(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 . Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex Brownian motion (Bt) 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 . Example: dyadic martingale-H1 In this example, Ω = [0, 1] and Σn is the finite field generated by the dyadic partition of [0, 1] into 2n intervals of length 2−n, for every n ≥ 0. If a function f on [0, 1] is represented by its expansion on the Haar system (hk) then the martingale-H1 norm of f can be defined by the L1 norm of the square function This space, sometimes denoted by H1(δ), is isomorphic to the classical real H1 space on the circle . The Haar system is an unconditional basis for H1(δ). Notes References Complex analysis Operator theory