Geometry of Measures: Harmonic Analysis meets Geometric Measure Theory
T. Toro ∗
1 Introduction
One of the central questions in Geometric Measure Theory is the extend to which the regularity of a measure determines the geometry of its support. This type of question was initially studied by Besicovitch, and then pursued by many authors among others Marstrand, Mattila and Preiss. These authors focused their attention on the extend to which the behavior of the density ratio of a m given Radon measure µ in R with respect to Hausdorff measure determines the regularity of the µ(B(x,r)) m support of the measure, i.e. what information does the quantity rs for x ∈ R , r > 0 and s > 0 encode. In the context of the study of minimizers of area, perimeter and similar functionals the correct density ratio is monotone, the density exists and its behavior is the key to study the regularity and structure of these minimizers.
This question has also been studied in the context of Complex Analysis. In this case, the Radon measure of interest is the harmonic measure. The general question is to what extend the structure of the boundary of a domain can be fully understood in terms of the behavior of the harmonic measure. Some of the first authors to study this question were Carleson, Jones, Wolff and Makarov.
The goal of this paper is to present two very different type of problems, and the unifying techniques that lead to a resolution of both. In Section 2 we look at the history of the question initially studied by Besicovitch in dimension 2, as well as some of its offsprings. In particular we focus our attention of Preiss’s work which addresses the higher dimensional case. In Section 3 we present a question which originated at the interface of Complex Analysis and Harmonic Analysis. We describe what happens when the question is translated to higher dimensions. Surprisingly analogues of the techniques introduced by Preiss to study the higher dimensional version of Besicovitch question come into play (see Section 4). It is remarkable that in dimensions greater than 2, these tools from GMT replace the techniques from Complex Analysis. Although somewhat unusual we devote Section 4 to some of the common technical results required in these two bodies of work. These powerful tools are not well-known but are very likely to have applications in other areas (see Section 5).
∗The author was partially supported by NSF grant DMS-0600915
1 2 Density - An indicator of regularity
The basic question underlying the work of Besicovitch ( [5], [6], [7]), Marstrand ( [25], [26], [27]), Mattila( [23]) and Preiss ( [28]) is to what extend the regular behavior of a Radon measure on balls determine the structure of the measure. More precisely they asked the following question: Assume m that for µ-a.e x ∈ R µ(B(x, r)) (2.1) 0 < θn(µ, x) = lim < ∞ (the n-density exists) r→0 rn where 0 ≤ n ≤ m, how regular is µ?
Besicovitch in the late 1930’s and then Marstrand in the mid 1950’s and early 1960’s provided some m n initial answers. Recall that a Radon measure µ in R is n-rectifiable, if µ H and there exits m an n-rectifiable Borel set E such that µ(R \E) = 0.
2 2 Theorem ( [5], [6], [7]) Let µ be a Radon measure in R and assume that for µ-a.e x ∈ R µ(B(x, r)) (2.2) 0 < θ(µ, x) = lim < ∞ r→0 r then µ is 1-rectifiable.
m Theorem ( [25], [27]) Let µ be a Radon measure in R if the n-density exists on a set of µ positive measure then n is an integer.
While the case n = 1 was settled in all co-dimensions, only partial results were available in other cases until 1986 when Preiss’s work appeared.
m m Theorem ( [28]) Let µ be a Radon measure in R and assume that for µ-a.e x ∈ R (2.3) 0 < θn(µ, x) < ∞, then µ is n-rectifiable.
Preiss’s paper contains several original and powerful ideas. He introduced the notion of tangent measures. He studied the connectivity properties of cones of Radon measures. He analyzed the structure of the support of uniform measures by means of their moments. Finally he proved, by a very delicate argument, that the space of tangent measures at a point is connected, in a suitable sense. Several of these ideas will be discussed in some detail in Section 4. Here we only sketch very briefly Preiss’s argument.
n m By a blow up procedure he shows that for a Radon measure µ if θ (µ, a) exists for µ-a.e a ∈ R , then m for µ-a.e a ∈ R all tangent measures are n-uniform. A tangent measure is to a measure like the derivative is to a function. It carries information about the local behavior of the measure. A measure
2 ν is n-uniform if there is a constant C > 0 so that for r > 0 and x ∈ spt ν, ν(B(x, r)) = Crn. One of the central arguments in Preiss’s paper consists in proving that if ν is n-uniform then either spt ν is an n-plane or it is very far away from any n-plane. Since at µ-a.e point tangent measures to tangent measures are tangent measures then µ a.e. point admits a tangent measure which is a multiple of the n-dimensional Hausdorff measure on an n-plane, i.e. µ a.e. point admits a flat tangent measure. A very general connectivity argument which requires as a hypothesis the fact that n-uniform measures are either flat or very far away from flat proves that at µ-a.e. point all tangent measures to µ are n-flat. From this and the fact that the n-density exists, a standard argument ensures that µ is n-rectifiable.
