Annals of Functional Analysis

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

E. R. Aragão-Costa

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In this article we give sufficient conditions for the hypoellipticity in the first level of the abstract complex generated by the differential operators Lj=tj+ϕtj(t,A)A, j=1,2,,n, where A:D(A)XX is a sectorial operator in a Banach space X, with σ(A)>0, and ϕ=ϕ(t,A) is a series of nonnegative powers of A1 with coefficients in C(Ω), Ω being an open set of Rn with nN 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=1Δ:W2,p(RN)Lp(RN)Lp(RN), for 1p2, which will allow us to solve the problem of points which do not belong to the elliptic region.

Article information

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

Received: 21 February 2018
Accepted: 23 August 2018
First available in Project Euclid: 22 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46Fxx: Distributions, generalized functions, distribution spaces [See also 46T30]
Secondary: 47Dxx: Groups and semigroups of linear operators, their generalizations and applications 47Fxx: Partial differential operators [See also 35Pxx, 58Jxx]

partial hypoellipticity complex of differential operators sectorial operator scale of fractional power spaces


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.

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