Partial hypoellipticity for a class of abstract differential complexes on Banach space scales

Abstract

In this article we give sufficient conditions for the hypoellipticity in the first level of the abstract complex generated by the differential operators $L_j=\frac{\partial }{\partial t_j}+\frac{\partial \varphi }{\partial t_j}(t,A)A$, $j=1,2,\dots ,n$, where $A:D(A)\subset X\to X$ is a sectorial operator in a Banach space $X$, with $\Re \sigma (A)>0$, and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of $A^{-1}$ with coefficients in $C^{\infty }(\Omega )$, $\Omega$ being an open set of ${\mathbb{R}}^n$ with $n\in \mathbb{N}$ arbitrary. Analogous complexes have been studied by several authors in this field, but only in the case $n=1$ and with $X$ a Hilbert space. Therefore, in this article, we provide an improvement of these results by treating the question in a more general setup. First, we provide sufficient conditions to get the partial hypoellipticity for that complex in the elliptic region. Second, we study the particular operator $A=1-\Delta :W^{2,p}({\mathbb{R}}^N)\subset L^p({\mathbb{R}}^N)\to L^p({\mathbb{R}}^N)$, for $1\le p\le 2$, which will allow us to solve the problem of points which do not belong to the elliptic region.