145 research outputs found
Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups
The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly
supported function on Euclidean n-space is controlled by the L 1-norm of its
gradient. The generalization to differential forms (due to Lanzani & Stein and
Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a
compactly supported differential h-form is controlled by the L 1-norm of its
exterior differential du and its exterior codifferential u (in special
cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend
this result to Heisenberg groups in the framework of an appropriate complex of
differential forms
-Poincar\'e and Sobolev inequalities for differential forms in Euclidean spaces
In this paper, we prove Poincar\'e and Sobolev inequalities for differential
forms in . The singular integral estimates that it is
possible to use for , , are replaced here with inequalities which go
back to Bourgain-Brezis.Comment: Accepted for publication in Science China Mathematics. arXiv admin
note: text overlap with arXiv:1902.0481
Sobolev-Poincaré inequalities for differential forms and currents
In this note we collect some results in R^n about (p,q) Poincaré and Sobolev inequalities for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the case p=1. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p=1 we obtain  Poincaré and Sobolev inequalities for Euclidean currents
The distributional divergence of horizontal vector fields vanishing at infinity on Carnot groups
We define a BV -type space in the setting of Carnot groups (i.e., simply
connected Lie groups with stratified nilpotent Lie algebra) that allows one to
characterize all distributions F for which there exists a continuous horizontal
vector field {\Phi}, vanishing at infinity, that solves the equation divH{\Phi}
= F. This generalize to the setting of Carnot groups some results by De Pauw
and Pfeffer, [12], and by De Pauw and Torres, [13], for the Euclidean setting.Comment: 24 page
-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups
In this paper, we prove interior Poincar{\'e} and Sobolev inequalities in
Euclidean spaces and in Heisenberg groups, in the limiting case where the
exterior (resp. Rumin) differential of a differential form is measured in L 1
norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree.
The singular integral estimates are replaced with inequalities which go back to
Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in
Heisenberg groups
L1-Poincar\ue9 and Sobolev inequalities for differential forms in Euclidean spaces
In this paper, we prove Poincar\ue9 and Sobolev inequalities for differential forms in L1(\u211dn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007)
Orlicz spaces and endpoint Sobolev-Poincaré inequalities for differential forms in Heisenberg groups
In this paper we prove Poincar´e and Sobolev inequalities for differential forms in the Rumin’s contact complex on Heisenberg groups. In particular, we deal with endpoint values of the exponents, obtaining finally estimates akin to exponential Trudinger inequalities for scalar function. These results complete previous results obtained by the authors away from the exponential case. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. They have also applications to regularity issues for partial differential equations
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