  Let $A = - \sum \partial_k \, c_{kl} \, \partial_l$ be a degenerate sectorial
differential operator with complex bounded mesaurable coefficients. Let $\Omega
\subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on
$\Omega$. Further, let $\chi \in C_{\rm b}^\infty(\mathds{R}^d)$ be such that
an $\varepsilon$-neighbourhood of $\supp \chi$ is contained in $\Omega$. Let
$\nu \in (0,1]$ and suppose that the ${c_{kl}}_{|\Omega} \in
C^{0,\nu}(\Omega)$. Then we prove (H\"older) Gaussian kernel bounds for the
kernel of the operator $u \mapsto \chi \, S_t (\chi \, u)$, where $S$ is the
semigroup generated by $-A$. Moreover, if $\nu = 1$ and the coefficients are
real, then we prove Gaussian bounds for the kernel of the operator $u \mapsto
\chi \, S_t u$ and for the derivatives in the first variable. Finally we show
boundedness on $L_p(\mathds{R}^d)$ of various Riesz transforms.
