Partial Gaussian bounds for degenerate differential operators II

TL;DR

This paper establishes Gaussian bounds for kernels of degenerate differential operators with Hölder continuous coefficients, extending known results and analyzing Riesz transforms on Lp spaces.

Contribution

It provides new Gaussian kernel bounds for degenerate sectorial operators with Hölder continuous coefficients and studies associated Riesz transforms.

Findings

Proves Hölder Gaussian kernel bounds for localized operators.

Establishes Gaussian bounds for kernels and derivatives when coefficients are real and Lipschitz.

Shows boundedness of Riesz transforms on Lp spaces.

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.