Unit disk in the context of Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) ${\displaystyle H^{p}}$ are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G. H. Hardy, because of the paper (Hardy 1915). In real analysis Hardy spaces are spaces of distributions on the real n-space ${\displaystyle \mathbb {R} ^{n}}$, defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the L spaces. For ${\displaystyle 1\leq p<\infty }$ these Hardy spaces are subsets of ${\displaystyle L^{p}}$ spaces, while for ${\displaystyle 0<p<1}$ the ${\displaystyle L^{p}}$ spaces have some undesirable properties, and the Hardy spaces are much better behaved. Hence, ${\displaystyle H^{p}}$ spaces can be considered extensions of ${\displaystyle L^{p}}$ spaces.

Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g. ${\displaystyle H^{\infty }}$ methods) and scattering theory.