We study the
L
p
{L^p}
-boundedness of pseudo-differential operators with the support of their symbols being contained in
E
×
R
n
E \times {{\mathbf {R}}^n}
, where
E
E
is a compact subset of
R
n
{{\mathbf {R}}^n}
, and their symbols have derivatives with respect to
x
x
only up to order
k
k
, in the Hölder continuous sense, where
k
>
n
/
2
k > n/2
(the case
1
>
p
⩽
2
1 > p \leqslant 2
) and
k
>
n
/
p
k > n/p
(the case
2
>
p
>
∞
2 > p > \infty
). We also give a new proof of the
L
p
{L^p}
-boundedness,
1
>
p
>
∞
1 > p > \infty
, of pseudo-differential operators of class
S
0
,
0
m
S_{0,0}^m
, where
m
=
m
(
p
)
=
−
n
|
1
/
p
−
1
/
2
|
m = m(p) = - n|1/p - 1/2|
, and
a
∈
S
0
,
0
m
a \in S_{0,0}^m
satisfies
|
∂
x
α
∂
ξ
β
a
(
x
,
ξ
)
|
⩽
C
α
,
β
⟨
ξ
⟩
m
|\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )| \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}
for
(
x
,
ξ
)
∈
R
n
×
R
n
,
|
α
|
⩽
k
(x,\xi ) \in {{\mathbf {R}}^n} \times {{\mathbf {R}}^n},|\alpha | \leqslant k
and
|
β
|
⩽
k
′
|\beta | \leqslant k’
, in the Hölder continuous sense, where
k
>
n
/
2
,
k
′
>
n
/
p
k > n/2,k’ > n/p
(the case
1
>
p
⩽
2
1 > p \leqslant 2
) and
k
>
n
/
p
,
k
′
>
n
/
2
k > n/p,k’ > n/2
(the case
2
>
p
>
∞
2 > p > \infty
).