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We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

Let G be any Carnot group. We prove that, if a subset of G is contained in a rectifiable curve, then it satisfies Peter Jones’ geometric lemma with some natural modifications. We thus prove one direction of the Traveling Salesman Theorem in G . Our proof depends on new Alexandrov-type curvature inequalities for the Hebisch–Sikora metrics. We also apply the geometric lemma to prove that, in every Carnot group, there exist - 1 -homogeneous Calderón–Zygmund kernels such that, if a set E ⊂ G is contained in a 1-regular curve, then the corresponding singular integral operators are bounded in L 2 ( E ) . In contrast to the Euclidean setting, these kernels are nonnegative and symmetric.

Let G be a Carnot group with homogeneous dimension Q ≥ 3 and let L be a sub-Laplacian on G . We prove that the critical dimension for removable sets of Lipschitz L -harmonic functions is ( Q - 1 ) . Moreover we construct self-similar sets with positive and finite H Q - 1 measure which are removable.

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L t are hypoelliptic. We study the removable sets for L -solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L -solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.

We construct a class of singular integral operators associated with homogeneous Calderón-Zygmund standard kernels on d-dimensional, d < 1, Sierpinski gaskets Ed. These operators are bounded in L2(μd) and their principal values diverge μd almost everywhere, where μd is the natural (d-dimensional) measure on Ed.

We initiate a classification of uniform measures in the first Heisenberg group \\(\\mathbb H\\) equipped with the Korányi metric \\(d_H\\), that represents the first example of a noncommutative stratified group equipped with a homogeneous distance. We prove that \\(1\\)-uniform measures are proportional to the spherical \\(1\\)-Hausdorff measure restricted to an affine horizontal line, while \\(2\\)-uniform measures are proportional to spherical \\(2\\)-Hausdorff measure restricted to an affine vertical line. It remains an open question whether \\(3\\)-uniform measures are proportional to the restriction of spherical \\(3\\)-Hausdorff measure to an affine vertical plane. We establish this conclusion in case the support of the measure is a vertically ruled surface. Along the way, we derive asymptotic formulas for the measures of small extrinsic balls in \\(({\\mathbb H},d_H)\\) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in \\(\\mathbb H\\).

The aim of the present paper is twofold. We study directed porosity in connection with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures \\(\\mu\\) and kernels \\(K\\), converge weakly in some dense subspaces of \\(L^2(\\mu)\\) when the support of \\(\\mu\\) belongs to a broad family of sets. This class contains many fractal sets like CIFS's limit sets.

We construct a class of singular integral operators associated with homogeneous Calder\\'{o}n-Zygmund standard kernels on \\(d\\)-dimensional, \\(d <1\\), Sierpinski gaskets \\(E_d\\). These operators are bounded in \\(L^2(\\mu_d)\\) and their principal values diverge \\(\\mu_d\\) almost everywhere, where \\(\\mu_d\\) is the natural (d-dimensional) measure on \\(E_d\\).

We consider the Calderón-Zygmund kernels \\(K_ {\\alpha,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\\alpha})_{i=1}^d\\) in \\(\\mathbb{R}^n\\) for \\(0<\\alpha\\leq 1\\) and \\(n\\in\\mathbb{N}\\). We show that, on the plane, for \\(0<\\alpha<1\\), the capacity associated to the kernels \\(K_{\\alpha,n}\\) is comparable to the Riesz capacity \\(C_{\\frac23(2-\\alpha),\\frac 3 2}\\) of non-linear potential theory. As consequences we deduce the semiadditivity and bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set \\(E\\subset\\mathbb{R}^n\\) with finite length the \\(L^2(\\mathcal{H}^1 \\lfloor E)\\)-boundedness of the singular integral associated to \\(K_{1,n}\\) implies the rectifiability of the set \\(E\\). We thus extend to any ambient dimension, results previously known only in the plane.