Hausdorff dimensin distortion by Sobolev maps in foliated spaces

Transcription

1 Hausdorff dimensin distortion by Sobolev maps in foliated spaces ZB, R. Monti, J. T. Tyson, K. Wildrick 1. Mai 2013 Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 1/17

2 v Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 2/17

3 v v Euclidean results: -, R. Monti, J. T. Tyson, Frequency of Sobolev and quasiconformal dimension distortion, J. Math. Pure. Appl Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 3/17

4 v v Euclidean results: -, R. Monti, J. T. Tyson, Frequency of Sobolev and quasiconformal dimension distortion, J. Math. Pure. Appl v Metric spaces: -, J. T. Tyson, K. Wildrick, Dimension distortion by Sobolev mappings in foliated metric spaces, Preprint, 2013 Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 3/17

5 v v Euclidean results: -, R. Monti, J. T. Tyson, Frequency of Sobolev and quasiconformal dimension distortion, J. Math. Pure. Appl v Metric spaces: -, J. T. Tyson, K. Wildrick, Dimension distortion by Sobolev mappings in foliated metric spaces, Preprint, 2013 v Heisenberg groups: -, J. T. Tyson, K. Wildrick, Frequency of Sobolev dimension distortion of horizontal subgroups of Heisenberg groups, Preprint 2013 Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 3/17

6 v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Euclidean results Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 4/17

7 Morrey-Sobolev estimate v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Proposition 0.1. Let f W 1,p (Ω,Y), p>n, and g f denote the minimal upper gradient for f. Then for all cubes Q compactly contained in Ω, wehave ( ) 1/p diam f(q) C(n, p)(diam Q) 1 n/p g p f. (0.1) Q Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 5/17

8 Morrey-Sobolev estimate v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Proposition 0.2. Let f W 1,p (Ω,Y), p>n, and g f denote the minimal upper gradient for f. Then for all cubes Q compactly contained in Ω, wehave ( ) 1/p diam f(q) C(n, p)(diam Q) 1 n/p g p f. (0.1) By the Morrey Sobolev embedding theorem, each supercritical mapping f W 1,p (Ω,Y), p>n, has a representative which is locally (1 n/p)-hölder continuous. In particular, if E Ω, dim E = t then dim f(e) tp p n. Q Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 5/17

9 Kaufman s theorem v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Theorem 0.3 (R. Kaufman 2000). E Ω; H t (E) <, 0 <t<n. f W 1,p (Ω,Y) for some p>n. Then f(e) has zero H pt/(p n+t) measure. This statement is sharp. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 6/17

10 Kaufman s theorem v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Theorem 0.4 (R. Kaufman 2000). E Ω; H t (E) <, 0 <t<n. f W 1,p (Ω,Y) for some p>n. Then f(e) has zero H pt/(p n+t) measure. This statement is sharp. In particular if dim E = t then dim f(e) tp p n + t. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 6/17

11 Kaufman s theorem v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Theorem 0.5 (R. Kaufman 2000). E Ω; H t (E) <, 0 <t<n. f W 1,p (Ω,Y) for some p>n. Then f(e) has zero H pt/(p n+t) measure. This statement is sharp. In particular if dim E = t then If V G(n, m). Then dim f(e) dim f(v a Ω) tp p n + t. pm p n + m. (0.2) Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 6/17

12 Main Result v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Let R n = V V and V a = a + V. Given a Sobolev map f : R n Y, how frequently can the intermediate values m α pm p n+m be exceeded by dim f(v a)? Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 7/17

13 Main Result v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Let R n = V V and V a = a + V. Given a Sobolev map f : R n Y, how frequently can the intermediate values m α pm p n+m be exceeded by dim f(v a)? Theorem 0.7 (-.R. Monti, J. Tyson). Let β = β(p, α) :=(n m) ( 1 m α ) p. f(v a Ω) has zero H α measure for H β -almost every a V. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 7/17

14 Main Result v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Let R n = V V and V a = a + V. Given a Sobolev map f : R n Y, how frequently can the intermediate values m α pm p n+m be exceeded by dim f(v a)? Theorem 0.8 (-.R. Monti, J. Tyson). Let β = β(p, α) :=(n m) ( 1 m α ) p. f(v a Ω) has zero H α measure for H β -almost every a V. In particular: dim{a V :dimf(v a ) α} β Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 7/17

15 Main Result v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H Let R n = V V and V a = a + V. Given a Sobolev map f : R n Y, how frequently can the intermediate values m α pm p n+m be exceeded by dim f(v a)? Theorem 0.9 (-.R. Monti, J. Tyson). Let β = β(p, α) :=(n m) ( 1 m α ) p. f(v a Ω) has zero H α measure for H β -almost every a V. In particular: dim{a V :dimf(v a ) α} β Note: α m β n m and α pm p n+m β 0 Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 7/17

