Marstrand's density theorem in the Heisenberg group

Chousionis, Vasilis; Tyson, Jeremy T.

doi:10.1112/blms/bdv056

Cited by 17 publications

(24 citation statements)

References 27 publications

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“…Uniform measures play an important role in geometric measure theory. For instance, if X = R n , then Marstrand [25] proved that for a non-trivial Q-uniform measure it necessarily holds that Q ∈ N (see also Chousionis-Tyson [5] for a discussion of Marstrand's theorem and uniform measures in the setting of Heisenberg groups). One of results of the celebrated paper by Preiss [28] stays that for Q = 1, 2 uniform measures are flat.…”

Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems.Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

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Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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“…The deep connections between the existence of densities and recti ability of sets and absolutely continuous measures in Euclidean space, as described above, have been explored in metric spaces beyond R n by several authors; see e.g. [53], [30], [37], [38], [12]. For perspectives on recti ability in metric spaces related to existence of tangents or projection properties, see e.g.…”

Section: Hausdor Densities and Recti Abilitymentioning

confidence: 99%

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Badger

Schul

2017

Analysis and Geometry in Metric Spaces

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A measure is 1-recti able if there is a countable union of nite length curves whose complement has zero measure. We characterize 1-recti able Radon measures µ in n-dimensional Euclidean space for all n ≥ in terms of positivity of the lower density and niteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which µ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between µ and 1-dimensional Hausdor measure H . We also characterize purely 1-unrecti able Radon measures, i.e. locally nite measures that give measure zero to every nite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L variant of P. Jones' traveling salesman construction, which is of independent interest.

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“…Proof of Theorem 1.5. Let µ be a 3-uniform measure on H. By [6], M = spt µ is a real analytic variety of topological dimension two and Hausdorff dimension three. The top dimensional stratum M (2) is a countable union of real analytic surfaces.…”

Section: -Uniform Measures and Smooth Fully Noncharacteristic Surfacesmentioning

confidence: 99%

“…Our choice of this metric stems from three facts. First, Marstrand's density theorem holds for the metric space (H, d H ) [6]; this fact ensures that supports of uniform measures in (H, d H ) are highly regular. Second, the rotational symmetry of the Korányi metric about the x 3 -axis simplifies the area formula for submanifolds in the Heisenberg group [21,19,18].…”

Section: Introductionmentioning

confidence: 99%

On uniform measures in the Heisenberg group

Chousionis

Magnani

Tyson

2020

Advances in Mathematics

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We initiate a classification of uniform measures in the first Heisenberg group H equipped with the Korányi metric dH , 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 (H, dH ) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.

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Marstrand's density theorem in the Heisenberg group

Cited by 17 publications

References 27 publications

Harmonic functions on metric measure spaces

Harmonic functions on metric measure spaces

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

On uniform measures in the Heisenberg group

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