitely many supersets of E i contains E i . Therefore, v i equals 1 on E i . It is easy to see that e g i is an upper gradient for v i . Passing to subsequences if necessary, we may assume that v i converges to u pointwise µ -almost everywhere, that e g i converges to g pointwise µ -almost everywhere and thatfor every i = 1 , , . . . we have || v i +1 − v i || L p,q ( X,µ ) + || e g i +1 − e g i || L p,q ( X,µ ) ≤ − i . (13)Since v i converges to u in L p,q ( X, µ ) and pointwise µ -almost everywhere on X while e g i converges to g in L p,q ( X, µ ) and pointwise µ -almost everywhere on X it follows viaCorollary 2.8 thatlim i →∞ || v i || L p,q ( X,µ ) = || u || L p,q ( X,µ ) and lim i →∞ || e g i || L p,q ( X,µ ) = || g || L p,q ( X,µ ) . (14)This, (11) and (12) yield || u || qL p,q ( X,µ ) + || g || qL p,q ( X,µ ) = lim i →∞ || v i || qN ,Lp,q ≤ L q/p + ε. (15)For j = 1 , , . . . we set w j = sup i ≥ j v i and b g j = sup i ≥ j e g i . It is easy to see that w j = 1 on E. We claim that b g j is a p, q -weak upper gradient for w j . Indeed, for every k > j, let w j,k = sup k ≥ i ≥ j v i . Via Lemma 3.15 and finite induction, it follows easily that b g j is a p, q -weak uppergradient for every w j,k whenever k > j. It is easy to see that w j = lim k →∞ w j,k pointwisein X. This and Lemma 3.10 imply that b g j is indeed a p, q -weak upper gradient for w j . Moreover, w j ≤ v j + ∞ X i = j | v i +1 − v i | and b g j ≤ e g j + k − X i = j | e g i +1 − e g i | (16)Thus || w j || L p,q ( X,µ ) ≤ || v j || L p,q ( X,µ ) + ∞ X i = j || v i +1 − v i || L p,q ( X,µ ) ≤ || v j || L p,q ( X,µ ) + 2 − j +1 and || b g j || L p,q ( X,µ ) ≤ || e g j || L p,q ( X,µ ) + ∞ X i = j || e g i +1 − e g i || L p,q ( X,µ ) ≤ || e g j || L p,q ( X,µ ) + 2 − j +1 , which implies that w j , b g j ∈ L p,q ( X, µ ) . Thus w j ∈ A ( E ) with p, q -weak upper gradient b g j . We notice that 0 ≤ g = inf j ≥ b g j µ -almost everywhere on X and 0 ≤ u = inf j ≥ w j µ -almost everywhere on X. Since w and b g are in L p,q ( X, µ ) , the absolute continuityof the p, q -norm (see Bennett-Sharpley [1, Proposition I.3.6] and the discussion afterDefinition 2.1) yieldslim j →∞ || w j − u || L p,q ( X,µ ) = 0 and lim j →∞ || b g j − g || L p,q ( X,µ ) = 0 . (17)By using (15), (17), and Corollary 2.8 we see thatCap p,q ( E ) q/p ≤ lim j →∞ || w j || qN ,Lp,q = || u || qL p,q ( X,µ ) + || g || qL p,q ( X,µ ) ≤ L q/p + ε. y letting ε → , we get the converse inequality so (ii) is proved.(iii) We can assume without loss of generality that ∞ X i =1 Cap p,q ( E i ) q/p < ∞ . For i = 1 , , . . . let u i ∈ A ( E i ) with upper gradient g i such that0 ≤ u i ≤ || u i || qN ,Lp,q < Cap p,q ( E i ) q/p + ε − i . Let u := ( P ∞ i =1 u qi ) /q and g := ( P ∞ i =1 g qi ) /q . We notice that u ≥ E. By repeatingthe argument from the proof of Theorem 3.2 (iii), we see that u, g ∈ L p,q ( X, µ ) and || u || qL p,q ( X,µ ) + || g || qL p,q ( X,µ ) ≤ ∞ X i =1 (cid:16) || u i || qL p,q ( X,µ ) + || g i || qL p,q ( X,µ ) (cid:17) ≤ ε + ∞ X i =1 Cap p,q ( E i ) q/p . We are done with the case 1 ≤ q ≤ p as soon as we show that u ∈ A ( E ) and that g is a p, q -weak upper gradient for u. It follows easily via Corollary 3.14 and finite inductionthat g is a p, q -weak upper gradient for e u n := ( P ≤ i ≤ n u qi ) /q for every n ≥ . Since u ( x ) = lim i →∞ e u i ( x ) < ∞ on X \ F, where F = { x ∈ X : u ( x ) = ∞} it follows fromLemma 3.10 combined with the fact that u ∈ L p,q ( X, µ ) that g is in fact a p, q -weakupper gradient for u. This finishes the proof for the case 1 ≤ q ≤ p. (iv) We can assume without loss of generality that ∞ X i =1 Cap p,q ( E i ) < ∞ . For i = 1 , , . . . let u i ∈ A ( E i ) with upper gradients g i such that0 ≤ u i ≤ || u i || pN ,Lp,q < Cap p,q ( E