aa r X i v : . [ m a t h . C A ] M a y On Newton-Sobolev spaces
Miguel Andrés Marcos ∗ Instituto de Matemática Aplicada del Litoral (CONICET-UNL)Departamento de Matemática (FIQ-UNL)
Abstract
Newton-Sobolev spaces, as presented by N. Shanmugalingam, describea way to extend Sobolev spaces to the metric setting via upper gradients,for metric spaces with ‘sufficient’ paths of finite length. Sometimes, as isthe case of parabolic metrics, most curves are non-rectifiable. As a courseof action to overcome this problem, we generalize some of these resultsto spaces where paths are not necessarily measured by arc length. Inparticular, we prove the Banach character of the space and the absolutecontinuity of these Sobolev functions over curves. Under the assumptionof a Poincaré-type inequality and an arc-chord property here defined, weobtain the density of some Lipschitz classes, relate Newton-Sobolev spacesto those defined by Hajłasz by means of Hajłasz gradients, and we alsoget some Sobolev embedding theorems. Finally, we illustrate some non-standard settings where these conditions hold, specifically by adding aweight to arc-length and specifying some conditions over it. If Ω is an open set in R n and f is a smooth function defined on Ω , the Funda-mental Theorem of Calculus for line integrals implies that for every piecewisesmooth path γ in Ω with endpoints x, y we get | f ( x ) − f ( y ) | ≤ ˆ γ |∇ f | d | s | . Nonnegative functions defined in Ω that satisfy this inequality for every x, y andevery γ joining them in place of |∇ f | are referred to as upper gradients (see forexample [HeK]).In the case Ω = R n , one can consider only segments parallel to the coordinateaxes instead of more general paths, and those are sufficient to describe partial ∗ The author was supported by Consejo Nacional de Investigaciones Científicas y Técnicas,Agencia Nacional de Promoción Científica y Tecnológica and Universidad Nacional del Litoral.Keywords and phrases: Newton-Sobolev spaces, Spaces of homogeneous type, Poincaréinequality, upper gradients2010 Mathematics Subject Classification: Primary 43A85. R with the parabolic metric defined further along this section, whereonly horizontal segments are rectifiable.In [Sh], N. Shanmugalingam describes, via upper gradients, a way to char-acterize Sobolev spaces W ,p in open sets of R n that extends to metric measurespaces, defining Newton-Sobolev spaces N ,p . If the space has ‘sufficient’ rectifi-able paths (in the sense that the set of rectifiable paths has nonzero p -modulus),an interesting theory of Sobolev functions can be developed, but if the set ofrectifiable paths is negligible, this ‘Sobolev space’ is just L p .Easy enough examples of metric measure spaces with no paths of dimension1 can be constructed. For instance, take X = R with d ( x, y ) = | x − y | / ,and we get that paths are either 0-dimensional (trivial paths) or 2-dimensional.While ‘classical’ Newton-Sobolev theory in such a space would be nonsensical,a good theory could be developed if we measured path ‘length’ by Hausdorff2-dimensional measure H with respect to the new distance d . Of course, H d coincides with H with respect to the Euclidean distance, and the above exampleseems to be just a change of parameters.In a more interesting scenario, we consider parabolic metrics associated to amatrix, see for instance [Gu]. Take an n × n diagonal matrix D with eigenvalues α , . . . , α n ≥ . For x ∈ R n and λ > , we define T λ x = e D log λ x = λ α . . . λ α n x ... x n . For a norm k · k in R n it can be shown that for x = 0 , k T λ x k is continuous,strictly increasing in λ , tends to 0 as λ → and tends to ∞ as λ → ∞ . Thenthere exists a unique < ρ ( x ) < ∞ such that k T /ρ ( x ) x k = 1 . If we define d ( x, y ) = ρ ( x − y ) for x = y and d ( x, x ) = 0 , then d is a traslation invariant metric that alsosatisfies d ( T λ x, T λ y ) = λd ( x, y ) and d ( x, y ) = 1 iff | x − y | = 1 , d ( x, y ) < iff | x − y | < , d ( x, y ) > iff | x − y | > . These metrics thus defined can havedifferent Hausdorff dimensions, see [A].The word parabolic refers to the case α = . . . = α n − = 1 and α n = 2 , whichprovides the right dilations for the heat equation and other partial differentialequations of parabolic type (see [Fa]). For example, if we