Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients

Abstract

Let $a_1,\dots ,a_{d^{\prime }}$ be an algebraic basis of rank $r$ in a Lie algebra $\mathfrak{g}$ of a connected Lie group $G$ and let $A_{\tau }$ be the left differential operator in the direction $a_i$ on the $L_p$-spaces with respect to the left, or right, Haar measure, where $p\in [1,\infty ]$. We consider $m$-th order operators

with complex variable bounded coefficients $c_{\alpha }$ which are subcoercive of step $r$, i.e., for all $g\in G$ the form obtained by fixing the $c_{\alpha }$ at $g$ is subcoercive of step $r$ and the ellipticity constant is bounded from below uniformly by a positive constant. If the principal coefficients are $m$-times differentiate in $L_{\infty }$ in the directions of  $a_1,\dots ,a_{d^{\prime }}$we prove that the closure of $H$ generates a consistent interpolation semigroup $S$ which has a kernel. We show that $S$ is holomorphic on a non-empty $p$-independent sector and if $H$ is formally self-adjoint then the holomorphy angle is $\pi /2$. We also derive ‘Gaussian’ type bounds for the kernel and its derivatives up to order $m-1$.