Yesterday in class, our professor mentioned the following:
$(*)$ If $u \ge 0$ and $\int_\Omega u(x,\cdot)\;dx=k \;$ then classical regularity theory implies ${\vert \vert u(\cdot,t) \vert \vert}_{L^\infty(\Omega)} \le \hat k \quad \forall t\ge 0$
where $\Omega$ is a connected, bounded, closed subset of $\mathbb R^2$. As I was searching for some appropriate reference for $(*)$, I came across with the answer of Davide Giraudo in another question asked here. he gave the following characterization of $L^1 \subset L^\infty$
$(**)$ Let $(X,F,\mu)$ a measure space. We have $L^1(\mu)\subset L^\infty(\mu)$ if and only if we can find a positive constant $c$ such that for $A\in F$, either $\mu(A)=0$ or $\mu(A)\ge c$.
I believe there might be a relation between $(*)$ and $(**)$ which I can not see at this moment. I would appreciate much if somebody could explain why $(*)$ holds and why this result is implied by classical theory.
Thanks in advance!