Young's inequality for convolutions states that for $g\in L^r(\mathbb{R}^d) $ and $f\in L^q(\mathbb{R}^d)$, we have $$\|g*f\|_p\leq \|g\|_r\|f\|_q$$ where $1+\frac{1}{p} = \frac{1}{r}+\frac{1}{q}.$
Now if we were to fix g, for example. Would there be conditions on $f$ that we could impose to ensure we get equality?
If it's not sharp, is it known whether there's an optimal constant in this case that makes it sharp?