The behavior of the density ratio of a Radon measure µ also yields information about higher regularity of its support. In [12] and [29] we prove the following results which were motivated by questions from Harmonic Analysis. The question we were studying at the time was whether the doubling properties of the harmonic measure of a domain determine the regularity of its boundary. Under the appropriate hypothesis the problem became a question about measures whose density ratio converges to a limit at a H¨olderrate.
m Definition 2.1. Let µ be a positive Radon measure supported on Σ ⊂ R . Let α ∈ (0, 1]. We say that the n-density ratio of µ is locally Cα if, for each compact set K ⊂ Σ, there is a constant CK > 0 such that
µ(B(x, r)) α (2.4) n − 1 ≤ CK r ωnr n for x ∈ K and 0 < r < 1. Here ωn denotes the Lebesgue measure of the unit ball in R . Theorem 2.1 ( [12], [29]) For each α > 0 there exists β = β(α) > 0 with the following property. m α If µ is a positive Radon measure supported on Σ ⊂ R whose n-density ratio is locally C , then:
1,β m (i) if n = 1, 2, Σ is a C submanifold of dimension n in R , 1,β m (ii) if n ≥ 3, Σ is a C submanifold of dimension n in R away from a closed set S such that Hn(S) = 0.
Theorem 2.1 combined with Kowalski and Preiss classification theorem for n-uniform measures in co-dimension 1(see [20] or Question 1 in Section 5) yields the following result.
Corollary 2.1 ( [29]) For each α > 0 there exists β = β(α) > 0 with the following property. If µ n+1 α is a positive Radon measure supported on Σ ⊂ R whose n-density ratio is locally C , then Σ is 1,β n+1 a C submanifold of dimension n in R away from a closed set S of dimension at most n − 3. If n = 3, S is discrete.
3 Harmonic Measure - Boundary structure and size
In joint work with Kenig and Kenig & Preiss ( [18], [19], [17]) it has surfaced that Preiss’s techniques described above yield information about the size and the structure of harmonic measure even in
3 the case when the density does not exist. In particular, in [17], Geometric Measure Theory takes the place that Complex Analysis occupies in dimension 2 when analyzing the size and structure of harmonic measure.
2 Let us briefly describe some of the 2-dimensional results. Let Ω ⊂ R be a regular domain for the Dirichlet problem, and let ω be the harmonic measure associated to Ω. Then Carleson ( [11]) and Jones & Wolff ( [16]) showed that
(3.1) H − dim ω ≤ 1
If Ω is simple connected Makarov ( [22]) proved Oskendal’s conjecture in dimension 2, i.e.
(3.2) H − dim ω = 1
Recall that the Hausdorff dimension of ω (denote by H − dim ω) is defined by
(3.3) H − dim ω = inf {k : there exists E ⊂ ∂Ω with Hk(E) = 0 and n ω(E ∩ K) = ω(∂Ω ∩ K) for all compact sets K ⊂ R }
2 Let Ω ⊂ R be a Jordan domain (i.e a simply connected domain bounded by a Jordan curve), and let ω be the interior harmonic measure associated to Ω. A combination of the works of Choi, Makarov, McMillan and Pommerenke (see [13] for precise references) shows that
(3.4) ∂Ω = G ∪ S ∪ N with
• In G, ω H1 ω
• ω(N) = 0
•H 1(S) = 0.
Moreover
• For ω-a.e Q ∈ G, the density of ω exits and
ω(B(Q, r)) (3.5) 0 < lim < ∞. r→0 r
• For ω-a.e Q ∈ S
ω(B(Q, r)) ω(B(Q, r)) (3.6) lim inf = 0 and lim sup = ∞ r→0 r r→0 r
4 T. Wolff [31] showed, by a deep example, that, for n ≥ 3, Oksendal’s conjecture (H−dim ω = n−1) 3 fails. He constructed what we will call “Wolff snowflakes”, domains in R , for which H − dim ω > 2 and others for which H − dim ω < 2. In Wolff’s construction, the domains have a certain weak regularity property, they are non-tangentially accessible domains (NTA), in the sense of [15], in fact, they are 2-sided NTA domains (i.e. Ω and int(Ωc) are both NTA) and this plays an important role in his estimates. Here, whenever we refer to a “Wolff snowflake,” we will mean a 2-sided NTA n domain in R , for which H − dim ω 6= n − 1. In [21], Lewis, Verchota and Vogel reexamined Wolff’s n construction and were able to produce “Wolff snowflakes” in R , n ≥ 3, for which both
(3.7) H − dim ω± > n − 1, and others for which
(3.8) H − dim ω± < n − 1.
