Approximation by uniform domains in doubling quasiconvex metric spaces
aa r X i v : . [ m a t h . M G ] J a n APPROXIMATION BY UNIFORM DOMAINS INDOUBLING QUASICONVEX METRIC SPACES
TAPIO RAJALA
Dedicated to Professor Pekka Koskela on the occasion of his 59th birthday.
Abstract.
We show that a bounded domain in a doubling quasiconvex metric space canbe approximated from inside and outside by uniform domains. Introduction
We provide an approximation of bounded domains from inside and from outside by uniformdomains in doubling quasiconvex metric spaces. A metric space (
X, d ) is called (metrically)doubling , if there exists a constant C d so that for all r >
0, any ball of radius r can be coveredby C d balls of radius r/
2. A metric space is called quasiconvex , if there exists a constant C q < ∞ such that any x, y ∈ X can be connected by a curve γ in X with the length bound ℓ ( γ ) ≤ C q d ( x, y ) . A domain Ω ⊂ X is called uniform , if there exists a constant C u < ∞ such that for every x, y ∈ Ω there exists a curve γ ⊂ Ω such that ℓ ( γ ) ≤ C u d ( x, y )and for all z ∈ γ it holds min { ℓ ( γ x,z ) , ℓ ( γ z,y ) } ≤ C u dist ( z, X \ Ω) , where γ x,z and γ z,y denote the shortest subcurves of γ joining z to x and y , respectively.With the definitions now recalled we can state the result of this paper. Theorem 1.1.
Let ( X, d ) be a doubling quasiconvex metric space and Ω ⊂ X a boundeddomain. Then for every ε > there exist uniform domains Ω I and Ω O such that Ω I ⊂ Ω ⊂ Ω O , Ω O ⊂ B (Ω , ε ) , and X \ Ω I ⊂ B ( X \ Ω , ε ) . Although there are characterizations of uniform domains in metric spaces, for instance viatangents [6], we are not aware of previous general existence results such as Theorem 1.1.The setting of Theorem 1.1 is motivated by Sobolev- and BV-extension domains in completedoubling metric measure spaces supporting a local Poincar´e inequality (PI-spaces for short):the local Poincar´e inequality [5] is known to imply quasiconvexity [3, 10]. In [2] it was shownthat uniform domains in PI-spaces are N ,p -extension domains, for 1 < p < ∞ , for the Date : January 7, 2020.2000
Mathematics Subject Classification.
Primary 30L99. Secondary 46E35, 26B30.
Key words and phrases.
Sobolev extension, uniform domain, quasiconvexity.The author acknowledges the support from the Academy of Finland, grant no. 314789.
Newtonian Sobolev spaces, and in [11] it was shown that bounded uniform domains in PI-spaces are BV-extension domains. See [13] for the definition of Newtonian Sobolev spaces and[1, 12] for the BV space. The main purpose of this paper is to increase the applicability ofthe results in [2, 11] by providing a large collection of uniform domains. As a straightforwardcorollary we have the following approximation result by extension domains.
Corollary 1.2.
Let ( X, d, µ ) be a complete, doubling metric measure space supporting a localPoincar´e inequality, and let Ω ⊂ X a bounded domain. Then Ω can be approximated (as inTheorem 1.1) by N ,p -extension and BV -extension domains. Notice also that in the case when Ω is unbounded, we can for example fix a point x ∈ Ωand for each i ∈ N approximate the connected component of B ( x , i ) ∩ Ω containing x frominside by Ω i using Theorem 1.1 with the choice ε = 1 /i , and thus obtainΩ = ∞ [ i =1 Ω i , with Ω i uniform for all i ∈ N .2. Construction of the uniform domains
In the Euclidean setting we could use closed dyadic cubes to construct the uniform domains.Using just the fact that a Euclidean cube is John (and not that it is in fact uniform), we couldstart with a finite union of cubes of some fixed side-length, then take all the neighbouringcubes with a constant c ∈ (0 ,
1) times smaller side-length than the original ones and continuetaking smaller and smaller cubes. The main thing one has to take care about is that twopoints near the boundary that are some small distance r from each other can be connected bygoing via cubes not much larger than r in side-length. This is handled by taking the constant c small enough because of the nice property of closed Euclidean dyadic cubes: if two cubesof side-length l do not intersect, then their distance is at least l .We will use the above idea in the metric setting. However, none of the dyadic cube con-structions that we have seen (for instance [4, 7, 8, 9]) take care about the separation ofnon-intersecting cubes, but only about other properties such as nestedness and size. Luckily,we do not need a nested structure, nor a decomposition, so we will work with coverings byballs having the needed separation property. The existence of such coverings is provided bythe next lemma. Lemma 2.1.
Let ( X, d ) be a doubling metric space. Then there exists a constant c > depending only on the doubling constant so that for every r > there exist r -separated points { x i } ⊂ X and radii r i ∈ [ r, r ] such that X ⊂ [ i B ( x i , r i ) and d ( x i , x j ) − r i − r j / ∈ (0 , cr ) for all i, j. Proof.