consider R with D = (cid:18) (cid:19) and the maximum norm, we obtain d (( x, y ) , ( x ′ , y ′ )) = max n | x − x ′ | , | y − y ′ | / o , and it can be shown balls have Hausdorff dimension 3 (in fact they are Ahlfors3-regular). Here, the only non-trivial rectifiable paths are horizontal segments,2o even though there are rectifiable paths, the space is not connected by them.Smooth non-horizontal paths have Hausdorff dimension 2, so we see that thismeasure is not rotation invariant.As another example of heterogeneity, we can consider adding a weight ω toarc-length by using the measure dµ = ωd H . In this case this path measure willnot necessarily be invariant under any kind of isometry.In this work, following the ideas in [Sh], we develop a more general theoryof Newton-Sobolev spaces by replacing Hausdorff 1-dimensional measure by anarbitrary measure µ as a way of measuring path ‘lengths’.In sections 2 and 3 we generalize all the machinery needed to constructNewton-Sobolev spaces. In section 4 we define these spaces and prove they arecomplete. In section 5 we call for some additional properties, such as Poincaréinequality, needed to prove some more interesting results, as Lipschitz densityor Sobolev embeddings. We also compare Newton-Sobolev spaces with anotherkind of Sobolev space in metric spaces: Hajłasz-Sobolev spaces. µ -arc length and upper gradients Classical definitions of arc length, length function, arc length parametrizationand line integrals in the metric setting can be found in [He]. In this sectionwe modify these concepts so they apply in more general ways to measure path‘lengths’.Given a metric space ( X, d ) and a (compact) path γ : [ a, b ] → X , i.e. acontinuous function from [ a, b ] into X , its length is defined as l ( γ ) = sup ( t i ) i X i d ( γ ( t i ) , γ ( t i +1 )) , where the supremum is taken over all partitions of [ a, b ] . We say that ˜ γ is asub-path of γ if it is the restriction of of γ to a subinterval of [ a, b ] . We say thata path (or subpath) is trivial if it is a constant path (for injective paths thismeans a = b ).The concept of arc length of a path is similar to, but not equal to, Hausdorffone-dimensional measure H of its image, but they do coincide for injectivepaths (see [Fl]). From this result, for injective paths and for Borel nonnegativemeasurable functions we get that ˆ γ gds = ˆ Im ( γ ) gd H , where dσ is arc-length, and from this we can think of exchanging the measure H for another Borel measure, as H s .Let µ be a non-atomic Borel measure in X (in the sense that µ ( { x } ) = 0 for each x ∈ X ). Define Γ µ as the set of all non trivial injective paths γ in X < µ ( Im (˜ γ )) < ∞ for all non trivial subpaths of γ . For nonnegativeBorel functions g : X → [0 , ∞ ] we define ˆ γ g = ˆ Im ( γ ) gdµ. Now, for a path γ : [ a, b ] → X in Γ µ , we define h ( γ ) = µ ( Im ( γ )) and its µ -arc length ν γ : [ a, b ] → R as ν γ ( x ) = h ( γ | [ a,x ] ) . Lemma 2.1.
For paths γ : [ a, b ] → X in Γ µ , we have that ν γ is strictly increas-ing, continuous, onto [0 , h ( γ )] , and besides h ( γ ) = h ( γ | [ a,x ] ) + h ( γ | [ x,b ] ) . Proof. ν γ is clearly increasing. Continuity follows from µ being non-atomic, andsurjectivity follows from it being continuous and increasing. The fact that ν γ isstrictly increasing follows from the fact that every non trivial subcurve of γ haspositive measure, as γ ∈ Γ µ . Theorem 2.2.
For γ : [ a, b ] → X in Γ µ , there is a unique γ h : [0 , h ( γ )] → X such that γ = γ h ◦ ν γ ,Im ( γ ) = Im ( γ h ) and ν ( γ h ) ( t ) = t in [0 , h ( γ )] (therefore γ h = γ h ◦ ν γ h ). We callthis the µ -arc length parametrization of γ .Proof. As ν γ : [ a, b ] → [0 , h ( γ )] is strictly increasing and onto, it is a bijectionbetween [ a, b ] and [0 , h ( γ )] and we can define γ h = γ ◦ ν − γ . We immediately see that Im ( γ ) = Im ( γ h ) , and ν ( γ h ) ( t ) = µ ( γ h ([0 , t ])) = µ ( γ ( ν − γ ([0 , t ]))) = µ ( γ ([ a, ν − γ ( t )]))= ν γ ( ν − γ ( t )) = t. Theorem 2.3. If γ : [0 , h ] → X is a path in Γ µ parametrized by µ -arc length,then for every Borel set B of [0 , h ] , we have µ ( γ ( B )) = l ( B ) . Furthermore, if g : X → R is nonnegative and Borel measurable, then for eachsubpath ˜ γ = γ | [ a,b ] we have ˆ ˜ γ g = ˆ ba g ◦ ˜ γ. Theorem 2.4.
