Differentiation bases for Sobolev functions on metric spaces

Publ. Mat. 48 (2004), 381–395 DIFFERENTIATION BASES FOR SOBOLEV FUNCTIONS ON METRIC SPACES Petteri Harjulehto and Juha Kinnunen Abstract We study Lebesgue points for Sobolev functions over other collections of sets than balls. Our main result gives several conditions for a differentiation basis, which characterize the existence of Lebesgue points outside a set of capacity zero. 1. Introduction By the classical Lebesgue differentiation theorem almost every point is a Lebesgue point for a locally integrable function f:Rn→[−∞,∞]. In particular, this implies that (1.1) lim r→0 1 |B(x, r)|ZB(x,r) f(y)dy =f(x) for almost every x∈Rn, where |·|denotes the Lebesgue measure and B(x, r) is a ball with the center xand the radius r > 0. It is a very interesting question to ask whether (1.1) holds for other collections of sets than balls. This question has been studied extensively in [dG1] and [dG2], see also Chapter 10 of [St]. It turns out that the existence of the Lebesgue points is equivalent to certain estimates for maximal functions and derivatives of the integrals as well as to some covering properties. The objective of this note is to study similar questions for functions which are more regular. Indeed, we are interested in functions, which belong to a first order Sobolev space. It is known that (1.1) holds for a Sobolev function outside a set of capacity zero, see [EG]. We are interested in limits of integral averages over other collections of sets than balls. Our main theorem gives several equivalent conditions for a differentiation basis, which characterize the existence of Lebesgue points outside a set of capacity zero. An interesting feature of the result is 2000 Mathematics Subject Classification. 46E35. Key words. Lebesgue points, capacity, maximal functions. 382 P. Harjulehto, J. Kinnunen that a qualitative result, as the existence of the Lebesgue points, implies quantitative estimates. Another interesting fact is that certain capacitary weak type estimates for maximal functions or derivatives of the integrals imply the existence of the Lebesgue points. These phenomena are visible already in [dG1] and [dG2]. In fact, our proofs are modifications of the corresponding arguments for integrable functions, but the proofs for Sobolev functions are somewhat more subtle. To emphasize the fact that our proof is based on a general principle we state and prove our main result in the context of metric measure spaces. Recently there has been some interest in defining the first order Sobolev spaces on a metric measure space, see [C], [FKS], [Ha], [HKM], [HeKo] and [Sh] suitable modifications our argument applies to any of these approaches. For simplicity, we have chosen the definition of Sobolev spaces on a metric measure space due to Haj lasz [Ha]. A general outline of the theory and further references can be found in [HaKo2]. The Lebesgue theorem with respect to balls for Sobolev functions on metric spaces has been studied in [KL]. In the final section we briefly comment on the Euclidean case. 2. Preliminaries In this section we recall the definition due to Haj lasz [Ha] of a first order Sobolev space on an arbitrary metric measure space. Let (X, d) be a metric space and let µbe a non-negative Borel regular outer measure on X. In the following, we keep the metric measure space (X, d, µ) fixed, and for short, we denote it by X. The Lebesgue space Lp(X) with 1 ≤p < ∞is the Banach space of all µ-a.e. defined µ-measurable functions f:X→[−∞,∞] with the norm kfkLp(X)=ZX |f|pdµ1/p . The space L∞(X) consists of essentially bounded functions. Let 1 < p ≤ ∞ and suppose that u∈Lp(X). We denote by D(u) the set of all µ-measurable functions g:X→[0,∞] such that (2.1) |u(x)−u(y)| ≤ d(x, y)g(x) + g(y) for