i ) + ε − i . Let u := sup i ≥ u i and g := sup i ≥ g i . We notice that u = 1 on E. Moreover, viaProposition 2.6 it follows that u, g ∈ L p,q ( X, µ ) with || u || pL p,q ( X,µ ) + || g || pL p,q ( X,µ ) ≤ ∞ X i =1 (cid:16) || u i || pL p,q ( X,µ ) + || g i || pL p,q ( X,µ ) (cid:17) ≤ ε + ∞ X i =1 Cap p,q ( E i ) . We are done with the case p < q ≤ ∞ as soon as we show that u ∈ A ( E ) and that g is a p, q -weak upper gradient for u. Via Lemma 3.15 and finite induction, it followsthat g is a p, q -weak upper gradient for e u n := max ≤ i ≤ n u i for every n ≥ . Since u ( x ) = lim i →∞ e u i ( x ) pointwise on X, it follows via Lemma 3.10 that g is in fact a p, q -weak upper gradient for u. This finishes the proof for the case p < q ≤ ∞ . (cid:3) Remark . We make a few remarks.(i) Suppose µ is nonatomic and 1 < q < ∞ . By mimicking the proof of Theorem 5.2(ii) and working with the ( p, q )-norm and the ( p, q )-capacity, we can also show thatlim i →∞ Cap ( p,q ) ( E i ) = Cap ( p,q ) ( E )whenever E ⊂ E ⊂ . . . ⊂ E = S ∞ i =1 E i ⊂ X. (ii) Moreover, if Cap p,q is an outer capacity then it follows immediately thatlim i →∞ Cap p,q ( K i ) = Cap p,q ( K ) henever ( K i ) ∞ i =1 is a decreasing sequence of compact sets whose intersection set is K. We say that Cap p,q is an outer capacity if for every E ⊂ X we haveCap p,q ( E ) = inf { Cap p,q ( U ) : E ⊂ U ⊂ X, U open } . (iii) Any outer capacity satisfying properties (i) and (ii) of Theorem 5.2 is called aChoquet capacity. (See Appendix II in Doob [9].)We recall that if A ⊂ X, then Γ A is the family of curves in Γ rect that intersect A andΓ + A is the family of all curves in Γ rect such that the Hausdorff one-dimensional measure H ( | γ | ∩ A ) is positive. The following lemma will be useful later in this paper. Lemma 5.4. If F ⊂ X is such that Cap p,q ( F ) = 0 , then Mod p,q (Γ F ) = 0 . Proof.
We follow Shanmugalingam [27]. We can assume without loss of generalitythat q = p. Since Cap p,q ( F ) = 0 , for each positive integer i there exists a function v i ∈ A ( F ) such that 0 ≤ v i ≤ || v i || N ,L ( p,q ) ≤ − i . Let u n := P ni =1 v i . Then u n ∈ N ,L ( p,q ) ( X, µ ) for each n, u n ( x ) is increasing for each x ∈ X, and for every m > n we have || u n − u m || N ,L ( p,q ) ≤ n X i = m +1 || v i || N ,L ( p,q ) ≤ − m → , as m → ∞ . Therefore the sequence { u n } ∞ n =1 is a Cauchy sequence in N ,L ( p,q ) ( X, µ ) . Since { u n } ∞ n =1 Cauchy in N ,L ( p,q ) ( X, µ ) , it follows that it is Cauchy in L p,q ( X, µ ) . Hence by passing to a subsequence if necessary, there is a function e u in L p,q ( X, µ ) towhich the subsequence converges both pointwise µ -almost everywhere and in the L ( p,q ) norm. By choosing a further subsequence, again denoted by { u i } ∞ i =1 for simplicity, wecan assume that || u i − e u || L ( p,q ) ( X,µ ) + || g i,i +1 || L ( p,q ) ( X,µ ) ≤ − i , where g i,j is an upper gradient of u i − u j for i < j. If g is an upper gradient of u , then u = u + ( u − u ) has an upper gradient g = g + g . In general, u i = u + i − X k =1 ( u k +1 − u k )has an upper gradient g i = g + i − X k =1 g k,k +1 for every i ≥ . For j < i we have || g i − g j || L ( p,q ) ( X,µ ) ≤ i − X k = j || g k,k +1 || L ( p,q ) ( X,µ ) ≤ i − X k = j − k ≤ − j → j → ∞ . Therefore { g i } ∞ i =1 is also a Cauchy sequence in L ( p,q ) ( X, µ ) , and hence converges inthe L ( p,q ) norm to a nonnegative Borel function g. Moreover, we have || g j − g || L ( p,q ) ( X,µ ) ≤ − j for every j ≥ . We define u by u ( x ) = lim i →∞ u i ( x ) . Since u i → e u µ -almost everywhere, it followsthat u = e u µ -almost everywhere and thus u ∈ L p,q ( X, µ ) . Let E = { x ∈ X : lim i →∞ u i ( x ) = ∞} . he function u is well defined outside of E. In order for the function