Given a function f : X → R and a path γ : [0 , h ] → X in Γ µ parametrized by µ -arc length, if there exists a Borel measurable nonnegative ρ : X → R satisfying | f ( γ ( s )) − f ( γ ( t )) | ≤ ˆ γ | [ s,t ] ρ < ∞ for every ≤ s < t ≤ h , then f ◦ γ : [0 , h ] → R is absolutely continuous.Proof. Let ǫ > . As ρ ∈ L ( Im ( γ ) , µ ) , by absolute continuity of the integralthere exists δ > such that for every E ⊂ Im ( γ ) with µ ( E ) < δ we have ´ E ρdµ < ǫ . Then if ≤ a < b < a < b < . . . < a n < b n ≤ h satisfy P i | b i − a i | < δ , µ ( ∪ i γ ([ a i , b i ])) = X i ν γ ( b i ) − ν γ ( a i ) = X i b i − a i < δ and therefore X i | f ◦ γ ( b i ) − f ◦ γ ( a i ) | ≤ X i ˆ γ | [ ai,bi ] ρ = ˆ ∪ i γ ([ a i ,b i ]) ρdµ < ǫ. Let now Γ ∗ be a subset of Γ µ , closed under taking subpaths (i.e. if γ ∈ Γ ∗ and ˜ γ is a non-trivial subpath of γ , then ˜ γ ∈ Γ ∗ ). A nonnegative Borel measurablefunction ρ satisfying | f ( x ) − f ( y ) | ≤ ˆ γ ρ for every γ ∈ Γ ∗ with endpoints x, y , for every pair of points x, y with f ( x ) , f ( y ) finite is called a µ -upper gradient for f with respect to Γ ∗ . As theorem 2.4shows, if a function f has an upper gradient with respect to Γ ∗ that is integrableover each path in Γ ∗ , then it is absolutely continuous over every path in Γ ∗ .Let R be equiped with the parabolic distance d discussed in the intro-duction, and let µ = H . If γ is a segment joining x = ( a, ka + b ) with y = ( a + h, k ( a + h ) + b ) for some h > , then its measure µ is just its height | k | h , while its length is √ k h so in fact we have dµ = k √ k dl over thesepaths (clearly when k → we get µ = 0 and when k → ∞ , µ = l ).Now, for f smooth, | f ( y ) − f ( x ) | ≤ ˆ √ k h (cid:12)(cid:12)(cid:12)(cid:12) ∇ f (cid:18) a + t √ k , b + t √ k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt = ˆ γ |∇ f | ds √ k k ˆ Im ( γ ) |∇ f | dµ, and the same bound can be shown in a similar way for h < .Therefore if we consider Γ ∗ k to be the set of all polygonal paths made up ofsegments of slope ± k for a fixed < k < ∞ , we obtain that √ k k |∇ f | is anupper gradient for f with respect to Γ ∗ k . The following picture illustrates a pathof Γ ∗ k for k = 1 . x yγ Now, if we consider X = R n with Euclidean distance, but dµ = ωd H where ω and ω are locally integrable with respect to H , we obtain Γ µ = Γ rect , where Γ rect is the set of all non-trivial injective rectifiable paths. For f smooth and γ ∈ Γ µ , | f ( y ) − f ( x ) | ≤ ˆ γ |∇ f | = ˆ Im ( γ ) |∇ f | d H = ˆ Im ( γ ) |∇ f | ω dµ, and in fact the same can be applied to any ‘classical’ upper gradient of a function f . There is clearly a one to one correspondence between upper gradients ρ with H and upper gradients of the form ρ/ω with measure µ . p -weak uppergradients Let now m be a Borel measure on X . As in [Sh], we adjust the definition ofmodulus of a set of measures in [Fu] to path families.For every family Γ ⊂ Γ µ and < p < ∞ , we define its p -modulus as M od p (Γ) = inf ˆ X g p dm g : X → R satisfying ˆ γ g ≥ for every γ ∈ Γ .The following results can be found in [Fu], we state them here in the languageof paths instead of measures. Theorem 3.1.
M od p is an outer measure on Γ µ . As expected, we say that a property holds for p -almost every path γ ∈ Γ µ ifthe set Γ where it does not hold has M od p (Γ) = 0 . A useful property of sets of p -modulus zero is the following. Lemma 3.2.
M od p (Γ) = 0 if and only if there exists a nonnegative Borelmeasurable function g satisfying ´ X g p dm < ∞ and ˆ γ g = ∞ for every γ ∈ Γ . We also need the following result.
Lemma 3.3. If ´ | g n − g | p dm → , there exists a subsequence ( g n k ) k such that ´ γ | g n k − g | → for p -almost every γ ∈ Γ µ . Given a set E ⊂ X we define Γ E = { γ ∈ Γ µ : Im ( γ ) ∩ E = ∅} , Γ + E = { γ ∈ Γ µ : µ ( Im ( γ ) ∩ E ) > } and we have the following lemma Lemma 3.4. If m ( E ) = 0 , then M od p (Γ + E ) = 0 .Proof. Trivial, as g = ∞ χ E satisfies g = 0 m -almost everywhere, but ´ γ g = ∞ for every γ ∈ Γ + E .A nonnegative Borel measurable function ρ satisfying | f ( x ) − f ( y ) | ≤ ˆ γ ρ for p -almost every γ ∈ Γ µ with endpoints x, y is called a p -weak upper gra-dient for f .As in Shanmugalingam’s case, we do not lose much by restricting ourselvesto weak upper gradients. 7 roposition 3.5. If ρ is a p -weak upper gradient for f and ǫ > , there existsan upper gradient ρ ǫ for f such that ρ ǫ ≥ ρ and k ρ − ρ ǫ k p < ǫ .Proof. Let Γ be the set of paths where the inequality for ρ does not hold( M od p (Γ) = 0 ). Then there exists g ≥ Borel measurable with ´ X g p dm < ∞ but ´ γ g = ∞ for every γ ∈ Γ . We define ρ ǫ = ρ + ǫ k g k p g, it is clear that ρ ǫ ≥ ρ , ´ γ ρ ǫ ≥ for every γ , so ρ ǫ is an upper gradient for f ,and finally k ρ ǫ − ρ k p = ǫ k g k p k g k p < ǫ. As seen in 2.4, functions with ‘small’ upper gradients are absolutely contin-uous on curves. We say that a function f is ACC p or absolutely continuousover p -almost every path if f ◦ γ h : [0 , h ( γ )] → R is absolutely continuousfor p -almost every γ . Lemma 3.6.
