[Paper Review] Poincar{é} and Sobolev inequalities for differential forms in Heisenberg groups

Annalisa Baldi, Bruno Franchi+1 more|arXiv (Cornell University)|Nov 27, 2017

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes sharp Poincaré and Sobolev inequalities for differential forms on Heisenberg groups using Rumin's complex, which replaces the de Rham complex to restore scale invariance under anisotropic dilations. The key contribution is a global homotopy formula for Rumin's complex on bounded geometry contact manifolds that improves the regularity of solutions, enabling optimal $L^p$ estimates for $d_c$-closed forms.

ABSTRACT

Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.

Motivation & Objective

• To establish $L^p$-Poincaré and Sobolev inequalities for differential forms on Heisenberg groups, addressing the lack of scale invariance in the standard de Rham complex.
• To overcome the breakdown of standard Sobolev and Poincaré inequalities on sub-Riemannian manifolds due to non-Euclidean geometry and anisotropic scaling.
• To construct a global homotopy formula for Rumin's complex on bounded geometry contact manifolds that enhances the differentiability of solutions to $d_c\phi = \omega$.
• To provide quantitative estimates for the solution operator $T_M$ and error term $S_M$ in terms of Sobolev norms, ensuring continuity in $L^p$ and Sobolev spaces.
• To extend the applicability of $L^p$-cohomological tools to sub-Riemannian contact manifolds by proving regularizing properties of the homotopy operator.

Proposed method

• Utilizes Rumin's complex, a second-order differential complex on contact manifolds, which restores scale invariance under anisotropic dilations $\delta_t$ by combining quotient and annihilator bundles.
• Applies a partition of unity subordinate to a finite cover by contact charts, each diffeomorphic to a Heisenberg ball, to localize the problem.
• Constructs a local homotopy formula on each Heisenberg chart using pullbacks of operators $T$ and $S$ defined via Rumin's calculus, ensuring $d_c T + T d_c + S = I$.
• Combines local operators via a partition of unity to define a global homotopy operator $T_M = \sum_{i=0}^{k-1} S^i T$, with error term $S_M = S^k$, ensuring regularizing properties.
• Establishes continuity of $T_M$ and $S_M$ between $L^p$ and Sobolev spaces $W^{m,p}$, using local estimates and the bounded geometry of the manifold.
• Relies on the fact that $\phi_j^\# \alpha$ and $\|\alpha\|_{W^{m,p}}$ are equivalent on $\mathbb{H}^n$ for $-k \leq m \leq k$, enabling transfer of estimates from the model space.

Experimental results

Research questions

• RQ1 Can Poincaré and Sobolev inequalities be established for differential forms on Heisenberg groups using Rumin's complex instead of the de Rham complex?
• RQ2 How can one construct a global homotopy formula for Rumin's complex on bounded geometry contact manifolds that improves the regularity of solutions to $d_c\phi = \omega$?
• RQ3 What are the optimal $L^p$-Sobolev estimates for the solution operator $T_M$ and error term $S_M$ in the homotopy decomposition $I = d_c T_M + T_M d_c + S_M$?
• RQ4 To what extent does the regularizing property of $S_M = S^k$ enhance the differentiability of solutions in Sobolev spaces?
• RQ5 How do the continuity properties of $T_M$ and $S_M$ depend on the form degree $h$ and the critical degrees $h = n$ and $h = n+1$?

Key findings

• The paper proves that for any bounded $C^k$-geometry sub-Riemannian contact manifold with $k \geq 2$, the identity operator decomposes as $I = d_c T_M + T_M d_c + S_M$, where $T_M = \sum_{i=0}^{k-1} S^i T$ and $S_M = S^k$.
• The solution operator $T_M$ maps $W^{-1,p}(M,E_0^{h+1})$ to $L^p(M,E_0^h)$ for $h \neq n$, and $W^{-2,p}(M,E_0^{n+1})$ to $L^p(M,E_0^n)$, ensuring control over weak solutions.
• For $h \neq n+1$, $T_M$ maps $L^p(M,E_0^h)$ to $W^{1,p}(M,E_0^{h-1})$, and for $h = n+1$, it maps $L^p(M,E_0^{n+1})$ to $W^{2,p}(M,E_0^n)$, showing improved regularity.
• The error term $S_M$ maps $L^p(M,E_0^h)$ to $W^{k,p}(M,E_0^h)$, providing maximal regularizing properties compatible with the $C^k$-regularity of the manifold.
• The continuity of $T_M$ and $S_M$ is established via local estimates on Heisenberg charts, using the equivalence of Sobolev norms under pullbacks and the bounded geometry assumption.
• The results extend to global $L^p$-Sobolev and Poincaré inequalities for $d_c$-closed forms on Heisenberg balls, with constants independent of the domain size due to the homotopy formula.