$L^p-L^q$ ESTIMATES FOR PARABOLIC SYSTEMS IN NON-DIVERGENCE FORM WITH VMO COEFFICIENTS

Abstract

Consider a parabolic $N\times N$-system of order $m$ on $\mathbb{R}^n$ with top-order coefficients $a_\alpha \in \mathrm{VMO} \cap L^\infty$. Let $1<p,q < \infty$ and let $\omega$ be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution $u$ satisfying

$$\|u'\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} + \|\mathcal{A} u\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} \le C \|f\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)},$$

where $\mathcal{A} u = \sum_{|\alpha| \le m}a_\alpha D^\alpha u$ and $J=[0,\infty)$. In particular, choosing $\omega =1$, the realization of $\mathcal{A}$ in $L^p({\mathbb{R}}^n)^N$ has maximal $L^p-L^q$ regularity.