اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدG^ateaux differentiability on infinite-dimensional Carnot groups
65
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper contributes to the generalization of Rademachers differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce the notion of metric scalable groups which are our infinite-dimensional analogues of Carnot groups. The groups in which we will mostly be interested are the ones that admit a dense increasing sequence of (finite-dimensional) Carnot subgroups. In fact, in each of these spaces we show that every Lipschitz function has a point of G^{a}teaux differentiability. We provide examples and criteria for when such Carnot subgroups exist. The proof of the main theorem follows the work of Aronszajn and Pansu.
قيم البحث
اقرأ أيضاً
We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set N such that every real-valued Lipschitz map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a max
imal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.
We develop aspects of geometric control theory on Lie groups G which may be infinite dimensional, and on smooth G-manifolds M modelled on locally convex spaces. As a tool, we discuss existence and uniqueness questions for differential equations on M
given by time-dependent fundamental vector fields which are L^1 in time. We then discuss the closures of reachable sets in M for controls in the Lie algebra of G, or within a compact convex subset of the Lie algebra. Regularity properties of the Lie group G play an important role.
We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
Some aspects of a mathematical theory of rigidity and flexibility are developed for general infinite frameworks and two main results are obtained. In the first sufficient conditions, of a uniform local nature, are obtained for the existence of a prop
er flex of an infinite framework. In the second it is shown how continuous paths in the plane may be simulated by infinite Kempe linkages.
We consider mappings $f:Gsupset Urightarrow G$ where $G$ and $G$ are Carnot groups and U is an open subset. We prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings, establishing (partial) rigidity or (partial)
regularity theorems, depending on the context. In particular, we prove the quasisymmetric rigidity conjecture for Carnot groups which are not rigid in the sense of Ottazzi-Warhurst.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
