Second order subelliptic operators on Lie groups I: complex uniformly continuous principal coefficients

Research Report MRR96-035

Second order subelliptic operators on Lie groups I: complex uniformly continuous principal 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

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

corresponding to a quadratic form with complex coefficients $c_{ij}$ , $c_{i}$ , $c'_{i}$ , $c_{0}\in L_{\infty}$ 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.

If the principal coefficients $(c_{ij})$ are right uniformly continuous then we prove that H generates a strongly continuous holomorphic semigroup S on with a kernel K which satisfies Gaussian bounds

$\begin{displaymath}|K_z(g\,;h)| \leq a \, |z|^{-D'/2} e^{\omega|z|} e^{-b(|gh^{-1}|')^2|z|^{-1}} \end{displaymath}$

for $g,h\in G$ and z in a subsector of the sector of holomorphy. Moreover, the kernel is Hölder continuous and for each $\nu\in\langle0,1\rangle$ and $\kappa>0$ one has estimates

for $g,h,k,l\in G$ and z in the subsector with $|k|'+|l|'\leq\kappa\,|z|^{1/2}+2^{-1}|gh^{-1}|'$ . These results are established by a blend of elliptic techniques in which De Giorgi estimates and Morrey-Campanato spaces play an important role.

If G is one-dimensional stronger results are obtained; Hölder continuity is valid for all $\nu\in\langle0,1]$ and no continuity of the coefficients $(c_{ij})$ is required.

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