If a function f has a p -weak upper gradient ρ ∈ L p , it is ACC p .Proof. Let Γ be the set of all paths γ such that | f ( x ) − f ( y ) | > ´ γ ρ and let Γ be the set of all paths with a subpath in Γ . As ρ is a weak upper gradient, M od p (Γ ) = 0 , but if g satisfies ´ γ g ≥ , it also satisfies ´ ˜ γ g ≥ for everysubpath ˜ γ of γ , and therefore M od p (Γ ) ≤ M od p (Γ ) = 0 . Let Γ be the set of all paths γ with ´ γ ρ = ∞ . Then as ρ ∈ L p , M od p (Γ ) = 0 .For paths not in Γ ∪ Γ , we can apply 2.4 and we conclude the lemma.We will also need the following lemma later on. Lemma 3.7. If f is ACC p and f = 0 m -almost everywhere, then the family Γ = { γ ∈ Γ µ : f ◦ γ } has p -modulus zero.Proof. Let E = { x : f ( x ) = 0 } , then m ( E ) = 0 and Γ = Γ E . As Γ + E hasmodulus zero (because m ( E ) = 0 ), we only need to see that Γ E \ Γ + E also hasmodulus zero. But if γ ∈ Γ E \ Γ + E , Im ( γ ) ∩ E = ∅ but µ ( Im ( γ ) ∩ E ) = 0 ,therefore γ − h ( E ) has length 0 in R and f ◦ γ h is nonzero in a set of length 0,and if E = ∅ this set is not empty and f ◦ γ h cannot be absolutely continuous.Therefore M od p (Γ E \ Γ + E ) = 0 . 8 Extended Newton-Sobolev spaces N ,p From now on, we will work on a fixed subset Γ ∗ ⊂ Γ µ , closed under takingsubpaths. Properties defined on the previous section, such as p -weak uppergradients or ACC p , can be easily adjusted to Γ ∗ instead of Γ µ . We will alsorequire that the space X be connected by paths belonging to Γ ∗ . This is the caseof Γ ∗ k in the example of R with the parabolic metric. For the Euclidean case,it is sufficient to consider piecewise linear paths made of segments parallel tothe coordinate axis instead of all rectifiable paths to obtain a theory of Sobolevspaces, but in general this need not be the case.We define the space ˜ N ,p as the space of all functions f having a p -weakupper gradient, both with finite p -norms. We define the N ,p norm as k f k N ,p = k f k p + inf ρ k ρ k p , where the infimum is taken over all p -weak upper gradients of f .It immediately follows from definition that ( ˜ N ,p , k · k N ,p ) is a semi-normedvector space. Moreover, if f, g ∈ ˜ N ,p , then | f | , min { f, g } , max { f, g } ∈ ˜ N ,p . As seen before, every function in ˜ N ,p is ACC p . ˜ N ,p is not a normed space, as two distinct functions can be equal almosteverywhere, but also because a function may be in ˜ N ,p while a function equalalmost everywhere to it may not. We do have the following as a corollary of 3.7. Corollary 4.1. If f, g ∈ ˜ N ,p and f = g m -a.e., then k f − g k N ,p = 0 . Finally, we define the equivalence relation f ∼ g iff k f − g k N ,p = 0 , and thequotient space N ,p = ˜ N ,p / ∼ , the generalized Newton-Sobolev space .In the case of R with the parabolic distance, µ = H and m = H definedin the introduction, we saw in section 2 that if we consider Γ ∗ k for a fixed k > as our path family, we obtain that √ k k |∇ f | is an upper gradient for f . Infact, one can show that N ,p = W ,p with equivalent norms. If we consider thewhole of Γ µ , this will not happen, as we can see by considering that, if ρ is abounded upper gradient for f , for paths γ joining ( x, y ) and ( x ′ , y ′ ) we obtain | f ( x, y ) − f ( x ′ , y ′ ) | ≤ ˆ γ ρ ≤ k ρ k ∞ µ ( Im ( γ )) . So for the case of segments in Γ ∗ k we would obtain | f ( x, y ) − f ( x ′ , y ′ ) | ≤ ˆ γ ρ ≤ k ρ k ∞ | y − y ′ | , and if we allow segments of arbitrarily small height we obtain that f mustbe cylindrical, f ( x, y ) = g ( y ) , so unless f ≡ or p = ∞ , we cannot obtain9 ∈ L p ( dm ) . For p = ∞ , we obtain that N , ∞ consists of cylindrical boundedfunctions f ( x, y ) = g ( y ) with g a Lipschitz-1 function (in the Euclidean sense).Back to the example of X = R n with Euclidean distance and dµ = ωd H ,where both ω and ω are locally integrable with respect to H , so for the measure dm = ω p dx (where dx is Lebesgue measure) we get that the space N ,p ( dm ) consists of those f ∈ L p ( dm ) such that |∇ f | ω ∈ L p ( dm ) , or in terms of Lebesguemeasure, N ,p = { f : ωf, |∇ f | ∈ L p ( dx ) } . For the particular case ω ( x ) = 1 + | x | , this space coincides with the Sobolev-Hermite space L p , as defined in [BT].We will show now, as [Sh], that N ,p is a Banach space, but first a lemma. Lemma 4.2.
