We prove
L
2
{L^2}
boundedness of the oscillatory singular integral
\[
T
f
(
x
,
y
)
=
∬
D
y
exp
(
2
π
i
N
(
y
)
x
′
)
x
′
y
′
f
(
x
−
x
′
,
y
−
y
′
)
d
x
′
d
y
′
Tf(x,y) = \iint \limits _{{D_y}} {\frac {{\operatorname {exp} (2\pi iN(y)x’)}}{{x’y’}}}f(x - x’,y - y’)dx’dy’
\]
where
N
(
y
)
N(y)
is an arbitrary integer-valued
L
∞
{L^\infty }
function and where nothing is assumed on the dependency upon
y
y
of the domain of integration
D
y
{D_y}
. We also prove
L
2
{L^2}
boundedness of the corresponding maximal opertaor. Operators of this kind appear in a problem of a.e. convergence of double Fourier series.