Hardy Spaces For The Unit Disk
For spaces of holomorphic functions on the open unit disk, the Hardy space H2 consists of the functions ƒ whose mean square value on the circle of radius r remains bounded as r → 1 from below.
More generally, the Hardy space Hp for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying
This class Hp is a vector space. The number on the left side of the above inequality is the Hardy space p-norm for f, denoted by It is a norm when p ≥ 1, but not when 0 < p < 1.
The space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm
For 0 < p ≤ q ≤ ∞, the class Hq is a subset of Hp, and the Hp-norm is increasing with p (it is a consequence of Hölder's inequality that the Lp-norm is increasing for probability measures, i.e. measures with total mass 1).
Read more about this topic: Hardy Space
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