every x, y ∈X\N,x6=y, with µ(N) = 0. In the metric setting, instead of having the gradient, we have a set D(u) of maximal gradients of u. A function u∈Lp(X) belongs to the Sobolev space M1,p(X) if D(u)∩Lp(X)6=∅. Differentiation Bases for Sobolev Functions 383 The Sobolev space M1,p(X) is equipped with the norm kukM1,p(X)=kukp Lp(X)+kukp L1,p(X)1/p, where kukL1,p(X)= infkgkLp(X):g∈D(u)∩Lp(X). We recall some basic properties of the Sobolev space M1,p(X). If u∈M1,p(X) and g∈D(u)∩Lp(X), then the Poincar´e inequality (2.2) ZB(x,r)u−uB(x,r)dµ ≤2rZB(x,r) g dµ holds for every x∈Xand r > 0. Here we use the standard notation fB(x,r)=ZB(x,r) f dµ =1 µ(B(x, r)) ZB(x,r) f dµ and B(x, r) denotes the open ball with the center xand the radius r > 0. The Poincar´e inequality is easily proved by integrating the pointwise inequality (2.1) twice over the ball. It follows immediately from the definition that if u∈M1,p(X) then |u| ∈ M1,p(X) with  |u| M1,p (X)≤ kukM1,p(X). If X=Rnwith the Euclidean metric and the Lebesgue measure, then M1,p(Rn) = W1,p(Rn),1< p ≤ ∞. Moreover, the norms are comparable (see [Ha]). Here W1,p(Rn) is the first order Sobolev space of those functions in Lp(Rn) whose first distributional derivatives belong to Lp(Rn) with the norm kukW1,p(Rn)=kukLp(Rn)+k∇ukLp(Rn). Indeed, if u∈W1,p(Rn), then we have the pointwise inequality |u(x)−u(y)| ≤ c|x−y|M|∇u|(x) + M|∇u|(y) for every x, y ∈Rn\Nwith |N|= 0. Here M|∇u|is the HardyLittlewood maximal function of |∇u|. The maximal operator is bounded in Lp(Rn) when 1 < p ≤ ∞. This shows that M|∇u| ∈ D(u)∩Lp(Rn) and hence W1,p(Rn)⊂M1,p(Rn). The reverse inclusion follows from the characterization of W1,p(Rn) with the integrated difference quotients, see 7.11 of [GT]. Since the maximal operator is not bounded in L1(Rn) the case p= 1 is excluded in the definition. This also suggests that g∈D(u) corresponds to the maximal function of the gradient of urather than the gradient. 384 P. Harjulehto, J. Kinnunen There is a natural capacity in the Sobolev space. For 1 < p < ∞, the Sobolev p-capacity of the set E⊂Xis the number Cp(E) = infkukp M1,p(X):u∈ A(E), where A(E) = u∈M1,p(X) : u≥1 on an open neighbourhood of E. If A(E) = ∅, we set Cp(E) = ∞. The Sobolev capacity is a monotone and countably subadditive set function, see Theorem 3.2 in [KM]. It is easy to see [KM, Remark 3.3] that the Sobolev capacity is an outer capacity, which means that Cp(E) = inf{Cp(O) : O⊃E, O open}. The capacity measures the exceptional sets for Sobolev functions. To be more precise, a function u:X→[−∞,∞] is p-quasi continuous in X if for every ε > 0 there is a set Esuch that Cp(E)< ε and the restriction of uto X\Eis continuous. By outer regularity, we may assume that E is open. Functions in M1,p(X) are defined only up to a set of measure zero, but the following result [KM, Corollary 3.7] shows that Sobolev functions are defined outside a set of capacity zero. Theorem 2.3. For each u∈M1,p(X)there is a p-quasi continuous function u∗∈M1,p(X)such that u=u∗µ-a.e. in X. Moreover, the p-quasi continuous representative is unique in the sense that if two p-quasi continuous functions coincide µ-almost everywhere, then they actually coincide outside a set of p-capacity zero. For a proof of this we refer to [Kil]. We say that a property holds p-quasi everywhere if it holds outside a set of p-capacity zero. There is a useful characterization of the capacity in terms of quasi continuous functions. Indeed (2.4) Cp(E) = infkukp M1,p(X):u∈ QA(E), where QA(E) = u∈M1,p(X) : uis p-quasi continuous and u≥1p-quasi everywhere on E. This will be a crucial fact for us later. For the proof we refer to Theorem 3.4 in [KKM]. Following [dG1] and [dG2] we say that a differentiation basis in X is a collection B=[B(x) : x∈X Differentiation Bases for Sobolev