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In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups
, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex
, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of
. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.
This chapter presents the case of a 63-year-old right-handed woman who was referred for language difficulties. Significant deficits were present in executive and visuospatial functions, as well as language. Language was fluent, characterized by frequent semantic paraphasias, jargonophasia, and occasional word finding difficulties, and decreased understanding. The initial clinical diagnostic impression was that of a progressive but primarily focal cognitive disorder involving predominantly language: primary progressive aphasia (PPA). Autopsy confirmed the presence of a large left frontal hematoma with intraventricular spillage. Diffuse staining for b-amyloid was found in meningeal and cortical arteries, consistent with severe cerebral amyloid angiopathy (CAA). The differential diagnosis for PPA includes variants of fronto-temporal dementia (FTD) and atypical Alzheimer's disease (AD). The clinical profile, with the early onset, strong family history, absence of severe memory complaints on history and amnestic deficits on testing, behavioral changes, and a predominance of executive dysfunction, swayed the diagnosis towards FTD.
In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.
We derive Sobolev--Poincaré inequalities that
estimate the $L^q(d\mu)$ norm of a function on
a metric ball when $\mu$ is an arbitrary Borel
measure. The estimate is in terms of the
$L^1(d\nu)$ norm on the ball of a vector field
gradient of the function, where $d\nu/dx$ is a
power of a fractional maximal function of $\mu$.
We show that the estimates are sharp in several
senses, and we derive isoperimetric inequalities
as corollaries. 1991 Mathematics Subject Classification:
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