≤ 1, a bounded measurable function f of compact support is in the Hardy space Hp if and only if all its moments whose order i1+ ... +in is at most n(1/p − 1), vanish. For example, the integral of f must vanish in order that f ∈ Hp, 0 < p ≤ 1, and as long as p > this is also sufficient. If in addition f has support in some ball B and is bounded by |B|−1/p then f is called an Hp-atom (here |B| denotes the Euclidean volume of B in Rn). The Hp-quasinorm of an arbitrary Hp-atom is bounded by a constant depending only on p and on the Schwartz function Φ. When 0 < p ≤ 1, any element f of Hp has an atomic decomposition as a convergent infinite combination of Hp-atoms, where the aj are Hp-atoms and the cj are scalars. On the line for example, the difference of Dirac distributions f = δ1−δ0 can be represented as a series of Haar functions, convergent in Hp-quasinorm when 1/2 < p < 1 (on the circle, the corresponding representation is valid for 0 < p < 1, but on the line, Haar functions do not belong to Hp when p ≤ 1/2 because their maximal function is equivalent at infinity to a x−2 for some a ≠ 0). Martingale Hp Let (Mn)n≥0 be a martingale on some probability space (Ω, Σ, P), with respect to an increasing sequence of σ-fields (Σn)n≥0. Assume for simplicity that Σ is equal to the σ-field generated by the sequence (Σn)n≥0. The maximal function of the martingale is defined by Let 1 ≤ p < ∞. The martingale (Mn)n≥0 belongs to martingale-Hp when M* ∈ Lp. If M* ∈ Lp, the martingale (Mn)n≥0 is bounded in Lp; hence it converges almost surely to some function f by the martingale convergence theorem. Moreover, Mn converges to f in Lp-norm by the dominated convergence theorem; hence Mn can be expressed as conditional expectation of f on Σn. 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 . The Burkholder–Gundy inequalities (when p > 1) and the Burgess Davis inequality