Hardy Space - Real Hardy Spaces For Rn

Real Hardy Spaces For Rn

In analysis on the real vector space Rn, the Hardy space Hp (for 0 < p ≤ ∞) consists of tempered distributions ƒ such that for some Schwartz function Φ with ∫Φ = 1, the maximal function

is in Lp(Rn), where ∗ is convolution and Φ_t(x) = t−nΦ(x/t). The Hp-quasinorm ||ƒ||_Hp of a distribution ƒ of Hp is defined to be the Lp norm of M_Φƒ (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). The Hp-quasinorm is a norm when p ≥ 1, but not when p < 1.

If 1 < p < ∞, the Hardy space Hp is the same vector space as Lp, with equivalent norm. When p = 1, the Hardy space H1 is a proper subspace of L1. One can find sequences in H1 that are bounded in L1 but unbounded in H1, for example on the line

The L1 and H1 norms are not equivalent on H1, and H1 is not closed in L1. The dual of H1 is the space BMO of functions of bounded mean oscillation. The space BMO contains unbounded functions (proving again that H1 is not closed in L1).

If p < 1 then the Hardy space Hp has elements that are not functions, and its dual is the homogeneous Lipschitz space of order n(1/p − 1). When p < 1, the Hp-quasinorm is not a norm, as it is not subadditive. The pth power ||ƒ||_Hpp is subadditive for p < 1 and so defines a metric on the Hardy space Hp, which defines the topology and makes Hp into a complete metric space.