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.
