Silvia Ghinassi
Abstract
We provide a sufficient condition in terms of Peter Jones’βnumbers for ad-dimensional Reifenberg flat set (with holes) in Rn to be parametrized by aC1,α map, withα∈[0,1]. The statement is quantitative in that the C1,α constants depend on a Bishop-Jones type square function.
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Outline of the paper and main results . . . 2
1.3 Motivation and previous work . . . 5
1.4 Further developments . . . 6
1.5 Acknowledgements . . . 6
2 Preliminaries 7 2.1 More definitions and statements of the main results . . . 7
2.2 Construction of the parametrization . . . 8
3 Proof of the main theorems 14 3.1 Proof of Theorem E . . . 14
3.2 Proof of Theorem F . . . 16
4 Improved sufficient conditions 18 4.1 A sufficient condition involvingβ∞ numbers . . . 18
4.2 A sufficient condition involvingβ1 numbers . . . 20
Bibliography 21
1
Introduction
1.1 Background
Recall that a set E in Rn is Lipschitz image d-rectifiable – countably d-rectifiable in
Federer’s terminology – if there exist countably many Lipschitz maps fi: Rd → Rn such that Hd(E\S
ifi(Rd)) =Hd E(Rn\ S
ifi(Rd)) = 0, where Hd denotes the d-dimensional
1
Hausdorff measure. In this paper, we will investigate sets that can be covered by images of more regular maps (see Section1.2for the statements of the main results and Section 1.3for motivations).
We say that a setE inRn isC1,α d-rectifiable if there exist countably many continuously
differentiable Lipschitz mapsfi:Rd→Rn withα-H¨older derivatives such that
(1.1) Hd E Rn\ [
i
fi(Rd)
! = 0.
For Lipschitz image rectifiability, we could replace the class of Lipschitz images with bi-Lipschitz images, C1 images, Lipschitz graphs, or C1 graphs without changing the class of rectifiable sets; see Theorem 3.2.29 in [Fed69] and [Dav99] for proofs of these equivalences.
On the contrary, rectifiability of orderC1,α does not imply rectifiability of order C1,β for any 0≤α < β≤1. More generally,Cs−1,1rectifiability is equivalent toCsrectifiability, while there areCs,αrectifiable sets that are not Ct,β rectifiable, whenevers, t≥1 and s+α < t+β (see G. Anzellotti and R. Serapioni, [AS94]).
While rectifiability of sets has been widely studied and characterized, see [Mat95] for an exposition, a quantitative theory of rectifiability was only developed in the early 1990s to study connections between rectifiable sets and boundedness of singular integral operators. Peter Jones in [Jon90] gives a quantitative control on the length of a rectifiable curve in terms of a sum of β numbers. These numbers capture, at a given scale and location, how far a set is from being a line. Jones’ proof was generalized to 1-dimensional objects in Rn by K. Okikioulu in [Oki92] and in Hilbert spaces by R. Schul in [Sch07].
In [DT12] G. David and T. Toro prove that Reifenberg flat sets (with holes) admit a bi-H¨older parametrization. Moreover, if one also assumes square summability of the β’s the parametrization is actually bi-Lipschitz. To better understand this, consider a variation of the usual snowflake. Start with the unit segment [0,1], and let this be step 0. At each step
iwe create an angle ofαi by adding to each segment of length 2−i−1 an isosceles triangle in the center, with basis 2−i+1/3 and height 2−i+1α
i/6 (since the αi’s are small we can use a first order approximation). Then the resulting curve is rectifiable (i.e. has finite length) if and only ifP
iα2i <∞ (see Exercise 10.16 in [BP17]).
Consider now a smoothened version of the snowflake where we stop after a finite number of iterations. This set is clearly C1,1 rectifiable. Our goal is to prove a quantitative bound on the Lipschitz constants in term of the quantitiyP
iα2i/2−i <∞. For a general Reifenberg flat set with holes E, this means that we can find a parametrization of E via a C1,1 map. The proof follows the steps of the proof in the paper [DT12]. However detailed knowledge of their paper will not be assumed. Instead specific references will be given for those interested in the proofs of the cited results.
