David S. Tartakoff
Global (and Local) Analyticity for Second Order Operators Constructed from Rigid Vector Fields on Products of Tori
The paper is published: Trans. Am. Math. Soc. 348, 7 (1996) 2577-2583

MSC:

58G07 Relations with hyperfunctions

35N15 $\overline\partial$-Neumann problem and generalizations; formal complexes, See also {32F20 and 58G05}

35B65 Smoothness/regularity of solutions of PDE

32F20 $\overline\partial$- and $\overline\partial_b$-Neumann problems, See also {35N15}

Abstract: We prove global analytic hypoellipticity on a product of tori
for partial differential operators which are constructed as
rigid (variable coefficient) quadratic polynomials in real
vector fields satisfying the H\"ormander condition
and where $P$ satisfies a `maximal' estimate.
We also prove an analyticity result that is local in some
variables and global in others for operators whose
prototype is
$$ P= \left({\partial \over {\partial
x_1}}\right)^2 + \left({\partial \over {\partial x_2}}\right)^2
+ \left(a(x_1,x_2){\partial \over {\partial t}}\right)^2.$$
(with analytic $a(x), a(0)=0,$ naturally, but not identically
zero). The results, because of the flexibility of the methods,
generalize recent work of Cordaro and Himonas in
\cite{Cordaro-Himonas 1994} and Himonas in
\cite{Himonas 199X} which showed that certain operators
known not to be locally analytic hypoelliptic (those of
Baouendi and Goulaouic
\cite{Baouendi-Goulaouic 1971}, Hanges and Himonas
\cite{Hanges-Himonas 1991}, and Christ \cite{Christ 1991a})
were {\it globally} analytic hypoelliptic on products of tori.

Keywords: global analytic hypoellipticity, product of tori, partial differential operators