Partial Gaussian bounds for degenerate differential operators II

Abstract

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.