Does the following hold?
There is a $C>0$ such that
$$\left|\left|\int_{\mathbb{R}^3} \frac{|f(y)|^2}{|x-y|} dy\right|\right|_{L^3(\mathbb{R}^3)} \leq C||f||^2_{L^2(\mathbb{R}^3)}$$ for all $f \in L^2(\mathbb{R}^3).$
Any advice would be appreciated.
Does the following hold?
There is a $C>0$ such that
$$\left|\left|\int_{\mathbb{R}^3} \frac{|f(y)|^2}{|x-y|} dy\right|\right|_{L^3(\mathbb{R}^3)} \leq C||f||^2_{L^2(\mathbb{R}^3)}$$ for all $f \in L^2(\mathbb{R}^3).$
Any advice would be appreciated.
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