A famous result is that $H^p\cap L^1$ is a dense subspace of $H^p$. We need the fact that $\displaystyle\sup_{0<s\leq M}|(P_s*P_t*f-P_s*f)(x)|$ is continuous for $t\in[0,1)$ to prove it, but I don't know how can we prove it. Or is there somebody who can tell me who is the first person proved the density( maybe by different method)?