Let F ⊂ X be such that inf n k f k N ,p : f ∈ ˜ N ,p ( X ) ∧ f | F ≥ o = 0 . Then
M od p (Γ F ) = 0 .Proof. For every n we take v n ∈ ˜ N ,p ( X ) with v n | F ≥ and k v n k N ,p < − n , and take weak upper gradients ρ n of v n with k ρ n k p < − n . Take u n = P n | v k | , g n = P n ρ k (each g n will be a weak upper gradient of u n ) and u = P | v n | (observe that u | F = ∞ ), g = P ρ n . Every u n turns to be in ˜ N ,p ,and ( u n ) , ( g n ) are Cauchy in L p , therefore convergent in L p to functions ˜ u, ˜ g respectively. Then u = ˜ u, g = ˜ g a.e. and we have ´ | u | p < ∞ . Let E = { x ∈ X : u ( x ) = ∞} , then m ( E ) = 0 (as ´ X | u | p < ∞ ) and F ⊂ E . If we take Γ = (cid:26) γ : ˆ γ g = ∞ ∨ ˆ γ g n ˆ γ g (cid:27) then M od p (Γ) = 0 from 3.2 and 3.3. If γ Γ ∪ Γ + E ( M od p (Γ + E ) = 0 ), then thereexists y ∈ Im ( γ ) \ E , and if x ∈ Im ( γ ) , | u n ( x ) | ≤ | u n ( y ) | + ˆ γ g n ≤ | u ( y ) | + ˆ γ g, therefore | u ( x ) | < ∞ and γ Γ E , and we have M od p (Γ F ) ≤ M od p (Γ E ) ≤ M od p (Γ ∪ Γ + E ) = 0 . Theorem 4.3. N ,p is Banach.Proof. Let ( u n ) be Cauchy in N ,p . By taking subsequences we can assume k u n − u n +1 k N ,p < − n p +1 p g n of u n − u n +1 with k g n k p < − n . Define E n = { x ∈ X : | u n ( x ) − u n +1 ( x ) | ≥ − n } , E = lim sup E n . If x E , then there exists n x such that | u n ( x ) − u n +1 ( x ) | < − n for n ≥ n x and therefore outside of E u ( x ) = lim u n ( x ) it is well defined.By Tchebyschev’s inequality, µ ( E n ) ≤ np k u n − u n +1 k pp ≤ − n , and µ ( E ) ≤ ∞ X n µ ( E k ) ≤ − n · , for every n , and on the other hand inf n k f k N ,p : f ∈ ˜ N ,p ( X ) ∧ f | E ≥ o ≤≤ ∞ X n inf n k f k N ,p : f ∈ ˜ N ,p ( X ) ∧ f | E n ≥ o ≤ ∞ X n np k u n − u n +1 k pN ,p ≤ − n · for every n .By the previous lemma, M od p (Γ E ) = 0 , and if we define u | E ≡ , as ( u n ) isCauchy in L p and u n → u a.e., we have ´ | u | p < ∞ . Finally for γ Γ E withendpoints x, y we have | ( u − u n )( x ) − ( u − u n )( y ) | ≤ ∞ X n | ( u k +1 − u k )( x ) − ( u k +1 − u k )( y ) |≤ ∞ X n ˆ γ g k , and we get that P ∞ n g k is a p -weak upper gradient of u − u n (which tends to 0in L p ), and we have u ∈ N ,p and k u − u n k N ,p ≤ k u − u n k p + k ∞ X n g k k p → . Poincaré Inequality
If there is no relationship between the ‘space measure’ m and the ‘path measure’ µ , most standard results about N ,p cannot be proven. The standard wayof relating them is by Poincaré inequality. In our case we will also need arelationship between the ‘path measure’ and the distance function.We say that X supports a (1 , p ) -Poincaré inequality of exponent β if thereexists C > , λ ≥ such that for every ball B and every pair f, ρ defined in B such that f ∈ L ( B ) and ρ is an upper gradient of f in B , we have B | f − f B | dm ≤ C diam ( B ) β (cid:18) λB ρ p dm (cid:19) /p . In Shanmugalingam’s case, this property suffices for proving that Lipschitzfunctions are dense in N ,p . One crucial fact for proving this is that the lengthof a path is always greater than or equal to the distance between any pair ofpoints over the curve, but in our context this may not be the case. We say thatthe family Γ ∗ has the µ -arc-chord property with exponent β if there exists C µ > such that for every γ ∈ Γ ∗ (and thus for every subpath of that γ , as Γ ∗ is closed under taking subpaths), we get thatdiam ( Im ( γ )) β ≤ C µ µ ( Im ( γ )) . Observe that the usual chord-arc property (see, for instance, [D]) means theopposite inequality: l ( γ ) ≤ Cd ( x, y ) if γ is a path joining x and y (which in turnimplies l ( γ ) ∼ d ( x, y ) , as the reverse inequality d ( x, y ) ≤ l ( γ ) always holds). Wedo not require this control over the measure of the curves in Γ ∗ , but the oppositeone (thus we reverse the word order in the definition).In this section we will prove some results that arise from these properties,and then we will go back to the example dµ = ωd H .First, we will prove a series of lemmas that will give us sufficient conditionsfor Lipschitz functions to be dense in N ,p . Lemma 5.1.
