An a.e. lower bound for Hausdorff dimension under vertical projections in the Heisenberg group

Let {πe:H→We:e∈S1} be the family of vertical projections in the first Heisenberg group H. We prove that if K⊂H is a Borel set with Hausdorff dimension dimH⁡K∈[0,2]∪{3}, thendimH⁡πe(K)≥dimH⁡K for H1 almost every e∈S1. This was known earlier if dimH⁡K∈[0,1].The proofs for dimH⁡K∈[0,2] and dimH⁡K=3 are based on different techniques. For dimH⁡K∈[0,2], we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl.To handle the case dimH⁡K=3, we introduce a point-line duality between horizontal lines and conical lines in R3. This allows us to transform the Heisenberg problem into a point-plate incidence question in R3. To solve the latter, we apply a Kakeya inequality for plates in R3, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets K⊂H with dimH⁡K∈(5/2,3).

An improved almost everywhere lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot‐Carathéodory metric. This improves the known lower bound, and answers a question of Fässler and Hovila. The approach uses the Euclidean Fourier transform, Basset's integral formula, and modified Bessel functions of the second kind.

This note concernsLoomis–Whitney inequalitiesin Heisenberg groups$$\mathbb {H}^n$$Hn:$$\begin{aligned} |K| \lesssim \prod _{j=1}^{2n}|\pi _j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb {H}^n. \end{aligned}$$|K|≲∏j=12n|πj(K)|n+1n(2n+1),K⊂Hn.Here$$\pi _{j}$$πj,$$j=1,\ldots ,2n$$j=1,…,2n, are thevertical Heisenberg projectionsto the hyperplanes$$\{x_j=0\}$${xj=0}, respectively, and$$|\cdot |$$|·|refers to a natural Haar measure on either$$\mathbb {H}^n$$Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group$$\mathbb {H}^1$$H1is a direct consequence of known$$L^p$$Lpimproving properties of the standard Radon transform in$$\mathbb {R}^2$$R2. In this note, we show how the Loomis–Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in$$\mathbb {H}^1$$H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound:$$\begin{aligned} \int _{\mathbb {H}^n} \prod _{j=1}^{2n} f_j(\pi _j(p))\;dp\lesssim \prod _{j=1}^{2n} \Vert f_j\Vert _{\frac{n(2n+1)}{n+1}} \end{aligned}$$∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1for all nonnegative measurable functions$$f_1,\ldots ,f_{2n}$$f1,…,f2non$$\mathbb {R}^{2n}$$R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in$$\mathbb {H}^n$$Hn, we mention the following sharper version of the classical geometric Sobolev inequality in$$\mathbb {H}^n$$Hn:$$\begin{aligned} \Vert u\Vert _{\frac{2n+2}{2n+1}} \lesssim \prod _{j=1}^{2n}\Vert X_ju\Vert ^{\frac{1}{2n}}, \qquad u \in BV(\mathbb {H}^n), \end{aligned}$$‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn),where$$X_j$$Xj,$$j=1,\ldots ,2n$$j=1,…,2n, are the standard horizontal vector fields in$$\mathbb {H}^n$$Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in$$\mathbb {H}^n$$Hn, where the Heisenberg vertical coordinate projections$$\pi _1,\ldots ,\pi _{2n}$$π1,…,π2nare replaced by more general families of mappings that allow us to apply the same inductive approach based on the$$L^{3/2}$$L3/2-$$L^3$$L3boundedness of an operator in the plane.

We explore the effect of vertical projections on the Hausdorff dimension of sets in the Heisenberg group endowed with the Koranyi distance. It is known that the dimension of an at most one-dimensional set generically does not decrease under such mappings. The proof uses a potential-theoretic approach which, for higher dimensional sets, only yields a trivial lower bound. In the present note, we provide an improved estimate for the dimension and thus prove that the previous trivial bound is not sharp. Moreover, for the larger family of projections onto cosets of vertical subgroups, we show that the potential-theoretic approach can be applied to establish almost sure dimension conservation for sets of dimension up to two.

We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.

We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, \mathbb{H}^n . We prove lower bounds for Hausdorff dimension distortion of sets under these mappings with respect to the natural quotient metric, which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical projections in \mathbb{H}^n for n \geq 2 .

In the first part of the dissertation we prove that, under quite general conditions on a cost function $c$ in $\RR^n$, the Hausdorff dimension of the singular set of a $c$-concave function has dimension at most $n-1$. Our result applies for non-semiconcave cost functions and has applications in optimal mass transportation. The purpose of the second part of the thesis is to extend a result of Alberti and Ambrosio about singularity sets of monotone multivalued maps to the sub-Riemannian setting of Heisenberg groups. We prove that the $k$-th horizontal singular set of a $H$-monotone multivalued map of the Heisenberg group $\HH^n$, with values in $\RR^{2n}$, has Hausdorff dimension at most $2n+2-k$.

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the { x o y } \{xoy\} -plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones.

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano.

The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.

We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.

We consider unions of lines in . These are lines of the form where . We show that if is a Kakeya set of lines, then the union has Hausdorff dimension 3. This answers a question of Wang and Zahl. The lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point‐line duality principle between horizontal and conical lines in , combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim.

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that ifAandBare Borel subsets of ℝ2nof dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images ofAandBunder orthogonal projections onto these planes have positive Hausdorffm-measure. In addition, ifAis a measurable set of Hausdorff dimension greater thanm, then there is a setB⊂ ℝ2nwith dimB⩽msuch that for allx∈ ℝ2n\Bthere is a positive measure set of isotropic m-planes for which the translate byxof the orthogonal complement of each such plane, intersectsAon a set of dimension dimA – m. These results are then applied to obtain analogous results on thenthHeisenberg group.

We study projectional properties of Poisson cut-out setsEin non-Euclidean spaces. In the first Heisenbeg group\[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection\[\pi (E)\](projection along the center of\[\mathbb{H}\]) almost surely equals\[\min \{ 2,{\dim _\operatorname{H} }(E)\} \]and that\[\pi (E)\]has non-empty interior if\[{\dim _{\text{H}}}(E) > 2\]. As a corollary, this allows us to determine the Hausdorff dimension ofEwith respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension\[{\dim _{\text{H}}}(E)\].We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere\[{{\text{S}}^3}\]endowed with the visual metricdobtained by identifying\[{{\text{S}}^3}\]with the boundary of the complex hyperbolic plane. In\[{{\text{S}}^3}\], we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in\[{{\text{S}}^3}\]satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.