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:
“Pessimism ... is, in brief, playing the sure game. You cannot lose at it; you may gain. It is the only view of life in which you can never be disappointed. Having reckoned what to do in the worst possible circumstances, when better arise, as they may, life becomes child’s play.”
—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)
““All men live in suffering
I know as few can know,
Whether they take the upper road
Or stay content on the low....””
—William Butler Yeats (1865–1939)
“Have you ever been up in your plane at night, alone, somewhere, 20,000 feet above the ocean?... Did you ever hear music up there?... It’s the music a man’s spirit sings to his heart, when the earth’s far away and there isn’t any more fear. It’s the high, fine, beautiful sound of an earth-bound creature who grew wings and flew up high and looked straight into the face of the future. And caught, just for an instant, the unbelievable vision of a free man in a free world.”
—Dalton Trumbo (1905–1976)