# When the following limit is zero

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.

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 $$=0$$.