TY - JOUR
AU1 - Ávila Silva, Fernando
AU2 - Gonzalez, Rafael
AU3 - Kirilov, Alexandre
AU4 - Medeira, Cleber
AB - We show that an obstruction of number-theoretical nature appears as a necessary condition for the global hypoellipticity of the pseudo-differential operator 
$$L=D_t+(a+ib)(t)P(D_x)$$



L
=

D
t

+

(
a
+
i
b
)


(
t
)

P

(

D
x

)




 on 
$$\mathbb {T}^1_t\times \mathbb {T}_x^{N}$$





T

t
1

×

T
x
N




. This condition is also sufficient when the symbol 
$$p(\xi )$$



p
(
ξ
)



 of 
$$P(D_x)$$



P
(

D
x

)



 has at most logarithmic growth. If 
$$p(\xi )$$



p
(
ξ
)



 has super-logarithmic growth, we show that the global hypoellipticity of L depends on the change of sign of certain interactions of the coefficients with the symbol 
$$p(\xi ).$$



p
(
ξ
)
.



 Moreover, the interplay between the order of vanishing of coefficients with the order of growth of 
$$p(\xi )$$



p
(
ξ
)



 plays a crucial role in the global hypoellipticity of L. We also describe completely the global hypoellipticity of L in the case where 
$$P(D_x)$$



P
(

D
x

)



 is homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.
TI - Global Hypoellipticity for a Class of Pseudo-differential Operators on the Torus
JF - Journal of Fourier Analysis and Applications
DO - 10.1007/s00041-018-09645-x
DA - 2018-10-12
UR - https://www.deepdyve.com/lp/springer-journals/global-hypoellipticity-for-a-class-of-pseudo-differential-operators-on-7m48m0KcX4
SP - 1717
EP - 1758
VL - 25
IS - 4
DP - DeepDyve
ER -