case. For p = 1, the real Hardy space H1(T) is a proper subspace of L1(T). The case of p = ∞ was excluded from the definition of real Hardy spaces, because the maximal function M f of an L∞ function is always bounded, and because it is not desirable that real-H∞ be equal to L∞. However, the two following properties are equivalent for a real valued function f the function f is the real part of some function g ∈ H∞(T) the function f and its conjugate H(f) belong to L∞(T). Real Hardy spaces for 0 < p < 1 When 0 < p < 1, a function F in Hp cannot be reconstructed from the real part of its boundary limit function on the circle, because of the lack of convexity of Lp in this case. Convexity fails but a kind of "complex convexity" remains, namely the fact that z → |z|q is subharmonic for every q > 0. As a consequence, if is in Hp, it can be shown that cn = O(n1/p–1). It follows that the Fourier series converges in the sense of distributions to a distribution f on the unit circle, and F(reiθ) =(f ∗ Pr)(θ). The function F ∈ Hp can be reconstructed from the real distribution Re(f) on the circle, because the Taylor coefficients cn of F can be computed from the Fourier coefficients of Re(f). Distributions on the circle are general enough for handling Hardy spaces when p < 1. Distributions that are not functions do occur, as is seen with functions F(z) = (1−z)−N (for |z| < 1), that belong to Hp when 0 < N p < 1 (and N an integer ≥ 1). A real distribution on the circle belongs to real-Hp(T) iff it is the boundary value of the real part of some F ∈ Hp. A Dirac distribution δx, at any point x of the unit circle, belongs to real-Hp(T) for every p < 1; derivatives δ′x belong when p < 1/2, second derivatives δ′′x when p < 1/3, and so on. Hardy spaces for the upper half plane It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on a complex half-plane (usually the right half-plane or upper half-plane) are used. The Hardy space Hp(H) on the upper half-plane H is defined to be the space of