Structure theorems for Radon measures

Transcription

1 Structure theorems for Radon measures Matthew Badger Department of Mathematics University of Connecticut Analysis on Metric Spaces University of Pittsburgh March 10 11, 2017 Research Partially Supported by NSF DMS ,

2 Decomposition of Measures Geometric Measure Theory: Understand a measure on a space through its interaction with canonical sets in the space Let (, M, µ) be a measure space Let N M be a family of measurable sets µ is carried by N if there exist countably many sets Γ N such that µ ( \ Γ ) = 0 µ is singular to N if µ(γ) = 0 for every Γ N Exercise (Decomposition Lemma) If µ is σ-finite, then µ can be written uniquely as µ N + µ N where µ N is carried by N and µ N is singular to N Proof of the Decomposition Theorem is abstract nonsense Identification Problem: Find measure-theoretic and/or geometric characterizations or constructions of µ N and µ N?

3 PSA: Don t Think About Support Three Measures Let > 0 be weights with =1 = 1 Let { : 1}, {l : 1}, { : 1} be a dense set of points, unit line segments, unit squares in the plane µ 0 = =1 δ µ 1 = =1 1 l µ 2 = =1 2 µ 0, µ 1, µ 2 are probability measures on R 2 µ is smallest closed set carrying µ; µ 0 = µ 1 = µ 2 = R 2 µ is carried by -dimensional sets (points, lines, squares) The support of a measure is a rough approximation that hides the underlying structure of a measure

4 Example: Atomic Measures vs Atomless Measures A Radon measure is a locally finite, Borel regular outer measure Let µ be a Radon measure on R Then we can write µ = µ 0 + µ 0, where µ 0 is carried by singletons (ie µ 0 is atomic) µ 0 is singular to singletons (ie µ 0 is atomless) How can you identify µ 0 and µ 0? µ 0 = µ { R : 0 µ( (, )) > 0} µ 0 = µ { R : 0 µ( (, )) = 0} µ denotes the restriction of µ to : (µ )( ) = µ( )

5 1-Rectifiable and Purely 1-Unrectifiable Measures A singleton { } has the following properties: { } is connected, compact, H 0 ({ }) < A candidate Γ for a 1-dimensional atom might satisfy: Γ is connected, compact, H 1 (Γ) < Theorem A set Γ R is connected, compact, and H 1 (Γ) < if and only if there exists a Lipschitz map : [0, 1] R such that Γ = ([0, 1]) A connected, compact set Γ R with H 1 (Γ) < is called a rectifiable curve or Lipschitz curve Decomposition: Let µ be a Radon measure on R Then µ = µ 1 + µ 1, where µ 1 is carried by rectifiable curves (µ 1 is 1-rectifiable) µ 1 is singular to rectifiable curves (µ 1 is purely 1-unrectifiable)

6 Identification Problem for µ 1 and µ 1 If R and H 1 ( ) <, then 1 H 1 ( (, )) H 1 -ae Solution for Hausdorff measures: Theorem (Besicovitch 1928,1938) Let µ = H 1, where R 2 and 0 < H 1 ( ) < Then µ 1 = µ { } R 2 µ( (, )) : = = µ { R 2 : has a tangent line at } µ 1 = µ = µ { R 2 µ( (, )) : 3 } { R 2 : does not have a tangent line at }

7 Identification Problem for µ 1 and µ 1 Let µ be a Radon measure on R The following are equivalent: µ H 1 (ie H 1 ( ) = 0 µ( ) = 0) µ( (, )) 0 < at µ-ae R 2 Solution for absolutely continuous measures: Theorem (Morse and Randolph 1944) Let µ be a Radon measure on R 2 with µ H 1 Then µ 1 = µ { } R 2 µ( (, )) : 0 < < 0 2 = µ { R 2 : has a µ-approximate tangent line at } µ 1 = µ = µ { } R 2 µ( (, )) : 1 µ( (, )) { R 2 : does not have a µ-approximate tangent line at }

