Partial Gaussian bounds for degenerate differential operators II

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

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