On Hessian measures for non-commuting vector fields

TL;DR

This paper extends Hessian-measure theory to the subelliptic setting with non-commuting vector fields of step 2, establishing the weak continuity of the 2-Hessian operator for locally integrable, $k$-convex functions (in particular the operator $F_2[u]+\alpha\sum_{i<j}[X_i,X_j]^2u$). The approach develops a divergence structure and a monotonicity formula via a functional $\mathscr F_2$, under hypotheses including anti-self-adjointness, the Hörmander condition, and vanishing second commutators, enabling the definition of a Hessian measure $\mu_2[u]$ and its weak continuity under $L^1_{loc}$ convergence. Gradient estimates for $k$-convex functions are derived by connecting $k$-convexity to subharmonicity with respect to the sub-Laplacian $\Delta_X$ and subelliptic $p$-Laplacians $\Delta_p$, yielding local $L^q$ and Hölder bounds with exponents depending on the homogeneous dimension $Q$ and on $m$ and $k$. The framework extends to general $\phi^k(\Omega)$, showing embeddings into $S^{1,q}_{loc}(\Omega)$ and, under stronger commutator conditions, that the symmetric Hessian $X_s^2u$ is a Radon measure and the full Hessian $X^2u$ is measure-valued with $\mu_2$ extending to $\phi^2(\Omega)$; in Carnot groups this yields almost everywhere twice differentiability with respect to $X$ for appropriate $k$.

Abstract

Previous results on Hessian measures by Trudinger and Wang are extended to the subelliptic case. Specifically we prove the weak continuity of the 2-Hessian operator, with respect to local L1 convergence, for a system of m vector fields of step 2 and derive gradient estimates for the corresponding k-convex functions, k=1,2....m, for arbitrary step.

Theorems & Definitions (10)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['thm11']}