Functions 385 of bounded measurable sets with positive measure such that for every x∈Xthere is a subfamily B(x) of sets of Bso that xis contained in every B∈ B(x) and B(x) contains sets of arbitrarily small diameter. Let f∈L1 loc(X). We define the upper derivative of fwith respect to the differentiation basis Bas Df :X→[0,∞] with (2.5) Df(x) = lim sup δx(B)→0ZB f(y)dµ(y), where δx(B) = diam(B) and B∈ B(x). Recall that ZB(x,r) f(y)dµ(y) = 1 µ(B(x, r)) ZB(x,r) f(y)dµ(y). The lower derivative Df(x) can be defined in the same way by replacing the limes superior by limes inferior. If Df(x) = Df(x), we say that the derivative (2.6) Df(x) = lim δx(B)→0ZB f(y)dµ(y) exists at x. Observe, that if fis continuous at x∈X, then the derivative exists and f(x) = Df(x). Definition 2.7. Suppose that 1 <p<∞. We say that a differentiation basis Bdifferentiates the Sobolev space M1,p(X) if for every u∈M1,p(X) the derivative Du(x) exists p-quasi everywhere in X and Du(x) = u∗(x) p-quasi everywhere in X, where u∗is a p-quasi continuous representative of u. Remarks 2.8.(1) By Theorem 1.2 it suffices to check that Du∗(x) = u∗(x) p-quasi everywhere in X, where u∗is a p-quasi continuous representative of u. (2) If Bdifferentiates M1,p(X), then for every u∈M1,p(X), we have (2.9) lim δx(B)→0ZB |u(y)−u∗(x)|dµ(y) = 0 for p-quasi every x∈Xwhere u∗is a p-quasi continuous representative of u. In other words, p-quasi every point is a Lebesgue point of u. By subadditivity it is enough to verify the claim in a ball B(z, r), where z∈Xand r > 0. Let φbe a Lipschitz continuous cut-off function such that 0 ≤φ≤1, φ= 1 in B(z, r) and φ= 0 in X\B(z, 2r). Let qk, 386 P. Harjulehto, J. Kinnunen k= 1,2,..., be an enumeration of rational numbers. For every qkthe function |φ(u∗−qk)|is a p-quasi continuous representative of |φ(u−qk)|. By Definition 2.7 there is a set Ek⊂Xof p-capacity zero such that (2.10) lim δx(B)→0ZB |φ(y)(u(y)−qk)|dy =|φ(x)(u∗(x)−qk)| for all x∈X\Ek. Let E=S∞ k=1 Ek. Then Cp(E) = 0 and (2.10) holds for every x∈X\E. Let F={x∈X:|u∗(x)|=∞}. By (2.4) we may use u∗as a test function and we obtain Cp(F)≤Cp({x∈X:|u∗(x)|> λ})≤λ−pkukp M1,p(X). Letting λtend to infinity we arrive at Cp(F)=0. Let x∈X\(E∪F) and let ε>0. Let qkbe a rational number such that |u∗(x)−qk|<ε/2. Then lim sup δx(B)→0ZB |φ(y)(u(y)−u∗(x))|dµ(y) ≤lim sup δx(B)→0ZB |φ(y)(u(y)−qk)|dµ(y) + lim sup δx(B)→0ZB |φ(y)(qk−u∗(x))|dµ(y) = 2|φ(x)(u∗(x)−qk)|< ε. Since ε > 0 is arbitrary, we obtain (2.9) for every x∈B(z, r)\(E∪F). The following theorem is our main result. Theorem 2.11. The following claims are equivalent: (i) Bdifferentiates M1,p(X). (ii) There is a constant csuch that Cp({x∈X:D|u|(x)> λ})≤cλ−pkukp M1,p(X) for every λ > 0and for every u∈M1,p(X). (iii) Let ui∈M1,p(X),i= 1,2,..., and suppose that kuikM1,p (X)→0 as i→ ∞. Then for each λ > 0we have Cp({x∈X:D|ui|(x)> λ})→0 as i→ ∞. (iv) For every u∈M1,p(X)we have Cp({x∈X:D|u|(x)> λ})→0 as λ→ ∞. Differentiation Bases for Sobolev Functions 387 Observe, that the claim (ii) is quantitative but the claims (i), (iii) and (iv) are qualitative. Remarks 2.12.(1) Recall that µis a doubling measure if there is a constant cµ≥1 so that µ(B(x, 2r)) ≤cµµ(B(x, r)) for every open ball B(x, r) in X. An iteration of the doubling property implies, that if B(y, R) is a ball in X,z∈B(y, R) and 0 < r ≤R < ∞, then µ(B(z, r)) µ(B(y, R)) ≥cr RQ for some c=c(cµ) and Q= log cµ/log 2. The exponent Qserves as a counterpart of dimension related to the measure and, for example, in Rn with the Lebesgue measure Qis equal to the dimension n. A result in [HaKo1] (see also Theorem 5.1 in [HaKo2]) shows that if µis doubling, then the Poincar´e inequality (2.2) implies a SobolevPoincar´e inequality. More precisely, if 1 < p < Q then for every κwith 1≤κ < Q/(Q−p) there is c=c(p, κ, cµ)>0 