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:
“The instincts of merry England lingered on here with exceptional vitality, and the symbolic customs which tradition has attached to each season of the year were yet a reality on Egdon. Indeed, the impulses of all such outlandish hamlets are pagan still: in these spots homage to nature, self-adoration, frantic gaieties, fragments of Teutonic rites to divinities whose names are forgotten, seem in some way or other to have survived mediaeval doctrine.”
—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)
“In time the scouring of wind and rain will wear down the ranges and plane off the region until it has the drab monotony of the older deserts. In the meantime—a two-million-year meantime—travelers may enjoy the cruel beauties of a desert in its youth,....”
—For the State of California, U.S. public relief program (1935-1943)