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Annalisa Baldi

Publications and source records attributed to Annalisa Baldi.

14 recordsLinked to original sources
arXiv2026

$L^p$-Hodge decomposition and global integral estimates on the Cartan group

The study of Sobolev and Poincar\'e inequalities for differential forms in Carnot groups and in the more general sub-Riemannian setting is still an open problem in its full generality. One may conjecture that, for general Carnot groups, these inequalities are expressed in terms of suitable graded Lebesgue norms. In recent years, many results have been obtained, both in the Euclidean setting and in the Heisenberg groups, as well as for contact manifolds with bounded geometry. There are also some results for general Carnot groups; however, these do not cover the problem in its full generality. In this paper, we consider a particular Carnot group, the so-called Cartan group (a free Carnot group, of step $3$ with $2$ generators), which provides a natural testing ground for these questions, since its step-three structure already exhibits several phenomena that do not occur in the Heisenberg groups. In this setting, we are able to prove global Poincar\'e and Sobolev-Gaffney inequalities for differential forms. With the aim of obtaining sharp estimates, we replace the de Rham complex of differential forms with the Rumin complex. The case $p>1$ is carried out after establishing an $L^p$-Hodge decomposition with homogeneous Sobolev classes. We are able to consider also the endpoint case $p=1$; however, as in Euclidean setting, when $p=1$, the operator we deal with provides only weak-type estimates which do not yield a Hodge decomposition analogous to the case $p>1$. Therefore, in this situation the proof follows a different approach, relying on a recent result proved in \cite{BT}.

math.AP↗
arXiv2025

$L^p$-Hodge Decomposition with Sobolev classes in Sub-Riemannian Contact Manifolds

Let $1<p<\infty$. In this article we establish an $L^p$-Hodge decomposition theorem on sub-Riemannian compact contact manifolds without boundary, related to the Rumin complex of differential forms. Given an $L^p$- Rumin's form, we adopt an approach in the spirit of Morrey's book to obtain a decomposition with higher regular ``primitives'' i.e. that belong to suitable Sobolev classes. Our proof relies on recent results obtained in [4] and [6].

math.AP↗
arXiv2024

A representation formula for regular functions on the characteristic plane of the second Heisenberg group

The aim of this paper is to study a Laplace-type operator and its fundamental solution on the characteristic plane in the Heisenberg group $\mathbb{H}^2$. We introduce a conformal version of the Laplacian and we prove that the distance induced by the immersion in the ambient space is a good approximation of its fundamental solution. We provide in particular a representation formula for smooth functions in terms of the gradient of the function and the gradient of the approximated fundamental solution. This representation formula in the plane is stable up to its characteristic point.

math.AP↗
arXiv2024

Primitives of volume forms in Carnot groups

In the Euclidean space it is known that a function $f\in L^2$ of a ball, with vanishing average,is the divergence of a vector field $F\in L^2$ with$$\| F\|\_{ L^2(B)} \le C \|f\|\_{L^2(B)}.$$In this Note we prove a similar result in any Carnot group $\mathbb{G}$ for a vanishing average $f\in L^p$, $1\le p < Q$, where $Q$ is the so-called homogeneous dimension of $\mathbb{G}$.

math.AP↗
arXiv2024

Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates

On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered by Rumin, who introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential $d_c$ is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.

math.AP↗
arXiv2022

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 Φ, vanishing at infinity, that solves the equation divHΦ = 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.

math.FA↗
arXiv2022

Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry

In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

math.DG↗
arXiv2021

Cohomology of annuli, duality and $L^\infty$-differential forms on Heisenberg groups

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups $\mathbb{H}^n$ for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that $d_c$ is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable $p$ and $q$, we mean that every exact differential form $ω$ in $L^p$ admits a primitive $ϕ$ in $L^q$ such that$\|ϕ\|_{L^{q}}\leq C\ \|ω\|_{L^{p}}$. The cases of the norm $L^p$, $p\ge 1$ and $q<\infty$ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where $q=\infty$: it is remarkable that, unlike in the scalar case, when the degree of the forms $ω$ is at least $2$, we can take $q=\infty$ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting $\mathbb{R}^N$ ($p=N$ and $q=\infty$) was proven by Bourgain and Brezis.

math.CA↗
arXiv2019

$L^1$-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincar{é} 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.

math.DG↗
arXiv2017

Schauder estimates at the boundary for sub-laplacians in Carnot groups

In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison, is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to built a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.

math.AP↗
arXiv2015

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 $δ$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.

math.DG↗