We study operators of the form
T
f
(
x
)
=
ψ
(
x
)
∫
f
(
γ
t
(
x
)
)
K
(
t
)
d
t
,
\begin{equation*} Tf(x)= \psi (x) \int f(\gamma _t(x))K(t)\,dt, \end{equation*}
where
γ
t
(
x
)
\gamma _t(x)
is a real analytic function of
(
t
,
x
)
(t,x)
mapping from a neighborhood of
(
0
,
0
)
(0,0)
in
R
N
×
R
n
\mathbb {R}^N \times \mathbb {R}^n
into
R
n
\mathbb {R}^n
satisfying
γ
0
(
x
)
≡
x
\gamma _0(x)\equiv x
,
ψ
(
x
)
∈
C
c
∞
(
R
n
)
\psi (x) \in C_c^\infty (\mathbb {R}^n)
, and
K
(
t
)
K(t)
is a “multi-parameter singular kernel” with compact support in
R
N
\mathbb {R}^N
; for example when
K
(
t
)
K(t)
is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth
γ
t
(
x
)
\gamma _t(x)
, in the single-parameter case when
K
(
t
)
K(t)
is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the
L
p
L^p
-boundedness of such operators. This paper shows that when
γ
t
(
x
)
\gamma _t(x)
is real analytic, the sufficient conditions of Street and Stein are also necessary for the
L
p
L^p
-boundedness of
T
T
, for all such kernels
K
K
.