Real-variable Techniques On The Unit Circle
Real-variable techniques, mainly associated to the study of real Hardy spaces defined on Rn (see below), are also used in the simpler framework of the circle. It is a common practice to allow for complex functions (or distributions) in these "real" spaces. The definition that follows does not distinguish between real or complex case.
Let Pr denote the Poisson kernel on the unit circle T. For a distribution f on the unit circle, set
where the star indicates convolution between the distribution f and the function ei θ → Pr(θ) on the circle. Namely, (f ∗ Pr)(ei θ) is the result of the action of f on the C∞-function defined on the unit circle by
For 0 < p < ∞, the real Hardy space Hp(T) consists of distributions f such that M f is in Lp(T).
The function F defined on the unit disk by F(r ei θ) = (f ∗ Pr)(ei θ) is harmonic, and M f is the radial maximal function of F. When M f belongs to Lp(T) and p ≥ 1, the distribution f "is" a function in Lp(T), namely the boundary value of F. For p ≥ 1, the real Hardy space Hp(T) is a subset of Lp(T).
Read more about this topic: Hardy Space
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