Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932
Note Mat. 31 (2011) n. 1, 1–13. doi:10.1285/i15900932v31n1p1
On the global C
∞
and Gevrey
hypoellipticity on the torus of some classes
of degenerate elliptic operators
Angela A. Albanese
Dipartimento di Matematica “E. De Giorgi”
Universit`a del Salento
Via Per Arnesano
I–73100 Lecce, Italy
angela.albanese@unisalento.it
Received: 06/06/2010 ; accepted: 14/11/2010.
Abstract. In this paper we prove the global C
∞
and Gevrey hypoellipticity on the multi-
dimensional torus for some classes of degenerate elliptic operators.
Keywords: Global hypoellipticity, Liouville number, Diophantine condition
MSC 2000 classification: 35D10, 35H10, 35J70, 46E10, 46F05, 58J99
Dedicated to the memory of V.B. Moscatelli
1 Introduction
In the last years many papers are concerned with the study of the global hypoellipticity
and solvability of linear partial differential operators on compact manifolds, e.g., on the torus,
in large scales of functional spaces (see e.g. [1–6,8–11,13–16,21–26,29] and the references listed
therein). It is well-known that the theory of global properties of differential operators is not
well-developped in comparison with the one of local properties. On the other hand, the local
and global hypoellipticity/solvability are rather different in general.
In this paper we are interested in the problem of global C
∞
and Gevrey hypoellipticity
for the following classes of linear partial differential operators on the multidimensional torus
T
N
= T
m+n
:
P1 = −
l
j=1
m
h=1
a
jh
(y)∂x
h
+
n
k=1
b
jk
(y)∂y
k
2
, (1)
P2 = −∆y −
m
j=1
aj (y)∂x
j
+
n
k=1
b
jk
(y)∂y
k
2
, (2)
where the coefficients a
jh
(y), aj (y) and b
jk
(y) are real–valued functions defined on T
n
. Pre-
cisely, in Theorems 1 and 2 (in Theorems 4 and 5) we give sufficient conditions for the global
C
∞
hypoellipticity (for the global Gevrey hypoellipticity) for the operator P1 defined in (1).
We point out that in Theorem 2 we partially answer to the following conjecture of Petroni-
lho [25]:
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