Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the ...fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
We introduce the H-type deviation of a step two Carnot group
G
. This quantity, denoted
δ
(
G
)
, measures the deviation of
G
from the class of H-type groups. More precisely,
δ
(
G
)
=
0
if and only ...if
G
carries a vertical metric which endows it with the structure of an H-type group. We compute the H-type deviation for several naturally occurring families of step two groups. In addition, we provide several analytic expressions which are comparable to the H-type deviation. As a consequence, we establish new analytic characterizations for the class of H-type groups. For instance, denoting by
N
(
x
,
t
)
=
(
|
|
x
|
|
h
4
+
16
|
|
t
|
|
v
2
)
1
/
4
the canonical Kaplan-type quasinorm in a step two group
G
with taming Riemannian metric
g
=
g
h
⊕
g
v
, we show that
G
is H-type if and only if
|
|
∇
0
N
(
x
,
t
)
|
|
h
2
=
|
|
x
|
|
h
2
/
N
(
x
,
t
)
2
in
G
\
{
0
}
. Similarly, we show that
G
is H-type if and only if
N
2
-
Q
is
L
-harmonic in
G
\
{
0
}
. Here
∇
0
denotes the horizontal differential operator,
L
the canonical sub-Laplacian, and
Q
the homogeneous dimension. Motivation for this work derives from a conjecture regarding polarizable Carnot groups. We formulate a quantitative stability conjecture regarding the fundamental solution for the sub-Laplacian on step two Carnot groups. Its validity would imply that all step two polarizable groups admit an H-type group structure. We confirm this conjecture for a sequence of anisotropic Heisenberg groups.
We use a Riemannnian approximation scheme to define a notion of
intrinsic Gaussian curvature
for a Euclidean
C
2
-smooth surface in the Heisenberg group
H
away from characteristic points, and a ...notion of
intrinsic signed geodesic curvature
for Euclidean
C
2
-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in
H
is provided.
Abstract
We prove that if $\mu $ is a Radon measure on the Heisenberg group $\mathbb {H}^n$ such that the density $\Theta ^s(\mu ,\cdot )$, computed with respect to the Korányi metric $d_H$, exists ...and is positive and finite on a set of positive $\mu $ measure, then $s$ is an integer. The proof relies on an analysis of uniformly distributed measures on $(\mathbb {H}^n,d_H)$. We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.
Heisenberg quasiregular ellipticity Fässler, Katrin; Lukyanenko, Anton; Tyson, Jeremy
Revista matemática iberoamericana,
01/2019, Volume:
35, Issue:
2
Journal Article
Peer reviewed
Open access
Following the Euclidean results of Varopoulos and Pankka–Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the ...sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $\mathbb{S}^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, an unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $\mathbb{S}^3\backslash L$. The main result is obtained by translating a growth condition on $\pi_1(M)$ into the existence of a supersolution to the 4-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.
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.
On transversal submanifolds and their measure Magnani, Valentino; Tyson, Jeremy T.; Vittone, Davide
Journal d'analyse mathématique (Jerusalem),
2015/1, Volume:
125, Issue:
1
Journal Article
Peer reviewed
Open access
We study the class of transversal submanifolds in Carnot groups. We characterize their blow-ups at transversal points and prove a negligibility theorem for their “generalized characteristic set”, ...with respect to the Carnot-Carathéodory Hausdorff measure. This set is made up of all points of non-maximal degree. In light of the fact that
C
1
submanifolds in Carnot groups are generically transversal, the previous results prove that the “intrinsic measure” of
C
1
submanifolds is generically equivalent to their Carnot-Carathéodory Hausdorff measure. As a result, the restriction of this Hausdorff measure to the submanifold can be replaced by a more manageable integral formula that should be seen as a “sub-Riemannian mass”. Another consequence of these results is an explicit formula, depending only on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of
C
1
transversal submanifolds.
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.
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