On second-order periodic elliptic operators in divergence form

We consider second-order, strongly elliptic, operators with complex coeffic ients in divergence form on R-d. We assume that the coefficients are all pe riodic with a common period. If the coefficients are continuous we derive G aussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its Holder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the L-p-spaces with p epsil on <1, infinity >. Secondly if the coefficients are Holder continuous we pr ove that the first-order derivatives of the kernel satisfy good Gaussian bo unds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, a nd only if, the second-order Riesz transforms are bounded. Finally if the t hird-order derivatives exist with good bounds then the coefficients must be constant.