Hardy Space - Hardy Spaces For The Upper Half Plane

Hardy Spaces For The Upper Half Plane

It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on a complex half-plane (usually the right half-plane or upper half-plane) are used.

The Hardy space on the upper half-plane is defined to be the space of holomorphic functions f on with bounded (quasi-)norm, the norm being given by

The corresponding is defined as functions of bounded norm, with the norm given by

Although the unit disk and the upper half-plane can be mapped to one another by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. However, for H2, one may still state the following theorem: Given the Möbius transformation with

then there is an isometric isomorphism

with

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