1.2 Outline of the paper and main results
Throughout the paper, we will prove three different versions of the two main theorems. For convenience we will now state only two of them. We state the more technical TheoremsEand
Fin Section2after a few more definitions. Let us recall the definition of the aforementioned
Definition 1.1. LetE ⊆Rn,x∈Rn, and r >0. Define
(1.2) β∞(x, r) = 1
rinfP (
sup y∈E∩B(x,r)
dist(y, P) )
,
if E ∩B(x, r) 6= ∅, where the infimum is taken over all d-planes P, and β∞(x, r) = 0 if
E∩B(x, r) =∅. Define
(1.3) β1(x, r) = inf P
( 1
rd Z
y∈E∩B(x,r)
dist(y, P)
r dH
d(y) )
,
forx∈Rn and r >0, where the infimum is taken over all d-planes P.
Next, we need to define what it is meant by Reifenberg flat with holes.
Definition 1.2. Let x ∈ Rn and r > 0. If E, F ⊆Rn both meet B(x, r) define normalized
Hausdorff distances to be the quantities
(1.4) dx,r(E, F) = 1
r max
( sup y∈E∩B(x,r)
dist(y, F), sup y∈F∩B(x,r)
dist(y, E) )
.
Definition 1.3. LetE ⊆Rn and let ε >0. Define E to be Reifenberg flat with holes if the
following conditions (1)-(2) hold.
(1) For x∈E, 0< r≤10 there is a d-planeP(x, r) such that
dist(y, P(x, r))≤ε, y∈E∩B(x, r).
(2) Moreover we require some compatibility between the P(x, r)’s:
dx,10−k(P(x,10−k), P(x,10−k+1))≤ε, x∈E, k≥0,
dx,10−k+2(P(x,10−k), P(y,10−k))≤ε, x, y∈E, |x−y| ≤10−k+2, k≥0.
Before we state our main results, let us recall some theorems of G. David and T. Toro [DT12].
Theorem 1.4. [G. David, T. Toro, Proposition 8.1 [DT12]] Let ε >0 small enough and let
E ⊆B(0,1), where B(0,1) denotes the unit ball in Rn and assume E is Reifenberg flat with
holes. Then we can construct a map f:Rd → Rn, such that E ⊂f(Rd) and f is bi-H¨older
(here we identifyΣ0 with Rd).
Setrk= 10−k.
Theorem 1.5. [G. David, T. Toro, Corollary 12.6 [DT12]] Let E be as above. Assume that
(1.5) X k≥0
β∞(x, rk)2≤M, for all x∈E.
Moreover,
Theorem 1.6. [G. David, T. Toro, Corollary 13.1 [DT12]] Let E be as above. Assume that
(1.6) X k≥0
β1(x, rk)2≤M, for all x∈E.
Thenf is bi-Lipschitz. Moreover the Lipschitz constants depend only onn, d, andM.
We are now ready to state our theorems.
Theorem A. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(1.7) X k≥0
β∞(x, rk)2/rk≤M, for allx∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only onn, d, and M.
Theorem B. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(1.8) X k≥0
β∞(x, rk)2/rαk ≤M, for allx∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have α-H¨older directional derivatives. In particular, f is continuously differentiable. Moreover the H¨older constants depend only onn, d, andM.
Even without assuming a higher regularity, such as Ahlfors regularity, we can prove a better sufficient condition involving the possibly smallerβ1 numbers.
Theorem C. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(1.9) X k≥0
β1(x, rk)2/rk≤M, for allx∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only onn, d, and M.
Theorem D. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(1.10) X k≥0
β1(x, rk)2/rαk ≤M, for allx∈E.
In Section 2, after stating TheoremsE andF, we construct a parametrization for our set
E using a so-called coherent collection of balls and planes. In Section 3 we proceed to prove Theorems E and F stated in Section 2. Finally, in Section 4 we provide proofs of the four theorems stated above (A,B,C, and D).
