Hardy Space - Martingale Hp

Martingale Hp

Let (M_n)_{n ≥ 0} be a martingale on some probability space (Ω, Σ, P), with respect to an increasing sequence of σ-fields (Σ_n)_{n ≥ 0}. Assume for simplicity that Σ is equal to the σ-field generated by the sequence (Σ_n)_{n ≥ 0}. The maximal function of the martingale is defined by

Let 1 ≤ p < ∞. The martingale (M_n)_{n ≥ 0} belongs to martingale-Hp when M∗ ∈ Lp.

If M∗ ∈ Lp, the martingale (M_n)_{n ≥ 0} is bounded in Lp, hence it converges almost surely to some function f  by the martingale convergence theorem. Moreover, M_n converges to f in Lp-norm by the dominated convergence theorem, hence M_n can be expressed as conditional expectation of f  on Σ_n. It is thus possible to identify martingale-Hp with the subspace of Lp(Ω, Σ, P) consisting of those f such that the martingale

belongs to martingale-Hp.

Doob's maximal inequality implies that martingale-Hp coincides with Lp(Ω, Σ, P) when 1 < p < ∞. The interesting space is martingale-H1, whose dual is martingale-BMO (Garsia 1973).

The Burkholder–Gundy inequalities (when p > 1) and the Burgess Davis inequality (when p = 1) relate the Lp-norm of the maximal function to that of the square function of the martingale

Martingale-Hp can be defined by saying that S(f) ∈ Lp (Garsia 1973).

Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex Brownian motion (B_t) in the complex plane, starting from the point z = 0 at time t = 0. Let τ denote the hitting time of the unit circle. For every holomorphic function F in the unit disk,

is a martingale, that belongs to martingale-Hp iff F ∈ Hp (Burkholder, Gundy & Silverstein 1971).