depending on such that ZB(z,r) |u−uB(z,r)|κp dµ!1/(κp) ≤cr ZB(z,5r) gpdµ!1/p for every g∈D(u)∩Lp(X). If p > Q, then |u(x)−u(y)| ≤ crQ/p d(x, y)1−Q/p ZB(z,5r) gpdµ!1/p for every x, y ∈B(z, r)\Nwith µ(N) = 0 and g∈D(u)∩Lp(X). In particular, this implies that, after a redefinition on a set of measure zero, functions in M1,p(X) with p > Q are H¨older continuous on bounded subsets of X. In the borderline case p=Qthere is an exponential estimate, but we do not need it here. This implies, in particular, that if the measure is doubling, then every differentiation basis differentiates M1,p(X) when p > Q. (2) A standard way to verify the conditions (ii), (iii) or (iv) for a differentiation basis Bis to consider the corresponding Hardy-Littlewood maximal function. The maximal function related to the differentiation basis Bof a f∈L1 loc(X) is defined as (2.13) Mf(x) = sup B∈B(x)ZB |f(y)|dµ(y). 388 P. Harjulehto, J. Kinnunen It is clear that D|f|(x)≤Mf(x) for every x∈Xso that, for instance, if we are able to prove the weak type estimate (2.14) Cp({x∈X:Mu(x)> λ})≤cλ−pkukp M1,p (X) for every λ > 0, then we obtain (ii). For the differentiation basis consisting of balls this estimate was proved in [KL]. In particular, this implies that the basis consisting of balls differentiates M1,p(X) when 1 < p < ∞ and the measure is doubling. 3. Proofs of the equivalencies The proofs are rather straightforward modifications of the corresponding results for the Lebesgue measure in [dG1] and [dG2]. However, the lack of measurable sets for the capacity has the effect that proofs are somewhat subtle. Lemma 3.1. The conditions (i) and (ii) are equivalent. Proof: First we show that the condition (i) implies (ii). Let u∈M1,p(X). Since Bdifferentiates M1,p(X) and |u| ∈ M1,p(X), we have D|u|(x) = |u|∗(x) p-quasi everywhere in Xfor every p-quasi continuous representative |u|∗ of |u|. In particular, this implies that D|u|is p-quasi continuous. By (2.4) we may use λ−1D|u|,λ > 0, as a test function and we obtain Cp({x∈X:D|u|(x)> λ})≤λ−p |u|∗  p M1,p (X) =λ−p |u|  p M1,p (X)≤λ−pkukp M1,p(X). This is the desired estimate. Then we show that (ii) implies (i). Suppose that u∈M1,p(X) and let u∗be a p-quasi continuous representative of u. By Remark 2.8 (1) it suffices to prove that Du∗(x) = u∗(x) for p-quasi every x∈X. Since continuous functions are dense in M1,p(X) there is a continuous function v∈M1,p(X) and a p-quasi continuous function w∈M1,p(X) such that u∗=v+wand that kwkM1,p (X)is as small as we please. From this we conclude that D|u∗−u∗(x)|(x)≤D|w−w(x)|(x)≤D|w|(x) + |w(x)| Differentiation Bases for Sobolev Functions 389 for every x∈Xand consequently Cp({x∈X:D|u∗−u∗(x)|(x)> λ}) ≤Cp({x∈X:D|w|(x)> λ/2}) + Cp({x∈X:|w(x)|> λ/2}). The assumption (ii) implies that Cp({x∈X:D|w|(x)> λ/2})≤cλ−p |w|  p M1,p (X)≤cλ−pkwkp M1,p (X). By (2.4) we may use was a test function and we obtain Cp({x∈X:|w(x)|> λ/2})≤2pλ−p |w|  p M1,p (X)≤2pλ−pkwkp M1,p (X). From these estimates we conclude that Cp({x∈X:D|u∗−u∗(x)|(x)> λ})≤cλ−pkwkp M1,p (X), where cis independent of uand kwkM1,p (X)is as small as we please. This shows that Cp({x∈X:D|u∗−u∗(x)|(x)> λ}) = 0 for every λ > 0. Since {x∈X:D|u∗−u∗(x)|(x)>0}= ∞ [ i=1 {x∈X:D|u∗−u∗(x)|(x)>1/i} subadditivity of the capacity implies that Cp({x∈X:D|u∗−u∗(x)|(x)>0}) = 0. Therefore D|u∗−u∗(x)|(x) = 0 for p-quasi every x∈Xand since ZB u∗(y)dy −u∗(x) ≤ZB |u∗(y)−u∗(x)|dy for every B∈ B(x) we conclude that Du∗(x) exists and Du∗(x) = u∗(x) for p-quasi every x∈X. This completes the proof. Lemma 3.2. The conditions (ii) and (iii) are equivalent. Proof: Since (ii) clearly implies (iii), it is enough to prove that (iii) implies (ii). Let u∈M1,p(X). Since continuous functions are dense in M1,p(X) there are continuous functions vi∈M1,p(X), i= 1,2,..., such that ku−vikM1,p (X)→0