16 Sharpness v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H pm p n+m Theorem 0.10 (-.R. Monti, J. Tyson). Let α satisfy m<α< and β = β(p, α) as above. There exists a E R n m s.th. H β (E) > 0 and for any integer N>α, there exists a map f W 1,p (R n, R N ) with the property that dim f({a} R m ) α, for H β -almost every a E. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 8/17

17 Sharpness v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H pm p n+m Theorem 0.12 (-.R. Monti, J. Tyson). Let α satisfy m<α< and β = β(p, α) as above. There exists a E R n m s.th. H β (E) > 0 and for any integer N>α, there exists a map f W 1,p (R n, R N ) with the property that dim f({a} R m ) α, for H β -almost every a E. Theorem 0.13 (S. Hencl, P. Honzik 2013). Let m<α<p n and β(p, α) :=(n m) ( 1 m α ) p. and f W 1,p (R n, R k ) be p quasicontinuous. Then dim{a V :dimf(v a ) α} β. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 8/17

18 Sharpness v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H pm p n+m Theorem 0.14 (-.R. Monti, J. Tyson). Let α satisfy m<α< and β = β(p, α) as above. There exists a E R n m s.th. H β (E) > 0 and for any integer N>α, there exists a map f W 1,p (R n, R N ) with the property that dim f({a} R m ) α, for H β -almost every a E. Theorem 0.15 (S. Hencl, P. Honzik 2013). Let m<α<p n and β(p, α) :=(n m) ( 1 m α ) p. and f W 1,p (R n, R k ) be p quasicontinuous. Then dim{a V :dimf(v a ) α} β. Note: α p β n p The above theorem is also sharp. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 8/17

19 Sharpness v Morrey-Sobolev estimate v Kaufaman s theorem v Main Result v Sharpness v Sobolev mappings between metric spaces v Prevalence of bad functions v Idea of Proof v Regular foliations v Frequency of distortion for metric foliations v Heisenberg foliation v Two types of foliations v Heisenberg Theorem v Sharpness in H pm p n+m Theorem 0.16 (-.R. Monti, J. Tyson). Let α satisfy m<α< and β = β(p, α) as above. There exists a E R n m s.th. H β (E) > 0 and for any integer N>α, there exists a map f W 1,p (R n, R N ) with the property that dim f({a} R m ) α, for H β -almost every a E. Theorem 0.17 (S. Hencl, P. Honzik 2013). Let m<α<p n and β(p, α) :=(n m) ( 1 m α ) p. and f W 1,p (R n, R k ) be p quasicontinuous. Then dim{a V :dimf(v a ) α} β. Note: α p β n p The above theorem is also sharp. Example: [P. Hajlasz, J. Tyson] Then there exists a continuous map f W 1,m (R n,l 2 ) s.th. dim f({a} [0, 1] m )= for all a [0, 1] n m. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 8/17

20 Metric space results Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 9/17

21 Metric case Theorem 0.18 (-J. T., K. W. ). X: proper, loc. Q-homogeneous, local Q-Poincaré inequality. If f : X Y is a continuous mapping that has an upper gradient in L p loc (X), p>q, then for any subset E X. dim Y f(e) p dim X E p Q +dim X E (0.3) Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 10/17

22 Metric case Theorem 0.20 (-J. T., K. W. ). X: proper, loc. Q-homogeneous, local Q-Poincaré inequality. If f : X Y is a continuous mapping that has an upper gradient in L p loc (X), p>q, then for any subset E X. dim Y f(e) p dim X E p Q +dim X E (0.3) Theorem 0.21 (-J. T., K. W. ). Let (X, d, µ), Ahlfors Q-regular. For 0 s Q and p>q, set α = ps p Q+s. Let E be a compact subset of X such that H s (E) > 0. Then for all N>α, there exists a continuous f : X R N such that f has an upper gradient in L p (X) and dim f(e) α. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 10/17

23 Prevalence Theorem 0.22 (-J. T., K. W. ). The set of functions S α := {f N 1,p (X; R N ):dimf(e) α} is a prevalent set in the Newtonian Sobolev space N 1,p (X; R N ). Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 11/17

24 Prevalence Theorem 0.23 (-J. T., K. W. ). The set of functions S α := {f N 1,p (X; R N ):dimf(e) α} is a prevalent set in the Newtonian Sobolev space N 1,p (X; R N ). Prevalence is an important notion in dynamics. Was introduced and developed by J. Yorke, V. Kaloshin, B. Hunt, W. Ott, T. Sauer Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 11/17

