Norm of convolution operator in L1

Question

Consider the convolution operator on $L^1(\Bbb R)$, $f→f∗g$, where $g$ is some $L1$ function. I need to show that the norm of this operator equals to $||g||_1$.
I have seen a question here which gives an answer to the case when $g$ is positive here:
Limit of convolution

I assume that the way of showing equality won't change for a negative function, but i have troubles with approach for a 'mixed' function, for which $Im(g) \in R$. Would be very glad for any help.

Answer 1

Let $f=\chi_{[0,1]}$.

$$\int_{R^d}[\int_{R^d}f(x-y)g(y)dy]dx = \int_{R^d}f(x-y)dx\int_{R^d}g(y)dy = \int_{R^d}g(y)dy$$ which is not neccessarily equal with $||g||_{L^1(R^d)}$
Actually what you can say is that $||*|| \leq ||g||_{L^1(R^d)}$