Subgroup coverings of some linear groups

Research Report MRR99-028

Subgroup coverings of some linear groups

R. A. Bryce, V. Fedri and L. Serena

Abstract: We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on $\Ri^d$. We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds with the correct small and large time asymptotic behaviour on the heat kernel and all its H{\"o}lder derivatives. Secondly if the coefficients are H{\"o}lder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds and the first-order Riesz transforms are bounded on the $L_p$-spaces with $p\in\langle1,\infty\rangle$. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant.

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