Mean-zero L2 functions on a cube embed continuously into H1

Given: Countable Choice, the fixed $\phi$ and $\tilde{N}$, a cube $Q$ of finite positive volume $\mathrm{\mid }Q\mathrm{\mid }$ (Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume) and a function $f\in L^2({\mathbb{R}}^n)$ with $\mathrm{supp}f\subseteq Q$ and ${\int }_{Q}f=0$.

The $H^1$ functional is $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^1}=\mathrm{\parallel }{M}_{\phi }^{0}g{\mathrm{\parallel }}_{L^1}$ for an admissible Schwartz function $\phi$ with $\int \phi \mathrm{\ne }0$, defined for tempered distributions (The real Hardy space $H^p$ defined by a radial maximal function), and every $L^2$ class has a representative defining a tempered distribution (Polynomial growth functions define tempered distributions).

If $a$ vanishes off a cube $Q$, has $\int a=0$ and $(\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }a{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}$, then $\mathrm{\parallel }a{\mathrm{\parallel }}_{H^1}\le C_{n,\tilde{N},\phi }$ with this constant independent of $Q$ and $a$ (L2-normalised H1 atoms have uniformly bounded H1 norm).

For $\lambda \in \mathbb{C}$ and a tempered distribution $g$, $M_{\phi }^0(\lambda g)=\mathrm{\mid }\lambda \mathrm{\mid }{M}_{\phi }^{0}g$ pointwise, because $(g\ast {\phi }_t)(y)=⟨g,{\phi }_t(y-\cdot )⟩$ is complex-linear in $g$; hence $\mathrm{\parallel }\lambda g{\mathrm{\parallel }}_{H^1}=\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\parallel }g{\mathrm{\parallel }}_{H^1}$ (The real Hardy space $H^p$ defined by a radial maximal function).

The $L^2$ classes of The space $L^p(\mu )$ as the quotient by null functions are used, so all statements below are insensitive to changes on null sets.