holomorphic functions f on H with bounded norm, the norm being given by The corresponding H∞(H) is defined as functions of bounded norm, with the norm given by Although the unit disk D and the upper half-plane H can be mapped to one another by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. However, for H2, one has the following theorem: if m : D → H denotes the Möbius transformation Then the linear operator M : H2(H) → H2(D) defined by is an isometric isomorphism of Hilbert spaces. Real Hardy spaces for Rn In analysis on the real vector space Rn, the Hardy space Hp (for 0 < p ≤ ∞) consists of tempered distributions f such that for some Schwartz function Φ with ∫Φ = 1, the maximal function is in Lp(Rn), where ∗ is convolution and . The Hp-quasinorm ||f ||Hp of a distribution f of Hp is defined to be the Lp norm of MΦf (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 ||f ||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. Atomic decomposition When 0 < p