We investigated properties of the Riesz transform and large time
behaviour of derivatives of the heat kernels corresponding to
operators with periodic coefficient. The obtained results are
interesting
and very surprising.
For example we proved that the
second-order Riesz transforms are bounded if, and only if,
the coefficients are divergence-free and third-order
Riesz transforms are bounded if, and only if,
the coefficients are constant.