1.3 Motivation and previous work
As mentioned before, Peter Jones [Jon90] proved that given a collection of points in the plane we can join them with a curve whose length is directly proportional to a sum of β
numbers (and the diameter). In particular, the length is independent of the number of points. This was the starting point of a series of results seeking to characterize, in a quantitative way, which sets are rectifiable. The motivation came from harmonic analysis, more specifically, the study of singular integral operators. It became clear that the classical notion of rectifiability does not capture quantititave aspects of the operators (such as boundedness) and a quan-titative notion of rectifiability was needed. A theory of uniform rectifiability was developed and it turned out that uniformly rectifiable sets are the natural framework for the study of
L2 boundedness of singular integral operators with an odd kernel (see [DS93;DS91;Tol14]). The theory is developed for sets of any dimension, but a necessary condition for a set to be uniformly rectifiable is that it isd-Ahlfors regular, whered∈N. That is, the d-dimensional
Hausdorff measure of a ball is comparable to its radius to thed-th power.
Peter Jones’ Traveling Salesman Theorem works only for 1-dimensional sets, but does not assume any regularity. Several attempts have been made to prove similar analogues for sets (or measures) of dimension more than 1. In [Paj96] a version for 2-dimensional sets is proved. Menger curvature was also introduced to characterize rectifiability (see [L´eg99;LW11;LW09; KS13;BK12;Kol11]). Other approaches can be found in [Mer16; Del08; San17]). In [AS17], J. Azzam and R. Schul estimate thed-dimensional Hausdorff measure of a set using a sum of
β numbers.
In [DKT01], G. David, C. Kenig, and T. Toro prove that a C1,α parametrization for Reifenberg flat sets (without holes) with vanishing constants can be achieved under a pointwise condition on theβ’s (their conditions are stronger than our conditions). D. Preiss, X. Tolsa, and T. Toro [PTT09] fully describe the H¨older regularity of doubling measures in Rn for
measures supported on any (integer) dimension.
Among the results involving Menger curvature, in [KS13], S. Kolasi´nski and M. Szuma´nska prove thatC1,αregularity, with appropriateα’s, implies finiteness of functionals closely related to Menger curvature. In [BK12], S. Blatt and S. Kolasi´nski prove that a compactC1manifold has finite integral Menger curvatures (a higher dimensional version of Menger curvature) if and only if it can be locally represented by the graph of some Sobolev type map.
In [Kol11], a bound on Menger curvature together with other regularity assumptions leads to a pointwise bound on β numbers; this is the same bound which appears in [DKT01]. If in addition the set isfine, which among other things implies Reifenberg flatness allowing for small holes, then the same conclusion as in [DKT01] holds, that is the set can be parametrized by aC1,α map.
impose any restrictions on the size of the holes.
In the last few years, C. Fefferman, A. Israel, and G.K. Luli [FIL16] have been investigat-ing Whitney type extension problems forCkmaps, finding conditions to fit smooth functions to data. Recently, N. Edelen, A. Naber, and D. Valtorta [ENV17] used β numbers to pro-vide interesting conditions on whether the support of a measure isclose to a d-dimensional subspace.
We say that a Radon measure µ on Rn is d-rectifiable if there exist countably many
Lipschitz mapsfi:Rd→Rnsuch that
µ Rn\[
i
fi(Rd)
! = 0.
Note that a setE is d-rectifiable if and only if Hd E is ad-rectifiable measure.
For measures which are absolutely continuous with respect to the Hausdorff measure, the above definition coincides whichLipschitz graphs rectifiability. That is, if we require the sets to be almost covered by Lipschitz graphs instead of images, we get an equivalent definition. J. Garnett, R. Kilip, and R. Schul [GKS10] proved that this is not true for general measures, even if we require the doubling condition (that is, the measure of balls is comparable if we double the radius). They exhibit a doubling measure supported inR2, singular with respect to
Hausdorff measure, which isLipschitz image rectifiable but is notLipschitz graph rectifiable. Given such distinctions it is natural to investigate different types of rectifiability (e.g. Lipschitz imageandLipschitz graph rectifiability,Ck andCk,αrectifiability). There has been some progress in this direction concerning rectifiability of sets (by e.g. [AS94]) but the tools involved rely heavily on the Euclidean structure ofHdand give qualitative conditions. There has been a great deal of interest in developing tools which allow further generalizations to rectifiability of measures which provide quantitative results. Using the techniques from [DT12] we develop such tools with the use ofβ numbers.