u to be in thespace N ,L p,q ( X, µ ) , the function u has to be defined on almost all paths by Proposition4.4. To this end it is shown that the p, q -modulus of the family Γ E is zero. Let Γ bethe collection of all paths from Γ rect such that R γ g = ∞ . Then we have via Theorem3.4 that Mod p,q (Γ ) = 0 since g ∈ L p,q ( X, µ ) . Let Γ be the family of all curves from Γ rect such that lim sup j →∞ R γ | g j − g | > . Since || g j − g || L p,q ( X, µ ) ≤ − j for all j ≥ , it follows via Theorem 3.6 that Mod p,q (Γ ) = 0 . Since u ∈ L p,q ( X, µ ) and E = { x ∈ X : u ( x ) = ∞} , it follows that µ ( E ) = 0 andthus Mod Γ + E = 0 . Therefore Mod p,q (Γ ∪ Γ ∪ Γ + E ) = 0 . For any path γ in the familyΓ rect \ (Γ ∪ Γ ∪ Γ + E ) , by the fact that γ is not in Γ + E , there exists a point in | γ | \ E. For any point x in γ, since g i is an upper gradient of u i , it follow that u i ( x ) − u i ( y ) ≤ | u i ( x ) − u i ( y ) | ≤ Z γ g i . Therefore u i ( x ) ≤ u i ( y ) + Z γ g i . Taking limits on both sides and using the fact that γ is not in Γ ∪ Γ , it follows thatlim i →∞ u i ( x ) ≤ lim i →∞ u i ( y ) + Z γ g = u ( y ) + Z γ g < ∞ , and therefore x is not in E. Thus Γ E ⊂ Γ ∪ Γ ∪ Γ + E and Mod p,q (Γ E ) = 0 . Therefore g is a p, q -weak upper gradient of u, and hence u ∈ N ,L p,q ( X, µ ) . For each x not in E we can write u ( x ) = lim i →∞ u i ( x ) < ∞ . If F \ E is nonempty, then u | F \ E ≥ u n | F \ E = n X i =1 v i | F \ E = n for arbitrarily large n, yielding that u | F \ E = ∞ . But this impossible, since x is not inthe set E. Therefore F ⊂ E, and hence Γ F ⊂ Γ E . This finishes the proof of the lemma. (cid:3)
Next we prove that ( N ,L p,q ( X, µ ) , || · || N ,L ( p,q ) ) is a Banach space. Theorem 5.5.
Suppose < p < ∞ and ≤ q ≤ ∞ . Then ( N ,L p,q ( X, µ ) , || · || N ,L ( p,q ) ) is a Banach space.Proof. We follow Shanmugalingam [27]. We can assume without loss of generality that q = p. Let { u i } ∞ i =1 be a Cauchy sequence in N ,L p,q ( X, µ ) . To show that this sequenceis convergent in N ,L p,q ( X, µ ) , it suffices to show that some subsequence is convergentin N ,L p,q ( X, µ ) . Passing to a further subsequence if necessary, it can be assumed that || u i +1 − u i || L ( p,q ) ( X,µ ) + || g i,i +1 || L ( p,q ) ( X,µ ) ≤ − i , where g i,j is an upper gradient of u i − u j for i < j. Let E j = { x ∈ X : | u j +1 ( x ) − u j ( x ) | ≥ − j } . Then 2 j | u j +1 − u j | ∈ A ( E j ) and henceCap p,q ( E j ) /p ≤ j || u j +1 − u j || N ,Lp,q ≤ − j . Let F j = ∪ ∞ k = j E k . ThenCap p,q ( E j ) /p ≤ ∞ X k = j Cap p,q ( E k ) /p ≤ − j . et F = ∩ ∞ j =1 F j . We notice that Cap p,q ( F ) = 0 . If x is a point in X \ F, there exists j ≥ x is not in F j = ∪ ∞ k = j E k . Hence for all k ≥ j, x is not in E k . Thus | u k +1 ( x ) − u k ( x ) | ≤ − k for all k ≥ j. Therefore whenever l ≥ k ≥ j we have that | u k ( x ) − u l ( x ) | ≤ − k . Thus the sequence { u k ( x ) } ∞ k =1 is Cauchy for every x ∈ X \ F. For every x ∈ X \ F, let u ( x ) = lim i →∞ u i ( x ) . For k < m,u m = u k + m − X n = k ( u n +1 − u n ) . Therefore for each x in X \ F,u ( x ) = u k ( x ) + ∞ X n = k ( u n +1 ( x ) − u n ( x )) . (18)Noting by Lemma 5.4 that Mod p,q (Γ F ) = 0 and that for each path γ in Γ rect \ Γ F equation (18) holds pointwise on | γ | , we conclude that P ∞ n = k g n,n +1 is a p, q -weak uppergradient of u − u k . Therefore || u − u k || N ,L ( p,q ) ≤ || u − u k || L ( p,q ) ( X,µ ) + ∞ X n = k || g n,n +1 || L ( p,q ) ( X,µ ) ≤ || u − u k || L ( p,q ) ( X,µ ) + ∞ X n = k − n ≤ || u − u k || L ( p,q ) ( X,µ ) + 2 − k → k → ∞ . Therefore the subsequence converges in the norm of N ,L p,q ( X, µ ) to u. This completesthe proof of the theorem. (cid:3) Density of Lipschitz functions in N ,L p,q ( X, µ )6.1.
