On second-order periodic elliptic operators in divergence form

Abstract

We consider second-order, strongly elliptic, operators with complex coecients in divergence form on R d . We assume that the coecients are all periodic with a common period. If the coecients are continu- ous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its Holder deriva- tives. Moreover, we show that the first-order Riesz transforms are bounded on the Lp-spaces with p 2 h1,1i. Secondly if the coe- cients are Holder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coecients 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 coecients must be constant.