1.4 Further developments
Clearly, it is interesting to ask whether there exist analogous necessary conditions for higher order rectifiability. The author believes similar results hold with an appropriate defi-nition of theβ’s in order to getCk,α regularity.
1.5 Acknowledgements
2
Preliminaries
2.1 More definitions and statements of the main results
Given a Reifenberg flat set with holes, we now want to construct a so-called coherent collection of balls and planes(CCBP) forE(for more details see the discussion after Theorem 12.1 in [DT12]).
Let E be as above and set rk = 10−k. Choose a net {xj,k} for E, that is a collection of points {xj,k}, j ∈ Jk such that |xi,k −xj,k| ≥ rk, for i, j ∈ Jk, i 6= j. Let Bj,k be the ball centered atxj,k with radiusrk. Forλ >1, set
(2.1) Vkλ= [ j∈Jk
λBj,k.
Because of our assumptions on the setEwe can assume that the initial points{xj,0}are close to ad-plane Σ0, that is dist(xj,0,Σ0)≤ε, for j∈J0. Moreover, for eachk≥0 andj∈Jk we assume that there exists adplane Pj,k through xj,k such that
dxj,k,100rk(Pi,k, Pj,k)≤εfork≥0 and i, j∈Jk such that xi,k−xj,k ≤100rk, (2.2)
dxi,0,100(Pi,0,Σ0)≤εfori∈J0,
(2.3)
dxi,k,20rk(Pi,k, Pj,k+1)≤εfork≥0,i∈Jk and j∈Jk+1 s.t. |xi,k−xj,k+1| ≤2rk. (2.4)
Definition 2.1. Acoherent collection of balls and planes forE is a pair (Bj,k, Pj,k) with the properties above. We assume thatε >0 is small enough, depending ondand n.
We will use this collection to construct the parametrization, as explained in the following section. Recall Theorem1.4:
Theorem 1.4. [G. David, T. Toro, Proposition 8.1 [DT12]] Let ε >0 small enough and let
E ⊆B(0,1), where B(0,1) denotes the unit ball in Rn and assume E is Reifenberg flat with
holes. Then we can construct a map f:Rd → Rn, such that E ⊂f(Rd) and f is bi-H¨older
(here we identifyΣ0 with Rd).
We now define the coefficients εk which differ from classic β numbers in that they take into account neighbouring points at nearby scales. In section4 the relationship between the two will be made explicit.
Definition 2.2. For k≥1 and y∈Vk10 define
(2.5) εk(y) = sup{dxi,l,100rl(Pj,k, Pi,l)|j∈Jk, l∈ {k−1, k}, i∈Jl y∈10Bj,k∩11Bi,k}
andεk(y) = 0, fory∈Rn\V10
k .
Theorem 2.3. [G. David, T. Toro, Proposition 8.3 [DT12]] Let ε >0 and E as above. If we also assume that there existsM <+∞ such that
(2.6) X k≥0
εk(f(z))2≤M, for all z∈Σ0.
then the mapf:Rd→Rn constructed in Theorem1.4is bi-Lipschitz. Moreover the Lipschitz
As said before, we are interested in finding a condition on theεk’s to improve the results on the mapf. The theorems we want to prove are the following.
Theorem E. Let E ⊆ B(0,1) as above, with ε > 0 small enough. Also assume that there existsM <+∞ such that
(2.7) X k≥0
εk(f(z))2
rk
≤M, for allz∈Σ0.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only onn, d, and M.
Theorem F. Let E ⊆ B(0,1) as above, with ε > 0 small enough. Also assume that there existsM <+∞ such that
(2.8) X k≥0
εk(f(z))2
rα k
≤M, for allz∈Σ0.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have α-H¨older directional derivatives. In particular, f is continuously differentiable. Moreover the H¨older constants depend only onn, d, andM.
2.2 Construction of the parametrization
As in [DT12] f will be constructed as a limit. To construct the sequence we need a partition of unity subordinate to{Bj,k}. Following the construction in Chapter 3 of [DT12], we can obtain functions θj,k(y) andψk(y) such that each θj,k is nonnegative and compactly supported in 10Bj,k, and ψk(y) = 0 onVk8. Moreover we have, for everyy∈Rn,
(2.9) ψk(y) + X
j∈Jk
θj,k(y) = 1.
