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1.
We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples. 相似文献
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Many results of classical Potential Theory are extended to sub-Laplacians ▵𝔾 on Carnot groups 𝔾. Some characterizations of ▵𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize
the Riesz-measure related to bounded-above ▵𝔾-subharmonic functions in ℝ
N
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Received: 21 June 2000 / Revised version: 12 March 2002 / Published online: 2 December 2002
RID="★"
ID="★" Investigation supported by University of Bologna. Funds for selected research topics.
Mathematics Subject Classification (2000): 31B05, 35J70, 35H20 相似文献
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《Comptes Rendus Mathematique》2008,346(23-24):1239-1243
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In this paper we initiate the study of quasiregular maps in a sub-Riemannian geometry of general Carnot groups. We suggest
an analytic definition for quasiregularity and then show that nonconstant quasiregular maps are open and discrete maps on
Carnot groups which are two-step nilpotent and of Heisenberg type; we further establish, under the same assumption, that the
branch set of a nonconstant quasiregular map has Haar measure zero and, consequently, that quasiregular maps are almost everywhere
differentiable in the sense of Pansu. Our method is that of nonlinear potential theory. We have aimed at an exposition accessible
to readers of varied background.
Dedicated to Seppo Rickman on his sixtieth birthday
J.H. was partially supported by NSF, the Academy of Finland, and the A. P. Sloan Foundation. I.H. was partially supported
by the EU HCM contract no. CHRX-CT92-0071. 相似文献
5.
Changyou Wang 《Proceedings of the American Mathematical Society》2005,133(4):1247-1253
We prove that any upper semicontinuous v-convex function in any Carnot group is h-convex.
6.
M. B. Karmanova 《Doklady Mathematics》2016,94(3):663-666
The polynomial sub-Riemannian differentiability of classes of mappings of Carnot groups and graphs is proved. Examples of polynomial sub-Riemannian differentials preserving Hausdorff dimension are given. 相似文献
7.
We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem. 相似文献
8.
We describe a procedure for constructing ”polar coordinates” in a certain class of Carnot groups. We show that our construction
can be carried out in groups of Heisenberg type and we give explicit formulas for the polar coordinate decomposition in that
setting. The construction makes use of nonlinear potential theory, specifically, fundamental solutions for the p-sub-Laplace operators. As applications of this result we obtain exact capacity estimates, representation formulas and an
explicit sharp constant for the Moser-Trudinger inequality. We also obtain topological and measure-theoretic consequences
for quasiregular mappings.
Received: 26 June 2001; in final form: 14 January 2002/Published online: 5 September 2002 相似文献
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Luca Brandolini Marco Magliaro 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2326-2337
This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form Δφu≥f(u)l(|∇0u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δφ is the φ-Laplace operator, a natural generalization of the p-Laplace operator which has recently been studied in the context of Carnot groups. We extend to general Carnot groups the results proved in Magliaro et al. (2011) [7] for the Heisenberg group, showing the validity of Liouville-type theorems under a suitable Keller-Osserman condition. In doing so, we also prove a maximum principle for inequality Δφu≥f(u)l(|∇0u|). Finally, we show sharpness of our results for a general φ-Laplacian. 相似文献
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Marius Măntoiu 《Archiv der Mathematik》2017,109(2):167-177
Let \(\textsf {G}\) be a Carnot group of homogeneous dimension M and \(\Delta \) its horizontal sublaplacian. For \(\alpha \in (0,M)\) we show that operators of the form \(H_\alpha :=(-\Delta )^\alpha +V\) have no singular spectrum, under generous assumptions on the multiplication operator V. The proof is based on commutator methods and Hardy inequalities. 相似文献
16.
Subelliptic harmonic maps from Carnot groups 总被引:1,自引:0,他引:1
For subelliptic harmonic maps from a Carnot group into a Riemannian manifold without boundary, we prove that they are smooth near any
-regular point (see Definition 1.3) for sufficiently small
. As a consequence, any stationary subelliptic harmonic map is smooth away from a closed set with zero HQ-2 measure. This extends the regularity theory for harmonic maps (cf. [SU], [Hf], [El], [Bf]) to this subelliptic setting.Received: 24 April 2002, Accepted: 30 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000):
35B65, 58J42 相似文献
17.
We prove some weighted Hardy and Rellich inequalities on general Carnot groups with weights associated to the norm constructed by Folland’s fundamental solution of the Kohn sub-Laplacian. 相似文献
18.
The research was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00378) and the International Science Foundation (Grant RAT000). 相似文献
19.
We generalize the classical Whitney theorem which describes the restrictions of functions of various smoothness to closed sets of a Carnot group. The main results of the article are announced in [1]. 相似文献