Hardy spaces are a class of function spaces that play a central role in complex analysis and several areas of harmonic analysis. They are primarily associated with functions that are analytic in a certain domain, typically within the unit disk in the complex plane, and have specific growth and boundary behavior.
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For , an analytic Hardy space controls the integral means of a holomorphic function as it approaches a domain boundary. On the disk, requires ; on the upper half-plane it requires . For use a supremum norm. These analytic spaces differ from the harmonic Hardy space of the upper half-space; the Hardy space of the circle is the boundary Hilbert-space realization of the disk space.