Distortion of dimension by Sobolev and quasiconformal mappings J. Tyson 17 June 2014 I. Introduction and overview, quasiconformal maps of R n and their effect on Hausdorff dimension II. Global quasiconformal dimension in R n III. Conformal dimension of metric spaces IV. Sobolev dimension distortion in R n and in metric spaces V. QC and Sobolev dimension distortion in the sub-Riemannian Heisenberg group 1
In this lecture we discuss Pansu’s conformal dimension . Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is qs C dim X = inf { dim Y : Y a metric space, Y ∼ X } . Recall: f : X → Y is η -quasisymmetric (qs) if | f ( x ) − f ( a ) | ≤ η ( t ) | f ( x ) − f ( b ) | whenever x , a , b ∈ X satisfy | x − a | ≤ t | x − b | and t > 0. 2
In this lecture we discuss Pansu’s conformal dimension . Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is qs C dim X = inf { dim Y : Y a metric space, Y ∼ X } . Recall: f : X → Y is η -quasisymmetric (qs) if | f ( x ) − f ( a ) | ≤ η ( t ) | f ( x ) − f ( b ) | whenever x , a , b ∈ X satisfy | x − a | ≤ t | x − b | and t > 0. Goal for today: estimates from below for C dim X . In particular, when is C dim X = dim X ? 3
In this lecture we discuss Pansu’s conformal dimension . Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is qs C dim X = inf { dim Y : Y a metric space, Y ∼ X } . Recall: f : X → Y is η -quasisymmetric (qs) if | f ( x ) − f ( a ) | ≤ η ( t ) | f ( x ) − f ( b ) | whenever x , a , b ∈ X satisfy | x − a | ≤ t | x − b | and t > 0. Goal for today: estimates from below for C dim X . In particular, when is C dim X = dim X ? From last time: C dim X = 0 if X ⊂ R n is a self-similar Cantor set or has dim < 1. 4
Lower bounds for conformal dimension The general philosophy is: lower bounds on C dim X arise from “well distributed” families of curves inside X . The model case is the foliation of R n by lines parallel to a fixed direction. T V V a V 5
Proposition A (after M. Bourdon, P. Pansu) Let ( X , d , µ ) be a doubling metric measure space and let 1 < p < ∞ . Let Γ be a family of curves in X equipped with a probability measure ν s.t. (i) the support of Γ is bounded, (ii) the elements of Γ have diameters uniformly bounded away from zero, and (iii) ∃ C > 0 s.t. ν { γ ∈ Γ : γ ∩ B ( x , r ) � = ∅} ≤ C µ ( B ( x , r )) 1 / p ∀ B ( x , r ) . Then C dim X ≥ p ′ , where p ′ = p p − 1 . 6
Proposition A (after M. Bourdon, P. Pansu) Let ( X , d , µ ) be a doubling metric measure space and let 1 < p < ∞ . Let Γ be a family of curves in X equipped with a probability measure ν s.t. (i) the support of Γ is bounded, (ii) the elements of Γ have diameters uniformly bounded away from zero, and (iii) ∃ C > 0 s.t. ν { γ ∈ Γ : γ ∩ B ( x , r ) � = ∅} ≤ C µ ( B ( x , r )) 1 / p ∀ B ( x , r ) . Then C dim X ≥ p ′ , where p ′ = p p − 1 . Remark If µ is assumed s -regular, then (iii) can be replaced by (iii’) there exists C > 0 s.t. ν { γ ∈ Γ : γ ∩ B ( x , r ) � = ∅} ≤ Cr s / p ∀ B ( x , r ). s − 1 then the conclusion is C dim X = dim X = s . In this case s s If p = p = s − 1. If we can foliate a piece of X (of positive measure) by a family of curves which is “uniformly 1-codimensional”, then X is minimal for conformal dimension. 7
Examples 1. X = R n ( s = n ), Γ foliation by parallel lines V a = V + a , ν Lebesgue measure n − 1 . Conclusion: C dim R n = dim R n = n . on V ⊥ , p = n 8
Examples 1. X = R n ( s = n ), Γ foliation by parallel lines V a = V + a , ν Lebesgue measure n − 1 . Conclusion: C dim R n = dim R n = n . on V ⊥ , p = n 2. X = H n Heisenberg group with left invariant Carnot–Carath´ eodory metric d cc ( s = 2 n + 2), Γ foliation by integral curves of a horiz vector field V , Γ can be s equipped with a measure ν s.t. the previous condition holds with p = s − 1 . 9
Examples 1. X = R n ( s = n ), Γ foliation by parallel lines V a = V + a , ν Lebesgue measure n − 1 . Conclusion: C dim R n = dim R n = n . on V ⊥ , p = n 2. X = H n Heisenberg group with left invariant Carnot–Carath´ eodory metric d cc ( s = 2 n + 2), Γ foliation by integral curves of a horiz vector field V , Γ can be s equipped with a measure ν s.t. the previous condition holds with p = s − 1 . V ∈ span { X 1 , Y 1 , . . . , X n , Y n } γ ∈ Γ satisfies γ ′ ( s ) = V ( γ ( s )) for all s � Γ length( γ ∩ A ) d ν ( γ ) for all A ⊂ H n ν satisfies | A | = Conclusion: C dim( H n , d cc ) = dim( H n , d cc ) = s . (due to Pansu) 10