Here ω± denotes the harmonic measure of Ω±, where Ω+ = Ω and Ω− = int(Ωc). They also observed, as a consequence of the monotonicity formula in [3], that if
(3.9) ω+ ω− ω+ then H − dim ω± ≥ n − 1.
Returning to the case of n = 2, when Ω a Jordan domain the work of Bishop, Carleson, Garnett & Jones [10] combined with (3.4) yields the following new decomposition
(3.10) ∂Ω = G± ∪ S± ∪ N ± = G ∪ S ∪ N, with
• ω+ H1 ω− ω+ in G = G+ ∩ G−. Moreover if E ⊂ G and ω±(E) > 0 then E is 1-rectifiable.
• N = N + ∩ N − and ω±(N) = 0.
• ω+ ⊥ ω− on S.
n − + In [9], motivated by this last result, Bishop asked whether in the case of R , n ≥ 3, if ω , ω are mutually absolutely continuous on a set E ⊂ ∂Ω, ω±(E) > 0, then ω± are mutually absolutely n−1 ± continuous with respect to H on E (modulo a set of ω measure zero) and hence dimH(E) = n − 1. Here dimH denotes the Hausdorff dimension. On the other hand, Lewis, Verchota and n ± Vogel [21] conjectured that there are “Wolff snowflakes” in R , n ≥ 3 with H − dim ω > n − 1, for which ω+, ω− are not mutually singular.
The study of these issues follows a similar pattern to the one in Preiss’s work on the rectifiability of measures, described in Section 2. It yields the following type of results (see [17]) in the case when n the domain Ω ⊂ R verify the weak regularity assumption of being 2-sided locally NTA (a condition satisfied by the Wolff snowflakes constructed both by Wolff and Lewis, Verchota & Vogel).
5 n Theorem 3.1 Let Ω ⊂ R be a 2-sided locally NTA domain. Then the boundary of Ω can be decomposed as follows:
(3.11) ∂Ω = G ∪ S ∪ N,
• In G, ω+ ω− ω+.
• In S, ω+ ⊥ ω−.
• ω+(N) = ω−(N) = 0.
Moreover
(3.12) dimH G ≤ n − 1.
Furthermore, if ω±(G) > 0 then
(3.13) dimH G = n − 1.
n n−1 Theorem 3.2 Let Ω ⊂ R be a 2-sided locally NTA domain such that H ∂Ω is a Radon measure. Then, as in Theorem 3.1, ∂Ω = G ∪ S ∪ N and G is (n − 1)-rectifiable.
The following theorem, which is a corollary of Theorem 3.1 proves that there are no Wolff snowflakes for which ω+ and ω− are mutually absolutely continuous, answering a question in [21].
Theorem Let Ω be a 2-sided locally NTA domain. Assume that ω+ and ω− are mutually absolutely continuous, then
(3.14) H − dim ω+ = H − dim ω− = n − 1.
Here the Hausdorff dimension of ω±, H − dim ω± is defined as in (3.3).
The study of these questions requieres three main ingredients:
1. Alt-Caffarelli-Friedman monotonicity formula: (see [3]) this yields Beurling’s inequality n in higher dimensions, i.e. given a compact set K ⊂ R , for Q ∈ ∂Ω ∩ K and r ∈ (0,RK ) where RK > 0 depends on K there exits a constant C > 0 depending on the NTA constants such that
(3.15) ω+(B(Q, r)) · ω−(B(Q, r)) ≤ Cr2(n−1).
2. Classification of the tangent measures to ω±: this is possible by means of a blow- up procedure compatible with the singularities of u±, the corresponding Green’s functions associated to ω±. This procedure simultaneously blows up the domains, their boundaries,
6 the harmonic measures and the corresponding Green’s functions in such a way that the sub- ± sequential limits are the harmonic measures ω∞ and the corresponding Green’s functions ± ± with pole at infinity u∞ of the blow-up domains Ω∞ ( [19]). This blow-up procedure has the additional property it lifts the singularities of u± at the limit. A remarkable feature is that the 2-sided locally NTA assumption ensures that at points where ω+ and ω− are mutually + − absolutely continuous all blow-up limits u∞ = u∞ − u∞ are harmonic polynomials. 3. Connectivity property of the cone of tangent measures to ω±: this resembles the one used by Preiss in [28] In this case the connectivity it is a consequence of the fact that u∞ is a harmonic polynomial.
4 Geometric Measure Theory Tools
Although it is somewhat unusual to focus on the techniques of proof, we believe that the underlying common ideas behind the two bodies of work described above are very powerful, not well known and will most likely have applications in other contexts. Thus we devote this section to these technical results. In both cases, the connectivity properties of the cone of tangent measures rely on two pillars, one of them is Theorem 4.1, the other is the understanding of the properties of the tangent measures which allows to very that condition (P) in Theorem 4.1 is satisfied.