Let { x i } be a maximal r -separated net of points in X . Because of the maximality ofthe net, the balls B ( x i , r i ) will cover X . We select the suitable radii by induction. Let r = r .Suppose that r , . . . , r k have been selected. Since x i are r -separated, by the metric doublingproperty of ( X, d ), there exists an integer
N > C d so that there exist at most N points x i ∈ { x , . . . , x k } with d ( x k +1 , x i ) ≤ r . Thus, at most PPROXIMATION BY UNIFORM DOMAINS IN DOUBLING QUASICONVEX METRIC SPACES 3 N of the real numbers d ( x i , x k +1 ) − r i , for i ∈ { , . . . , k } , are on the interval [ r, r ]. Therefore,we may select a radius r k +1 ∈ [ r, r ] so that d ( x i , x k +1 ) − r i − r k +1 / ∈ (0 , r/N )for all i ∈ { , . . . , k } . (cid:3) With the replacement of the Euclidean dyadic cubes by balls given in Lemma 2.1 we cannow follow the idea presented for the Euclidean case to prove the metric version.
Proof of Theorem 1.1.
We start by noting that since our space (
X, d ) is quasiconvex, theinduced length distance d g ( x, y ) = inf { ℓ ( γ ) : the curve γ joins x to y } satisfies d g ≤ d ≤ C q d g with the quasiconvexity constant C q . By the generalized Hopf-RinowTheorem we know that d g is geodesic. Because the property of being a uniform domain isinvariant under a biLipschitz change of the distance, we may then assume that ( X, d ) is ageodesic space.
Construction:
The constructions of Ω I and Ω O are similar. The only difference is thestarting point of the construction. Fix a point x ∈ Ω and let τ ∈ (0 , min { dist ( x , ∂ Ω) , } ).For constructing Ω O we simply start with the set E = Ω , and for Ω I we take E to be the connected component of { x ∈ X : dist ( x, X \ Ω) > τ } (2.1)containing the fixed point x . Let us consider the case Ω I . Thus E is defined via (2.1).Let c > δ = min (cid:26) c
20 + c , τ τ (cid:27) . We construct Ω I using induction as follows. Suppose E k has been defined for a k ∈ N . Let { x i } and { r i } be the points and radii given by Lemma 2.1 for the choice r = δ k , and define B k = n B ( x i , r i ) : B ( x i , r i ) ∩ B ( E k , δ k ) = ∅ o . We then set E k +1 = [ B ( x,r ) ∈ B k B ( x, r ) . Finally, we define Ω I = ∞ [ k =1 E k . Uniformity:
Let us next show that Ω I is uniform. Take x, y ∈ Ω I . Let k x and k y be thesmallest integers such that x ∈ E k x and y ∈ E k y . Without loss of generality we may assume k x ≤ k y .Suppose first that d ( x, y ) < cδ . Let n ∈ N be such that14 cδ n +1 ≤ d ( x, y ) < cδ n . TAPIO RAJALA
Notice that since in each construction step k + 1 we take a neighbourhood δ k of the previousset E k , we have that dist ( E k , X \ Ω I ) ≥ ∞ X i = k δ i = δ k − δ ≥ δ k . (2.2)Therefore, if k x < n , then dist ( x, ∂ Ω I ) > δ k x > cδ n > d ( x, y ) , and, consequently, we can take the geodesic from x to y to be the curve γ for the uniformitywith constant C u = 1.If k x ≥ n , we first connect x and y to E n . We do this as follows. Starting with x , let B ( z, r ) ∈ B k x − be such that x ∈ B ( z, r ), which exists by the definitions of k x and E k x . Nexttake w ∈ E k x − with d ( z, w ) < r + δ k x − , which we have by the definition of B k x − . Now theconcatenation γ xk x of the geodesics from x to z and from z to w has the length bound ℓ ( γ xk x ) < r + r + δ k x − ≤ δ k x − , and for the distance to the boundary of Ω I we can estimatedist ( γ xk x , ∂ Ω I ) > δ k x (2.3)by the fact that in the construction of E k x +1 we take a δ k x -neighbourhood of E k x and thecurve γ xk x is contained in E k x . We then continue inductively connecting w to E k x − by γ xk x − and so on, until we have connected x to a point x ′ in E n .The curve γ x,x ′ obtained by concatenating the previous curves γ xk x , γ xk x − , . . . , γ xn has thelength bound ℓ ( γ x,x ′ ) ≤ k x − X i = n δ i ≤ δ n − δ ≤ cδ n − . (2.4)With a similar construction, we connect y to a point y ′ ∈ E n by a curve γ y,y ′ with lengthbounded from above by cδ n − /