Examples nski carpet SC ⊂ R 2 3. X = Sierpi´ s = log 8 log 3 ≈ 1 . 893 . . . 11
Examples nski carpet SC ⊂ R 2 3. X = Sierpi´ s = log 8 log 3 ≈ 1 . 893 . . . Γ family of horiz lines (parameterized by 1 3 Cantor set C along y -axis) C , p = log 8 / log 3 ν = H log 2 / log 3 log 2 / log 3 = 3. 12
Examples nski carpet SC ⊂ R 2 3. X = Sierpi´ s = log 8 log 3 ≈ 1 . 893 . . . Γ family of horiz lines (parameterized by 1 3 Cantor set C along y -axis) log 2 / log 3 = 3. Conclusion: C dim SC ≥ p ′ = 3 C , p = log 8 / log 3 ν = H log 2 / log 3 2 > 1. 13
The estimate can be improved. Consider X 1 = C × [0 , 1] ⊂ SC . The measure t = log 2 µ 1 = H t × L 1 X 1 ≃ H t +1 X 1 log 3 is Ahlfors regular on X 1 , and ν ( { γ ∈ Γ : γ ∩ B ( x , r ) � = ∅} ) ≤ C µ 1 ( B ( x , r )) 1 / p with p = 1+ t t . 14
The estimate can be improved. Consider X 1 = C × [0 , 1] ⊂ SC . The measure t = log 2 µ 1 = H t × L 1 X 1 ≃ H t +1 X 1 log 3 is Ahlfors regular on X 1 , and ν ( { γ ∈ Γ : γ ∩ B ( x , r ) � = ∅} ) ≤ C µ 1 ( B ( x , r )) 1 / p t . Hence C dim X 1 ≥ p ′ = 1 + t and with p = 1+ t C dim SC ≥ C dim X 1 = 1 + log 2 log 3 > 3 2 . 15
Lower bounds for conformal dimension can also be obtained using moduli of curve families. Definition Let Γ be a family of curves in a metric measure space ( X , d , µ ) and let p ≥ 1. The p-modulus of Γ is � ρ p d µ Mod p (Γ) = inf X where the infimum is taken over all admissible Borel functions ρ : X → [0 , ∞ ], � i.e., γ ρ ds ≥ 1 for all locally rectifiable γ ∈ Γ. 16
Lower bounds for conformal dimension can also be obtained using moduli of curve families. Definition Let Γ be a family of curves in a metric measure space ( X , d , µ ) and let p ≥ 1. The p-modulus of Γ is � ρ p d µ Mod p (Γ) = inf X where the infimum is taken over all admissible Borel functions ρ : X → [0 , ∞ ], � i.e., γ ρ ds ≥ 1 for all locally rectifiable γ ∈ Γ. Example Let ( Z , d , ν ) be any compact mms and X = Z × [0 , h ] with the usual product metric and measure µ = ν ⊗ L 1 . Let Γ = { γ z : z ∈ Z } , γ z : [0 , h ] → X , γ z ( s ) = ( z , s ). Mod p (Γ) = ν ( Y ) h p − 1 . “ ≤ ”: ρ ( z , s ) = 1 h is admissible “ ≥ ”: apply Fubini’s theorem and H¨ older’s inequality 17
Proposition B Let ( X , d , µ ) be a doubling metric measure space satisfying the upper mass bound µ ( B ( x , r )) ≤ r s for all x ∈ X and r > 0 . Assume that there exists a curve family Γ in X s.t. Mod p (Γ) > 0 for some 1 < p ≤ s. Then C dim X ≥ s. 18
Proposition B Let ( X , d , µ ) be a doubling metric measure space satisfying the upper mass bound µ ( B ( x , r )) ≤ r s for all x ∈ X and r > 0 . Assume that there exists a curve family Γ in X s.t. Mod p (Γ) > 0 for some 1 < p ≤ s. Then C dim X ≥ s. Corollary Assume that X is Ahlfors s-regular and supports a curve family Γ in X s.t. Mod s (Γ) > 0 . Then C dim X = dim X = s. 19
Minimal sets for global qc dimension For any t ∈ (0 , n − 1) choose a compact t -regular set Z ⊂ R n − 1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0 , 1] ⊂ R n equipped with product metric and measure µ = H t × L 1 ≃ H s , s = t + 1. 20
Minimal sets for global qc dimension For any t ∈ (0 , n − 1) choose a compact t -regular set Z ⊂ R n − 1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0 , 1] ⊂ R n equipped with product metric and measure µ = H t × L 1 ≃ H s , s = t + 1. Then X is s -regular and supports a curve family Γ with Mod p (Γ) > 0 for any p . Alternatively, the criteria of Proposition A hold with ν = H t s Z and p = s − 1 . 21
Minimal sets for global qc dimension For any t ∈ (0 , n − 1) choose a compact t -regular set Z ⊂ R n − 1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0 , 1] ⊂ R n equipped with product metric and measure µ = H t × L 1 ≃ H s , s = t + 1. Then X is s -regular and supports a curve family Γ with Mod p (Γ) > 0 for any p . Alternatively, the criteria of Proposition A hold with ν = H t s Z and p = s − 1 . Hence C dim X = dim X = s . In particular, GQC dim R n X = s . 22
Recommend
More recommend