8 Examples I For 1, let be a subset of a rectifiable curve Γ R µ = =1 H1 is a finite 1-rectifiable measure, µ H 1 Let be the Cantor middle-third set and let µ = H {0}, where = (2)/ (3) is the Hausdorff dimension of µ is 1-rectifiable measure with µ( (, )) for all µ and 0 < 1; and µ H 1 µ( (, )) and 0 2 = µ-ae

9 Examples II Lebesgue measure on R is purely 1-unrectifiable for all 2 (This is obvious!) Let R 2 be the 4 corners Cantor set, = =0 is Ahlfors regular: H 1 ( (, )) for all, 0 < 1 Every rectifiable curve Γ = ([0, 1]) R 2 intersects in a set of zero H 1 measure H 1 is a purely 1-unrectifiable measure on R 2

10 Examples III Let λ R 2 be the generalized 4 corners Cantor set, where 0 < λ 1/2 is the scaling factor (λ = 1/5) has Hausdorff dimension = (4)/ (1/λ) H ( λ (, )) for all λ and 0 < 1 When λ = 1/2, = 2 and H is just Lebesgue measure restricted to the unit square If 1/4 λ 1/2, then H λ is purely 1-unrectifiable If 0 < λ < 1/4, then H λ is 1-rectifiable see eg Martin and Mattila (1988)

11 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

12 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

13 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

14 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

15 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

16 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

17 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

18 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable

19 Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable µ( (, )) 0 = µ-ae 2 1 ( ) 1 µ( (, )) 2 < µ-ae 0 (see B-Schul 2016) µ(γ) = 0 whenever Γ = ([0, 1]) and : [0, 1] R 2 is bi-lipschitz Nevertheless there exist Lipschitz maps ( : [0, 1] R 2 such ) that µ R 2 \ ([0, 1]) = 0 =1

20 Identification Problem for µ 1 and µ 1 A Radon measure µ on R is doubling if µ( (, 2 )) µ( (, )) for all µ and > 0 Solution for doubling measures with connected supports: Theorem (Azzam-Mourgoglou 2016) Let µ be a doubling measure on R such that µ is connected Then { } µ 1 = µ R µ( (, )) : > µ 1 = µ { } R µ( (, )) : = Azzam and Mourgoglou s main result applies to doubling measures with connected supports on arbitrary metric spaces

21 Non-homogeneous 2 Jones β numbers Let µ be a Radon measure on R For every cube, define β 2 (µ, 3 ) = line β 2 (µ, 3, ) [0, 1], where β 2 (µ, 3, ) 2 = 3 ( (, ) 3 ) 2 µ( ) µ(3 ) Non-homogeneous refers to the normalization 1/µ(3 )

22 Identification Problem for µ 1 and µ 1 A Radon measure µ on R is pointwise doubling if 0 µ( (, 2 )) µ( (, )) < at µ-ae R Solution for pointwise doubling measures: Theorem (B and Schul 2017) Let µ be a pointwise doubling measure on R Then { µ 1 = µ R : } 2 (µ, ) < { µ 1 = µ R : } 2 (µ, ) = Here 2 (µ, ) is a non-homogeneous 2 Jones square function: 2 (µ, ) = dyadic 1 β 2 (µ, 3 ) 2 µ( ) χ ( )

23 Interpretation Let µ be a pointwise doubling measure on R Then µ is 1-rectifiable iff 2(µ, ) = dyadic 1 β 2(µ, 3 ) 2 µ( ) χ ( ) < µ-ae Extreme Behaviors Suppose 0 < 0 µ( (, )) 2 0 µ( (, )) 2 Then µ is 1-rectifiable if and only if β 2(µ, 3 ) 2 χ ( ) < µ-ae < µ-ae dyadic 1 When µ = H 1, R compact, this was proved by Pajot (1997) Suppose µ has is badly linearly approximable in the sense β 2(µ, 3 ) > 0 µ-ae Then µ is 1-rectifiable if and only if dyadic 1 µ( ) χ ( ) < µ-ae

