Grushin Operator on Infinite Dimensional Homogeneous Lie Groups
M. E. Egwe, J. I. Opadara
TL;DR
This work extends Grushin-type sub-Riemannian analysis to infinite-dimensional homogeneous Lie groups. By framing the problem on a locally convex manifold with an infinite family of complete, skew-adjoint vector fields satisfying Hörmander’s condition, it constructs a robust framework using representation theory to lift to a simply connected Lie group and derive Gaussian heat kernel bounds and a doubling condition. The main theorem shows global Poincaré inequalities, two-sided Gaussian heat-kernel bounds, and L^p bounds for the Riesz transform, along with spectral multiplier results, thereby generalizing finite-dimensional sub-Riemannian results to the infinite-dimensional setting. The results have potential implications for analysis on infinite-dimensional spaces and for stochastic and geometric problems in such contexts.
Abstract
A collection of infinite dimensional complete vector fields $\left\{V_i\right\}_{i=1}^{\infty}$ acting on a locally convex manifolds $M$ on which a smooth positive measure $μ$ is defined was considered. It was assumed that the vector fields generates an infinite dimensional Lie algebra $\mathfrak{g}$ and satisfies H$\ddot{o}$rmander's condition. The sum of squares of Grushin operators related to the vector fields was examined and the operator is then considered as the generalized Grushin operator. The paramount proofs were Poincar$\acute{e}$ inequality, Gaussian two-bounded estimate for the related heat kernels and the doubling condition for the metric defined by the underlying vector fields.