Rectifiability of measures Some results m = 1 m 1 Raanan Schul - PowerPoint PPT Presentation

Rectifiability of measures R. Schul Length and curvature 1-Rectfiability Background Rectifiability of measures Some results m = 1 m ≥ 1 Raanan Schul (Stony Brook) Other notions end August, 2019 1 / 48

Rectifiability of Length measures R. Schul Length and curvature 1-Rectfiability Let f : [ 0 , 1 ] → R be Lipschitz function. Let G ⊂ R 2 be the Background graph of f . Then the length of G is Some results m = 1 � � � � � � 1 � 1 2 2 � � � � m ≥ 1 df df H 1 ( G ) = � � � � 1 + dx ≈ 1 + c dx Other notions � � � � dx dx 0 0 end Key player: � � � df 2 � � � dx 2 2 / 48

Rectifiability of Curvature measures ◮ Let J be a dyadic interval, and J = J L ∪ J R be its R. Schul decomposition into its left and right parts. Length and curvature ◮ Let � � 1-Rectfiability H J ( x ) = | J | − 1 χ J L ( x ) − χ J R ( x ) 2 Background Some results Then { H J } J ∈ ∆ is an orthonormal basis for L 2 ( R ) . m = 1 ( ∆ = all dyadic intervals) m ≥ 1 ◮ Extend f as a constant right of 1 and left of 0. Write Other notions � end df dx = a J H J ( x ) . � � � � df 2 � � | a J | 2 . 2 = � dx J ∈ ∆ ◮ What does | a J | mean? If J=[0,1] � � � �� � 1 � � 1 a [ 0 , 1 ] = � df dx , H [ 0 , 1 ] � = f − f ( 0 ) − f ( 1 ) − f 2 2 = “ change in slope between the two halves ” 3 / 48

Rectifiability of Length and Curvature measures R. Schul Length and curvature 1-Rectfiability 2 = � | a J | 2 = L 2 quantity which measures ◮ � df dx � 2 Background Some results curvature. m = 1 ◮ Length ‘ = ’ m ≥ 1 diam + L 2 quantity which measures curvature. Other notions ◮ The above is a quantitative connection between end length and curvature. It comes into play when working on qualitative questions. (if you fall asleep now, then at least remember that) 4 / 48

Rectifiability of Curvature - II measures R. Schul ◮ Let Length and curvature b ( x , y , z ) := | f ( x ) − f ( y ) | + | f ( y ) − f ( z ) |−| f ( x ) − f ( z )) | 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end ◮ If no edge is much larger than the other two, then h 2 b ( x , y , z ) diam 4 ∼ 1 ∼ diam 3 R 2 where R = R ( x , y , z ) is radius of circle through f ( x ) , f ( y ) , f ( z ) (Menger curvature := 1 R ). ◮ Note: we don’t need f anymore to make these definitions. 5 / 48

Rectifiability of Curvature - III measures R. Schul ◮ Non-paramtric version of b : Length and curvature b min ( P 1 , P 2 , P 3 ) := � � 1-Rectfiability | P σ ( 1 ) − P σ ( 2 ) | + | P σ ( 2 ) − P σ ( 3 ) |−| P σ ( 1 ) − P σ ( 3 ) | min Background σ ∈ S 3 Some results m = 1 m ≥ 1 Other notions end ◮ If no edge is much larger than the other two, then diam ( A , B , C ) 3 ∼ h 2 b min ( A , B , C ) diam 4 ∼ 1 min R 2 6 / 48

Rectifiability of Length and Curvature - II measures R. Schul Suppose G is a graph of an L-Lipschitz function f . Then Length and � curvature H 1 ( G ) h 2 ( J ) / | J | ∼ diam ( G ) + c L 1-Rectfiability J ��� Background b min ∼ diam ( G ) + c diam 3 Some results m = 1 m ≥ 1 Other notions end ◮ � : over dyadic intervals J . For J = [ a , b ] , � � h ( J ) := sup f ( z ) , line dist z ∈ [ a , b ] where for each J we choose a line minimizing h ( J ) . ◮ ��� : over all triples in G , ( d length ) 3 . True in much more generality... (many contributors) 7 / 48

Rectifiability of 1-Rectfiability measures R. Schul Length and Slightly non-standard way of saying it curvature ◮ Let µ be a measure on R n . We say that µ is 1-Rectfiability Background 1-rectifiable if there is a countable collection of Some results Lipschitz curves m = 1 f i : [ 0 , 1 ] → R n m ≥ 1 Other notions end such that � ∞ � � R n \ µ f i [ 0 , 1 ] = 0 . i = 1 ◮ If E ⊂ R n and µ = H 1 | E then E is called a “1-rectifiable set”. ◮ m -rectifiability uses [ 0 , 1 ] m as domain... 8 / 48

Rectifiability of Questions measures R. Schul Length and curvature 1-Rectfiability Let µ be a measure on R n . We say that µ is 1-rectifiable if there is a countable collection of ◮ Lipschitz curves Background f i : [ 0 , 1 ] → R n Some results such that ∞ m = 1 R n \ � � � f i [ 0 , 1 ] = 0 . µ m ≥ 1 i = 1 Other notions ◮ When is µ 1-rectifiable? end ◮ When is one curve enough to capture all of µ ? ◮ When does one curve capture a significant part of µ ? The case µ = H 1 | E (or µ ≪ H 1 | E ) is very well studied, and the case and µ ⊥ H 1 is not. 9 / 48

Rectifiability of Example 1 measures R. Schul Length and curvature 1-Rectfiability Background ◮ Let µ = Lebesgue measure on [ 0 , 1 ] 2 ⊂ R 2 . Some results m = 1 ◮ For any f : [ 0 , 1 ] → R 2 Lipschitz, m ≥ 1 � � Other notions µ f [ 0 , 1 ] = 0 . end ◮ µ is NOT 1-rectifiable. 10 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end 11 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 12 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 13 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 14 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 15 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 16 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 17 / 48

Rectifiability of Example 2 measures R. Schul ◮ Let µ be a more eccentric version of Example 1: Length and curvature 1-Rectfiability Background Some results m = 1 m ≥ 1 Other notions end pic by M. Badger 18 / 48

Rectifiability of Example 2 - continued measures R. Schul Length and curvature (If ǫ = 1 3 we recover 2-dim. Leb. meas.) 1-Rectfiability If ǫ > 0 is small enough, then Background ◮ µ ⊥ H 1 | E for any E ⊂ R 2 with H 1 ( E ) < ∞ . Some results ◮ µ is doubling on R 2 . µ ( L ) = 0 for any line L . m = 1 m ≥ 1 ◮ µ ( G ) = 0 for any G , an isometric copy of a Lipschitz Other notions graph . end ◮ µ is 1-rectifiable (Theorem [Garnett-Killip-S. 2009]) A measure µ is “doubling on R n ” if there is a C > 0 such that for any x ∈ R n and r > 0 we have µ ( B ( x , 2 r )) < C µ ( B ( x , r )) . 19 / 48

Rectifiability of Rectifiable Measures measures R. Schul Length and curvature { m -rectifiable measures µ on R n } 1-Rectfiability Background � Some results m = 1 { m -rectifiable measures µ on R n such that µ ≪ H m } m ≥ 1 Other notions end � { m -rectifiable measures µ on R n of the form µ = H m | E } ◮ How do you tell if a ‘generic’ measure is 1-rectifiable? ◮ What about 2-rectifiable? m -rectifiable? 20 / 48

µ ≪ H 1 | E Rectifiability of measures R. Schul Lower and upper (Hausdorff) m -density: Length and µ ( B ( x , r )) µ ( B ( x , r )) curvature m ( µ, x ) = lim sup D m ( µ, x ) = lim inf D 1-Rectfiability c m r m c m r m r ↓ 0 r ↓ 0 Background Some results Write D m ( µ, x ) , the m -density of µ at x , if m = 1 m ( µ, x ) . D m ( µ, x ) = D m ≥ 1 Other notions Theorem 1 (Mattila 1975) end Suppose that E ⊂ R n is Borel and µ = H m | E is locally finite. Then µ is m-rectifiable if and only if D m ( µ, x ) = 1 µ -a.e. Theorem 2 (Preiss 1987) Suppose that µ is a locally finite Borel measure on R n . Then µ is m-rectifiable and µ ≪ H m if and only if 0 < D m ( µ, x ) < ∞ µ -a.e. 21 / 48

µ ≪ H 1 | E Rectifiability of measures ◮ For s > 0, a ∈ R n and P an m -plane in R n (through R. Schul 0) define the two sided cone Length and curvature X ( a , P , s ) = { x ∈ R n : d ( x − a , P ) < s | x − a |} . 1-Rectfiability ◮ We say that P above is an approximate tangent of E Background m ( µ, a ) > 0, and for all s ≥ 0 Some results at a if for µ = H m | E , D m = 1 µ ( B ( a , r ) \ X ( a , P , s )) m ≥ 1 lim = 0 r m Other notions r ↓ 0 end Theorem 3 (Marstrand-Mattila) Suppose that E ⊂ R n is Borel and µ = H m | E is locally finite. TFAE ◮ µ is m-rectifiable ◮ For µ a.e. a ∈ E there is a unique approx. tangent plane for E at a. ◮ For µ a.e. a ∈ E there is a some approx. tangent plane for E at a. 22 / 48