In complex analysis, the Hardy spaces (or Hardy classes are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are spaces of distributions on the real , defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the Lp spaces. these Hardy spaces are subsets of L^p spaces spaces.

Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g. H^{\infty} methods) and scattering theory.

Definition

On the unit disk

is the class of holomorphic functions on the open unit disk

is defined as the vector space of bounded holomorphic functions on the unit disk, with norm

is a subset (it is a consequence of Hölder's inequality-norm is increasing for probability measures, i.e. measures with total mass 1) .

is a Hilbert space.

On the unit circle

The Hardy spaces can also be viewed as closed vector subspaces of the complex Lp spaces on the unit circle. This connection is provided by the following theorem such that , one then has that for p≥1,

are the Fourier coefficients is a Banach space.

, 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 is in Hp(T is in Hp(T has Fourier coefficients (an)n∈Z with an = 0 for every n < 0, then the associated holomorphic function f of Hp 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 from below. Thus, the space H2 is seen to sit naturally inside L2 space, and is represented by infinite sequences indexed by N; whereas L2 consists of bi-infinite sequences indexed by Z.

On the upper half plane

The Hardy space on the upper half-plane The unit disk is isomorphic to the upper half-plane by means of a Möbius transformation is an isometric isomorphism of Hardy spaces.

A similar approach applies to, e.g., the right half-plane.

On the real vector space

) consists of tempered distributions such that for some Schwartz function, the maximal function

-quasinorm.

; for example, on the line

of functions of bounded mean oscillation).

into a complete metric space.

Atomic decomposition

if and only if all its moments

, this is also sufficient.

is called an -atom.

has an atomic decomposition-atoms,

are scalars.

can be represented as a series of Haar functions.)

Link between real- and complex-variable Hardy spaces

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. 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 eiθ → Pr(θ) on the circle. 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 Mf is in Lp(T).

The function F defined on the unit disk by F(reiθ) = (f ∗ Pr)(eiθ) is harmonic, and Mf is the radial maximal function of F. When Mf belongs to Lp(T) and p1, 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).

Conjugate function

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). When 1 p< ∞, the following are equivalent for a real valued integrable function f on the unit circle:

• the function f is the real part of some function g ∈ Hp(T)

• the function f and its conjugate H(f) belong to Lp(T)

• the radial maximal function M f belongs to Lp(T).

When 1 < p < ∞, H(f) belongs to Lp(T) when f ∈ Lp(T), hence the real Hardy space Hp(T) coincides with Lp(T) in this case. For p = 1, the real Hardy space H1(T) is a proper subspace of L1(T).

The case of p = ∞ was excluded from the definition of real Hardy spaces, because the maximal function Mf of an L∞ function is always bounded, and because it is not desirable that real-H∞ be equal to L∞. However, the two following properties are equivalent for a real valued function f

• the function f is the real part of some function g ∈ H∞(T)

• the function f and its conjugate H(f) belong to L∞(T).

For 0 < p < 1

When 0 < p < 1, a function F in Hp cannot be reconstructed from the real part of its boundary limit function on the circle, because of the lack of convexity of Lp in this case. Convexity fails but a kind of "complex convexity" remains, namely the fact that z → |z|q is subharmonic for every q > 0. As a consequence, if

is in Hp, it can be shown that cn = O(n1/p–1). It follows that the Fourier series

converges in the sense of distributions to a distribution f on the unit circle, and F(reiθ)=(f∗Pr)(θ). The function F ∈ Hp can be reconstructed from the real distribution Re(f) on the circle, because the Taylor coefficients cn of F can be computed from the Fourier coefficients of Re(f).

Distributions on the circle are general enough for handling Hardy spaces when p<1. Distributions that are not functions do occur, as is seen with functions F(z) = (1−z)−N (for |z| < 1), that belong to Hp when 0< Np< 1 (and N an integer ≥1).

A real distribution on the circle belongs to real-Hp(T) iff it is the boundary value of the real part of some F ∈ Hp. A Dirac distribution δx, at any point x of the unit circle, belongs to real-Hp(T) for every p < 1; derivatives δ′x belong when p < 1/2, second derivatives δ′′x when p < 1/3, and so on.

Beurling factorization

For 0<p≤∞, every non-zero function f in Hp can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below . This "Beurling factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions.

One says that G(z) is an outer (exterior) function if it takes the form

for some complex number c with |c is integrable on the circle, G is in H1 because the above takes the form of the Poisson kernel . This implies that

for almost every θ.

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. In particular, h is in H∞. The inner function can be further factored into a form involving a Blaschke product.

The function f, decomposed as f = Gh, is in Hp if and only if φ belongs to Lp(T), where φ is the positive function in the representation of the outer function G.

Let G be an outer function represented as above from a function φ on the circle. Replacing φ by φα, α > 0, a family (Gα) of outer functions is obtained, with the properties:

1. G1= G, Gα+β = GαGβ and |Gα| = |G|α almost everywhere on the circle.

It follows that whenever 0 < p, q, r < ∞ and 1/r = 1/p + 1/q, every function f in Hr can be expressed as the product of a function in Hp and a function in Hq. For example: every function in H1 is the product of two functions in H2; every function in Hp, p < 1, can be expressed as product of several functions in some Hq, q>1.

Martingale Hp

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. The maximal function of the martingale is defined by

Let 1 ≤ p < ∞. The martingale (Mn)n≥0 belongs to martingale-Hp when M* ∈ Lp.

If M* ∈ Lp, the martingale (Mn)n≥0 is bounded in Lp; hence it converges almost surely to some function f by the martingale convergence theorem. Moreover, Mn converges to f in Lp-norm by the dominated convergence theorem; hence Mn can be expressed as conditional expectation of f on Σn. It is thus possible to identify martingale-Hp with the subspace of Lp(Ω,Σ,P) consisting of those f such that the martingale

belongs to martingale-Hp.

Doob's maximal inequality implies that martingale-Hp coincides with Lp(Ω,Σ,P) when 1 < p < ∞. The interesting space is martingale-H1, whose dual is martingale-BMO .

The Burkholder–Gundy inequalities (when p>1) and the Burgess Davis inequality (when p = 1) relate the Lp-norm of the maximal function to that of the square function of the martingale

Martingale-Hp can be defined by saying that S(f)∈ Lp .

Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex Brownian motion (Bt) in the complex plane, starting from the point z = 0 at time t = 0. Let τ denote the hitting time of the unit circle. For every holomorphic function F in the unit disk,

is a martingale, that belongs to martingale-Hp iff F∈Hp .

Example

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 . The Haar system is an unconditional basis for H1(δ).

See also

• H2

• H∞ methods

• Paley–Wiener theorem

Notes

References