Finally we have that
(2.10) |∇mθj,k(y)| ≤Cm/rkm, |∇mψk(y)| ≤Cm/rmk.
We define the sequence{fk} inductively by
(2.11) f0(y) =y and fk+1 =σk◦fk,
where
(2.12) σk(y) =ψk(y)y+ X
j∈Jk
θj,k(y)πj,k(y).
whereπj,k denotes the orthogonal projection fromRntoPj,k. In the future we denote by πj,k⊥ the projection onto the d-plane perpendicular to Pj,k.
Remark 2.6. Note that while πj,k is an affine map,π⊥j,k is a linear map. We also denote by ˙
πj,k the projection onto thed-plane parallel toPj,k passing through the origin. From now on we will use the following convention for notation: if u is a direction in Rn, ∂πj,k
∂u = ˙πj,k(u). Moreover ifvis another direction, then ∂2πj,k
∂u ∂v = 0n×n(the zero matrix), as we are considering ∂πj,k
∂u as a map from Rn to L(Rn,Rn), linear maps fromRn to itself, which is identified with
Rn2. As such, ∂πj,k
∂u is constant and thus has zero derivative in every direction.
Remark 2.7. Remark2.6 also applies toσk as it is a convex combination of linear maps.
By differentiating (2.12), we get that fory∈V10 0n×n (where we intend this as discussed in Remark2.6).
Lemma 2.8. Let u, v∈Rn, andy ∈Vk10. We have will work under this assumption.
Remark 2.10. As the computation at the beginning of the proof shows, the map g is just ∂2σ
k
∂u ∂v(y) withπi,k replacingπj,k.
have ately from the previous lemma.
We now want to collect some more estimates. Let Σk be the image of fk, i.e. Σk =
fk(Σ0) =σk−1◦ · · · ◦σ0(Σ0). First, we need to recall some results from [DT12]. The main result is a local Lipschitz description of the Σk’s. For convenience we introduce the following notation for boxes.
Definition 2.12 (Chapter 5, [DT12]). If x ∈ Rn, P is a d-plane through x and R > 0, we
define the boxD(x, P, R) by
Recall that for a Lipschitz map A: P → P⊥ the graph of A overP is Γ
such that aroundxi,j Σk coincides with the graph of Aj,k, that is
(2.23) Σk∩D(xj,k, Pj,k,49r, k) = ΓAj,k ∩D(xj,k, Pj,k,49r, k).
Moreover, we have that
(2.24) |σk(y)−y| ≤Cεrk for y∈Σk
Using Proposition2.13 we can get estimates on the second derivatives of theσk’s.
Proposition 2.14. For allk≥0, j∈Jk,y∈Σk∩45Bj,k,u, v∈Rn,|u|=|v|= 1, we have
We want to control
B =g(y)−∂ψk
In the construction of the coherent families of balls and planes, since y ∈ 45Bj,k ∩10Bj,k, (2.2) says that
and so, foru∈Rn,|u|= 1,
Recalling also that By the results in Proposition2.13, we also have
Lemma 2.17. For k≥0, u, v∈Rn,|u|=|v|= 1 and y∈Σk∩Vk8 we have
Using the Pythagorean theorem we can improve Lemma 2.17with the following lemma.
Lemma 2.18. For u, v∈Rn,|u|=|v|= 1
By the Pythagorean Theorem we get
(2.45)
Before proceeding to prove the main theorems let us collect some estimates on the inverses
∂u is bounded below. Note that we can improve the lower bound in the same way we improved the upper bound. Recalling thatσk◦σk−1(x) =x for every x ∈ Σk and by taking the directional derivative we can conclude that 2.46 holds. Similarly, by taking one more derivative we get
(2.48) ∂
from which2.47 follows after a simple computation and using the estimates collected above.
3
Proof of the main theorems
3.1 Proof of Theorem E
TheoremE. Suppose that there exists M <+∞ such that for every m,
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. That is, ∂f∂u: Σ0∩B(0,1)→Rn is Lipschitz and so is ∂f∂u−1. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only on n, d, and M. the definition of thefk’s we have
We want to estimate Am. If 0< x <1 clearly (1 +x)2≤1 + 3x, so we have
Recalling that A0 = 0, and using the fact that we can bound every term of the form
εm(zm)/rm we get
2.3holds and in particularQ
0≤i≤m
Now, we want to prove that
Then there is a C2 curve γ:I → Σ
m that goes from x to y with length bounded above by (1 +Cε)|x−y|.
