Distortion of Dimension by Sobolev and Quasiconformal Mappings

Distortion of dimension by Sobolev and quasiconformal mappings

J. Tyson

18 June 2014

I. Introduction and overview, quasiconformal maps of Rn and their eﬀect on Hausdorﬀ dimension II. Global quasiconformal dimension in Rn III. Conformal dimension of metric spaces IV. Sobolev dimension distortion in Rn and in metric spaces V. QC and Sobolev dimension distortion in the sub-Riemannian Heisenberg group

1 In this ﬁnal lecture we will survey known results on distortion of dimension by Sobolev and quasiconformal mappings of the sub-Riemannian Heisenberg group Hn.

n We equip H with the Carnot-Carath´eodory metric dcc and recall that the n Hausdorﬀ dimension of (H , dcc ) is Q = 2n + 2.

A homeomorphism f :Ω → Ω0 between domains in Hn is (metrically) K-quasiconformal if

sup{dcc (f (p), f (q)) : dcc (p, q) = r} lim sup 0 0 ≤ K ∀ p ∈ Ω. r→0 inf{dcc (f (p), f (q )) : dcc (p, q ) = r}

2 A major diﬃculty, however, lies in the lack of eﬀective methods for constructing Heisenberg quasiconformal maps. For instance,

• naive versions of the radial stretch map x 7→ |x|a−1x fail to be quasiconformal (cf. recent work of Balogh–F¨assler–Platis), • PL techniques are not obviously available, • no theory of Heisenberg quasiuniform convexity exists to date.

Many results of the Euclidean theory of quasiconformal maps carry over to the Heisenberg group (and even to more general metric measure spaces).

3 Many results of the Euclidean theory of quasiconformal maps carry over to the Heisenberg group (and even to more general metric measure spaces).

A major diﬃculty, however, lies in the lack of eﬀective methods for constructing Heisenberg quasiconformal maps. For instance,

4 1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn → Hn qc s.t. dim E = s and dim f (E) = t.

2. Gehring–Väisälätheorem: for each 0 < s < Q and K ≥ 1 ∃ 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc.

Distortion of dimension by quasiconformal maps in Hn

Z. M. Balogh, “Hausdorﬀ dimension distribution of quasiconformal mappings on the Heisenberg group”, J. d’Analyse Math. (2001)

5 2. Gehring–Väisälätheorem: for each 0 < s < Q and K ≥ 1 ∃ 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc.

Distortion of dimension by quasiconformal maps in Hn

Z. M. Balogh, “Hausdorﬀ dimension distribution of quasiconformal mappings on the Heisenberg group”, J. d’Analyse Math. (2001)

1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn → Hn qc s.t. dim E = s and dim f (E) = t.

6 Distortion of dimension by quasiconformal maps in Hn

Z. M. Balogh, “Hausdorﬀ dimension distribution of quasiconformal mappings on the Heisenberg group”, J. d’Analyse Math. (2001)

1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn → Hn qc s.t. dim E = s and dim f (E) = t.

2. Gehring–Väisälätheorem: for each 0 < s < Q and K ≥ 1 ∃ 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc. 7 Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector field X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon–Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r∈C ∂BH (o, r) be a Cantor set’s worth of Korányi n spheres ∂BH (o, r) = {p ∈ H : dH (o, p) = r}. Apply a version of the Bourdon–Pansu criterion for such foliations.

4. (T, 2008) Kovalev’s theorem holds for the Assouad dimension on Hn.

n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA,H E = 0.

Question: Kovalev’s theorem in Hn for Hausdorﬀ dimension?

n n 3. For each s ∈ [1, Q] ∃ E ⊂ H compact s.t. GQCdimH E = dim E = s.

8 4. (T, 2008) Kovalev’s theorem holds for the Assouad dimension on Hn.

n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA,H E = 0.

Question: Kovalev’s theorem in Hn for Hausdorﬀ dimension?

n n 3. For each s ∈ [1, Q] ∃ E ⊂ H compact s.t. GQCdimH E = dim E = s.

Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector ﬁeld X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon–Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r∈C ∂BH (o, r) be a Cantor set’s worth of Kor´anyi n spheres ∂BH (o, r) = {p ∈ H : dH (o, p) = r}. Apply a version of the Bourdon–Pansu criterion for such foliations.

9 Question: Kovalev’s theorem in Hn for Hausdorﬀ dimension?

n n 3. For each s ∈ [1, Q] ∃ E ⊂ H compact s.t. GQCdimH E = dim E = s.

4. (T, 2008) Kovalev’s theorem holds for the Assouad dimension on Hn.

n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA,H E = 0.

