Derivative Formulae and Poincar\'e Inequality for Kohn-Laplacian Type Semigroups

TL;DR

This paper extends derivative formulae and Poincaré inequalities to a class of semigroups generated by subelliptic operators related to the Kohn-Laplacian, generalizing results from the Heisenberg group setting.

Contribution

It introduces new Bismut-type and Driver-type derivative formulae for these semigroups, with applications to gradient estimates and coupling properties, extending prior work on the Heisenberg group.

Findings

Established new derivative formulae for subelliptic semigroups.

Derived gradient estimates and coupling properties.

Extended results from the Heisenberg group to more general settings.

Abstract

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator $L:=\ff 1 2 \sum_{i=1}^m X_i^2$ on $\R^{m+d}:= \R^m\times\R^d$ is investigated, where $$X_i(x,y)= \sum_{k=1}^m \si_{ki} \pp_{x_k} + \sum_{l=1}^d (A_l x)_i\pp_{y_l},\ \ (x,y)\in\R^{m+d}, 1\le i\le m$$ for $\si$ an invertible $m\times m$-matrix and $\{A_l\}_{1\le l\le d}$ some $m\times m$-matrices such that the H\"ormander condition holds. We first establish Bismut-type and Driver-type derivative formulae with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.