Then we can proceed exactly as in the first part of the proof. Write
(3.9) ∂f
Now recall the estimates2.46 and 2.47 and use the fact that ∂fk
∂u is bounded by a constant, which follows from the proof of Lemma 2.3. Because of that we can carry the proof exactly as before, to get
(3.10)
3.2 Proof of Theorem F
Theorem F. Let E ⊆ B(0,1) as above, with ε > 0 small enough. Also assume that there
Then the map f constructed in Theorem 1.4 is invertible and differentiable, and both f and its inverse haveα-H¨older directional derivatives. That is, ∂f∂u: Σ0∩B(0,1)→Rn is α-H¨older and so is ∂f∂u−1. In particular, f is continuously differentiable. Moreover the H¨older constants depend only on n, d, and M.
Lemma 3.1. Suppose gj is a sequence of continuous functions onB(0,1), that satisfy
The lemma is analogous to Lemma 2.8, Chapter 7 in [SS05].
Proof. First note that g(x) is the limit of the uniformly convergent series
(3.15) g(x) =g1(x) +
By the triangle inequality we get
(3.17) |g(x)−g(y)| ≤ |g(x)−gj(x)|+|gj(x)−gj(y)|+|g(y)−gj(y)| ≤C(Aj|x−y|+B−j).
Now, for fixedx 6= y we want to choosej so that the two terms on the right hand side are comparable. We want to choosej such that
(3.18) (AB)j|x−y| ≤1 and 1≤(AB)j+1|x−y|.
Letj=−⌊logAB|x−y|⌋. Then the two inequalitites are clearly satisfied. The first one gives
Aj|x−y| ≤B−j and by raising the second one to the power γ, recalling that (AB)γ =B by definition, we get thatB−j ≤ |x−y|γ. This gives
(3.19) |g(x)−g(y)| ≤C(Aj|x−y|+B−j)≤CB−j ≤C|x−y|γ,
which is what we wanted to prove.
Then we can apply Lemma3.1, withgj = ∂f∂uj,ak=εm(xm)2,A= 101−α, andB = 10α, since we know, by (2.7), that
X
k≥j
εk(xk)2
rα k
≤M
X
k≥j
εk(xk)2rαj rα
k
≤M rαj
X
k≥j
εk(xk)2 ≤M rjα.
Thenγ = log 10log(10)α =α and the lemma hence gives that ∂f∂u isα-H¨older. In a similar fashion, we can prove that ∂f∂u−1 isα-H¨older.
4
Improved sufficient conditions
4.1 A sufficient condition involving β∞ numbers
Note that the sufficient conditions in TheoremsEand Frely on the parametrization. We proceed to remove such dependence and in order to do so, we use some results from [DT12]. Recall that
(4.1) β∞(x, rk) = 1
rk inf
P (
sup y∈E∩B(x,rk)
dist(y, P) )
,
if E∩B(x, rk) 6= ∅, where the infimum is taken over all d-planes P, and β∞(x, rk) = 0 if
E∩B(x, rk) =∅. Now recall Theorem 1.5:
Theorem1.5. Let E be Reifenberg flat with holes. Assume that
(4.2) X k≥0
β∞(x, rk)2≤M, for all x∈E.
Then the mapf constructed in Theorem1.4is bi-Lipschitz. Moreover the Lipschitz constants depend only on n, d, and M.
Let us define, as in Chapter 12 of [DT12], new coefficientsγk(x) as follows
(4.3) γk(x) =dx,rk(Pk+1(x), Pk(x)) + sup y∈E∩B(x,35rk)
dx,rk(Pk(x), Pk(y)).
Then define, forx∈E,
(4.4) Jˆα,γ(x) = X
k≥0
γk(x)2/rαk.
Proposition 4.1. [Corollary 12.5, [DT12]] If in addition to the hypotheses of Theorem 1.4
we have that
(4.5) Jˆ1,γ(x)≤M, for all x∈E,
then constructed in Theorem1.4is bi-Lipschitz. Moreover the Lipschitz constants depend only onn, d, and M.
