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^p 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.

The Hardy space ${\displaystyle H^{p}}$ for ${\displaystyle 0<p<\infty }$ is the class of holomorphic functions ${\displaystyle f}$ on the open unit disk ${\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}$ satisfying $${\displaystyle \sup _{0\,\leqslant \,r\,<\,1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .}$$ If ${\displaystyle p\geq 1}$, this coincides with the definition of the Hardy space ${\displaystyle p}$-norm, denoted by ${\displaystyle \|f\|_{H^{p}}.}$

The space ${\displaystyle H^{\infty }}$ is defined as the vector space of bounded holomorphic functions on the unit disk, with norm

For ${\displaystyle 0<p\leq q\leq \infty }$, the class ${\displaystyle H^{q}}$ is a subset of ${\displaystyle H^{p}}$, and the ${\displaystyle H^{p}}$-norm is increasing with ${\displaystyle p}$ (it is a consequence of Hölder's inequality that the ${\displaystyle L^{p}}$-norm is increasing for probability measures, i.e. measures with total mass 1) (Rudin 1987, Def 17.7).

${\displaystyle H^{2}}$ is a Hilbert space, and it is unitarily equivalent to ${\displaystyle \ell ^{2}(\mathbb {N} )}$ via the unitary map ${\displaystyle \sum _{n=0}^{\infty }a_{n}z^{n}\leftrightarrow (a_{n})_{n=0}^{\infty }}$.

The Hardy spaces can also be viewed as closed vector subspaces of the complex L^p spaces on the unit circle ${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}$. This connection is provided by the following theorem (Katznelson 1976, Thm 3.8): Given ${\displaystyle f\in H^{p}}$ with ${\displaystyle p\geq 1}$, the radial limit $${\displaystyle {\tilde {f}}\!\left(e^{i\theta }\right)=\lim _{r\to 1}\,f\!\left(re^{i\theta }\right)}$$ exists for almost every ${\displaystyle \theta }$ and ${\displaystyle {\tilde {f}}\in L^{p}(\mathbb {T} )}$ such that ^{[clarification needed]} $${\displaystyle {\|{\tilde {f}}\|}_{L^{p}}={\|f\|}_{H^{p}}.}$$ Denote by ${\displaystyle H^{p}(\mathbb {T} )}$ the vector subspace of ${\displaystyle L^{p}(\mathbb {T} )}$ consisting of all limit functions ${\displaystyle {\tilde {f}}}$, when ${\displaystyle f}$ varies in ${\displaystyle H^{p}}$, one then has that for p ≥ 1,(Katznelson 1976)

where the ${\displaystyle {\hat {g}}_{n}}$ are the Fourier coefficients defined as $${\displaystyle {\hat {g}}_{n}={\frac {1}{2\pi }}\int _{0}^{2\pi }g\left(e^{i\phi }\right)e^{-in\phi }\,\mathrm {d} \phi ,\quad \forall n\in \mathbb {Z} .}$$ The space ${\displaystyle H^{p}(\mathbb {T} )}$ is a closed subspace of ${\displaystyle L^{p}(\mathbb {T} )}$. Since ${\displaystyle L^{p}(\mathbb {T} )}$ is a Banach space (for ${\displaystyle 1\leq p\leq \infty }$), so is ${\displaystyle H^{p}(\mathbb {T} )}$.