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 2 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 2 R µ(B(x; r)) (2.1) 0 < θn(µ, x) = lim < 1 (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 nE) = 0. 2 2 Theorem ( [5], [6], [7]) Let µ be a Radon measure in R and assume that for µ-a.e x 2 R µ(B(x; r)) (2.2) 0 < θ(µ, x) = lim < 1 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 2 R (2.3) 0 < θn(µ, x) < 1; 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 2 R , then m for µ-a.e a 2 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 2 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 2 (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 !nr n for x 2 K and 0 < r < 1. Here !n 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 ! be the harmonic measure associated to Ω. Then Carleson ( [11]) and Jones & Wolff ( [16]) showed that (3.1) H − dim ! ≤ 1 If Ω is simple connected Makarov ( [22]) proved Oskendal's conjecture in dimension 2, i.e. (3.2) H − dim ! = 1 Recall that the Hausdorff dimension of ! (denote by H − dim !) is defined by (3.3) H − dim ! = inf fk : there exists E ⊂ @Ω with Hk(E) = 0 and n !(E \ K) = !(@Ω \ K) for all compact sets K ⊂ R g 2 Let Ω ⊂ R be a Jordan domain (i.e a simply connected domain bounded by a Jordan curve), and let ! 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, ! H1 ! • !(N) = 0 •H 1(S) = 0.