Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions with bounded support and average 0 are dense in $H^1$ (simple functions = finite linear combinations of characteristic functions). But I grew suspicious since I can not find any mention of this fact in the literature. If true this would be very useful, e.g. it implies a short proof that the complex interpolation between $H^1$ and $L^\infty$ is an $L^p$ space.

So, can anywone confirm/disprove/point at the relevant literature?

Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions with bounded support and average 0 are dense in $H^1$ (simple functions = finite linear combinations of characteristic functions). But I grew suspicious since I can not find any mention of this fact in the literature. If true this would be very useful, e.g. it implies a short proof that the complex interpolation between $H^1$ and $L^\infty$ is an $L^p$ space.

So, can anyone confirm/disprove/point at the relevant literature?