Second-order strongly elliptic operators on Lie groups with H{\"o}lder continuous coefficients

Research Report MRR96-038

Second-order strongly elliptic operators on Lie groups with Hölder continuous coefficients

A.F.M. ter Elst and Derek W. Robinson

Abstract: Let G be a connected Lie group with Lie algebra $\gotg$ and $a_1,\ldots,a_{d'}$ an algebraic basis of $\gotg$. Further let

denote the generators of left translations, acting on the

-spaces $L_p(G\,;dg)$ formed with left Haar measure dg, in the directions

. We consider second-order operators

$\begin{displaymath}-\sum_{i,j=1}^{d'} A_i \, c_{ij} \, A_j + \sum_{i=1}^{d'} (c_i \, A_i + A_i \, c'_i) + c_0 \, I \end{displaymath}$

in divergence form corresponding to a quadratic form with complex coefficients bounded Hölder continuous principal coefficients $c_{ij}$ and lower order coefficients $c_{i}$ , $c'_{i}$ , $c_{0}\in L_{\infty}$ such that the matrix $C=(c_{ij})$ of principal coefficients satisfies the subellipticity condition

$\begin{displaymath}\Re C = 2^{-1}\Big(C+C^*\Big)\geq \mu I>0 \end{displaymath}$

uniformly over G.

We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the -spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.

Similar theorems are proved for operators

$\begin{displaymath}-\sum_{i,j=1}^{d'} c_{ij} \, A_i \, A_j + \sum_{i=1}^{d'} c_i \, A_i + c_0 \, I \end{displaymath}$

in nondivergence form for which the principal coefficients are at least once differentiable.

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