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:
“If all hearts were open and all desires known—as they would be if people showed their souls—how many gapings, sighings, clenched fists, knotted brows, broad grins, and red eyes should we see in the market-place!”
—Thomas Hardy (1840–1928)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)
“Give me the islands of the upper air,
all mountains
and the towering mountain trees.”
—Hilda Doolittle (1886–1961)
“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)