A fundamental solution for a subelliptic operator in Finsler geometry

TL;DR

The paper develops a nonlinear PDE framework at the crossroads of Finsler and sub-Riemannian geometry by introducing an anisotropic Minkowski gauge $\Theta$ and its dual $\Theta^0$ on $\mathbb{R}^N=\mathbb{R}^m\times\mathbb{R}^k$. It derives how the associated operator $\mathscr L_{\alpha,p}$ acts on radial profiles $F(\rho)$ with $\rho=\Theta^0$, establishing explicit one-dimensional reductions and constructing fundamental solutions $\mathscr G_{\alpha,p}$ that generalize the Euclidean and horizontal subelliptic kernels. The main results show that $\mathscr G_{\alpha,p}(z,\sigma)=C_{\alpha,p}\,\rho^{-(Q-p)/(p-1)}$ for $p\neq Q$, with a logarithmic form when $p=Q$, and identify the limiting behaviors as $\alpha\to 0$ (classical $p$-Laplacian in $\mathbb{R}^N$) and $\alpha\to 1$ (Korányi–Kaplan-type gauge on a Heisenberg-type group). This framework provides a unified approach to nonlinear diffusion in mixed Finsler-sub-Riemannian settings and yields explicit kernels useful for geometric PDEs and CR-type problems. The results connect classical Euclidean, sub-Riemannian, and conformal geometries via precise anisotropic scaling and dual-transform constructions.

Abstract

We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge $Θ$ for which we introduce a suitable notion of Legendre transform $Θ^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $Θ^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof