E. R. Aragão-Costa. Source: Annals of Functional Analysis, Volume 10, Number 2, 262--276.Abstract:
In this article we give sufficient conditions for the hypoellipticity in the first level of the abstract complex generated by the differential operators $L_{j}=\frac{\partial}{\partial t_{j}}+\frac{\partial\phi}{\partialt_{j}}(t,A)A$ , $j=1,2,\ldots,n$ , where $A:D(A)\subset X\longrightarrow X$ is a sectorial operator in a Banach space $X$ , with $\Re\sigma(A)\gt 0$ , and $\phi=\phi(t,A)$ 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 $A=1-\Delta:W^{2,p}({\mathbb{R}}^{N})\subset L^{p}({\mathbb{R}}^{N})\longrightarrow L^{p}({\mathbb{R}}^{N})$ , for $1\leq p\leq2$ , which will allow us to solve the problem of points which do not belong to the elliptic region.