When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$? 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?
 A: Let $\theta_f = \frac{|f|}{f}$ (with any convention for $\frac{0}{0}$, it doesn't matter), let $A_\epsilon = \{t \in \mathbb{R}: |f|(t) \geq \epsilon\}$, and let $g_s$ be the shifted function $g_s(t) = g(s - t)$.
The condition is: For every $\epsilon > 0$ there exists $s \in \mathbb{R}$ and a scalar $a$ of modulus 1 such that $g_s$ is uniformly within $\epsilon$ of $a\cdot\|g\|_\infty\cdot\theta_f$ on a subset of $A_\epsilon$ of measure at least $\mu(A_\epsilon) - \epsilon$.
It's easy to see that if $f$ and $g$ satisfy this condition then there will be points $s \in \mathbb{R}$ at which $|f*g|(s)$ gets arbitrarily close to $\|f\|_1\|g\|_\infty$. Conversely, if there are such points then $g$ must satisfy the stated condition: if $g_s$ is too far from $\|g\|_\infty\cdot \theta_f$ on $A_\epsilon$ then there will be cancellations which keep us away from $\|f\|_1\|g\|_\infty$. But it would take a bit of work to write this out.
A: 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.)