10 n n 3. For each s ∈ [1, Q] ∃ E ⊂ H compact s.t. GQCdimH E = dim E = s.

4. (T, 2008) Kovalev’s theorem holds for the Assouad dimension on Hn.

n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA,H E = 0.

Question: Kovalev’s theorem in Hn for Hausdorff dimension? 11 Examples: ϕ = 1, ϕ = xk , ϕ = yk , k = 1,..., n generate left translation flows P ϕ = 2t V = 2tT + j (xj Xj + yj Yj ) generates dilation flow p0 7→ δes (p0)

Quasiconformal ﬂows

n Recall the standard left invariant vector ﬁelds X1, Y1,..., Xn, Yn generating HH .

∂ ∂ ∂ ∂ Xj (p) = + 2yj , Yj (p) = − 2xj ∂xj ∂t ∂yj ∂t

Kor´anyiand Reimann (1995) give conditions on a vector ﬁeld V which guarantee that the time s map p(0) 7→ p(s) associated to the ODEp ˙ = V (p) is K(s)-qc. Basic structure is

n 1 X V = ϕ T + (X ϕ Y − Y ϕ X ) for ϕ ∈ C ∞( n) 4 j j j j H j=1

with additional restrictions on the second horiz derivatives of ϕ.

12 Quasiconformal ﬂows

n Recall the standard left invariant vector ﬁelds X1, Y1,..., Xn, Yn generating HH .

∂ ∂ ∂ ∂ Xj (p) = + 2yj , Yj (p) = − 2xj ∂xj ∂t ∂yj ∂t

Kor´anyiand Reimann (1995) give conditions on a vector ﬁeld V which guarantee that the time s map p(0) 7→ p(s) associated to the ODEp ˙ = V (p) is K(s)-qc. Basic structure is

n 1 X V = ϕ T + (X ϕ Y − Y ϕ X ) for ϕ ∈ C ∞( n) 4 j j j j H j=1

with additional restrictions on the second horiz derivatives of ϕ.

Examples: ϕ = 1, ϕ = xk , ϕ = yk , k = 1,..., n generate left translation ﬂows P ϕ = 2t V = 2tT + j (xj Xj + yj Yj ) generates dilation ﬂow p0 7→ δes (p0)

13 f

Fix a ball B(p, R) and a family of disjoint balls Bj = B(pj , rj ) ⊂ B(p, r).

Choose potentials ϕj generating contractive similarities with fixed points at the P pj , and define ϕ = j χj ϕj for suitable smooth cutoff functions χj .

For suitable s, the time s map F associated to ϕ is qc, ﬁxes ∂B(p, r), and contracts each Bj . Moreover, F is conformal outside of B and inside each Bj . Iterate.

As the ﬂow method only works for smooth maps, use compactness of the space of normalized K-qc maps to pass to the limit.

14 In particular, it follows that f is a.e. diﬀ’ble in horizontal directions, and that f is n n n a generalized contact map, dhfp = (Dfp)∗|Hp H maps HpH to Hf (p)H a.e.

Intuition: The CC ball at p of radius 0 < r 1 is comparable to the box [−r, r]2n × [−r 2, r 2], sheared so that the [−r, r]2n factor lies along the subspace n HpH . Metric quasiconformality ⇒ at a point of diﬀerentiability, the size r and size r 2 directions cannot be interchanged by f .

Theorem (Pansu) Every qc map f of domains in Hn is diﬀerentiable almost everywhere in the following sense: for a.e. p ∈ Ω the limit

−1 δ1/r ◦ `f (p) ◦ f ◦ `p ◦ δr

n converges locally uniformly to a grading preserving automorphism Dfp of H .

15 Theorem (Pansu) Every qc map f of domains in Hn is diﬀerentiable almost everywhere in the following sense: for a.e. p ∈ Ω the limit

−1 δ1/r ◦ `f (p) ◦ f ◦ `p ◦ δr

n converges locally uniformly to a grading preserving automorphism Dfp of H .

In particular, it follows that f is a.e. diﬀ’ble in horizontal directions, and that f is n n n a generalized contact map, dhfp = (Dfp)∗|Hp H maps HpH to Hf (p)H a.e.

16 Deﬁnition Given a metric space Y , ﬁx an isometric embedding κ of Y into a Banach space V . Then

W 1,p(Ω : Y ) := {f ∈ W 1,p(Ω : V ): f (p) ∈ κ(Y ) a.e.}

For instance when Y = Hn we can choose V = `∞ (Fr´echetembedding).