First we recall two families of “distances” between Radon measures in Euclidean space which are compatible with weak convergence. They were initially introduced in [28].
n n Definition 4.1. Let Φ and Ψ be Radon measures in R . Let K be a compact set in R define
Z c i) FK (Φ) = dist (z, K ) dΦ(z) and Fr(Φ) = FB(0,r)(Φ).
ii) Z Z
(4.1) FK (Φ, Ψ) = sup fdΦ − fdΨ : spt f ⊂ K, f ≥ 0, f Lipschitz, Lipf ≤ 1 .
n n n n Remark 4.1. Let Φ be a Radon measure in R . For x ∈ R and r > 0 define Tx,r : R → R by the formula Tx,r(z) = (z − x)/r. Note that:
−1 i) Tx,r[Φ](B(0, s)) := Φ(Tx,r (B(0, s))) = Φ(B(x, sr)) for every s > 0. Z Z z − x ii) f(z)dT [Φ](z) = f dΦ(z) whenever at least one of these integrals is defined x,r r
iii) FB(x,r)(Φ) = rF1(Tx,r[Φ])
iv) FB(x,r)(Φ, Ψ) = rF1(Tx,r[Φ],Tx,r[Ψ])
7 n Definition 4.2. Let µ, µ1, µ2,... be Radon measures on R . We say that µi → µ or limi→∞ µi = µ if
n i) lim supi→∞ FK (µi) < ∞ for every compact set K ⊂ R n ii) limi→∞ FK (µi, µ) = 0 for every compact set K ⊂ R .
n Lemma 4.1 ( [28], Proposition 1.11) Let µ1, µ2 ... and µ be Radon measures on R such that n lim supi→∞ µi(K) < ∞ for each compact set K in R . Then µi → µ if and only if µi * µ ( i.e. if µi converges weakly to µ as Radon measures).
n Lemma 4.2 ( [24], Lemma 14.13) Let µ1, µ2,... and µ be Radon measures on R . Then µi → µ if and only if
(4.2) lim Fr(µi, µ) = 0 ∀ r > 0. i→∞
We now introduce a scale invariant relative of Fr, which behaves well under weak convergence, multiplication by a positive constant and scaling.
n Definition 4.3 ( [28], §2) i) A set M of non-zero Radon measures in R will be called a cone if cΨ ∈ M whenever Ψ ∈ M and c > 0.
ii) A cone M will be called a d-cone if T0,r[Ψ] ∈ M whenever Ψ ∈ M and r > 0.
n iii) Let M be a d-cone, and let Φ a Radon measure in R such that for s > 0, 0 < Fs(Φ) < ∞ then we define the distance between Φ and M by Φ (4.3) ds(Φ, M) = inf Fs , Ψ :Ψ ∈ M and Fs(Ψ) = 1 . Fs(Φ) We also define
(4.4) ds(Φ, M) = 1 if Fs(Φ) = 0 or Fs(Φ) = +∞.
Remark 4.2. Note that if M is a d-cone and Φ is a Radon measure then
i) ds(Φ, M) ≤ 1
ii) ds(Φ, M) = d1(T0,s[Φ], M)
iii) if µ = lim µi and Fs(µ) > 0 then ds(µ, M) = lim ds(µi, M). i→∞ i→∞
n n Definition 4.4. i) Let η be a Radon measure in R . Let x ∈ R , a non-zero Radon measure ν n in R is said to be a tangent measure of η at x if there are sequences rk & 0 and ck > 0 such that ν = lim ckTx,r [η]. k→∞ k ii) The set of all tangent measures to η at x is denoted by Tan (η, x).
8 Definition 4.5. The basis of a d-cone M of Radon measures is the set {Ψ ∈ M : F1(Ψ) = 1}. We say that M has a closed (respectively compact) basis, if its basis is closed (respectively compact) in the topology induced by the metric ∞ X −p (4.5) 2 min{1,Fp(Φ, Ψ)} p=0 defined for Radon measures Ψ and Φ.
n Remark 4.3. For η a non-zero Radon measure and x ∈ R , Tan (η, x) is a d-cone with closed basis (see [28], 2.3).
n Proposition 4.1 ( [28] Proposition 1.12) The set of Radon measures on R with the metric above is a complete separable metric space.
Remark 4.4. i) As indicated in [28] (see 1.9(4) and proposition 1.11 and 1.12) the notion of convergence in the metric above (see 4.5) coincides with the notion of weak convergence of Radon measures.
n ii)A d-cone of Radon measures in R has a closed basis if and only if it is a relatively closed n subset of the set of non-zero Radon measures in R . Proposition 4.2 ( [28] Proposition 2.2) Let M be a d-cone of Radon measures. M has a compact basis if and only if for every λ > 1 there is τ > 1 such that Fτr(Ψ) ≤ λFr(Ψ) for every Ψ ∈ M and every r > 0. In this case 0 ∈ spt Φ for all Ψ ∈ M, where
n spt Ψ = {x ∈ R : Ψ(B(x, r)) > 0, ∀r > 0}.