Let f be ACC p such that f | F = 0 m -a.e., for F a closed subsetof X . If ρ is an upper gradient of f , then ρχ X \ F is a p -weak upper gradient of f .Proof. Let Γ be the set of paths for which f ◦ γ h is not absolutely continuous,and let E = { x ∈ F : f ( x ) = 0 } , so M od p (Γ ∪ Γ + E ) = 0 . Now, if γ Γ ∪ Γ + E has endpoints x, y , • If Im ( γ ) ⊂ ( X \ F ) ∪ E , then | f ( x ) − f ( y ) | ≤ ´ γ ρ = ´ γ ρχ X \ F as µ ( Im ( γ ) ∩ E ) = 0 . • If x, y ∈ F \ E , then f ( x ) = f ( y ) = 0 and | f ( x ) − f ( y ) | ≤ ´ γ ρχ X \ F holdstrivially. 12 If x ∈ ( X \ F ) ∪ E (or the same for y ) but Im ( γ ) is not completely in ( X \ F ) ∪ E , as ( f ◦ γ h ) − ( { } ) is a closed set of [0 , h ( γ )] ( f ◦ γ h is continu-ous), it has a minimum a and maximum b (with f ◦ γ h ( a ) = f ◦ γ h ( b ) = 0 ).Then, | f ( x ) − f ( y ) | ≤≤ | f ( x ) − f ( γ h ( a )) | + | f ( γ h ( a )) − f ( γ h ( b )) | + | f ( γ h ( b )) − f ( y ) |≤ ˆ γ h | [0 ,a ] ρ + ˆ γ h | [ b,h ( γ )] ρ ≤ ˆ γ ρχ X \ F as γ h ([0 , a ]) and γ h ([ b, h ( γ )]) intersect F in a set of µ -measure zero. Lemma 5.2. If Γ ∗ has the µ -arc-chord property with exponent β , then everyLipschitz- β function is absolutely continuous over every curve of Γ ∗ .Proof. Let γ : [0 , h ] → X be a path in Γ ∗ parametrized by µ -arc length, and let f : X → R be Lipschitz with constant L . If ǫ > and ≤ a < b < a < b < · · · < a n < b n ≤ h satisfies P i | b i − a i | < ǫLC µ , then X i | f ( γ ( b i )) − f ( γ ( a i )) | ≤ L X i d ( γ ( b i ) , γ ( a i )) β ≤ L X i diam ( γ ([ a i , b i ])) β ≤ LC µ X i µ ( γ ([ a i , b i ])) = LC µ X i | b i − a i | < ǫ. Lemma 5.3. If Γ ∗ has the µ -arc-chord property with exponent β and f : X → R is a Lipschitz- β function with constant L , then C µ Lχ supp ( f ) is an upper gradientof f . In particular if supp ( f ) is compact we have f ∈ ˜ N ,p .Proof. Let γ : [ a, b ] → X have endpoints x, y . Consider the following cases: • Im ( γ ) ⊂ supp ( f ) . Then | f ( x ) − f ( y ) | ≤ Ld ( x, y ) β ≤ C µ Lµ ( Im ( γ )) = ´ γ LC = ´ γ CLχ supp ( f ) . • Im ( γ ) ∩ supp ( f ) = ∅ . Then | f ( x ) − f ( y ) | = 0 = ´ γ CLχ supp ( f ) . • x ∈ supp ( f ) but Im ( γ ) supp ( f ) . Then as ( f ◦ γ ) − ( { } ) is closed in [ a, b ] , it has minimum a > a and maximum b ≤ b . We have that γ ([ a, a ]) and γ ([ b , b ]) are subsets of supp ( f ) and f ( γ ( a )) = f ( γ ( b )) = 0 so, | f ( x ) − f ( y ) | ≤≤ | f ( x ) − f ( γ ( a )) | + | f ( γ ( a )) − f ( γ ( b )) | + | f ( γ ( b )) − f ( y ) | Ld ( x, γ ( a )) β + Ld ( γ ( b ) , y ) β ≤ LC µ µ ( γ ([ a, a ])) + LC µ µ ( γ ([ b , b ]) ≤ ˆ γ LC µ χ supp ( f ) . Finally if supp ( f ) is compact, f, CLχ supp ( f ) ∈ L p ( m ) for every p .With the previous results, and also requiring the measure m to be doubling,we get the following. Theorem 5.4. If m is doubling, X supports a (1 , p ) -Poincaré inequality ofexponent β ≤ and Γ ∗ satisfies the µ -arc-chord property with exponent β , thenLipschitz- β functions are dense in N ,p .Proof. Let f ∈ ˜ N ,p and let g ∈ L p be an upper gradient of f . Assume f isbounded (bounded functions are clearly dense in N ,p ). We define E k = { x ∈ X : M g p ( x ) > k p } , where M is the noncentered Hardy-Littlewood maximal function. As m is dou-bling, M is weak type , , and m ( E k ) ≤ Ck p ˆ X g p → as k → ∞ . Let F k = X \ E k (which is closed as E k is open). If x ∈ F k , r > and B = B ( x, r ) , B | f − f B | ≤ Cr β ( B g p ) /p ≤ Cr β ( M g p ( x )) /p ≤ Cr β k. Then if we define f n ( x ) = f B ( x, − n r ) , we have | f n + j ( x ) − f n ( x ) | ≤ j X i =1 | f n + i +1 ( x ) − f n + i ( x ) |≤ j X i =1 B ( x, − ( n + i +1) r ) | f − f B ( x, − ( n + i ) r ) |≤ C j X i =1 B ( x, − ( n + i ) r ) | f − f B ( x, − ( n + i ) r ) |≤ Ckr β (2 β ) − n j X i =1 − i ≤ Ckr β − nβ , and therefore f n ( x ) is