Following the proof of Corollary 12.5 in [DT12], it is easy to check that under the as-sumption that ˆJα is uniformly bounded, the sufficient conditions in Theorems E and F are satisfied. More specifically, we have (see page 71 of [DT12]),
Lemma 4.2. Let z∈Σ0 and let x∈E such that
(4.6) |x−f(z)| ≤2 dist(f(z), E).
Then
(4.7) εk(fk(z))≤C(γk(x) +γk−1(x)).
Using the lemma, the following results follow immediately.
Proposition 4.3. If in addition to the hypotheses of Theorem 1.4we have that (4.8) Jˆ1,γ(x)≤M, for all x∈E,
then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only onn, d, and M.
Proposition 4.4. If in addition to the hypotheses of Theorem 1.4we have that (4.9) Jˆα,γ,(x)≤M, for all x∈E,
then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have α-H¨older directional derivatives. In particular, f is continuously differentiable. Moreover the H¨older constants depend only onn, d, andM.
We want to replace ˆJα,γ with a more explicit Bishop-Jones type function involving β∞’s. Define
(4.10) Jˆα,∞(x) =X k≥0
β∞(x, rk)2/rkα.
Finally, we can state the following theorems, which are an improved version of TheoremsE
andF.
Theorem A. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(4.11) Jˆ1,∞(x) =X k≥0
β∞(x, rk)2/rk≤M, for all x∈E.
Theorem B. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(4.12) Jˆα,∞(x) =X k≥0
β∞(x, rk)2/rαk ≤M, for allx∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have α-H¨older directional derivatives. In particular, f is continuously differentiable. Moreover the H¨older constants depend only onn, d, andM.
The proof of Corollary 12.6 in [DT12], which we restated as Theorem 1.5, can be used directly to prove the two theorems above, which are obtained as corollaries of Theorems E
andF, respectively.
4.2 A sufficient condition involving β1 numbers
We would now like to replace ˆJα,∞ with ˆJα,1 based on an L1 version of the β numbers. Usually such coefficients are used when the Hausdorff measure restricted to the setEis Ahlfors regular. We will not need to assume such regularity, after observing that Reifenberg flatness implies lower regularity. The following is Lemma 13.2 in [DT12]. Let E⊂Rn and define
(4.13) β1(x, r) = inf P
( 1
rd Z
y∈E∩B(x,r)
dist(y, P)
r dH
d(y) )
,
forx∈Rn and r >0, where the infimum is taken over all d-planes P.
Lemma 4.5. [Lemma 13.2, [DT12]]LetE⊆B(0,1)be a Reifenberg flat set with holes. Then, for x∈E and for smallr >0,
(4.14) Hd(E∩B(x, r))≥(1−Cε)ωdrd,
where ωd denotes the measure of the unit ball inRd.
Moreover, recall Theorem1.6:
Theorem1.6. Let E and f be as in the previous section. Assume that
(4.15) X k≥0
β1(x, rk)2≤M, for all x∈E.
Then the mapf constructed in Theorem1.4is bi-Lipschitz. Moreover the Lipschitz constants depend only on n, d, and M.
The following lemma is implied by the proof of Corollary 13.1 in [DT12].
Lemma 4.6. By changing the net xj,k if necessary, we have that εk(xk)≤β1(z, rk−3), where
z∈E is chosen appropriately.
Theorem C. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(4.16) Jˆ1,1(x) = X
k≥0
β1(x, rk)2/rk≤M, for allx∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have Lipschitz directional derivatives. In particular, f is continuously differentiable. Moreover the Lipschitz constants depend only onn, d, and M.
Theorem D. Let E ⊆ B(0,1) be a Reifenberg flat set with holes. Also assume that there existsM <+∞ such that
(4.17) Jˆα,1(x) = X
k≥0
β1(x, rk)2/rαk ≤M, for all x∈E.
Then the mapf constructed in Theorem1.4is invertible and differentiable, and bothf and its inverse have α-H¨older directional derivatives. In particular, f is continuously differentiable. Moreover the H¨older constants depend only onn, d, andM.
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Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA
