\(L^1\)-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups (Q2309115)

The celebrated Sobolev inequality, called ($p$-Sobolev), claims that, for every $1\leq p<n$, there exists a constant $C(n,p)$ such that all smooth compactly supported functions $u$ on $\mathbb{R}^{n}$ abide by \[ \left\Vert u\right\Vert_{q}\leq C(n,p) \left\Vert \nabla u\right\Vert_{p}, \] provided that $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$. The celebrated Poincaré inequality, called ($p$-Poincaré), claims that there exists a constant $c_{u}$ such that \[ \left\Vert u-c_{u}\right\Vert_{q}\leq C(n,p) \left\Vert \nabla u\right\Vert_{p}, \] provided that $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$. The principal objective in this paper is to generalize these inequalities to differential forms. The question considered here is to ask whether, given a closed differential $h$-form $\varpi$ in $L^{p}(\mathbb{R}^{n})$, there exists an $(h-1)$-form $\phi$ in $L^{q}(\mathbb{R}^{n})$ with $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$ such that $d\phi=\varpi$ and \[ \left\Vert \phi\right\Vert_{q}\leq C(n,p,h) \left\Vert\varpi\right\Vert_{p}. \] This paper establishes ($1$-Poincaré) for $h$-forms of degree $h<n$ in de Rham complex $(\Omega^{\bullet},d)$, relying on \textit{L. Lanzani} and \textit{E. M. Stein}'s observation [Math. Res. Lett. 12, No. 1, 57--61 (2005; Zbl 1113.26015)] that the duality estimate underlying \textit{J. Bourgain} and \textit{H. Brezis}' result [J. Eur. Math. Soc. (JEMS) 9, No. 2, 277--315 (2007; Zbl 1176.35061)] descends $(n-1)$-forms to forms of lower degrees, and the resulting Gagliardo and Nirenberg inequalities [\textit{L. Nirenberg}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 13, 115--162 (1959; Zbl 0088.07601); C.I.M.E., Principio di Minimo e sue Applicazioni alle Equazioni funzionali 1--48 (1960; Zbl 0108.10001); \textit{E. Gagliardo}, Ric. Mat. 8, 24--51 (1959; Zbl 0199.44701)]. The approach generalizes to the non-commutative Heisenberg groups $\mathbb{H}^{n}$ endowed with \textit{M. Rumin}'s complex [J. Differ. Geom. 39, No. 2, 281--330 (1994; Zbl 0973.53524)] $(E_{0}^{\bullet},d_{c})$. As for Heisenberg groups, one can exploit Lanzani and Stein's type arguments [\textit{A. Baldi} and \textit{B. Franchi}, J. Funct. Anal. 265, No. 10, 2388--2419 (2013; Zbl 1282.35391); \textit{A. Baldi} et al., Matematiche 75, No. 1, 167--194 (2020; Zbl 1436.58003)], relying on \textit{S. Chanillo} and \textit{J. Van Schaftingen}'s inequality [Math. Res. Lett. 16, No. 2--3, 487--501 (2009; Zbl 1184.35119)]. The main results, namely global Poincaré and Sobolev inequalities (Theorem 1.1) and their local verions (Theorem 1.2), are stated for both Euclidean and Heisenberg cases. Smoothing homotopies on Riemannian or contact subRiemannian manifolds of bounded geometry are also constructed.