If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f\*g\rVert\_p\leq\lVert f\rVert\_1\lVert g\rVert\_p\qquad (f\in L^1(\mathbb{R}^n),\,g\in L^p(\mathbb{R}^n)),$$ for complex $f$ and $g$. But I cannot seem to determine one for $p=\infty$ nor find a reference that contains such a condition. Is there a natural equality condition in this case?



When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?
Here is a fairly simple condition.


It uses the following notion: A family of functions $f\_t\in L^1$ depending on a parameter $t$ in a measure space $X$ is said to *tend to $f$ somewhere* if the essential infimum of $\lVert f\_t-f\rVert\_1$ over $X$ is $0$.


Put $g\_s(t)=g(t-s)$. The condition is that $afg\_s\to\lvert f\rvert\lVert g\rVert\_\infty$ somewhere, for some constants $a=a(s)$ with $|a|=1$.


Clearly, this condition implies equality. For the converse, choose $a$ so that $\lvert\int fg\_s\rvert=\int\Re(afg\_s)$. Then equality implies that $\Re(afg\_s)\to\lvert f\rvert\lVert g\rVert\_\infty$ somewhere, and the condition follows via the inequality $\lVert\Im f\rVert\_1^2\leq2\lVert f\rVert\_1\lVert\lvert f\rvert-\Re f\rVert\_1$. ($\Re$ and $\Im$ denote real and imaginary parts.)


