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
Read more about this topic: Hardy Space
Famous quotes containing the words hardy, spaces, upper and/or plane:
“A lover without indiscretion is no lover at all. Circumspection and devotion are a contradiction in terms.”
—Thomas Hardy (1840–1928)
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)
“But that beginning was wiped out in fear
The day I swung suspended with the grapes,
And was come after like Eurydice
And brought down safely from the upper regions;
And the life I live now’s an extra life
I can waste as I please on whom I please.”
—Robert Frost (1874–1963)
“As for the dispute about solitude and society, any comparison is impertinent. It is an idling down on the plane at the base of a mountain, instead of climbing steadily to its top.”
—Henry David Thoreau (1817–1862)