On bilinear Littlewood-Paley square functions

On the real line, let the Fourier transform of $k_n$ be $\hat k_n(\xi)=\hat k(\xi-n)$ where $\hat k(\xi)$ is a smooth compactly supported function. Consider the bilinear operators $ S_n(f,g)(x)=\int f(x+y)g(x-y)k_n(y)\,dy$. If $2\le p,q\le\infty$, with $1/p+1/q=1/2$, I prove that
$$
\sum_{n=-\zI}^\zI\|S_n(f,g)\|_2^2\le{}C^2\|f\|_p^2 \|g\|_q^2\,.
$$
The constant $C$ depends only upon $k$.