I am trying to find a bound for the $L^p$ norm, $1<p\leq \infty$ of the Riesz transform of $f^2$, where $f \in S$.
$$\mathcal{F}[{\mathcal{R}_x f}](\xi,\eta) = -i \frac{\xi}{|(\xi,\eta)|}\hat{f}(\xi,\eta).$$
I am looking for a a bound on $L^p$ for the term $\partial_x [\mathcal{R}_x(f^2)](x,y))$.
$$\int \partial_x (\mathcal{R}_x (f^2(x,y)) dxdy = \int \frac{\xi_1^2}{|(\xi,\eta)|} \hat{f}^2(\xi,\eta) d\xi d\eta.$$
The domain of your integration is not written! If $x$ is in the real line I can not see any bounds, however, if $x$ belongs to a bounded set or torus then the fraction will be bounded by the sub-norm and the expression is bounded by $L^2$. As
$$\int_\mathbb{T \times \mathbb{R}} \partial_x (\mathcal{R}_x (f^2(x,y)) dxdy = \int_\mathbb{T \times \mathbb{R}} \frac{\xi_1^2}{|(\xi,\eta)|} \hat{f}^2(\xi,\eta) d\xi d\eta \lesssim \left\| \frac{\xi_1^2}{|(\xi,\eta)|}\right\|_{L^\infty} \left\|f\right\|^2_{L^2}$$