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 triumphs of peace have been in some proximity to war. Whilst the hand was still familiar with the sword-hilt, whilst the habits of the camp were still visible in the port and complexion of the gentleman, his intellectual power culminated; the compression and tension of these stern conditions is a training for the finest and softest arts, and can rarely be compensated in tranquil times, except by some analogous vigor drawn from occupations as hardy as war.”
—Ralph Waldo Emerson (1803–1882)
“We should read history as little critically as we consider the landscape, and be more interested by the atmospheric tints and various lights and shades which the intervening spaces create than by its groundwork and composition.”
—Henry David Thoreau (1817–1862)
“The enemy are no match for us in a fair fight.... The young men ... of the upper class are kind-hearted, good-natured fellows, who are unfit as possible for the business they are in. They have courage but no endurance, enterprise, or energy. The lower class are cowardly, cunning, and lazy. The height of their ambition is to shoot a Yankee from some place of safety.”
—Rutherford Birchard Hayes (1822–1893)
“It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.”
—Anna Howard Shaw (1847–1919)