While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \leq c|B|^{1/2}\left\Vert g \right\Vert_{L^2}$ where the $L^2$ norm is on a ball which contains the support of g: $$ \left\Vert g \right\Vert_{L^2} = \left( \int_B \left\vert g \right\vert^2 dx \right)^{1/2}. $$
He does not provide a proof of this inequality but references the following on page 112: the size condition $|a| \leq |B|^{-1/p}$ can be replaced by the weaker condition $$ \left( \frac{1}{|B|} \int_B |a|^q dx \right)^{1/q} \leq |B|^{-1/p}$$
with $q > 1$ if $p=1$, and with $q = 1$ if $p < 1$.
My question is how to prove the inequality $\left\Vert g \right\Vert_{H^1} \leq c|B|^{1/2}\left\Vert g \right\Vert_{L^2}$ using the $H^1$ atomic norm: $$\left\Vert g \right\Vert_{H^1} = \inf \left\{\sum_{i=1}^\infty |\lambda_i| \colon g = \sum_{i=1}^\infty \lambda_i a_i \right\} $$ where $a$ is an atom and the infimum is taken over all representations of $g$ as a linear combination of atoms? I have seen a proof using the maximal function characterization of the $H^1$ norm but was wondering how you would do it with this one.
Following Stein, fix a ball $B$ and consider the space $L^2(B)$ of square integrable functions supported on $B$. Write $L_0^2(B)$ for the closed subspace of $L^2(B)$ consisting of the functions whose integrals on $B$ are zero. By construction, each function in this subspace is a scalar multiple of a $2$-atom.
Fix a function $g$ in $L_0^2(B)$ and set $g = \lambda \mathfrak{a}$ for some scalar $\lambda$ and some $2$-atom $\mathfrak{a}$ supported on $B$.
We know that there is a bound $c_2$, independent of the $2$-atom $\mathfrak{a}$ such that $\Vert \mathfrak{a} \Vert_{H^1} \leq c_2$. (See Stein and Shakarchi Volume 4, Page 81 for a proof.) Using this, we have $$\Vert g \Vert_{H^1} \leq c_2 \vert \lambda \vert. \quad (1)$$
In particular, when $\lambda = \vert B \vert^\frac{1}{2} \Vert g \Vert_{L_0^2(B)}$ and $\mathfrak{a} = \vert B \vert^{-\frac{1}{2}} \Vert g \Vert_{L_0^2(B)}^{-1} g$, we get that $g = \lambda \mathfrak{a}$ is an atomic decomposition of $g$ and (1) gives $$ \Vert g \Vert_{H^1} \leq c_2 \vert B \vert^{\frac{1}{2}}\Vert g \Vert_{L_0^2(B)},$$ which is the inequality you want for this proof.