The following theorem is in the same vein as Theorem 2.6 in [28].
Theorem 4.1 ( [17]) Let F and M be d-cones. Assume that F ⊂ M, that F is relatively closed in M with respect to the weak convergence of Radon measures and that M has a compact basis. Furthermore suppose that the following property holds: ∃ > 0 such that ∀ ∈ (0, ) there exists no µ ∈ M satisfying (P) 0 0 dr(µ, F) ≤ ∀r ≥ r0 > 0 and dr0 (µ, F) = . Then for a Radon measure η and x ∈ spt η if (4.6) Tan (η, x) ⊂ M and Tan (η, x) ∩ F 6= ∅ then Tan (η, x) ⊂ F.
Corollary 4.1 ( [17]) Let F and M be d-cones. Assume that F ⊂ M, that F is relatively closed in M with respect to the weak convergence of Radon measures and that M has a compact basis. Furthermore suppose that following property holds:
(P’) ∃ 0 > 0 such that if dr(µ, F) < 0 for all r ≥ r0 > 0 then µ ∈ F. Then for a Radon measure η and x ∈ spt η if (4.7) Tan (η, x) ⊂ M and Tan (η, x) ∩ F 6= ∅ then Tan (η, x) ⊂ F.
9 Note that the condition (P’) stated in Corollary 4.1 is stronger than condition (P).
Proof of Theorem 4.1: We proceed by contradiction; i.e. assume that Tan (η, x) ⊂ M, Tan (η, x) ∩ F 6= ∅ but there exists ν ∈ Tan (η, x)\F. Since F is closed there exists 1 ∈ 1 (0, 2 min{0, 1}) such that d1(η, F) > 21. Moreover there exist si & 0 and ci > 0 such that ciTx,si [η] → ν. Since Tan (η, x) ∩ F= 6 ∅ there also exist δi > 0 and ri & 0 such that δiTx,ri [η] → νe ∈ F. Thus for i large enough
(4.8) d1(Tx,ri [η], F) = d1(δiTx,ri [η], F) < 1, and d1(Tx,si [η], F) > 1 Without loss of generality we may assume that s < r . Let τ ∈ si , 1 be the largest number such i i i ri that τiri = ρi satisfies
(4.9) d1(Tx,ρi [η], F) = 1.
Hence for all α ∈ (τi, 1)
(4.10) d1(Tx,αri [η], F) = dα/τi (Tx,ρi [η], F) < 1.
We claim that τi → 0 as i → ∞. In fact, otherwise there exists a subsequence τik → τ ∈ (0, 1), and δi Tx,ρ [η] = δi Tx,τ r [η] → T0,τ [ν] ∈ F, which implies that d1(Tx,ρ [η], F) → 0 as ik → ∞ k ik k ik ik e ik which contradicts (4.9). Therefore (4.9) and (4.10) yield
(4.11) lim d1(Tx,ρ [η], F) = 1 i→∞ i and for every r > 1,
(4.12) lim sup dr(Tx,ρi [η], F) ≤ 1. i→∞ Note that F (T [η]) = 1 F (η) ∈ (0, ∞) for x ∈ spt η. Moreover a simple calculation shows r x,ρi ρi B(x,rρi) that for i large enough r rρ (4.13) 0 < η B x, i ≤ F (T [η]) ≤ rη(B(x, rρ )) ≤ rη(B(x, r)) < ∞. 2 2 r x,ρi i Since < 1, λ = 2 > 1 and by Proposition 4.2 there is τ > 1 so that F (Ψ) ≤ λF (Ψ) for 1 1+1 τr r every Ψ ∈ M and every r > 0. For r ≥ 1 and i large enough there is Ψ ∈ M so that Fτr(Ψ) = 1 and Tx,ρi [η] Tx,ρi [η] (4.14) Fr , Ψ ≤ Fτr , Ψ ≤ 1. Frτ (Tx,ρi [η]) Frτ (Tx,ρi [η]) Hence
Fr(Tx,ρi [η]) 1 + 1 1 − 1 (4.15) ≥ Fr(Ψ) − 1 ≥ Fτr(Ψ) − 1 = . Fτr(Tx,ρi [η]) 2 2 Thus for p = 1, 2,... (4.15) yields for i large enough
−p F p (T [η]) 1 − (4.16) τ x,ρi ≤ 1 . F1(Tx,ρi [η]) 2
10 Combining (4.13) and (4.16) we conclude that for p = 1, 2, ··· , τ > 1 (as above), and i large enough
T [η](B(0, τ p/2)) 2 p 2 p (4.17) x,ρi ≤ 2 τ −p ≤ 2 . F1(Tx,ρi [η]) 1 − 1 (1 − 1)τ Thus for any s > 0, (4.17) ensures that
T [η](B(0, s)) (4.18) lim sup x,ρi < ∞. i→∞ F1(Tx,ρi [η])
Tx,ρ [η] By the compactness theorem for Radon measures there exists a subsequence i such that ik k F1(Tx,ρ [η]) ik converges to a Radon measure Φ ∈ M (as M has a closed basis), satisfying F1(Φ) = 1. Therefore Fr(Φ) > 0 for r ≥ 1.
