Does there exist an integrable function f such that norm of f*g is equals that of g for all g? Does there exist $f \in L^{1} (\mathbb R)$ such that $||f*g||_1 =||g||_1$ for all  $g \in L^{1} (\mathbb R)$? I read somewhere (long ago) that no such function exists. It is easy to see that $L^{1} (\mathbb R)$ has no unit under convolution, but this question is much harder and I still have no idea how one proves it. Thanks in advance for any hints or solution.

No. If $\tau_x f(t)=f(t-x)$ we know that $$\lim_{x\to0}||f-\tau_xf||_1=0.$$It follows easily that if $$g_n=n(\chi_{(0,1/n)}-\chi_{(1/n.2/n)})$$then $$\lim_{n\to\infty}||f*g_n||_1=0.$$