1,Q n Each f qc lies in the metric space-valued Sobolev space Wloc (Ω : H ), in fact, n n 1,p n ∃ p(H , K) > Q s.t. f K-qc in H lies in Wloc for all p < p(H , K).

We’ll consider the exceptional sets problem for W 1,p maps, p > Q, for generic elements in foliations of Hn by left or right cosets of horizontal subgroups.

For f = (f1,..., f2n+1) quasiconformal, we have

Q ||dhf || ≤ K det Df a.e.

1,Q In particular, each component fj lies in the local horiz Sobolev space Wloc (Ω).

17 We’ll consider the exceptional sets problem for W 1,p maps, p > Q, for generic elements in foliations of Hn by left or right cosets of horizontal subgroups.

For f = (f1,..., f2n+1) quasiconformal, we have

Q ||dhf || ≤ K det Df a.e.

1,Q In particular, each component fj lies in the local horiz Sobolev space Wloc (Ω). Deﬁnition Given a metric space Y , ﬁx an isometric embedding κ of Y into a Banach space V . Then

W 1,p(Ω : Y ) := {f ∈ W 1,p(Ω : V ): f (p) ∈ κ(Y ) a.e.}

For instance when Y = Hn we can choose V = `∞ (Fr´echetembedding).

1,Q n Each f qc lies in the metric space-valued Sobolev space Wloc (Ω : H ), in fact, n n 1,p n ∃ p(H , K) > Q s.t. f K-qc in H lies in Wloc for all p < p(H , K).

18 For f = (f1,..., f2n+1) quasiconformal, we have

Q ||dhf || ≤ K det Df a.e.

1,Q In particular, each component fj lies in the local horiz Sobolev space Wloc (Ω). Deﬁnition Given a metric space Y , ﬁx an isometric embedding κ of Y into a Banach space V . Then

W 1,p(Ω : Y ) := {f ∈ W 1,p(Ω : V ): f (p) ∈ κ(Y ) a.e.}

For instance when Y = Hn we can choose V = `∞ (Fr´echetembedding).

1,Q n Each f qc lies in the metric space-valued Sobolev space Wloc (Ω : H ), in fact, n n 1,p n ∃ p(H , K) > Q s.t. f K-qc in H lies in Wloc for all p < p(H , K).

We’ll consider the exceptional sets problem for W 1,p maps, p > Q, for generic elements in foliations of Hn by left or right cosets of horizontal subgroups.

19 A smooth submanifold Σ ⊂ Hn of (topological) dimension p ≤ n has CC Hausdorﬀ dimension p if and only if Σ is horizontal. Otherwise, it has CC dimension p + 1. On the other hand, every submanifold Σ of topological dimension > n satisﬁes dim Σ = p + 1.

For example, a (smooth) curve γ in H1 is horizontal if and only if dim γ = 1, while any surface has CC dimension 3.

20 Left coset foliation {a ∗ = (pL )−1(a): a ∈ }: V W W ﬁbers are all 1-dimensional, (W, dcc ) is (2n + 1)-dimensional, pL :( n, d ) → ( , d ) is not Lipschitz. W H cc W cc Right coset foliation { ∗ a = (pR )−1(a): a ∈ }: V W W ﬁbers are generically 2-dimensional, W can be equipped with a quotient metric d w.r.t. which is (2n)-dimensional, pR :( n, d ) → ( , d) is Lipschitz. W W H cc W Lemma: pL :( n, d ) → ( , d ) and pR :( n, d ) → ( , d) are both locally W H cc W E W H cc W David–Semmes 2-regular.

Recall: π : X → W is locally s-regular if π is locally Lipschitz, onto, and for each −1 −s compact K ∃ C > 0 s.t. for every Br ⊂ W , π (B) ∩ K is covered by ≤ Cr balls of radius Cr.

Let V be a left invariant horiz vector ﬁeld, V = {exp(sV ): s ∈ R} the corresponding one-parameter horizontal subgroup.

21 Right coset foliation { ∗ a = (pR )−1(a): a ∈ }: V W W ﬁbers are generically 2-dimensional, W can be equipped with a quotient metric d w.r.t. which is (2n)-dimensional, pR :( n, d ) → ( , d) is Lipschitz. W W H cc W Lemma: pL :( n, d ) → ( , d ) and pR :( n, d ) → ( , d) are both locally W H cc W E W H cc W David–Semmes 2-regular.

Recall: π : X → W is locally s-regular if π is locally Lipschitz, onto, and for each −1 −s compact K ∃ C > 0 s.t. for every Br ⊂ W , π (B) ∩ K is covered by ≤ Cr balls of radius Cr.

