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Danqing
Danqing
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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 itthe continuity of this function. Or is there somebody who can tell me who is the first person proved the density( maybe by different method)?

The dense subspace of $H^p$

A famous result is that $H^p\cap L^1$ is a dense subspace of $H^p$. We need the fact that $\displaystyle\sup_{0<s\leq M}|(P_s*P_t*f-P_s*f)(x)|$ 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)?

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)?

Source Link
Danqing
Danqing
  • 231
  • 2
  • 6

The dense subspace of $H^p$

A famous result is that $H^p\cap L^1$ is a dense subspace of $H^p$. We need the fact that $\displaystyle\sup_{0<s\leq M}|(P_s*P_t*f-P_s*f)(x)|$ 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)?