## Annals of Functional Analysis

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

E. R. Aragão-Costa

#### 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.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 262-276.

Dates
Accepted: 23 August 2018
First available in Project Euclid: 22 March 2019

https://projecteuclid.org/euclid.afa/1553241620

Digital Object Identifier
doi:10.1215/20088752-2018-0023

Mathematical Reviews number (MathSciNet)
MR3941387

Zentralblatt MATH identifier
07083894

#### Citation

Aragão-Costa, E. R. Partial hypoellipticity for a class of abstract differential complexes on Banach space scales. Ann. Funct. Anal. 10 (2019), no. 2, 262--276. doi:10.1215/20088752-2018-0023. https://projecteuclid.org/euclid.afa/1553241620

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