Cauchy for each x ∈ F k . Now, we define for x ∈ F k , f k ( x ) = lim f n ( x ) . f in F k we have f k ( x ) = f ( x ) . Let’s verifythat f k is Lipschitz- β . Given x, y ∈ F k , take r = d ( x, y ) , B n = B ( x, − n r ) , B ′ n = B ( y, − n r ) , and | f k ( x ) − f k ( y ) | ≤≤ ∞ X n =0 | f n ( x ) − f n +1 ( x ) | + | f ( x ) − f ( y ) | + ∞ X n =0 | f n ( y ) − f n +1 ( y ) |≤ ∞ X n =0 C B n | f − f B n | + C B | f − f B | + ∞ X n =0 C B ′ n | f − f B ′ n |≤ Ckr β ∞ X n =0 − nβ + Cr β k ≤ Ckr β = Ckd ( x, y ) β . Now, f k can be extended to all of X as a Lipschitz- β function with the sameLipschitz constant, and we can assume it is bounded by Ck (see [A]). Then ˆ X | f − f k | p = ˆ E k | f − f k | p ≤ C ˆ E k | f | p + Ck p m ( E k ) → as k → ∞ , for m ( E k ) → and the weak type of the Hardy-Littlewood maximalimplies k p m ( E k ) = k p m ( M ( g p ) > k p ) ≤ k p m ( M ( g p χ { g p >k p / } ) > k p / ≤ C ˆ { g p >k p / } g p → . So f k tends to f in L p . As f y f k are ACC p , ( g + ˜ Ck ) χ E k is a p -weak uppergradient of f − f k , and as it is in L p and tends to 0 when k → ∞ , f − f k ∈ N ,p for every k and k f − f k k N ,p → .If X is doubling and supports a (1 , q ) Poincaré inequality of exponent β for some ≤ q < p , then we have that every function in N ,p has a Hajłaszgradient in L p , i.e. N ,p ֒ → M β,p with k · k M β,p ≤ C k · k N ,p (see [Ha], [KM],[Sh], we define M β,p to be the space M ,p for the metric d β ). The converseembedding holds true in general for Shanmugalingam’s case. In our case weneed the µ -arc-chord property. Lemma 5.5.
Assume Γ ∗ satisfies the µ -arc-chord property and let f be a con-tinuous function satisfying | f ( x ) − f ( y ) | ≤ d ( x, y ) β ( g ( x ) + g ( y )) for every x, y , for some nonnegative measurable function g . Then there exists C > such that Cg is an upper gradient for f .Proof. Let γ : [0 , h ] → X be a path in Γ ∗ parametrized by µ -arc length withendpoints x, y . If ´ γ g = ∞ we are done. Otherwise, for each n we take γ i = γ | [ in , i +1 n ] , ≤ i ≤ n − , as γ is a µ -arc length parametrization we have that15 ( | γ i | ) = µ ( Im ( γ )) /n = h/n . For each i , there exists x i ∈ | γ i | with g ( x i ) ≤ ffl γ i g ,and the µ -arc-chord property implies that d ( x i , x i +1 ) β ≤ Cµ ( | γ i | ) , then | f ( x ) − f ( x n − ) | ≤ X i | f ( x i ) − f ( x i +1 ) |≤ X i d ( x i , x i +1 ) β ( g ( x i ) + g ( x i +1 )) ≤ C X i ˆ γ i g + ˆ γ i +1 g ! ≤ C ˆ γ g. Taking n → ∞ , x → x, x n − → y and | f ( x ) − f ( y ) | ≤ C ˆ γ g and we have what we needed. Corollary 5.6. If Γ ∗ satisfies the µ -arc-chord property with exponent β andcontinuous functions are dense in M β,p (which happens for instance if β ≤ ),then M ,p ֒ → N ,p , with k · k N ,p ≤ C k · k M β,p . Theorem 5.7. If X is doubling and supports a (1 , q ) Poincaré inequality withexponent β ≤ for some ≤ q < p , and Γ ∗ satisfies the µ -arc-chord propertywith exponent β , then M ,p = N ,p , with equivalent norms. As in [Sh], we have the following versions of the classical Sobolev embeddingtheorems. In Shanmugalingam’s case they are proven for β = 1 , but the sameproof can be applied for other β in our case. Theorem 5.8. If m is doubling and satisfies m ( B ( x, r )) ≥ Cr N for C, N independent of x ∈ X, < r < diam ( X ) , and if X supports a (1 , p ) Poincaré inequality of exponent β ≤ for p > N/β , then functions in N ,p areLipschitz- α with α = β − N/p . Theorem 5.9. If X is bounded and satisfies cr N ≤ m ( B ( x, r )) ≤ Cr N with c, C, N independent of x ∈ X, < r < diam ( X ) (i.e. X is Ahlfors N -regular), and if X supports a (1 , q ) Poincaré inequality of exponent β for q > /β , then for p satisfying q < p < N q , p ∗ = p − Nq we have that every f ∈ N ,p with upper gradient g , k u − u X k p ∗ ≤ C diam ( X ) β − /q k g k p .