Let V be a left invariant horiz vector ﬁeld, V = {exp(sV ): s ∈ R} the corresponding one-parameter horizontal subgroup.

22 Lemma: pL :( n, d ) → ( , d ) and pR :( n, d ) → ( , d) are both locally W H cc W E W H cc W David–Semmes 2-regular.

Recall: π : X → W is locally s-regular if π is locally Lipschitz, onto, and for each −1 −s compact K ∃ C > 0 s.t. for every Br ⊂ W , π (B) ∩ K is covered by ≤ Cr balls of radius Cr.

Let V be a left invariant horiz vector ﬁeld, V = {exp(sV ): s ∈ R} the corresponding one-parameter horizontal subgroup.

The Euclidean complement W of V is a normal subgroup of Hn, and Hn admits two semidirect decompositions: Hn = W n V or Hn = V o W. These decompositions induce projection maps pL : n → and pR : n → . W H W W H W Left coset foliation {a ∗ = (pL )−1(a): a ∈ }: V W W ﬁbers are all 1-dimensional, (W, dcc ) is (2n + 1)-dimensional, pL :( n, d ) → ( , d ) is not Lipschitz. W H cc W cc Right coset foliation { ∗ a = (pR )−1(a): a ∈ }: V W W ﬁbers are generically 2-dimensional, W can be equipped with a quotient metric d w.r.t. which is (2n)-dimensional, pR :( n, d ) → ( , d) is Lipschitz. W W H cc W

23 Let V be a left invariant horiz vector ﬁeld, V = {exp(sV ): s ∈ R} the corresponding one-parameter horizontal subgroup.

Recall: π : X → W is locally s-regular if π is locally Lipschitz, onto, and for each −1 −s compact K ∃ C > 0 s.t. for every Br ⊂ W , π (B) ∩ K is covered by ≤ Cr balls of radius Cr. 24 From the metric space results of Lecture IV we obtain conclusions about generic dimension increase for images of cosets of horizontal subgroups. These results are stated for the quotient metric d on W in the right coset foliation V o W and for the Euclidean metric on W in the left coset foliation W n V. Corollary Let Ω ⊂ H and let f ∈ W 1,p(Ω : Y ), p > 4, be continuous. For the horizontal subgroup V = {(x, 0, 0) : x ∈ R} with complementary vertical subgroup 2p W = {(0, y, t): y, t ∈ R}, and for 2 < α ≤ p−2 we have

2 dimd {a ∈ W : dim f (V ∗ a) ≥ α} ≤ 2 − p(1 − α ), 2 dimdE {a ∈ W : dim f (a ∗ V) ≥ α} ≤ 2 − p(1 − α ).

Exceptional sets for Hn Sobolev dimension distortion

joint work with Z. Balogh and K. Wildrick

25 Exceptional sets for Hn Sobolev dimension distortion

joint work with Z. Balogh and K. Wildrick

From the metric space results of Lecture IV we obtain conclusions about generic dimension increase for images of cosets of horizontal subgroups. These results are stated for the quotient metric d on W in the right coset foliation V o W and for the Euclidean metric on W in the left coset foliation W n V. Corollary Let Ω ⊂ H and let f ∈ W 1,p(Ω : Y ), p > 4, be continuous. For the horizontal subgroup V = {(x, 0, 0) : x ∈ R} with complementary vertical subgroup 2p W = {(0, y, t): y, t ∈ R}, and for 2 < α ≤ p−2 we have

2 dimd {a ∈ W : dim f (V ∗ a) ≥ α} ≤ 2 − p(1 − α ), 2 dimdE {a ∈ W : dim f (a ∗ V) ≥ α} ≤ 2 − p(1 − α ).

26 Theorem p Same assumptions as above. For 1 ≤ α ≤ p−3 we have

( p 1 p 2 − 2 (1 − α ) 1 ≤ α < p−2 dimdE {a ∈ W : dim f (a ∗ V) ≥ α} ≤ 1 p p 3 − p(1 − α ) p−2 ≤ α ≤ p−3

2 2

1 1

α α 1 p p 2p 1 p p 2p p−2 p−3 p−2 p−2 p−3 p−2

p 1,p 2 Example: For p > 4 and 1 < α < p−2 , ∃ F ⊂ W compact and f ∈ W (H : R ) continuous s.t. dim f (a ∗ ) ≥ α ∀ a ∈ F and 0 < H2−p(1−1/α)(F ) < ∞. V dE

The dimension estimate for left coset foliations on the previous slide is not optimal (since the David–Semmes regularity exponent of the foliation diﬀers from the ﬁber dimension).