Poincar´e inequality.
Now we define the weak (1 , L p,q )- Poincar´e inequality . Pod-brdsky in [26] introduced a stronger Poincar´e inequality in the case of Banach-valuedNewtonian Lorentz spaces.
Definition 6.1.
The space (
X, d, µ ) is said to support a weak (1 , L p,q ) -Poincar´e in-equality if there exist constants C > σ ≥ B with radius r, all µ -measurable functions u on X and all upper gradients g of u we have1 µ ( B ) Z B | u − u B | dµ ≤ Cr µ ( σB ) /p || gχ σB || L p,q ( X,µ ) . (19)Here u B = 1 µ ( B ) Z B u ( x ) dµ ( x )whenever u is a locally µ -integrable function on X. In the above definition we can equivalently assume via Lemma 3.9 and Corollary2.8 that g is a p, q -weak upper gradient of u. When p = q we have the weak (1 , p )-Poincar´e inequality. For more about the Poincar´e inequality in the case p = q seeHaj lasz-Koskela [14] and Heinonen-Koskela [17]. measure µ is said to be doubling if there exists a constant C ≥ µ (2 B ) ≤ Cµ ( B )for every ball B = B ( x, r ) in X. A metric measure space ( X, d, µ ) is called doubling ifthe measure µ is doubling. Under the assumption that the measure µ is doubling, itis known that ( X, d, µ ) is proper (that is, closed bounded subsets of X are compact) ifand only if it is complete.Now we prove that if 1 ≤ q ≤ p, the measure µ is doubling, and the space( X, d, µ ) carries a weak (1 , L p,q )-Poincar´e inequality, the Lipschitz functions are densein N ,L p,q ( X, µ ) . In order to prove this we need a few definitions and lemmas.
Definition 6.2.
Suppose (
X, d ) is a metric space that carries a doubling measure µ. For 1 < p < ∞ and 1 ≤ q ≤ ∞ we define the noncentered maximal function operatorby M p,q u ( x ) = sup B ∋ x || uχ B || L p,q ( X,µ ) µ ( B ) /p , where u ∈ L p,q ( X, µ ) . Lemma 6.3.
Suppose ( X, d ) is a metric space that carries a doubling measure µ. If ≤ q ≤ p, then M p,q maps L p,q ( X, µ ) to L p, ∞ ( X, µ ) boundedly and moreover, lim λ →∞ λ p µ ( { x ∈ X : M p,q u ( x ) > λ } ) = 0 . Proof.
We can assume without loss of generality that 1 ≤ q < p. For every
R > M Rp,q be the restricted maximal function operator defined on L p,q ( X, µ ) by M Rp,q u ( x ) = sup B ∋ x, diam ( B ) ≤ R || uχ B || L p,q ( X,µ ) µ ( B ) /p . Denote G λ = { x ∈ X : M p,q u ( x ) > λ } and G Rλ = { x ∈ X : M Rp,q u ( x ) > λ } . It is easyto see that G R λ ⊂ G R λ if 0 < R < R < ∞ and G Rλ → G λ as R → ∞ . Fix
R > . For every x ∈ G Rλ , λ > , there exists a ball B ( y x , r x ) with diameter atmost R such that x ∈ B ( y x , r x ) and such that || uχ B ( y x ,r x ) || pL p,q ( X,µ ) > λ p µ ( B ( y x , r x )) . We notice that B ( y x , r x ) ⊂ G Rλ . The set G Rλ is covered by such balls and by Theo-rem 1.2 in Heinonen [15] it follows that there exists a countable disjoint subcollection { B ( x i , r i ) } ∞ i =1 such that the collection { B ( x i , r i ) } ∞ i =1 covers G Rλ . Hence µ ( G Rλ ) ≤ ∞ X i =1 µ ( B ( x i , r i )) ≤ C ∞ X i =1 µ ( B ( x i , r i )) ! ≤ Cλ p ∞ X i =1 || uχ B ( x i ,r i ) || pL p,q ( X,µ ) ! ≤ Cλ p || uχ G Rλ || pL p,q ( X,µ ) . The last inequality in the sequence was obtained by applying Proposition 2.4. (Seealso Chung-Hunt-Kurtz [5, Lemma 2.5].)Thus µ ( G Rλ ) ≤ Cλ p || uχ G Rλ || pL p,q ( X,µ ) ≤ Cλ p || uχ G λ || pL p,q ( X,µ ) or every R > . Since G λ = S R> G Rλ , we obtain (by taking the limit as R → ∞ ) µ ( G λ ) ≤ Cλ p || uχ G λ || pL p,q ( X,µ ) . The absolute continuity of the p, q -norm (see the discussion after Definition 2.1), the p, q -integrability of u and the fact that G λ → ∅ µ -almost everywhere as λ → ∞ yieldthe desired conclusion. (cid:3) Question 6.4.
