Affiliation:
1. School of Mathematical Sciences, Zhejiang Normal University , Jinhua 321004 , China
Abstract
Abstract
In this note, we prove that the toroidal pseudo-differential operator is bounded from
L
∞
(
T
n
)
{L}^{\infty }\left({{\mathbb{T}}}^{n})
to
BMO
(
T
n
)
{\rm{BMO}}\left({{\mathbb{T}}}^{n})
if the symbol belongs to the toroidal Hörmander class
S
ρ
,
δ
n
(
ρ
−
1
)
∕
2
(
T
n
×
Z
n
)
{S}_{\rho ,\delta }^{n\left(\rho -1)/2}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n})
with
0
<
ρ
≤
1
0\lt \rho \le 1
and
0
≤
δ
<
1
0\le \delta \lt 1
. As a corollary, we obtain a result of toroidal pseudo-differential operators on
L
p
{L}^{p}
when
2
<
p
<
∞
2\lt p\lt \infty
for symbols in the class
S
ρ
,
δ
m
(
T
n
×
Z
n
)
{S}_{\rho ,\delta }^{m}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n})
with
m
≤
n
(
ρ
−
1
)
1
2
−
1
p
+
n
p
min
{
0
,
ρ
−
δ
}
m\le n\left(\rho -1)\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-\frac{1}{p}\right)+\frac{n}{p}\min \left\{0,\rho -\delta \right\}
.