25 Prevalence Theorem 0.24 (-J. T., K. W. ). The set of functions S α := {f N 1,p (X; R N ):dimf(e) α} is a prevalent set in the Newtonian Sobolev space N 1,p (X; R N ). Prevalence is an important notion in dynamics. Was introduced and developed by J. Yorke, V. Kaloshin, B. Hunt, W. Ott, T. Sauer There exists a compactly supported Borel probability measure λ on N 1,p (X; R N ) such that λ(n 1,p (X; R N ) \ (S α + g)) = 0, g N 1,p (X; R N ). Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 11/17

26 Idea of Proof Idea: let µ be a Frostman measure on E and find first h N 1,p (X; R N ) with I α (h µ) < i.e. 1 dµ(x)dµ(y) <. h(x) h(y) α E E Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 12/17

27 Idea of Proof Idea: let µ be a Frostman measure on E and find first h N 1,p (X; R N ) with I α (h µ) < i.e. 1 dµ(x)dµ(y) <. h(x) h(y) α E E Let M = {M M(N N,R), M 1} and ν to be the normalized N 2 -dimensional Lebesgue measure on M. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 12/17

28 Idea of Proof Idea: let µ be a Frostman measure on E and find first h N 1,p (X; R N ) with I α (h µ) < i.e. 1 dµ(x)dµ(y) <. h(x) h(y) α E E Let M = {M M(N N,R), M 1} and ν to be the normalized N 2 -dimensional Lebesgue measure on M. Consider Φ:M N 1,p (X; R N ), Φ(M) :=M h, and define: λ := Φ ν. Let g N 1,p (X; R N ) and f M = M h + g. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 12/17

29 Idea of Proof Idea: let µ be a Frostman measure on E and find first h N 1,p (X; R N ) with I α (h µ) < i.e. 1 dµ(x)dµ(y) <. h(x) h(y) α E E Let M = {M M(N N,R), M 1} and ν to be the normalized N 2 -dimensional Lebesgue measure on M. Consider Φ:M N 1,p (X; R N ), Φ(M) :=M h, and define: λ := Φ ν. Let g N 1,p (X; R N ) and f M = M h + g. Have to prove: 1 dµ(x)dµ(y)dν(m) <. f M (x) f M (y) α M E E Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 12/17

30 Regular foliations Definition[G. David, S. Semmes] A surjection π : X W is called s-regular if for any compact K X v π K is Lipschitz v π 1 (B) K can be covered by at most Cr s balls in X of radius Cr. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 13/17

31 Regular foliations Definition[G. David, S. Semmes] A surjection π : X W is called s-regular if for any compact K X v π K is Lipschitz v π 1 (B) K can be covered by at most Cr s balls in X of radius Cr. Note that H s X (π 1 (a) K) C, in particular, for the leaves π 1 (a) it follows dim π 1 (a) s, and this inequality can be sometimes strict. The triple (X, W, π) will be called an s-foliation of X. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 13/17

32 Dimension distortion for metric foliations Theorem Let (X, d X,µ) be a proper, loc. Q-homogeneous, with local Q-Poincaré inequality, and is equipped with an s-foliation (X, W, π). Iff : X Y is a continuous mapping with upper gradient in L p loc (X), p>q, then dim{a W :dim(f(π 1 (a))) α} (Q s) (1 s α ) for s<α ps p Q+s. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 14/17

33 Dimension distortion for metric foliations Theorem Let (X, d X,µ) be a proper, loc. Q-homogeneous, with local Q-Poincaré inequality, and is equipped with an s-foliation (X, W, π). Iff : X Y is a continuous mapping with upper gradient in L p loc (X), p>q, then dim{a W :dim(f(π 1 (a))) α} (Q s) (1 s α ) for s<α ps p Q+s. If s =dimπ 1 (a) then this is a good theorem to apply. Examples: Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 14/17

34 Dimension distortion for metric foliations Theorem Let (X, d X,µ) be a proper, loc. Q-homogeneous, with local Q-Poincaré inequality, and is equipped with an s-foliation (X, W, π). Iff : X Y is a continuous mapping with upper gradient in L p loc (X), p>q, then dim{a W :dim(f(π 1 (a))) α} (Q s) (1 s α ) for s<α ps p Q+s. If s =dimπ 1 (a) then this is a good theorem to apply. Examples: 1. X = R n,w = R m n with the orthogonal projection π : R n W defines a regular s = m-foliation. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 14/17

35 Dimension distortion for metric foliations Theorem Let (X, d X,µ) be a proper, loc. Q-homogeneous, with local Q-Poincaré inequality, and is equipped with an s-foliation (X, W, π). Iff : X Y is a continuous mapping with upper gradient in L p loc (X), p>q, then dim{a W :dim(f(π 1 (a))) α} (Q s) (1 s α ) for s<α ps p Q+s. If s =dimπ 1 (a) then this is a good theorem to apply. Examples: 1. X = R n,w = R m n with the orthogonal projection π : R n W defines a regular s = m-foliation. 2. X = H n = V V, V-horizontal subgroup of H n. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 14/17