Is Lemma 6.3 true when p < q < ∞ ?The following proposition is necessary in the sequel. Proposition 6.5.
Suppose < p < ∞ and ≤ q < ∞ . If u is a nonnegative functionin N ,L p,q ( X, µ ) , then the sequence of functions u k = min( u, k ) , k ∈ N , converges inthe norm of N ,L p,q ( X, µ ) to u as k → ∞ . Proof.
We notice (see Lemma 3.16) that u k ∈ L p,q ( X, µ ) . That lemma also yields easily u k ∈ N ,L p,q ( X, µ ) and moreover || u k || N ,Lp,q ≤ || u || N ,Lp,q for all k ≥ . Let E k = { x ∈ X : u ( x ) > k } . Since µ is a Borel regular measure, there exists anopen set O k such that E k ⊂ O k and µ ( O k ) ≤ µ ( E k ) + 2 − k . In fact the sequence ( O k ) ∞ k =1 can be chosen such that O k +1 ⊂ O k for all k ≥ . Since µ ( E k ) ≤ C ( p,q ) k p || u || pL p,q ( X,µ ) , itfollows that µ ( O k ) ≤ µ ( E k ) + 2 − k ≤ C ( p, q ) k p || u || pL p,q ( X,µ ) + 2 − k . Thus lim k →∞ µ ( O k ) = 0 . We notice that u = u k on X \ O k . Thus 2 gχ O k is a p, q -weakupper gradient of u − u k whenever g is an upper gradient for u and u k . See Lemma4.6. The fact that O k → ∅ µ -almost everywhere and the absolute continuity of the( p, q )-norm yieldlim sup k →∞ || u − u k || N ,L ( p,q ) ≤ k →∞ (cid:16) || uχ O k || L ( p,q ) ( X,µ ) + || gχ O k || L ( p,q ) ( X,µ ) (cid:17) = 0 . (cid:3) Counterexample 6.6.
For q = ∞ Proposition 6.5 is not true. Indeed, let n ≥ < p ≤ n be fixed. Let X = B (0 , \ { } ⊂ R n , endowed with theEuclidean metric and the Lebesgue measure.Suppose first that 1 < p < n. Let u p and g p be defined on X by u p ( x ) = | x | − np − g p ( x ) = np − ! | x | − np . It is easy to see that u p , g p ∈ L p, ∞ ( X, m n ) . Moreover, (see for instance Haj lasz [13,Proposition 6.4]) g p is the minimal upper gradient for u p . Thus u p ∈ N ,L p, ∞ ( X, m n ) . For every integer k ≥ u p,k and g p,k on X by u p,k ( x ) = ( k if 0 < | x | ≤ ( k + 1) pp − n , | x | − np − k + 1) pp − n < | x | < g p,k ( x ) = (cid:16) np − (cid:17) | x | − np if 0 < | x | < ( k + 1) pp − n k + 1) pp − n ≤ | x | < . e notice that u p,k ∈ N ,L p, ∞ ( X, m n ) for all k ≥ . Moreover, via [13, Proposition6.4] and Lemma 4.6 we see that g p,k is the minimal upper gradient for u p − u p,k forevery k ≥ . Since g p,k ց X as k → ∞ and || g p,k || L p, ∞ ( X,m n ) = || g p || L p, ∞ ( X,m n ) = C ( n, p ) > k ≥ , it follows that u p,k does not converge to u p in N ,L p, ∞ ( X, m n )as k → ∞ . Suppose now that p = n. Let u n and g n be defined on X by u n ( x ) = ln 1 | x | and g n ( x ) = 1 | x | . It is easy to see that u n , g n ∈ L p, ∞ ( X, m n ) . Moreover, (see for instance Haj lasz [13,Proposition 6.4]) g n is the minimal upper gradient for u n . Thus u n ∈ N ,L n, ∞ ( X, m n ) . For every integer k ≥ u n,k and g n,k on X by u n,k ( x ) = ( k if 0 < | x | ≤ e − k , ln | x | if e − k < | x | < g n,k ( x ) = ( | x | if 0 < | x | < e − k e − k ≤ | x | < . We notice that u n,k ∈ N ,L n, ∞ ( X, m n ) for all k ≥ . Moreover, via [13, Proposition6.4] and Lemma 4.6 we see that g n,k is the minimal upper gradient for u n − u n,k forevery k ≥ . Since g n,k ց X as k → ∞ and || g n,k || L p, ∞ ( X,m n ) = || g n || L n, ∞ ( X,m n ) = C ( n ) > k ≥ , it follows that u n,k does not converge to u n in N ,L n, ∞ ( X, m n )as k → ∞ . The following lemma will be used in the paper.
