Absolute value of functions in $H^1(\mathbb R^n)$

Question

Let $f$ be a function in the atomic Hardy space $H^1_{at}(\mathbb R^n)$. That is, there exists a sequence of atoms $a_j$ satisfying

1. supp $a_j \subset B_j$ for some ball $B_j$,
2. $\int a_j dx = 0$,
3. $\Vert a_j\Vert_\infty \leq |B_j|^{-1}$,

such that $f = \sum_j \lambda_j a_j$ and $\sum|\lambda_j| < +\infty$.

It is easy to see that $H^1 \subset L^1$ and that $f\in H^1 \implies \int f dx = 0$.

My question is, what can we say about the absolute value of a function in $H^1$? Can we say something more than $|f| \in L^1$? What happens if, moreover, $f$ has compact support?

EDIT: I've found that if $f\in H^1$ and $f \geq 0$ in $B$, then $f \in L \log L$ in any compact inside $B$.