Stratified Lie groups and potential theory for their sub-Laplacians

By Andrea Bonfiglioli, E. Lanconelli, Francesco Uguzzoni. 2007

900 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/63c977e2af22a66b94b98cc37b9ca92f369733e7

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups (2011, methodology): “The explicit expression of the group operation ·, determined by the Campbell–Hausdorff formula (see [6] or [16]), has the form p · q = p+ q +Q(p,q), for all p,q ∈RN ≡G (1) where Q(p,q)= (Q1(p, q), .”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “The problem of relating the sub-Riemannian distance to the heat-kernel is an old problem (see for instance [6, 13, 18, 21, 22, 23, 30, 36, 38, 39, 44, 50, 52]).”
The Cα regularity of a class of non-homogeneous ultraparabolic equations (2007, methodology): “For the more details of the subject, we refer to Cinti and Polidoro [4], Kogoj and Lanconelli [8], or Bonfiglioli, Lanconelli and Uguzzoni[1].”
Attractors for a class of semi-linear degenerate parabolic equations (2013, background): “Denoting by HN the canonical sub-Laplacian on HN , we have ( x (1) + 1 4 |x (1)|2 x (2) ) u = HN u, for every smooth function u : RN −→ R which is radially symmetric in the variable x (1) (see [5], p.”
Quantization on Nilpotent Lie Groups (2016, methodology): “We refer to the book [BLU07] by Bonfiglioli, Lanconelli and Uguzonni for a detailed consideration of these groups and of their sub-Laplacians as well as related operators.”
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups (2008, background): “Since then, many estimates and properties of the kernel in terms of the sub-Riemannian distance have been provided (see [2, 4, 6, 7, 10, 19, 23, 28, 31] and references therein).”
Precise estimates for the subelliptic heat kernel on H-type groups (2008, background): “A standard argument, paraphrased from [2], where many more details can be fo und, is as follows.”
Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications ✩ (2012, background): “Our proof does not rely on the special structure of the Heisenberg group and applies to much more general cases including the stratified groups [13], Euclidean spaces, complete Riemannian manifolds, even appropriate metric measure spaces, etc.”
Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs (2014, neutral): “For analysis on Carnot (stratified) groups, we refer to [29,30].”
On the Hausdorff volume in sub-Riemannian geometry (2010, background): “A deep study of Hausdorff measure for hypersurfaces in sub-Riemannian geometry, in particular in the context of Carnot groups, can be found in [8, 10, 11, 14, 19, 22, 32, 36] and references therein.”
A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities (2010, background): “Carnot groups Here, we quote some facts on Carnot groups and refer the interested reader to [28,29,6] for a more detailed information on this structures.”
Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups (2017, background): “We recall that the L-gauge (see [4]) is a symmetric homogeneous (quasi-) norm on the homogeneous Carnot group G = (R , ◦, δλ), that is, • d(x) > 0 if and only if x = 0, • d(δλ(x)) = λd(x) for all λ > 0 and x ∈ G, • d(x−1) = d(x) for all x ∈ G.”
Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group (2012, positive): “3), and the regularity estimates and asymptotic behavior for solutions to the Euler–Lagrange equations to the |u| weighted inequality hold in the more general setting of stratified groups (we refer to [25,26] for an introduction of stratified groups).”
Compensated compactness for differential forms in Carnot groups and applications (2010, background): “[3], [26], [28]), which play versus subriemannian spaces the role played by Euclidean spaces (considered as tangent spaces) versus Riemannian manifolds.”