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

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`$\sup_{s>0}|(P_s*f)(x)|\in L^p$`

– Danqing Aug 29 '12 at 3:57