24 Examples V Theorem (Martikainen and Orponen, arxiv 2016) For all ε > 0, there exists a probability measure µ on R 2 with µ [0, 1] 2 such that 1 pointwise non-doubling: 0 µ( (, 2 )) µ( (, )) = µ-ae µ( (, )) 2 vanishing lower density: 0 = 0 µ-ae 2 3 uniformly bounded square function: 2 (µ, ) = dyadic 1 2 β 2 (µ, 3 ) µ( ) χ ( ) ε for all µ Interpretation: vanishing lower density implies that µ is purely 1-unrectifiable cannot hope to characterize rectifiability of a Radon measure using only non-homogeneous square function 2 (µ, )

25 Anisotropic 2 Jones β numbers Given dyadic cube in R, ( ) denotes a subdivision of 1600 into overlapping dyadic cubes of same / previous (larger) generation as For every Radon measure µ on R and every dyadic cube, we define β 2 (µ, ) 2 = line ( ) β 2(µ, 3, ) 2 3 [0, 1], where β 2(µ, 3, ) 2 3 = 3 ( ) 2 ( ) (, ) µ(3 ) µ( ) 1, 3 3 µ(3 )

26 Identification Problem for µ 1 and µ 1 Solution for Radon measures: Theorem (B and Schul 2017) Let µ be a Radon measure on R Then { } µ 1 = µ R µ( (, )) : > 0 and 2 (µ, ) < 0 2 µ 1 = µ { } R µ( (, )) : = 0 or 2 (µ, ) = 0 2 Here 2 (µ, ) is an anisotropic 2 Jones square function: 2 (µ, ) = dyadic 1 β 2(µ, ) 2 µ( ) χ ( )

27 Main Takeaway Corollary A Radon measure µ on R is 1-rectifiable if and only if at µ-ae, µ( (, )) 0 > 0 and 2 2 (µ, ) = dyadic 1 β 2 (µ, ) 2 µ( ) χ ( ) < A Radon measure µ on R is purely 1-rectifiable if and only if at µ-ae, µ( (, )) 0 = 0 or 2 2 (µ, ) = dyadic 1 β 2 (µ, ) 2 µ( ) χ ( ) = Takeaway: To understand geometric properties of non-doubling measures (such as rectifiability) using multiscale analysis, it may be convenient or necessary to incorporate anisotropic normalizations

28 Proof Ingredient: Drawing Rectifiable Curves Theorem (B and Schul 2017) Let 2, let > 1, let 0 R, and let 0 > 0 Let ( ) =0 be a sequence of nonempty finite subsets of ( 0, 0 ) such that 1 distinct points, are uniformly separated: 2 0 ; 2 for all, there exists such that +1 < 2 0 ; and, 3 for all ( 1), there exists 1 1 such that 1 < 2 0 Suppose that for all 1 and for all we are given a straight line l, in R and a number α, 0 such that (, l, ) α, 2 0 (1) ( 1 ) (, ) and α 2, 2 0 < (2) =1 Then the sets converge in the Hausdorff metric to a compact set ( 0, 0 ) and there exists a compact, connected set Γ ( 0, 0 ) such that Γ and H 1 (Γ), 0 + α 2, 2 0 (3) =1 This is a flexible criterion for drawing a rectifiable curve through the leaves of a tree; extends P Jones Traveling Salesman construction (Inventiones 1990), which required +1 Our write-up separates relatively simple description of the curve from the intricate length estimates

29 Sample of Higher Dimensional Results I A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R That is, a Radon measure is -rectifiable provided it is carried by Lipschitz images of -cubes Theorem (Preiss 1987) Assume 0 µ( (, )) < µ-ae (or equivalently, µ H ) Then µ is -rectifiable if and only if 0 < 0 µ( (, )) < Preiss introduced tangent measures and studied global geometry of -uniform measures in R New examples of 3-uniform measures in R have been announced by Nimer on the arxiv in August 2016!!!!!

