Plenty of big projections imply big pieces of Lipschitz graphs
I prove that a closed $n$-regular set $E \subset \mathbb{R}^{d}$ with plenty of big projections has big pieces of Lipschitz graphs. This answers a question of David and Semmes.
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arXiv subjects
Tuomas Orponen
Publications and source records attributed to Tuomas Orponen.
On the dimension of visible parts
I prove that the visible parts of a compact set in $\mathbb{R}^{n}$, $n \geq 2$, have Hausdorff dimension at most $n - \tfrac{1}{50n}$ from almost every direction.
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Rickman rugs and intrinsic bilipschitz graphs
This paper studies the geometry of bilipschitz maps $f \colon \mathbb{W} \to \mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W} \subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images $f(\mathbb{W})$ of such maps are called Rickman rugs in the Heisenberg group. The main theorem states that a Rickman rug in the Heisenberg group admits a corona decomposition by intrinsic bilipschitz graphs. As a corollary, Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs. Here, an intrinsic bilipschitz graph is an intrinsic Lipschitz graph, which is simultaneously a Rickman rug. General intrinsic Lipschitz graphs need not be Rickman rugs, even locally, by an example of Bigolin and Vittone.
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Combinatorial proofs of two theorems of Lutz and Stull
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$.
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Sub-elliptic boundary value problems in flag domains
A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn-Laplacian $\bigtriangleup^{\flat} = X^{2} + Y^{2}$ in flag domains $Ω\subset \mathbb{R}^{3}$, with $L^{2}$-boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order $L^{2}$-Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.
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Singular integrals on regular curves in the Heisenberg group
Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all $3$-dimensional horizontally odd kernels yield $L^{2}$ bounded operators on Lipschitz flags in $\mathbb{H}$. This was known earlier for only one specific operator, the $3$-dimensional Riesz transform. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.
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Absolute continuity in families of parametrised non-homogeneous self-similar measures
Let $μ$ be a planar self-similar measure with similarity dimension exceeding $1$, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections $π_{e\sharp}(μ)$ are absolutely continuous for all $e \in S^{1} \setminus E$, where the exceptional set $E$ has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with $μ$ have a common contraction ratio.
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Planar incidences and geometric inequalities in the Heisenberg group
We prove that if $P,\mathcal{L}$ are finite sets of $δ$-separated points and lines in $\mathbb{R}^{2}$, the number of $δ$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\mathcal{L}|^{2/3} \cdot δ^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$ |K| \lesssim |π_{x}(K)|^{2/3} \cdot |π_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}. $$ Here $π_{x}$ and $π_{y}$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\cdot|$ refers to natural Haar measure on either $\mathbb{H}$, or one of the planes. Finally, as a corollary of the Loomis-Whitney inequality, we deduce that $$ \|f\|_{4/3} \lesssim \sqrt{\|Xf\| \|Yf\| }, \qquad f \in BV(\mathbb{H}), $$ where $X,Y$ are the standard horizontal vector fields in $\mathbb{H}$. This is a sharper version of the classical geometric Sobolev inequality $\|f\|_{4/3} \lesssim \|\nabla_{\mathbb{H}}f\|$ for $f \in BV(\mathbb{H})$.
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Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group
A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets.
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Quantitative absolute continuity of planar measures with two independent Alberti representations
We study measures $μ$ on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A-C-P. Assuming that the representations of $μ$ are bounded from above, in a natural way to be defined in the introduction, we prove that $μ\in L^{2}$. If the representations are also bounded from below, we show that $μ$ satisfies a reverse Hölder inequality with exponent $2$, and is consequently in $L^{2 + ε}$ by Gehring's lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal.
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On the Assouad dimension of projections
Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} π_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This is a strong variant of Marstrand's projection theorem for Assouad dimension, whose analogue is not true for other common notions of fractal dimension, such as Hausdorff or packing dimension.
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Quantitative Alberti representations in spaces of bounded geometry
A metric measure space $(X,d,μ)$ is said to be $A_{\infty}$ on curves if there exist constants $τ< 1$ and $θ> 0$ with the following property. For every $x \in X$, $0 < r \leq \mathrm{diam}(X)$, and a Borel set $S \subset B(x,r)$ with $μ(S) > τμ(B(x,r))$, there exists a continuum $γ\subset X$ of length $\leq r$ satisfying $\mathcal{H}^{1}_{\infty}(γ\cap S) \geq θr$. I first observe that spaces of $Q$-bounded geometry, $Q > 1$, are $A_{\infty}$ on curves. Then, I show that any complete, doubling, and quasiconvex space $(X,d,μ)$ which is $A_{\infty}$ on curves has Alberti representations with $L^{p}$-densities for some $p > 1$, depending only on the doubling and $A_{\infty}$-constants. More precisely, any normalised restriction of $μ$ to a ball $B \subset X$ can be written as $μ_{B} = f_{B} \, dν_{B}$, where $ν_{B}$ is a convex combination of measures of linear growth supported on continua of length $\le \mathrm{diam}(B)$, and $\|f_{B}\|_{L^{p}(ν_{B})} \leq C$ for some constant $C \geq 1$ independent of $B$.