Lemma 6.7.
Let f ∈ N ,L p,q ( X, µ ) be a bounded Borel function with p, q -weak uppergradient g ∈ L p,q ( X, µ ) and let f be a bounded Borel function with p, q -weak uppergradient g ∈ L p,q ( X, µ ) . Then f := f f ∈ N ,L p,q ( X, µ ) and g := | f | g + | f | g is a p, q -weak upper gradient of f . Proof.
It is easy to see that f and g are in L p,q ( X, µ ) . Let Γ ⊂ Γ rect be the family ofcurves on which R γ ( g + g ) = ∞ . Then it follows via Theorem 3.4 that Mod p,q (Γ ) = 0because g + g ∈ L p,q ( X, µ ) . Let Γ ,i ⊂ Γ rect , i = 1 , | f i ( γ (0)) − f i ( γ ( ℓ ( γ ))) | ≤ Z γ g i is not satisfied. Then Mod Γ ,i = 0 , i = 1 , . Let Γ ⊂ Γ rect be the family of curves thathave a subcurve in Γ , ∪ Γ , . Then F (Γ , ∪ Γ , ) ⊂ F (Γ ) and thus Mod p,q (Γ ) ≤ Mod p,q (Γ , ∪ Γ , ) = 0 . We notice immediately that Mod p,q (Γ ∪ Γ ) = 0 . Fix ε > . By using the argument from Lemma 1.7 in Cheeger [4], we see that | f ( γ (0)) − f ( γ ( ℓ ( γ ))) | ≤ Z ℓ ( γ )0 ( | f ( γ ( s )) | + ε ) g ( γ ( s )) + ( | f ( γ ( s )) | + ε ) g ( γ ( s )) ds for every curve γ in Γ rect \ (Γ ∪ Γ ) . Letting ε → (cid:3) ix x ∈ X. For each integer j > η j ( x ) = d ( x , x ) ≤ j − ,j − d ( x , x ) if j − < d ( x , x ) ≤ j, d ( x , x ) > j. Lemma 6.8.
Suppose ≤ q < ∞ . Let u be a bounded function in N ,L p,q ( X, µ ) . Thenthe function v j = uη j is also in N ,L p,q ( X, µ ) . Furthermore, the sequence v j convergesto u in N ,L p,q ( X, µ ) . Proof.
We can assume without loss of generality that u ≥ . Let g ∈ L p,q ( X, µ ) be anupper gradient for u. It is easy to see by invoking Lemma 4.6 that h j := χ B ( x ,j ) \ B ( x ,j − is a p, q -weak upper gradient for η j and for 1 − η j . By using Lemma 6.7, we see that v j ∈ N ,L p,q ( X, µ ) and that g j := uh j + gη j is a p, q -weak upper gradient for v j . Byusing Lemma 6.7 we notice that e h j := uh j + g (1 − η j ) is a p, q -weak upper gradient for u − v j . We have in fact0 ≤ u − v j ≤ uχ X \ B ( x ,j − and e h j ≤ ( u + g ) χ X \ B ( x ,j − . (20)for every j > . The absolute continuity of the ( p, q )-norm when 1 ≤ q < ∞ (see thediscussion after Definition 2.1) together with the p, q -integrability of u, g and (20) yieldthe desired claim. (cid:3) Now we prove the density of the Lipschitz functions in N ,L p,q ( X, µ ) when 1 ≤ q < p. The case q = p was proved by Shanmugalingam. (See [27] and [28].) Theorem 6.9.
Let ≤ q ≤ p < ∞ . Suppose that ( X, d, µ ) is a doubling metric measurespace that carries a weak (1 , L p,q ) -Poincar´e inequality. Then the Lipschitz functions aredense in N ,L p,q ( X, µ ) . Proof.
We can consider only the case 1 ≤ q < p because the case q = p was provedby Shanmugalingam in [27] and [28]. We can assume without loss of generality that u is nonnegative. Moreover, via Lemmas 6.5 and 6.7 we can assume without loss ofgenerality that u is bounded and has compact support in X. Choose
M > ≤ u ≤ M on X. Let g ∈ L p,q ( X, µ ) be a p, q -weak upper gradient for u. Let σ ≥ , L p,q )-Poincar´e inequality.Let G λ := { x ∈ X : M p,q g ( x ) > λ } . If x is a point in the closed set X \ G λ , then forall r > µ ( B ( x, r )) Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ Cr || gχ B ( x,σr ) || L p,q ( X,µ ) µ ( B ( x, σr )) /p ≤ CrM p,q g ( x ) ≤ Cλr.