30 Sample of Higher Dimensional Results II A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R Theorem (Azzam-Tolsa + Tolsa 2015) µ( (, )) Assume 0 < 0 < µ-ae Then µ is -rectifiable if and only if the homogeneous 2 Jones function 2 (µ, ) = 1 0 β ( ) 2 (µ, (, )) 2 µ( (, )) < µ-ae One ingredient in Azzam-Tolsa s proof is David and Toro s version of the Reifenberg algorithm for sets with holes Edelen-Naber-Valtorta announced an extension of Azzam-Tolsa on the arxiv in December 2016

31 Understanding higher-dim rectifiability is hard A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R This definition is due to Federer (1947) When 2, it is not clear if Lipschitz images of [0, 1] is the correct family N of -dimensional sets The catastrophe: If : [0, 1] R is Lipschitz, then Γ = ([0, 1] ) is connected, compact, and H (Γ) < But the converse is false when 2! Open Problem: Find additional metric, geometric, and/or topological conditions which ensure that a compact, connected set R with H 2 ( ) < is contained in the image of a Lipschitz map : [0, 1] 2 R

32 Related developments Azzam and Schul, The Analyst s Traveling Salesman Theorem for sets of dimension larger than one, arxiv 2016 Novel definition of higher-dimensional beta numbers of sets using the Choquet integral Results include a characterization of subsets of Reifenberg vanishing bi-lipschitz surfaces that is similar to Jones TST K Rajala, Uniformization of two-dimensional metric surfaces, Inventiones 2016 Gives an intrinsic characterization of metric spaces with locally finite H 2 measure that are quasiconformally equivalent to R 2 Does not immediately extend to higher dimensions

33 Current Project (w/ Vellis): Non-integral Dimensions For each [1, ], let N denote all (1/ )-Hölder curves in R, ie all images Γ of (1/ )-Hölder continuous maps : [0, 1] R Decomposition: Every Radon measure µ on R can be uniquely written as µ = µ N + µ N, where µ N is carried by (1/ )-Hölder curves µ N Notes is singular to (1/ )-Hölder curves Every measure µ on R is carried by (1/ )-Hölder curves (space-filling curves) A measure µ is carried by 1-Hölder curves iff µ is 1-rectifiable If µ is -rectifiable, then µ is carried by (1/ )-Hölder curves Martín and Mattila (1988,1993,2000) studied this concept for measures µ of the form µ = H, where 0 < H ( ) <

34 Measures with extreme lower densities Theorem (B-Vellis, in preparation) Let µ be a Radon measure on R and let (1, ) Then the measure } µ { := µ R µ( (, )) : = is singular to (1/ )-Hölder curves, ie µ (Γ) = 0 whenever Γ is a (1/ )-Hölder 0 curve; and, the measure 1 } µ { := µ R µ( (, 2 )) : < and µ( (, )) 0 µ( (, )) < 0 is carried by H null sets of (1/ )-Hölder curves, ie there exist (1/ )-Hölder curves and Borel sets Γ with H ( ) = 0 such that µ (R \ =1 ) = 0 1 The condition 0 µ( (, )) < implies µ( (, )) 0 = The theorem is also true when = 1 by B-Schul (2015, 2016)

35 Further results Corollary (B-Vellis, in preparation) Let µ be a Radon measure on R, let [1, ), and let [0, ) Then the measure µ + := µ { } R µ( (, )) µ( (, )) : 0 < < 0 0 is carried by H null sets of (1/ )-Hölder curves Theorem (B-Vellis, in preparation) Let µ be a Radon measure on R and let [0, 1) Then the measure µ + is carried by H 1 null sets of bi-lipschitz curves, ie there exist bi-lipschitz curves Γ and Borel sets Γ with H 1 ( ) = 0 st µ +(R \ =1 ) = 0 Martìn and Mattila (1988): If 0 < H ( ) < for some [0, 1) and 0 H ( (, )) > 0 at H -ae, then H is 1-rectifiable

36 Thank you!