Let $f$ be a smooth real-valued function on $\mathbb{R}^d$ with support contained in a compact rectangle $R$. Write $f=f^{+} - f^{-}$ and $|f|=f^{+} + f^{-}$. Do we have a bound $$ \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{f^{+}(x)f^{-}(y)}{|x-y|} dx dy \leq C \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{f(x)f(y)}{|x-y|} dx dy $$ with a constant $C$ independent of $f$ (but possibly dependent on $R$)?
Note $$ \int\int \frac{g(x)h(y)}{|x-y|}dy = \int(g \ast k_1)(y)h(y)dy $$ where $k_1(x) = |x|^{-1}$ is a Riesz kernel.