are spaces of holomorphic functions on the unit disk or upper half plane.
, defined (in the sense of distributions) as boundary values of the holomorphic functions.
[2] Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g.
then the equation coincides with the definition of the Hardy space p-norm, denoted by
The space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm For 0 < p ≤ q ≤ ∞, the class Hq is a subset of Hp, and the Hp-norm is increasing with p (it is a consequence of Hölder's inequality that the Lp-norm is increasing for probability measures, i.e. measures with total mass 1) (Rudin 1987, Def 17.7).
Denote by Hp(T) the vector subspace of Lp(T) consisting of all limit functions
, with p ≥ 1, one can regain a (harmonic) function f on the unit disk by means of the Poisson kernel Pr: and f belongs to Hp exactly when
In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions.
For example, the Hardy space H2 consists of functions whose mean square value remains bounded as
The unit disk is isomorphic to the upper half-plane by means of a Möbius transformation.
In analysis on the real vector space Rn, the Hardy space Hp (for 0 < p ≤ ∞) consists of tempered distributions f such that for some Schwartz function Φ with ∫Φ = 1, the maximal function is in Lp(Rn), where ∗ is convolution and Φt (x) = t −nΦ(x / t).
The Hp-quasinorm ||f ||Hp of a distribution f of Hp is defined to be the Lp norm of MΦf (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms).
One can find sequences in H1 that are bounded in L1 but unbounded in H1, for example on the line The L1 and H1 norms are not equivalent on H1, and H1 is not closed in L1.
The dual of H1 is the space BMO of functions of bounded mean oscillation.
When 0 < p ≤ 1, a bounded measurable function f of compact support is in the Hardy space Hp if and only if all its moments whose order i1+ ... +in is at most n(1/p − 1), vanish.
If in addition f has support in some ball B and is bounded by |B|−1/p then f is called an Hp-atom (here |B| denotes the Euclidean volume of B in Rn).
The Hp-quasinorm of an arbitrary Hp-atom is bounded by a constant depending only on p and on the Schwartz function Φ.
Real-variable techniques, mainly associated to the study of real Hardy spaces defined on Rn, 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.
Namely, (f ∗ Pr)(eiθ) 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).
To every real trigonometric polynomial u on the unit circle, one associates the real conjugate polynomial v such that u + iv extends to a holomorphic function in the unit disk, This mapping u → v extends to a bounded linear operator H on Lp(T), when 1 < p < ∞ (up to a scalar multiple, it is the Hilbert transform on the unit circle), and H also maps L1(T) to weak-L1(T).
is integrable on the circle, G is in H1 because the above takes the form of the Poisson kernel (Rudin 1987, Thm 17.16).
One says that h is an inner (interior) function if and only if |h| ≤ 1 on the unit disk and the limit exists for almost all θ and its modulus is equal to 1 a.e.
[clarification needed] The inner function can be further factored into a form involving a Blaschke product.
Let (Mn)n≥0 be a martingale on some probability space (Ω, Σ, P), with respect to an increasing sequence of σ-fields (Σn)n≥0.
Assume for simplicity that Σ is equal to the σ-field generated by the sequence (Σn)n≥0.
Doob's maximal inequality implies that martingale-Hp coincides with Lp(Ω, Σ, P) when 1 < p < ∞.
For every holomorphic function F in the unit disk, is a martingale, that belongs to martingale-Hp iff F ∈ Hp (Burkholder, Gundy & Silverstein 1971).
In this example, Ω = [0, 1] and Σn is the finite field generated by the dyadic partition of [0, 1] into 2n intervals of length 2−n, for every n ≥ 0.
If a function f on [0, 1] is represented by its expansion on the Haar system (hk) then the martingale-H1 norm of f can be defined by the L1 norm of the square function This space, sometimes denoted by H1(δ), is isomorphic to the classical real H1 space on the circle (Müller 2005).