Combining iii) in Remark 4.2 with (4.11) and (4.12) we have that
(4.19) d1(Φ, F) = 1 and
(4.20) dr(Φ, F) ≤ 1 for all r > 1.
Since 1 < 0/2 (4.19) and (4.20) contradict condition (P). This concludes the proof of Theorem 4.1.
The next couple of results, from [28] and [24], provide additional information about Tan (Φ, x) for a Radon measure Φ and x ∈ spt Φ. The first result yields conditions that ensure that Tan (Φ, x) has a compact basis. These conditions are satisfied both by a Radon measure whose density exist or by the harmonic measure of an NTA domain. Therefore their cones of tangent measures have compact basis. The second result states that at almost every point tangent measures to tangent measures to Φ are tangent measures to Φ. This result is used to ensure that at almost every point of the support of a Radon measure whose density exist or at almost every point in the set where ω+ and ω− are mutually absolutely continuous there exists a flat tangent measure.
n Theorem 4.2 ( [28] Corollary 2.7) Let Φ be a Radon measure in R , and x ∈ spt Φ. Tan (Φ, x) has a compact basis if and only if
Φ(B(x, 2r)) (4.21) lim sup < ∞. r→0 Φ(B(x, r))
n n Theorem 4.3 ( [24], Theorem 14.16) Let Φ be a Radon measure in R , Φ a.e. a ∈ R , if Ψ ∈ Tan (Φ, a) then
i) Tx,ρ[Ψ] ∈ Tan (Φ, a) for all x ∈ spt Ψ and all ρ > 0 ii) Tan (Ψ, x) ⊂ Tan (Φ, a) for all x ∈ spt Ψ.
11 We finish this section by giving a detailed sketch of the proof of Theorem 3.1 using the results discussed above.
n Let Ω ⊂ R be a 2-sided locally NTA domain. Let Q ∈ ∂Ω and {rj}j≥1 be a sequence of positive numbers such that limj→∞ rj = 0. Consider the domains
± 1 ± ± 1 ± (4.22) Ωj = (Ω − Q) with ∂Ωj = (∂Ω − Q), rj rj the functions
± ± u (rjX + Q) n−2 (4.23) uj (X) = ± rj ω (B(Q, rj)) and the measures
± ± ω (rjE + Q) n (4.24) ωj (E) = ± for E ⊂ R a Borel set. ω (B(Q, rj))
n Note that Lemma 4.8 in [15] ensures that given a compact set K ⊂ R containing Q, for j large enough (depending only on K)
± ± −1 u (A (Q, rj)) n−2 (4.25) CK ≤ ± rj ≤ CK . ω (B(Q, rj))
± Here CK is a constant that only depends on K and A (Q, rj) denote the non-tangential points ± associated to Q at radius rj in Ω (see definition in [15]). The boundary Harnack principle (see Lemma 4.4 in [15]) yields that for N > 1, X ∈ B(0,N) and j large enough depending only on N
± ± ± (4.26) u (rjX + Q) ≤ CN,K u (A (Q, rj)).
Thus combining (4.23) and (4.25) we obtain that
± (4.27) sup sup uj (X) ≤ CN,K < ∞. j≥1 X∈B(0,N)
Furthermore since ω± are locally doubling (see Lemmata 4.8, 4.11 in [15])
± (4.28) sup ωj (B(0,N)) ≤ CN,K < ∞. j≥1
For Q ∈ ∂Ω let h(Q) denote the Radon-Nikodym derivative of ω− with respect to ω+ when it exists, i.e.
dω− ω−(B(Q, r)) (4.29) h(Q) := (Q) = lim dω+ r→0 ω+(B(Q, r))
Define
(4.30) Λ = {Q ∈ ∂Ω : 0 < h(Q) < ∞} ,
12 and
(4.31) Γ = Q ∈ Λ: Q density point of Λ with respect to ω±, Z Z + + o h(Q) = lim ¨B(Q,r)h dω , lim ¨ |h(P ) − h(Q)|dω (P ) = 0 . r→0 r→0 B(Q,r)
Note that ω±(Λ\Γ) = 0.