Hence for s ∈ [ r/ , r ] one has that | u B ( x,s ) − u B ( x,r ) | ≤ µ ( B ( x, s )) Z B ( x,s ) | u − u B ( x,r ) | dµ ≤ µ ( B ( x, r )) µ ( B ( x, s )) · µ ( B ( x, r )) Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ Cλr henever x is in X \ G λ . For a fixed s ∈ (0 , r/
2) there exists an integer k ≥ − k r ≤ s < − k +1 r. Then | u B ( x,s ) − u B ( x,r ) | ≤ | u B ( x,s ) − u B ( x, − k r ) | ++ k − X i =0 | u B ( x, − i − r ) − u B ( x, − i r ) | ≤ Cλ k X i =0 − i r ! ≤ Cλr.
For any sequence r i ց u B ( x,r i ) ) ∞ i =1 is a Cauchy sequence for everypoint x in X \ G λ . Thus on X \ G λ we can define the function u λ ( x ) := lim r → u B ( x,r ) . We notice that u λ ( x ) = u ( x ) for every Lebesgue point x in X \ G λ . For every x, y in X \ G λ we consider the chain of balls { B i } ∞ i = −∞ , where B i = B ( x, i d ( x, y )) , i ≤ B i = B ( y, − i d ( x, y )) , i > . For every two such points x and y we have that they are Lebesgue points for u λ byconstruction and hence | u λ ( x ) − u λ ( y ) | ≤ ∞ X i = −∞ | u B i − u B i +1 | ≤ Cλd ( x, y ) , where C depends only on the data on X. Thus u λ is Cλ -Lipschitz on X \ G λ . Byconstruction it follows that 0 ≤ u λ ≤ M on X \ G λ . Extend u λ as a Cλ -Lipschitzfunction on X (see McShane [25]) and denote the extension by v λ . Then v λ ≥ X since u λ ≥ X \ G λ . Let w λ := min( v λ , M ) . We notice that w λ is a nonnegative Cλ -Lipschitz function on X since v λ is. Moreover, w λ = v λ = u λ on X \ G λ whenever λ > M. Since u = w λ µ -almost everywhere on X \ G λ whenever λ > M we have || u − w λ || L p,q ( X,µ ) = || ( u − w λ ) χ G λ || L p,q ( X,µ ) ≤ || uχ G λ || L p,q ( X,µ ) + C ( p, q ) λµ ( G λ ) /p whenever λ > M. The absolute continuity of the p, q -norm when 1 ≤ q ≤ p togetherwith Lemma 6.3 imply that lim λ →∞ || u − w λ || L p,q ( X,µ ) = 0 . Since u − w λ = 0 µ -almost everywhere on the closed set G λ , it follows via Lemma4.6 that ( Cλ + g ) χ G λ is a p, q -weak upper gradient for u − w λ . By using the absolutecontinuity of the p, q -norm when 1 ≤ q ≤ p together with Lemma 6.3, we see thatlim λ →∞ || ( Cλ + g ) χ G λ || L p,q ( X,µ ) = 0 . This finishes the proof of the theorem. (cid:3)
Theorem 6.9 yields the following result.
Proposition 6.10.
Let ≤ q ≤ p < ∞ . Suppose that ( X, d, µ ) satisfies the hypothesesof Theorem 6.9. Then Cap p,q is an outer capacity.
In order to prove Proposition 6.10 we need to state and prove the following propo-sition, thus generalizing Proposition 1.4 from Bj¨orn-Bj¨orn-Shanmugalingam [3]. roposition 6.11. (See [3, Proposition 1.4] ) Let < p < ∞ and ≤ q < ∞ . Supposethat ( X, d, µ ) is a proper metric measure space. Let E ⊂ X be such that Cap p,q ( E ) = 0 . Then for every ε > there exists an open set U ⊃ E with Cap p,q ( U ) < ε. Proof.
