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 $Ω\subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on $\Omega$. Further, let $χ\in C_{\rm b}^\infty(\mathds{R}^d)$ be such that an $\epsilon$-neighbourhood of $\supp χ$ is contained in $\Omega$. Let $\nu \in (0,1]$ and suppose that the ${c_{kl}}_{|\Omega }\in C^{0,\nu }(\Omega )$. Then we prove (Hölder) 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.