Singular points of Hölder asymptotically optimally doubling measures

Abstract

We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered in [DKT] for codimension 1 under a crucial assumption of flatness, and later in [PTT] in higher codimension. However, the studies leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here answer the question (in an almost classical sense) for codimension-1 Hölder doubling measures in R 4 .

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

“…Let U = U(n, m) denote the collection of supports of m-uniform measures in R n . The support of an masymptotically optimally doubling measure in R n is locally well approximated by U (see [21,Theorem 3.8]). …”

“…Uniform measures in codimension n − m 2 have resisted a complete classification, but partial descriptions of them have been given by Preiss [30], Kirchheim and Preiss [16], and Tolsa [34]. The structure of m-asymptotically optimally doubling measures µ whose doubling characteristic decays locally uniformly at a Hölder rate (that is R(µ, K , r ) C K r α for all 0 < r r K ) was studied by David et al [7], Preiss et al [29], and Lewis [21]. In [7,29], it was proved that the support of µ is an m-dimensional C 1,β submanifold of R n away from a closed set of zero m-dimensional Hausdorff measure.…”

“…Furthermore, Preiss, Tolsa, and Toro prove that when m = n − 1 the support of µ is an m-dimensional C 1,β submanifold away from a closed set of Hausdorff dimension at most n − 4. Recently, in [21], Lewis has proved that when n = 4 and m = 3 the support of µ admits local C 1,β parameterizations at every point either by open subsets of a three-dimensional plane or by open subsets of the light cone C.…”

“…Our first application is a new decomposition theorem for the support of an (n−1)-asymptotically optimally doubling measure in R n (defined above in Example 1.6). Note that, in contrast with previous work [7,21,29], we do not impose any restriction on the rate of convergence of R(µ, K , r ) → 0 as r → 0. Even more, we obtain a qualitatively better dimension bound on the singular set (see Remark 9.2).…”

“…Now assume that µ is an (n − 1)-asymptotically optimally doubling measure in R n . Then spt µ is locally bilaterally well approximated by U(n, n − 1) by [21,Theorem 3.8]. But G(n, n − 1) points are detectable in U(n, n − 1) by [21, Lemma 2.5].…”