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?



Dense subspaces of the Hardy space $H^1$
It seems to be true. Due to the atomic decomposition, it is enough to approximate in $H^1$ any "$(1,\infty)$"-atom, that is, a function $a$ such that $$\mathrm{supp}\, a \subset Q,\quad \|a\|\_{L^\infty}\leq |Q|^{-1}, \quad \int\_Q a=0$$
for a cube $Q$.


In order to do this, we can cut $Q$ into $N=k^n$ smaller equal cubes $Q\_1, Q\_2, \ldots, Q\_N$ for a large $k$. We put
$$
b\_N(x)=\sum\_{j=1}^N \Big( \frac{1}{|Q\_j|}\int\_{Q\_j}a(y)\,dy\Big) \chi\_{Q\_j}(x).
$$
Put $g\_N=a-b\_N$. Due to Lebesgue differentiation theorem we have
$$
\lim\limits\_{N\to \infty}\int |g\_N(x)|^2dx = 0.
$$


Then for big $N$ the following holds:
$$
\mathrm{supp}\, g\_N\subset Q, \quad \|g\_N\|\_{L^2}\leq \varepsilon |Q|^{-1/2}, \quad \int\_Q g\_N = 0.
$$


It means that $\varepsilon^{-1} g\_N$ is a $(1,2)$-atom. Hence, $\|g\_N\|\_{H^1}\leq C\varepsilon$.


The atomic decomposition in $H^1$ is described in the book "Weighted norm inequalities and related topics" by Garcia-Querva and Rubio de Francia. Another reference is "Modern Fourier Analysis" by Grafakos.