n Theorem 4.4 Let Ω ⊂ R be a 2-sided locally NTA domain. Using the notation above, we have that there exists a sequence (which we relabel), satisfying as j → ∞
± ± (4.32) Ωj → Ω∞ in the Hausdorff distance sense uniformly on compact sets ± ± (4.33) ∂Ωj → ∂Ω∞ in the Hausdorff distance sense uniformly on compact sets
± + − ± n where Ω∞ are unbounded NTA domains with ∂Ω∞ = ∂Ω∞. Moreover, there exist u∞ ∈ C(R ) such that
± ± (4.34) uj → u∞ uniformly on compact sets,
± ± ∆u∞ = 0 in Ω∞ ± ± (4.35) u∞ = 0 on ∂Ω∞ ± ± u∞ > 0 in Ω∞, and
± ± (4.36) ωj * ω∞ weakly as Radon measures.
± ± ± Here ω∞ are the harmonic measures of Ω∞ with pole at infinity, corresponding to u∞, i.e. ∀ϕ ∈ ∞ n Cc (R ), Z Z ± ± (4.37) u∞∆ϕ = ϕ dω∞. ± ± Ω∞ ∂Ω∞ Furthermore if Q ∈ Γ,
+ − (4.38) ω∞ = ω∞ + − n (4.39) u∞ = u∞ − u∞ is a harmonic polynomial in R .
For the proof of this theorem see §4 in [19] and §3 in [17].
Next we state a lemma about harmonic polynomials which ensures that hypothesis (P’) in Corollary 4.1 is satisfied. For the proof see §4 in [17].
13 n Let F be the set of (n − 1) flat measures in R , i.e. (4.40) F = {cHn−1 V : c ∈ (0, ∞); V ∈ G(n, n − 1)}.
n Here G(n, n − 1) denotes the Grassmanian of n − 1 planes in R . G(n, n − 1) is compact which implies that F has a compact basis, and it is closed under weak convergence in the space of non-zero Radon measures.
n Lemma 4.3 ( [17]) Let h be a harmonic polynomial in R such that h(0) = 0 and {h > 0} and {h < 0} are unbounded NTA domains. Let ω be the corresponding harmonic measure, i.e. ∞ n ∀ϕ ∈ Cc (R ) Z Z Z (4.41) h∆ϕ = h−∆ϕ = ϕdω. {h>0} {h<0} {h=0}
There exists 0 > 0 (depending on the NTA constant of {h > 0} and on n) such that if for some r0 > 0
(4.42) dr(ω, F) < 0 for r ≥ r0, then ω ∈ F.
It is interesting to note that the previous lemma holds for any harmonic polynomial not only for those such that {h > 0} and {h < 0} are unbounded NTA domains. In the general case 0 depends on n, on the degree of the polynomial and on the leading coefficient.
The following lemma is a simple geometric measure theory fact which allows us to give an estimate on the Hausdorff dimension of sets which approach (n − 1)-planes locally (see [30] or [17] for a proof).
n Lemma 4.4 Let Σ ⊂ R be such that ∀Q ∈ Σ d(y, L) (4.43) lim βΣ(Q, r) = 0 where βΣ(Q, r) = inf sup . r→0 L∈G(n,n−1) y∈B(Q,r)∩Σ r
Then
(4.44) dimH Σ ≤ n − 1.
We are now ready to describe the set G which appears in Theorem 3.1 and state some of its properties. The proof of of the following theorem illustrates how to combine all the ingredients discussed above.
Theorem 4.5 Let Ω be a 2-sided locally NTA domain. Let Γ be as in (4.31), and
(4.45) G = Q ∈ Γ : Tan (ω±,Q) ∩ F 6= ∅ .
Then for Q ∈ G, Tan (ω±,Q) ⊂ F. In particular, all blow-ups of ∂Ω at Q ∈ G are (n − 1)-planes, ± and dimH G ≤ n − 1. Furthermore Γ0 = Γ\G satisfies ω (Γ0) = 0.
