As in E. Steins's "Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals" consider the space $H^p$ (with $p < 1$), which contains all tempered distributions $f \in \mathcal{S}'$ such that $M_\Phi f \in L^p(\mathbb{R}^d)$. Here $M_\Phi f(x) = \sup_{t > 0} |f * \Phi_t(x)|$ and $\Phi_t(x) = t^{-d} \Phi(\frac{x}{t})$. Now, equip $H^p$ with a metric given by $$ d(f,g) = \| M_\Phi(f-g) \|_{L^p(\mathbb{R}^d)}^p. $$
Is this space a Banach space? Stein says (on page 88) that $H^p$ is not a Banach space for $p<1$, but I do not know which metric he is referring to. On the other hand, an exercise told me to show the converse and I have come up with the following proof. I would be happy if someone could check whether it makes sense and possibly point out the mistake.
Let $(f_n)_n$ be a Cauchy sequence in $H^p$. We want to show that it has a limit $f \in H^p$.
Step 1 (a distributional limit for $(f_n)_n$)
Let $\varphi \in \mathcal{S}$. Then we show that for all $g \in \mathcal{S}'$, $$ |\langle g, \varphi \rangle | \leq c \, M_{\mathcal{F}}f(x) \qquad x \in B(0,1), $$ where the constant $c$ may depend on $\varphi$ and $M_{\mathcal{F}}$ is a grand maximal function suitable for defining $H^p$ (see Stein). By averaging over the Ball $B(0,1)$ and using that $ \| M_\Phi g \|_{L^p} \sim \| M_{\mathcal{F}} g \|_{L^p}$, we derive $$ |\langle f_n, \varphi \rangle |^p \leq c \, \| M_\Phi f(x) \|_{L^p}. $$ So $\langle f_n, \varphi \rangle$ is a Cauchy sequence in $\mathcal{C}$ and we write $\langle f, \varphi \rangle$ for its limit. So as operators on $\mathcal{S}$ the $f_n$ converge pointwise to $\langle f, \cdot \rangle$ and from functional analysis we know that then the latter is a bounded operator as well, i.e. $f \in \mathcal{S}'$.
Step 2 (Convergence in $H^p$)
Write $g_n := f_{n+1} - f_n$. Wlog $f_1 = 0$ and $\| M_\Phi g_n \|_{L^p}^p \leq 2^{-n}$. Then \begin{align*} d(f, f_n) &= \| M_\Phi \left( \sum_{k=n}^\infty g_k \right) \|_{L^p}^p \\ &= \int_{\mathbb{R}^d} \left| M_\Phi \left( \sum_{k=n}^\infty g_k \right)(x) \right|^p dx \\ &\leq \int_{\mathbb{R}^d} \left| \sum_{k=n}^\infty M_\Phi g_k (x) \right|^p dx && \text{(taking the supremum outside of the series)} \\ &\leq \int_{\mathbb{R}^d} \sum_{k=n}^\infty \left| M_\Phi g_k (x) \right|^p dx && p \leq 1 \\ &\leq \sum_{k=n}^\infty \int_{\mathbb{R}^d} \left| M_\Phi g_k (x) \right|^p dx && \text{(Fatou)} \\ &\leq \sum_{k=n}^\infty 2^{-k} \rightarrow 0 \quad (n \rightarrow \infty). \end{align*}
In particular this shows that $f \in H^p$ since $f_1 = 0$.
To not make this too complicated I have omitted many details but sketched the main argument. It would be helpful enough to know which parts of the argument are most likely to contain a mistake.
Any help is appreciated!