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Register a new userCharacterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups
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In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Holder properties along the projections of left-invariant vector fields on the graph. We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group. As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.
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We prove that in arbitrary Carnot groups $mathbb G$ of step 2, with a splitting $mathbb G=mathbb Wcdotmathbb L$ with $mathbb L$ one-dimensional, the graph of a continuous function $varphicolon Usubseteq mathbb Wto mathbb L$ is $C^1_{mathrm{H}}$-regular precisely when $varphi$ satisfies, in the distributional sense, a Burgers type system $D^{varphi}varphi=omega$, with a continuous $omega$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $varphi$ to a Burgers type system $D^{varphi}varphi=omega$, with $omega$ continuous, is actually a broad solution to $D^{varphi}varphi=omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{varphi}varphi=omega$, with $omega$ continuous, enjoy $1/2$-little Holder regularity along vertical directions.
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analysts Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $mathbb{R}^2$ (P. Jones, 1990), in $mathbb{R}^n$ (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones $beta$-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in $mathbb{R}^n$ that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.
We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
In this paper, we construct Holder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $mathbb{H}$. Pansu and Gromov observed that any surface embedded in $mathbb{H}$ has Hausdorff dimension at least 3, so there is no $alpha$-Holder embedding of a surface into $mathbb{H}$ when $alpha>frac{2}{3}$. Zust improved this result to show that when $alpha>frac{2}{3}$, any $alpha$-Holder map from a simply-connected Riemannian manifold to $mathbb{H}$ factors through a metric tree. In the present paper, we show that Zusts result is sharp by constructing $(frac{2}{3}-epsilon)$-Holder maps from $D^2$ and $D^3$ to $mathbb{H}$ that do not factor through a tree. We use these to show that if $0<alpha < frac{2}{3}$, then the set of $alpha$-Holder maps from a compact metric space to $mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $mathbb{R}^3$ to $mathbb{H}$ with Holder exponents arbitrarily close to $frac{2}{3}$.
In this paper we prove the one-dimensional Preiss theorem in the first Heisenberg group $mathbb H^1$. More precisely we show that a Radon measure $phi$ on $mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps $Asubseteq mathbb Rtomathbb H^1$. The previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove in arbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normal complement, our criterion applies in the particular case of one-dimensional horizontal subgroups. These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measure on a Carnot group is $mathscr{P}_h$-rectifiable, for $hinmathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of a homogeneous subgroup of Hausdorff dimension $h$. We also prove several structure properties of $mathscr{P}_h$-rectifiable measures. First, we compare $mathscr{P}_h$-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups and we realize that it is strictly weaker than them. Furthermore, we show that a $mathscr{P}_h$-rectifiable measure has almost everywhere positive and finite $h$-density whenever the tangents admit at least one complementary subgroup.
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