14 Proof. For Q ∈ Γ the blow-up procedure described in Theorem 4.4 always yields a harmonic polynomial. Let h be a tangent harmonic polynomial of u at Q, and ν the corresponding harmonic measures to h±. By [14] the zero set of h, i.e. ∂{h > 0} decomposes into a disjoint union of the embedded C1 submanifold h−1{0} ∩ {|Dh| > 0}, together with a closed set h−1{0} ∩ |Dh|−1{0} which is countably (n − 2)-rectifiable. Furthermore spt ν = h−1{0}. For Y ∈ h−1{0} ∩ {|Dh| > 0} n and X ∈ R h(rX + Y ) (4.46) hY,r(X) = −→hDh(Y ),Xi r r→0
−1 Dh(Y ) ⊥ uniformly on compact sets. Thus r (∂{h > 0} − Y ) → h |Dh(Y )| i = V as r → 0, in the Hausdorff −(n−1) n−1 −1 distance sense and r TY,r[ν] → |Dh(Y )|H V . Therefore, for Y ∈ h {0}∩{|Dh| > 0} all non-zero tangent measures of ν at Y are flat, i.e. Tan (ν, Y ) ⊂ F. By Theorem 4.3 for ω = ω± a.e. Q ∈ Γ if ν ∈ Tan (ω, Q), then for all Y ∈ spt ν, Tan (ν, Y ) ⊂ Tan (ω, Q). Thus, for ω a.e. Q ∈ Γ, ± F ∩ Tan (ω, Q) 6= ∅, which proves that ω (Γ0) = 0. Our goal is to use Corollary 4.1 combined with Lemma 4.3 to show that for Q ∈ G, Tan (ω, Q) ⊂ F. Let M = F ∪ Tan (ω, Q). Recall that F the set of all (n−1) flat measures is a d-cone with compact basis. Since ω is a doubling Radon measure Theorem 4.2 ensures that for Q ∈ Γ, Tan (ω, Q) is a d-cone with compact basis. Hence M is also a d-cone with compact basis. Moreover F ⊂ M, and F is relatively closed with respect to weak convergence of Radon measures. By Lemma 4.3 there exists 0 > 0 such that if dr(µ, F) < 0 for all r ≥ r0, then µ ∈ F. Corollary 4.1 ensures then that for Q ∈ G, Tan (ω, Q) ⊂ F. Thus all blow ups of ∂Ω at Q converge in the Hausdorff distance sense to an (n − 1)-plane. By Lemma 4.4 we conclude that dimH G ≤ n − 1.
5 Open questions
m Question 1: Classification of n-uniform measures in R . Preiss raised this question in [28]. The case when m = n + 1 was settled by Kowalski and Preiss n+1 [20]. They showed that if ν is an n-uniform measure in R , i.e. there exists C > 1 such that ν(B(x, r)) = Crn for all x ∈ spt ν and r > 0 then either spt ν is an n-plane or if n ≥ 3 modulo n+1 2 2 2 2 rotation and translation spt ν = {x ∈ R : x4 = x1 + x2 + x3}.
Question 2: Structure of measures with locally H¨olderdensities in co-dimension 1.
Is it true that for each α > 0 there exists β = β(α) > 0 with the following property: if µ is a n+1 α positive Radon measure supported on Σ ⊂ R whose n-density ratio is locally C , then locally 1,β n+1 2 2 2 2 Σ is a C image of an n-plane or the cone {x ∈ R : x4 = x1 + x2 + x3}? An answer to this question would provide insight into some of the issues that arise in Question 1. In particular it would indicate how the moments of a measure yield precise geometric information about its support in the non-flat case.
Question 3: Size and structure of the singular set of measures with locally H¨older densities.
15 m α If µ is a positive Radon measure supported on Σ ⊂ R whose n-density ratio is locally C , Theorem 1,β m 2.1 ensures that for n ≥ 3, Σ is a C submanifold of dimension n in R away from a closed set S with Hn(S) = 0. What is the Hausdorff dimension of S? Is S rectifiable?
Question 4: Regularity of the set of mutual absolute continuity for ω±.
n − + In [9], Bishop asked whether in the case of R , n ≥ 3, if ω , ω are mutually absolutely continuous on a set E ⊂ ∂Ω and ω±(E) > 0, then ω± are mutually absolutely continuous with respect to Hn−1 on E (modulo a set of ω± measure zero). The work described above proves that if Ω is a 2-sided locally NTA domain then dimH(E) = n − 1. It answers Bishop’s question under the additional assumption that Hn−1 ∂Ω is a Radon measure. In this case E is (n − 1)-rectifiable. Is it truth that, in general, if Ω is a 2-sided locally NTA domain and ω−, ω+ are mutually absolutely continuous on a set E ⊂ ∂Ω and ω±(E) > 0 then E is (n − 1)-rectifiable?
Question 5: Find a larger class of domains for which Theorem 3.1 holds.
One of the major questions in this area is whether the 2-sided locally NTA assumption can be removed from the hypothesis of Theorem 3.1. In particular it would be very interesting to know to what extend the decomposition theorem holds for John domains (see [2]).
Question 6: Applications of Theorem 4.1
Theorem 4.1 is a very general tool, can it be applied to other situations? For example could it be adapted to the question studied to Ambrosio, Kleiner & LeDonne [1] to prove if E is a set of locally finite perimeter in a Carnot group then for almost every point (with respect to the perimeter measure) all tangents of E are vertical half-spaces?
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