Let us consider a real Lévy process $X$ whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of $X$ before any deterministic time $t$ is absolutely continuous on $(0,\infty)$. We show that its density $f_t(x)$ is continuous on $(0,\infty)$ if and only if the potential density $h'$ of the upward ladder height process is continuous on $(0,\infty)$. Then we prove that $f_t$ behaves at 0 as $h'$. We also describe the asymptotic behaviour of $f_t$, when $t$ tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.