27 2

α 1 p p 2p p−2 p−3 p−2

p 1,p 2 Example: For p > 4 and 1 < α < p−2 , ∃ F ⊂ W compact and f ∈ W (H : R ) continuous s.t. dim f (a ∗ ) ≥ α ∀ a ∈ F and 0 < H2−p(1−1/α)(F ) < ∞. V dE

The dimension estimate for left coset foliations on the previous slide is not optimal (since the David–Semmes regularity exponent of the foliation diﬀers from the ﬁber dimension). Theorem p Same assumptions as above. For 1 ≤ α ≤ p−3 we have

( p 1 p 2 − 2 (1 − α ) 1 ≤ α < p−2 dimdE {a ∈ W : dim f (a ∗ V) ≥ α} ≤ 1 p p 3 − p(1 − α ) p−2 ≤ α ≤ p−3

α 1 p p 2p p−2 p−3 p−2

28 The dimension estimate for left coset foliations on the previous slide is not optimal (since the David–Semmes regularity exponent of the foliation diﬀers from the ﬁber dimension). Theorem p Same assumptions as above. For 1 ≤ α ≤ p−3 we have

( p 1 p 2 − 2 (1 − α ) 1 ≤ α < p−2 dimdE {a ∈ W : dim f (a ∗ V) ≥ α} ≤ 1 p p 3 − p(1 − α ) p−2 ≤ α ≤ p−3

2 2

1 1

α α 1 p p 2p 1 p p 2p p−2 p−3 p−2 p−2 p−3 p−2

p 1,p 2 Example: For p > 4 and 1 < α < p−2 , ∃ F ⊂ W compact and f ∈ W (H : R ) continuous s.t. dim f (a ∗ ) ≥ α ∀ a ∈ F and 0 < H2−p(1−1/α)(F ) < ∞. V dE 29 The estimate recovers the borderline value Q − m − 1 = 2n + 1 − m = dimdE W when α → m, and reaches zero when α tends to the universal upper dimension pm estimate p−Q+m .

The statement is only on the level of dimension, not measure. Also, there is a gap between the positive result and the H1 example which we are not able to resolve. Such a result would follow, for instance, from a positive answer to the following:

Question: Does there exist a locally David–Semmes 1-regular surjection from Hn?

For left cosets of m-dimensional horizontal subgroups the result reads as follows: Theorem Let Ω ⊂ Hn and let f ∈ W 1,p(Ω : Y ), p > Q, be continuous. For m ∈ {1,..., n}, pm an m-dimensional horizontal subgroup V, and m ≤ α ≤ p−Q+m we have

( p m pm (Q − m − 1) − 2 (1 − α ) m ≤ α < p−2 dimdE {a : dim f (a ∗ V) ≥ α} ≤ m pm pm (Q − m) − p(1 − α ) p−2 ≤ α ≤ p−Q+m

Again, we used the Eucl metric on W to estimate the size of the exceptional set.

30 Question: Does there exist a locally David–Semmes 1-regular surjection from Hn?

( p m pm (Q − m − 1) − 2 (1 − α ) m ≤ α < p−2 dimdE {a : dim f (a ∗ V) ≥ α} ≤ m pm pm (Q − m) − p(1 − α ) p−2 ≤ α ≤ p−Q+m

Again, we used the Eucl metric on W to estimate the size of the exceptional set.

The estimate recovers the borderline value Q − m − 1 = 2n + 1 − m = dimdE W when α → m, and reaches zero when α tends to the universal upper dimension pm estimate p−Q+m .

31 For left cosets of m-dimensional horizontal subgroups the result reads as follows: Theorem Let Ω ⊂ Hn and let f ∈ W 1,p(Ω : Y ), p > Q, be continuous. For m ∈ {1,..., n}, pm an m-dimensional horizontal subgroup V, and m ≤ α ≤ p−Q+m we have

( p m pm (Q − m − 1) − 2 (1 − α ) m ≤ α < p−2 dimdE {a : dim f (a ∗ V) ≥ α} ≤ m pm pm (Q − m) − p(1 − α ) p−2 ≤ α ≤ p−Q+m

Again, we used the Eucl metric on W to estimate the size of the exceptional set.

The estimate recovers the borderline value Q − m − 1 = 2n + 1 − m = dimdE W when α → m, and reaches zero when α tends to the universal upper dimension pm estimate p−Q+m .

Question: Does there exist a locally David–Semmes 1-regular surjection from Hn? 32 Thanks for your attention!