Real-variable Techniques On The Unit Circle
Real-variable techniques, mainly associated to the study of real Hardy spaces defined on Rn (see below), are also used in the simpler framework of the circle. It is a common practice to allow for complex functions (or distributions) in these "real" spaces. The definition that follows does not distinguish between real or complex case.
Let Pr denote the Poisson kernel on the unit circle T. For a distribution f on the unit circle, set
where the star indicates convolution between the distribution f and the function ei θ → Pr(θ) on the circle. Namely, (f ∗ Pr)(ei θ) is the result of the action of f on the C∞-function defined on the unit circle by
For 0 < p < ∞, the real Hardy space Hp(T) consists of distributions f such that M f is in Lp(T).
The function F defined on the unit disk by F(r ei θ) = (f ∗ Pr)(ei θ) is harmonic, and M f is the radial maximal function of F. When M f belongs to Lp(T) and p ≥ 1, the distribution f "is" a function in Lp(T), namely the boundary value of F. For p ≥ 1, the real Hardy space Hp(T) is a subset of Lp(T).
Read more about this topic: Hardy Space
Famous quotes containing the words techniques, unit and/or circle:
“It is easy to lose confidence in our natural ability to raise children. The true techniques for raising children are simple: Be with them, play with them, talk to them. You are not squandering their time no matter what the latest child development books say about “purposeful play” and “cognitive learning skills.””
—Neil Kurshan (20th century)
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)
“... in any war a victory means another war, and yet another, until some day inevitably the tides turn, and the victor is the vanquished, and the circle reverses itself, but remains nevertheless a circle.”
—Pearl S. Buck (1892–1973)