We adjust the proof of Proposition 1.4 in Bj¨orn-Bj¨orn-Shanmugalingam [3] tothe Lorentz setting with some modifications. It is enough to consider the case when q = p. Due to the countable subadditivity of Cap p,q ( · ) /p we can assume without lossof generality that E is bounded. Moreover, we can also assume that E is Borel.Since Cap p,q ( E ) = 0 , we have χ E ∈ N ,L p,q ( X, µ ) and || χ E || N ,Lp,q = 0 . Let ε ∈ (0 , g ∈ L p,q ( X, µ ) suchthat g is an upper gradient for χ E and || g || L p,q ( X,µ ) < ε. By adapting the proof ofthe Vitali-Carath´eodory theorem to the Lorentz setting (see Folland [10, Proposition7.14]) we can find a lower semicontinuous function ρ ∈ L p,q ( X, µ ) such that ρ ≥ g and || ρ − g || L p,q ( X,µ ) < ε. Since Cap p,q ( E ) = 0 , it follows immediately that µ ( E ) = 0 . By using the outer regularity of the measure µ and the absolute continuity of the( p, q )-norm, there exists a bounded open set V ⊃ E such that || χ V || L p,q ( X,µ ) + || ( ρ + 1) χ V || L p,q ( X,µ ) < ε . Let u ( x ) = min ( , inf γ Z γ ( ρ + 1) ) , where the infimum is taken over all the rectifiable (including constant) curves connect-ing x to the closed set X \ V. We notice immediately that 0 ≤ u ≤ X and u = 0on X \ V. By Bj¨orn-Bj¨orn-Shanmugalingam [3, Lemma 3.3] it follows that u is lowersemicontinuous on X and thus the set U = { x ∈ X : u ( x ) > } is open. We noticethat for x ∈ E and every curve connecting x to some y ∈ X \ V, we have Z γ ( ρ + 1) ≥ Z γ ρ ≥ χ E ( x ) − χ E ( y ) = 1 . Thus u = 1 on E and E ⊂ U ⊂ V. From [3, Lemmas 3.1 and 3.2] it follows that( ρ + 1) χ V is an upper gradient of u. Since 0 ≤ u ≤ χ V and u is lower semicontinuous,it follows that u ∈ N ,L p,q ( X, µ ) . Moreover, 2 u ∈ A ( U ) and thusCap p,q ( U ) /p ≤ || u || N ,Lp,q ≤ || u || L p,q ( X,µ ) + || ( ρ + 1) χ V || L p,q ( X,µ ) ) ≤ || χ V || L p,q ( X,µ ) + || ( ρ + 1) χ V || L p,q ( X,µ ) ) < ε. This finishes the proof of Proposition 6.11. (cid:3)
Now we prove Proposition 6.10.
Proof.
We start the proof of Proposition 6.10 by showing that every function u in N ,L p,q ( X, µ ) is continuous outside open sets of arbitrarily small p, q -capacity. (Sucha function is called p, q -quasicontinuous.) Indeed, let u be a function in N ,L p,q ( X, µ ) . From Theorem 6.9 there exists a sequence { u j } ∞ j =1 of Lipschitz functions on X suchthat || u j − u || N ,Lp,q < − j for every integer j ≥ . For every integer j ≥ E j = { x ∈ X : | u j +1 ( x ) − u j ( x ) | > − j } . hen all the sets E j are open because the all functions u j are Lipschitz. By letting F = ∩ ∞ j =1 ∪ ∞ k = j E k and applying the argument from Theorem 5.5 to the sequence { u k } ∞ k =1 which is Cauchy in N ,L p,q ( X, µ ) , we see that Cap p,q ( F ) = 0 and the sequence { u k } converges in N ,L p,q ( X, µ ) to a function e u whose restriction on X \ F is continuous.Thus || u − e u || N ,Lp,q = 0 and hence if we let E = { x ∈ X : u ( x ) = e u ( x ) } , we haveCap p,q ( E ) = 0 . Therefore Cap p,q ( E ∪ F ) = 0 and hence, via Proposition 6.11 we havethat u = e u outside open supersets of E ∪ F of arbitrarily small p, q -capacity. Thisshows that u is quasicontinuous.Now we fix E ⊂ X and we show thatCap p,q ( E ) = inf { Cap p,q ( U ) , E ⊂ U ⊂ X, U open } . For a fixed ε ∈ (0 ,
1) we choose u ∈ A ( E ) such that 0 ≤ u ≤ X and such that || u || N ,Lp,q < Cap p,q ( E ) /p + ε. We have that u is p, q -quasicontinuous and hence there is an open set V such thatCap p,q ( V ) /p < ε and such that u | X \ V is continuous. Thus there exists an open set U such that U \ V = { x ∈ X : u ( x ) > − ε } \ V ⊃ E \ V. We see that U ∪ V = ( U \ V ) ∪ V is an open set containing E ∪ V = ( E \ V ) ∪ V. ThereforeCap p,q ( E ) /p ≤ Cap p,q ( U ∪ V ) /p ≤ Cap p,q ( U \ V ) /p + Cap p,q ( V ) /p ≤ − ε || u || N ,Lp,q + Cap p,q ( V ) /p ≤ − ε (Cap p,q ( E ) /p + ε ) + ε. Letting ε → (cid:3) Theorems 5.2 and 6.9 together with Proposition 6.10 and Remark 5.3 yield immedi-ately the following capacitability result. (See also Appendix II in Doob [9].)
Theorem 6.12.
Let < q ≤ p < ∞ . Suppose that ( X, d, µ ) satisfies the hypotheses ofTheorem 6.9. The set function E Cap p,q ( E ) is a Choquet capacity. In particular,all Borel subsets (in fact, all analytic subsets) E of X are capacitable, that is Cap p,q ( E ) = sup { Cap p,q ( K ) : K ⊂ E, K compact } whenever E ⊂ X is Borel (or analytic).Remark . Counterexample 6.6 gives also a counterexample to the density resultfor N ,L p, ∞ in the Euclidean setting for 1 < p ≤ n and q = ∞ . References [1] C. Bennett and R. Sharpley.
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