# The dense subspace of Hardy Space $H^p$

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)?

-
What exactly is $H^p$ here? –  Nate Eldredge Aug 29 '12 at 3:32
It means the Harday spaces: bounded distribution $f$ s.t. $\sup_{s>0}|(P_s*f)(x)|\in L^p$ –  Danqing Aug 29 '12 at 3:57