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Suppose that $f\in L_1(\mathbb{R})\cap L_2(\mathbb{R})$ and $g(y)=\int_\mathbb{R}f(x)f(x-y) dx$. I would like to know when $$\lim_{\|f\|\to 0}\frac{\left\|g\right\|}{\|f\|}=0,$$ where $\|\|$ is the $L_2$ norm

By the Young's inequlality $$\|g\|\leq \|f\|_1\|f\|,$$ so the question reduces to showing that $$\lim_{\|f\|\to 0}\|f\|_1=0,$$ and I'm not sure if this is the case.

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What is the parameter for the family of $f$'s? If there is a limiting function say $f_0$, with $||f_0||_1=0$, then $|f_0|=0$ almost everywhere and the $L_2$ norm $=0$.

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