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Wikipedia

In complex analysis, the Hardy spaces (or Hardy classes) H^p are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them for G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (more or less) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the Lp spaces of functional analysis. For 1 < p < ∞ these real Hardy spaces H^p are essentially the same as L^p, while for p ≤ 1 the L^p spaces have some undesirable properties, and the Hardy spaces are much better behaved.

There are also higher dimensional generalizations, consisting of certain holomorphic functions on tube domains in the complex case, or certain spaces of distributions on Rⁿ in the real case.

Hardy spaces for the unit disk

For spaces of holomorphic functions on the open unit disc, the Hardy space H² consists of the functions ƒ whose mean square value on the circle of radius r remains finite as r → 1 from below.

More generally, the Hardy space H^p for $0 is the class of holomorphic functions on the open unit disc satisfying$

sup_{0

The number on the left side of the above inequality is the Hardy space

$p$ -norm for $f$ , denoted by $|f|_{H^p}.$

For $0, it can be shown that H$ ^q is a subset of H^p.

Applications

Such spaces have a number of applications in mathematical analysis itself, and also to control theory and scattering theory. A space H² may sit naturally inside an L² space as a 'causal' part, for example represented by infinite sequences indexed by N, where L² consists of bi-infinite sequences indexed by Z.

Factorization

For $pgeq 1$ , every function $f in H^p$ can be written as the product ƒ = Gh where G is an outer function and h is an inner function, as defined below.

One says that h(z) is an inner (interior) function if and only if $|h(z)|leq 1$ on the unit disc and the limit

lim_{rrightarrow 1^-} h(re^{itheta})

exists for almost all $theta$ and its modulus is equal to 1.

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

G(z)=expleft[iphi+frac{1}{2pi} int_0^{2pi}

frac{e^{itheta}+z}{e^{itheta}-z} g(e^{itheta}) dtheta right]

for some real value $,phi$ and some real-valued function g(z) that is integrable on the unit circle.

The inner function can be further factored into a form involving a Blaschke product.

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. See, for example, the cited book of Hoffman.

The Hardy space H^p on the upper half plane is defined to be the space of holomorphic functions F on the upper half plane such that

int|F(x+iy)|^p,dx

is bounded over all y > 0. The Hardy quasi-norm ||F||_Hp is defined to be the pth root of the supremum over y > 0.

Real Hardy spaces for Rⁿ

In analysis on the real vector space Rⁿ, the Hardy space H^p (for 0 < p ≤ ∞) consists of distributions ƒ such that for some Schwartz function Φ with ∫Φ = 1, the maximal function

(M_Phi f)(x)=sup_{t>0}|(f*Phi_t)(x)|

is in L^p, where * is convolution and Φ_t(x) = t⁻ⁿΦ(x/t).

If 1<p≤∞ then the Hardy space H^p is essentially the same as L^p. When p=1, the Hardy space H¹ is a proper subspace of L¹, and its dual is the space of functions of bounded mean oscillation. If p < 1 then the Hardy space H^p has elements that are not functions, and its dual is the homogeneous Lipschitz space of order n(1/p − 1).

The H^p-quasinorm ||ƒ||_Hp of a distribution ƒ of H^p is defined to be the L^p norm of M_Φƒ. (This depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms.) Strictly speaking this quasinorm is not a norm in the usual sense of Banach spaces for p < 1 as it is not subadditive, but is still often called a norm. The pth power ||ƒ||_Hp^p is subadditive for p ≤ 1 and so defines a metric on the Hardy space H^p, which defines the topology and makes H^p into a complete metric space. (Warning: if p < 1 then H^p is not a Banach space, as ||ƒ||_Hp^p is not a norm: it is not homogeneous of degree 1.)

A bounded function ƒ of compact support is in the Hardy space H^p if and only if all its moments

int f(x)x_1^{i_1}cdots x_n^{i_n},dx

whose order i₁ + ... + i_n is at most n(1/p − 1) vanish. If in addition ƒ has support in some ball B and is bounded by |B|^−1/p then ƒ is called an H^p-atom. Moreover any element of H^p has an atomic decomposition as a convergent infinite sum of H^p-atoms.