Let
(
t
,
γ
(
t
)
)
(t,\gamma (t))
be a plane curve. Set
H
γ
f
(
x
,
y
)
=
p.v.
∫
f
(
x
−
t
,
y
−
γ
(
t
)
)
d
t
/
t
{H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t
for
f
∈
C
0
∞
(
R
2
)
f \in C_0^\infty ({R^2})
. For a large class of curves, the authors prove
‖
H
γ
f
‖
p
⩽
A
p
‖
f
‖
p
,
5
/
3
>
p
>
5
/
2
{\left \| {{H_\gamma }f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p},5/3 > p > 5/2
. Various examples are given to show that some condition on the curve
(
t
,
γ
(
t
)
)
(t,\gamma (t))
is necessary.