Hardy Space - Hardy Spaces For The Upper Half Plane

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:

    “Ethelberta breathed a sort of exclamation, not right out, but stealthily, like a parson’s damn.”
    —Thomas Hardy (1840–1928)

    “Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
    —Jean Baudrillard (b. 1929)

    “When my old wife lived, upon
    This day she was both pantler, butler, cook,
    Both dame and servant, welcomed all, served all,
    Would sing her song and dance her turn, now here
    At upper end o’the table, now i’the middle,
    On his shoulder, and his, her face afire
    With labor, and the thing she took to quench it
    She would to each one sip.”
    —William Shakespeare (1564–1616)

    “At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
    —Friedrich Nietzsche (1844–1900)