4. We can bound the distance between x ′ and y ′ by d ( x ′ , y ′ ) ≤ d ( x ′ , x ) + d ( x, y ) + d ( y, y ′ ) < cδ n − + 14 cδ n + 14 cδ n − < cδ n − . (2.5)Now we use the crucial separation property given by Lemma 2.1. Let B ( z x , r x ) , B ( z y , r y ) ∈ B n − be such that x ′ ∈ B ( z x , r x ) and y ′ ∈ B ( z y , r y ). Since the collection B n − was definedvia Lemma 2.1 with the radius δ n − , we have d ( z x , z y ) − r x − r y / ∈ (0 , cδ n − ) , whereas (2.5) gives d ( z x , z y ) − r x − r y ≤ d ( z x , x ′ ) + d ( x ′ , y ′ ) + d ( y ′ , z y ) − r x − r y ≤ d ( x ′ , y ′ ) < cδ n − . Therefore, d ( z x , z y ) ≤ r x + r y and thus we can connect x ′ to y ′ by a curve γ x ′ ,y ′ defined bygoing first with a geodesic from x ′ to z x , then to z y and from there to y ′ . This curve has thelength bound ℓ ( γ x ′ ,y ′ ) ≤ r x + 2 r y ≤ δ n − , (2.6)and its distance to the boundary of Ω I has the bounddist ( γ x ′ ,y ′ , ∂ Ω I ) > δ n . (2.7) PPROXIMATION BY UNIFORM DOMAINS IN DOUBLING QUASICONVEX METRIC SPACES 5
Now, the curve γ obtained by concatenating γ x,x ′ , γ x ′ y ′ and γ y,y ′ has, by (2.4) and (2.6),length at most ℓ ( γ ) ≤ cδ n − + 8 δ n − + 14 cδ n − ≤ δ n − = 36 cδ · cδ n +1 ≤ cδ d ( x, y ) . (2.8)Let us check the uniformity for this curve. Let z ∈ γ . Suppose first that z ∈ γ x ′ ,y ′ . Thenby (2.7) and (2.8), we getmin { ℓ ( γ x,z ) , ℓ ( γ z,y ) } ≤ ℓ ( γ ) ≤ δ n − = 92 δ δ n ≤ δ dist ( z, X \ Ω I ) . (2.9)By symmetry it then remains to check the case z ∈ γ x,x ′ . Then there exists k ≥ n such that z ∈ γ xk . Then by (2.3) and the same estimate as in (2.4), we getmin { ℓ ( γ x,z ) , ℓ ( γ z,y ) } ≤ δ k − δ ≤ δ k ≤
10 dist ( z, X \ Ω I ) . (2.10)By combining the estimates (2.8), (2.9) and (2.10) we see that γ satisfies the uniformitycondition with the constant C u = 36 / ( cδ ).We are still left with proving the uniformity in the case d ( x, y ) ≥ cδ . For this we firstobserve that we can connect x to a point x ′ ∈ E , and y to a point y ′ ∈ E by curves havinglengths bounded from above by c/ x ′ to y ′ with a curve whose length is bounded by a constant (independent of x ′ and y ′ ) from above and whose distance to the boundary of Ω I is bounded by another constantfrom below. This is achieved directly by compactness: on one hand, any two points in thecompact set E can be joined by a rectifiable curve inside B ( E , δ/ ⊂ Ω I and the infimumover the lengths of curves joining two given points is a continuous function in terms of theendpoints. Thus, there exists the needed constant upper bound for the lengths of curves. Onthe other hand, the distance of these curves to the boundary of Ω I is at least δ/ Closeness:
Let us then show that for every ε > τ > τ in the construction above we get X \ Ω I ⊂ B ( X \ Ω , ε ).In order to have the dependence on τ , write now E ( τ ) to be the connected component of { x ∈ X : dist ( x, X \ Ω) > τ } containing x . Since any point x ∈ Ω can be connected to x by a curve inside Ω, we have that Ω = [ τ> E ( τ ) . Since X \ B ( X \ Ω , ε ) is compactly contained in Ω, by the above there exists τ > X \ B ( X \ Ω , ε ) ⊂ E ( τ ), and consequently, X \ Ω I ⊂ X \ E ( τ ) ⊂ B ( X \ Ω , ε ) . The final thing we still need to observe is that Ω I ⊂ Ω. By the construction procedure, wehave E k +1 ⊂ B ( E k , δ k )for every k ∈ N . Thus, by the choice of δ we getΩ I ⊂ B ( E , ∞ X k =1 δ k ) ⊂ B ( E , τ ) ⊂ Ω . TAPIO RAJALA
This completes the proof for Ω I . The proof for Ω O goes almost verbatim. Only the argumentfor closeness becomes easier in this case. In particular, for Ω O can then take τ = ε . (cid:3) Acknowledgments
The author thanks Nageswari Shanmugalingam for bringing this question to his attention,and for the related discussions during the IMPAN conference
Latest in Geometric Analysis.Celebration of Pekka Koskela’s 59th birthday in November 2019.
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