Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm $$ \|X\|_{\mathcal{H}_{\mathscr{S}}^p}:= \inf\big\{ \mathbb{E}\big[\big( [M]_{\infty}^{1/2} + \int_0^{\infty} |dA|_s \big)^p\big]^{1/p} \big\} $$ is finite, where the infimum is taken over all decompositions of $X$ into a local martingale with $M_0=0$ and a process $A$ of finite variation with $A_0=X_0=\Delta A_0$. It is known (Lemma 16.2.25 of this book) that every semi-martingale in ${\mathcal{H}_{\mathscr{S}}^p}$ is special and the above norm can be written as $$ \|X\|_{\mathcal{H}_{\mathscr{S}}^p} = \mathbb{E}[([\tilde{M}]^{1/2})^p]^{1/p} + \mathbb{E}[(\int_0^{\infty} |d\tilde{A}_s|)^p]^{1/p}, $$ where $X=\tilde{M}+\tilde{A}$ is the canonical decomposition of $X$.

We call $\mathcal{H}_{\mathscr{S}}^p$ the *semi-martingale-hardy space*.

Clearly, $\mathcal{H}_{\mathscr{S}}^p$ is a separable Banach space and it is a Hilbert space when $p=2$.

**Question:** What is an explicit Schauder basis of this space?