# Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty$.

Consider a filtration generated by a 2-d (complex) Brownian Motion $B$. The martingale hardy space $\mathcal{H}^p$ defined on some time interval $[0,T]$, say, is the set of martingales $M$ such that $M^* = \sup_{t \in [0,T]} |M_t| \in L^p$. This definition is mostly interesting for $p=1$, as for $p>1$, $\mathcal{H}^p$ can be associated with a regular $L^p$ space of martingales.

If $B$ starts at zero, let $\tau$ be the hitting time of the boundary of $\mathbb{D}$. Then a connection between these two spaces is the following: for $f$ analytic on the unit disk, $f(B_{t \wedge \tau}) \in \mathcal{H}^p$ if and only if $f \in H^p$, and this mapping is continuous.

This allows you to associate $H^p$ to a subspace of $\mathcal{H}^p$. For studying $\mathcal{H}^p$, it would be useful to have a more complete representation of part of $\mathcal{H}^p$ in terms of functions evaluated on $B$. Specifically, for martingales that run on the whole time interval. Can this be obtained by using another hardy space, such as the Hardy space $h^p$ on $\mathbb{R}^2$? Can anything else be said relating hardy spaces of martingales and hardy spaces of functions?

• maybe helpful arxiv.org/abs/1411.5407 – john mangual Jun 3 '15 at 17:28
• The standard text on this subject is "Isomorphisms of $H^1$ spaces" by P.F.X. Müller. – priel Aug 20 '15 at 12:40

Charles Fefferman proved that the space of functions of bounded mean oscillation $BMO$ is the dual of the Hardy space $H^1$. The key was the observation that the classical Hardy space $H^1$ is a natural substitution for $L^1$, and $BMO$ is a natural substitution for $L^{\infty}$.
One result of comparing classical Hardy and martingale Hardy spaces is that we can establish that the dual of the martingale Hardy space $\mathcal{H}^1$ is martingale $BMO$.