36 Heisenberg group results Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 15/17

37 Heisenberg foliations H n R 2n+1 p =(x 1,y 1,...,x n,y n,t)=(z, t), ω(z, z )= n i=1 (x iy i x i y i), p p =(z + z,t+ t +2ω(z, z )) d H n(p, p )= p 1 p H n, p H n =( z 4 + t 2 ) 1/4. δ r p =(rz, r 2 t), Q=2n +2. Ahlfors, Poincare OK Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 16/17

38 Heisenberg foliations H n R 2n+1 p =(x 1,y 1,...,x n,y n,t)=(z, t), ω(z, z )= n i=1 (x iy i x i y i), p p =(z + z,t+ t +2ω(z, z )) d H n(p, p )= p 1 p H n, p H n =( z 4 + t 2 ) 1/4. δ r p =(rz, r 2 t), Q=2n +2. Ahlfors, Poincare OK Isotropic subspace of V R 2n : ω(u, v) =0 u, v V horizontal homogenous subgroups of dim V = m n. Vertical complementary subspace: V = V R is a normal subgroup generating the semidirect product. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 16/17

39 Heisenberg foliations H n R 2n+1 p =(x 1,y 1,...,x n,y n,t)=(z, t), ω(z, z )= n i=1 (x iy i x i y i), p p =(z + z,t+ t +2ω(z, z )) d H n(p, p )= p 1 p H n, p H n =( z 4 + t 2 ) 1/4. δ r p =(rz, r 2 t), Q=2n +2. Ahlfors, Poincare OK Isotropic subspace of V R 2n : ω(u, v) =0 u, v V horizontal homogenous subgroups of dim V = m n. Vertical complementary subspace: V = V R is a normal subgroup generating the semidirect product. H n = V V, p = p V p V defines two types of folitations: π V : H n V,π 1 V (a) =V a and π V : H n V,π 1 (a) =a V V Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 16/17

40 Two types of foliations Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 17/17

41 Two types of foliations Good news: (H n, V,π V ) is a regular (Q m)- foliation. Moreover: dim(π 1 V (a)) = dim(v a) =Q m Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 17/17

42 Two types of foliations Good news: (H n, V,π V ) is a regular (Q m)- foliation. Moreover: Bad news: is NOT a regular m- foliation. dim(π 1 V (a)) = dim(v a) =Q m (H n, V,π V ) Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 17/17

43 Two types of foliations Good news: (H n, V,π V ) is a regular (Q m)- foliation. Moreover: Bad news: dim(π 1 V (a)) = dim(v a) =Q m (H n, V,π V ) is NOT a regular m- foliation. Improvement: (H n, (V,d E ),π V ) is a regular m +1- foliation, but s = m +1>m=dim(π 1 V (a)), which for m α Qm p Q + m and β =(Q s) p(1 s α ) does not imply sharp results at endpoints. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 17/17

44 Heisenberg Foliation Distortion Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 18/17

45 Heisenberg Foliation Distortion Redefine β as: (Q 1 m) p ( ) 2 1 m α β(p, m, α) = (Q m) p ( ) 1 m α α α [ m, pm p 2 [ pm p 2, ], pm p (Q m) ]. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 18/17

46 Heisenberg Foliation Distortion Redefine β as: (Q 1 m) p ( ) 2 1 m α β(p, m, α) = (Q m) p ( ) 1 m α α α [ m, pm p 2 [ pm p 2, ], pm p (Q m) ]. Theorem Let f W 1,p loc (Hn ; Y ), p>q. Given a horizontal subgroup V of H n of dimension 1 m n, and it holds that m α pm p (Q m), dim R {a V :dimf(a V) α} β(p, m, α). Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 18/17

47 Sharpness in H Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 19/17

48 Sharpness in H Theorem Let V x denote the horizontal subgroup defined by the x-axis in H, and let p>4. For each 1 <α< p p 2 < p p 3 there is a compact set E α V x and a continuous mapping f W 1,p (H; R 2 ) such that 0 < H 2 p (1 α) 1 R (E 3 α ) < and dim f(a V) α for every a E α. Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 19/17

49 Sharpness in H Theorem Let V x denote the horizontal subgroup defined by the x-axis in H, and let p>4. For each 1 <α< p p 2 < p p 3 there is a compact set E α V x and a continuous mapping f W 1,p (H; R 2 ) such that 0 < H 2 p (1 α) 1 R (E 3 α ) < and dim f(a V) α for every a E α. Note: 2 p(1 1 α ) <β=2 p 2 (1 p(1 1 α ), however the example is asymptotically sharp 2 p(1 1 ) 2, if α 1. α Zoltán M. Balogh Non-smooth Workshop// May 2013 IMPAN UCLA 19/17