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B V spaces and rectifiability for Carnot-Carathéodory metrics: an introduction

Franchi, Bruno (2003)

Nonlinear Analysis, Function Spaces and Applications

This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...

Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform

Detlev Poguntke (2010)

Colloquium Mathematicae

For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group....

Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields

Mitya Boyarchenko, Sergei Levendorski (2006)

Annales de l’institut Fourier

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on L 2 ( ℝ n ) with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain “algebraic integrals,” studied by Nilsson. By using the direct variational method, we prove that the...

BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

Sean Li (2015)

Analysis and Geometry in Metric Spaces

Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that BZcan be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of [14], which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not admitting such...

Blow-up of regular submanifolds in Heisenberg groups and applications

Valentino Magnani (2006)

Open Mathematics

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence...

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