16e finish this work with the example X = R n with Euclidean distance, dµ = ωd H , dm = ω p dx where ω and ω are locally integrable. First we considerwhen a Poincaré inequality holds.If ω is bounded, as Poincaré inequality is true for dx , we get B | f − f B | dm ≤ B | f − f B,dx | dm + | f B − f B,dx |≤ B | f − f B,dx | dm ≤ (cid:18) B | f − f B,dx | p dm (cid:19) /p ≤ (cid:18) | B | m ( B ) (cid:19) /p (cid:18) | B | ˆ B | f − f B,dx | p ω p dx (cid:19) /p ≤ C (cid:18) | B | m ( B ) (cid:19) /p k ω k ∞ diam ( B ) (cid:18) | B | ˆ B |∇ f | p dx (cid:19) /p = C k ω k ∞ diam ( B ) (cid:18) B (cid:18) |∇ f | ω (cid:19) p dm (cid:19) /p , where f B,dx = ffl B f dx .Instead of asking for ω to be bounded, we may use a two-weight Poincaréinequality as found in [Hr]. Let < p < n , ω p ∈ A ∞ and | Q | q ( p − n ) ˆ Q ω p dx ≤ C for each cube Q , with C independent of Q , and some q such that p − n ≤ q < p .If p = q = 2 this would be Fefferman-Phong’s condition (see [FP]).In our case, the pair , ω p satisfies condition A /np,q , where we say two weights w , w satisfy condition A αp,q if there exists C > such that (cid:18) ˆ Q w − p ′ /p (cid:19) /p ′ (cid:18) ˆ Q w (cid:19) /q ≤ C | Q | − α for each cube Q , for ≤ α < , < p, q < ∞ , /p − α ≤ /q .E. Harboure proves in [Hr] that these conditions imply there exists constants C > and δ > (depending on the A ∞ and A /np,q constants) such that thefollowing Poincaré inequality holds ˆ Q | f − f Q,dx | p ω p dx ≤ C (cid:18) ˆ Q ω p dx (cid:19) δ ˆ Q |∇ f | p dx. From this condition, our (1 , p ) Poincaré inequality follows, Q | f − f Q | dm ≤ C m ( Q ) /p (cid:18) ˆ Q | f − f Q,dx | p ω p dx (cid:19) /p C m ( Q ) /p (cid:18) m ( Q ) δ ˆ Q (cid:18) |∇ f | ω (cid:19) p dm (cid:19) /p = Cm ( Q ) δ/p (cid:18) Q (cid:18) |∇ f | ω (cid:19) p dm (cid:19) /p ≤ C diam ( Q ) β (cid:18) Q (cid:18) |∇ f | ω (cid:19) p dm (cid:19) /p , for β = δqp (cid:16) np − (cid:17) , where the last inequality follows from the fact that, by ourassumption, as dm = ω p dx , m ( Q ) = ˆ Q ω p dx ≤ C | Q | q ( p − n ) = C diam ( Q ) q ( n/p − . As an example of such ω , we may consider ω ( x ) = | x | λ , for some ≤ λ < .Then ω p ∈ A ∞ if pλ < n and the pair , ω p satisfies condition A /np,q for q = n − λpn − p p : for Q = Q (0 , R ) , | Q | q (1 /p − /n ) ˆ Q ω p dx = CR − q n − pp ˆ Q (0 ,R ) | x | λp dx ∼ R − q n − pp R n − λp = C and for Q = Q ( x , R ) with x = 0 , we consider two cases. If R > | x | , then Q ( x , R ) ⊂ Q (0 , R ) , so | Q | q (1 /p − /n ) ˆ Q ω p dx ≤ CR − q n − pp ˆ Q (0 , R ) | x | λp dx ≤ C ; on the other hand if R ≤ | x | , then for x ∈ Q we have | x | ∼ | x | , so | Q | q (1 /p − /n ) ˆ Q ω p dx ∼ R − q n − pp | x | λp R n ≤ C. As a special case, we can consider λ = 0 , so the weight ω = 1 , which givesclassical Sobolev spaces W ,p , is included in our result.With Poincaré inequality, theorems 5.8 and 5.9 hold, provided the otherconditions are met. We also obtain one half of theorem 5.7, as a Poincaréinequality is sufficient to obtain N ,p ֒ → M β,p .If there exists c > such that ω ( x ) ≥ c for all x , we also get the arc-chordproperty, diam ( Im ( γ )) ≤ H ( Im ( γ )) = ˆ Im ( γ ) d H ≤ c ˆ Im ( γ ) ωd H = 1 c µ ( Im ( γ )) . For example, the weight ω ( x ) = | x | λ satisfies this restriction if λ = 0 or if X = Q for some fixed cube Q , here we consider only cubes Q ⊂ Q (that may18ontain the origin, so ω is not necessarily bounded), and as it also satisfies the A /np,q condition restricted to those cubes. This case allows for both a Poincaréinequality and an arc-chord property, even though the exponents in each casemay not coincide. References []A Aimar, H.
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