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1.
First-order regularity of convex functions on Carnot Groups 总被引:1,自引:0,他引:1
Matthieu Rickly 《Journal of Geometric Analysis》2006,16(4):679-702
We prove that h-convex functions on Carnot groups of step two are locally Lipschitz continuous with respect to any intrinsic
metric. We show that an additional measurability condition implies the local Lipschitz continuity of h-convex functions on
arbitrary Carnot groups.
To the Memory of Q. G. 相似文献
2.
In this note we prove a recent conjecture of Hasson [11]: we show that, for a locally
integrable function u, a sufficient condition to be harmonic is that $ \lim\limits_{r\to 0^+} r^{-2}(M_{r}u-u) = 0 $ in the
weak sense of distributions (M
r
being the averaging operator on balls of radius r). We also extend
this and other results to the setting of sub-Laplacians on Carnot groups.Investigation supported by University of Bologna. Funds for selected research topics. 相似文献
3.
Annalisa Baldi Bruno Franchi 《Calculus of Variations and Partial Differential Equations》2003,16(3):283-298
Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski
content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals.
Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002
Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors
are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening
with patience and for many unvaluable suggestions. 相似文献
4.
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.
Alessandro Ottazzi 《Mathematische Zeitschrift》2008,259(3):617-629
In this article we consider contact mappings on Carnot groups. Namely, we are interested in those mappings whose differential
preserves the horizontal space, defined by the first stratum of the natural stratification of the Lie algebra of a Carnot
group. We give a sufficient condition for a Carnot group G to admit an infinite dimensional space of contact mappings, that
is, for G to be nonrigid. A generalization of Kirillov’s Lemma is also given. Moreover, we construct a new example of nonrigid
Carnot group.
This research was partly supported by the Swiss National Science Foundation. The author would like to thank H. M. Reimann
for the helpful advices and the constant support. 相似文献
6.
Gabriella Arena Raul Serapioni 《Calculus of Variations and Partial Differential Equations》2009,35(4):517-536
We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs.
G. Arena is supported by MIUR (Italy), by INDAM and by University of Trento.
R. Serapioni is supported by MIUR (Italy), by GALA project of the Sixth Framework Programme of European Community and by University
of Trento. 相似文献
7.
In this paper we establish a decomposition theorem for a positive regular measure on an orthoalgebra with values in an ordered
topological group not necessarily commutative. We deduce from it the A. D. Alexandroff’s classical first decomposition theorem
and we discuss its uniqueness in the setting of metric spaces.
This research is partially supported by the project Analisi Reale of Ministero dell’Università e della Ricerca Scientifica
e Tecnologica (Italy) and by Gruppo Nazionale per l’Analisi Funzionale e Applicazioni (Italy). 相似文献
8.
A. Kaplan F. Levstein L. Saal A. Tiraboschi 《Annali di Matematica Pura ed Applicata》2008,187(1):67-91
We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of
the canonical distribution (“lagrangians”) and illustrate some notable ones of small dimension. An infinitesimal classification
of the arbitrary maximal horizontal submanifolds follows as a consequence.
This work was supported by CONICET, Antorchas, FONCyT and Secyt (UNC). 相似文献
9.
Vittorio Barone Adesi Francesco Serra Cassano Davide Vittone 《Calculus of Variations and Partial Differential Equations》2007,30(1):17-49
In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Carathéodory (CC) structures. More precisely
we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group . In particular we will positively answer to the Bernstein problem in the case n = 1 and we will provide counter examples when .
V.B.A. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy.
F.S.C. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy.
D.V. is supported by MIUR, Italy, GNAMPA of INDAM and Scuola Normale Superiore, Italy. Part of the work was done while D.V.
was a visitor at the University of Trento. He wishes to thank the Department of Mathematics for its hospitality. 相似文献
10.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x∈G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups:
they show in Math. Ann. 321, 479–531, 2001 and J. Geom. Anal. 13, 421–466, 2003 that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.
The second author was partially supported by NSF grant DMS-0701515. 相似文献
11.
The problem of common extension ofcharges (finitely additive measures) is generalised to include group-valued functions defined on a system of sets (u-systems). To eachu-systemU an Abelian groupH(U) is attached. Every Abelian group is isomorphic to one of the formH(U). The groupH(U) is an indicator for extendability of charges fromU to the Boolean algebra generated byU. AllG-valued measures extend if and only if Ext(H(U),G)=0, for instance.
Supported as van Vleck visiting professor at Wesleyan University, Connecticut in 1993.
Partially supported by the Graduierten KollogTheoretische und experimentelle Methoden der reinen Mathematik of Essen University, a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development
and by the German Academic Exchange, DAAD 1994. 相似文献
12.
In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ?e in the direction of e for which dimH(?e∩F)?α. It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets.The main difficulty we had to overcome, was that if Hh(F)=0, there always exists g?h such that Hg(F)=0 (here ? refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α=0. 相似文献
13.
We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type. 相似文献
14.
Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type 总被引:1,自引:0,他引:1
We prove the hypoellipticity for systems of H?rmander type with constant coefficients in Carnot groups of step 2. This result
is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for
solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish
estimates of the Hausdorff dimension of the singular set.
Received October 23, 2000 / final version received February 5, 2002?Published online May 15, 2002
The work of the first author was partially supported by NSF Grant No. DMS-9800794
The work of the second author was partially supported by NSF Grants No. DMS-9706892 and DMS-0070492 相似文献
15.
One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved. 相似文献
16.
We prove that property (β) of Rolewicz implies normal structure of the dual space and we characterize spaces which are duals
of spaces with property (β).
Research of the second author was supported in part by the M.U.R.S.T. of Italy. 相似文献
17.
The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors [14],
[29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems,
as well as, the spectral mapping theorem relative to some other classical spectra.
We also prove that ifT orT* has the single-valued extension property some of the more important spectra originating from Fredholm theory coincide. This
result is extended, always in the caseT orT* has the single valued extension property, tof(T), wheref is an analytic function defined on an open disc containing the spectrum ofT. In the last part we improve a recent result of Curto and Han [10] by proving that for every transaloid operatorT a-Weyl’s theorem holds forf(T) andf(T)*.
The research was supported by the International Cooperation Project between the University of Palermo (Italy) and Conicit-Venezuela. 相似文献
18.
Elias Flytzanis 《Monatshefte für Mathematik》1995,119(4):267-273
For a bounded linear operator in a complex separable Hilbert space we show that it accepts an invariant Borel probability measure of square integrable norm whose essential support spans the space, iff its eigenvectors with unimodular eigenvalues span the space.This research has been supported by the Economic Research Center (KOE) of the Athens University of Economics and Business. 相似文献
19.
Todor Gramchev Petar Popivanov Massafumi Yoshino 《Annali di Matematica Pura ed Applicata》1996,170(1):103-131
Summary
In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund. 相似文献
20.
If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation
theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as
translates of the same ball.
Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di
Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica
e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva,
Israel. 相似文献