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Vertical versus horizontal Sobolev spaces
Let $α\geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2α}(\mathbb{H}^{n})$, then $φf$ belongs to the Euclidean Sobolev space $S^{p}_α(\mathbb{R}^{2n + 1})$ for any test function $φ$. In short, $S^{p}_{2α}(\mathbb{H}^{n}) \subset S^{p}_{α,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2α}^{p}(\mathbb{H}^{n})$ is continuously contained in the vertical Sobolev space $V^{p}_α(\mathbb{H}^{n})$. Our search for the sharper result was motivated by the following two applications. First, combined with a short additional argument, it implies that bounded Lipschitz functions on $\mathbb{H}^{n}$ have a $\tfrac{1}{2}$-order vertical derivative in $\mathrm{BMO}(\mathbb{H}^{n})$. Second, it yields a fractional order generalisation of the (non-endpoint) vertical versus horizontal Poincaré inequalities of V. Lafforgue and A. Naor.
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Riesz transform and vertical oscillation in the Heisenberg group
We study the $L^{2}$-boundedness of the $3$-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. Inspired by the notion of vertical perimeter, recently defined and studied by Lafforgue, Naor, and Young, we first introduce new scale and translation invariant coefficients $\operatorname{osc}_Ω(B(q,r))$. These coefficients quantify the vertical oscillation of a domain $Ω\subset \mathbb{H}$ around a point $q \in \partial Ω$, at scale $r > 0$. We then proceed to show that if $Ω$ is a domain bounded by an intrinsic Lipschitz graph $Γ$, and $$\int_{0}^{\infty} \operatorname{osc}_Ω(B(q,r)) \, \frac{dr}{r} \leq C < \infty, \qquad q \in Γ,$$ then the Riesz transform is $L^{2}$-bounded on $Γ$. As an application, we deduce the boundedness of the Riesz transform whenever the intrinsic Lipschitz parametrisation of $Γ$ is an $ε$ better than $\tfrac{1}{2}$-Hölder continuous in the vertical direction. We also study the connections between the vertical oscillation coefficients, the vertical perimeter, and the natural Heisenberg analogues of the $β$-numbers of Jones, David, and Semmes. Notably, we show that the $L^{p}$-vertical perimeter of an intrinsic Lipschitz domain $Ω$ is controlled from above by the $p^{th}$ powers of the $L^{1}$-based $β$-numbers of $\partial Ω$.
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Dorronsoro's theorem in Heisenberg groups
A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincaré inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.
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An integralgeometric approach to Dorronsoro estimates
A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro's theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one, and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher's theorem for parabolic Lipschitz functions.
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Metric currents and the Poincaré inequality
We show that a complete doubling metric space $(X,d,μ)$ supports a weak $1$-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points $s,t \in X$. This notion was introduced by S. Semmes in the 90's, and has been previously known to be a sufficient condition for the weak $1$-Poincaré inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining $s$ and $t$ is a normal $1$-current $T$, in the sense of Ambrosio and Kirchheim, with boundary $\partial T = δ_{t} - δ_{s}$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\|T\| \ll μ$, with $$\frac{d\|T\|}{dμ}(y) \lesssim \frac{d(s,y)}{μ(B(s,d(s,y)))} + \frac{d(t,y)}{μ(B(y,d(t,y)))}.$$ We show that the $1$-Poincaré inequality implies the existence of GPCs joining any pair of points in $X$. Then, we deduce the existence of PCs from a recent decomposition result for normal $1$-currents due to Paolini and Stepanov.
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On the dimension and smoothness of radial projections
This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with $\dim_{\mathrm{H}} K > 0$. Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if $\dim_{\mathrm{H}} E > 0$, and $E$ does not lie on a line, then there exists a point in $x \in E$ such that the radial projection $π_{x}(K)$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. Applying the result with $E = K$ gives the following corollary: if $K \subset \mathbb{R}^{2}$ is Borel set, which does not lie on a line, then the set of directions spanned by $K$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. For the second result, let $d \geq 2$ and $d - 1 < s < d$. Let $μ$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections of $μ$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus \operatorname{spt} μ$, outside an exceptional set of dimension at most $2(d - 1) - s$. In fact, for $x$ outside an exceptional set as above, the proof shows that $π_{x\sharp}μ\in L^{p}(S^{d - 1})$ for some $p > 1$. The dimension bound on the exceptional set is sharp.
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