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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 (Riesz 1923), who named them after G. H. Hardy, because of the paper (Hardy 1915). In real analysis

There are also higher-dimensional generalizations, consisting of certain holomorphic functions on tube domains in the complex case, or certain spaces of distributions on **R**^{n} in the real case.

Hardy spaces have a number of applications in mathematical analysis itself, as well as in control theory (such as *H*^{∞} methods) and in scattering theory.

For spaces of holomorphic functions on the open unit disk, the Hardy space *H*^{2} consists of the functions *f* whose mean square value on the circle of radius *r* remains bounded as *r* → 1 from below.

More generally, the Hardy space *H ^{p}* for 0 <

This class *H ^{p}* is a vector space. The number on the left side of the above inequality is the Hardy space

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 *H ^{q}* is a subset of

The Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex *L ^{p}* spaces on the unit circle. This connection is provided by the following theorem (Katznelson 1976, Thm 3.8): Given

exists for almost every θ. The function belongs to the *L ^{p}* space for the unit circle,

Denoting the unit circle by **T**, and by *H ^{p}*(

where the *ĝ*(*n*) are the Fourier coefficients of a function *g* integrable on the unit circle,

The space *H ^{p}*(

The above can be turned around. Given a function , with *p* ≥ 1, one can regain a (harmonic) function *f* on the unit disk by means of the Poisson kernel *P _{r}*:

and *f* belongs to *H ^{p}* exactly when is in

In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions.^{[clarification needed]} Thus, the space *H*^{2} is seen to sit naturally inside *L*^{2} space, and is represented by infinite sequences indexed by **N**; whereas *L*^{2} consists of bi-infinite sequences indexed by **Z**.

When 1 ≤ *p* < ∞, the *real Hardy spaces* *H ^{p}* discussed further down

For 0 < *p* < 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid. For example, consider the function

Then *F* is in *H ^{p}* for every 0 <

exists for a.e. *θ* and is in *H ^{p}*(

For the same function *F*, let *f _{r}*(e

For 0 < *p* ≤ ∞, every non-zero function *f* in *H ^{p}* can be written as the product

One says that *G*(*z*)^{[clarification needed]} is an **outer (exterior) function** if it takes the form

for some complex number *c* with |*c*| = 1, and some positive measurable function on the unit circle such that is integrable on the circle. In particular, when is integrable on the circle, *G* is in *H*^{1} because the above takes the form of the Poisson kernel (Rudin 1987, Thm 17.16). 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*^{∞}.^{[clarification needed]} The inner function can be further factored into a form involving a Blaschke product.

The function *f*, decomposed as *f* = *Gh*,^{[clarification needed]} is in *H ^{p}* if and only if φ belongs to

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

*G*_{1}=*G*,*G*_{α+β}=*G*_{α}*G*_{β}and |*G*_{α}| = |*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 *H ^{r}* can be expressed as the product of a function in

Real-variable techniques, mainly associated to the study of *real Hardy spaces* defined on **R**^{n} (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 *P _{r}* denote the Poisson kernel on the unit circle

where the *star* indicates convolution between the distribution *f* and the function e^{iθ} → *P _{r}*(θ) on the circle. Namely, (

For 0 < *p* < ∞, the *real Hardy space* *H ^{p}*(

The function *F* defined on the unit disk by *F*(*re*^{iθ}) = (*f* ∗ *P _{r}*)(e

To every real trigonometric polynomial *u* on the unit circle, one associates the real *conjugate polynomial* *v* such that *u* + i*v* extends to a holomorphic function in the unit disk,

This mapping *u* → *v* extends to a bounded linear operator *H* on *L ^{p}*(

- the function
*f*is the real part of some function*g*∈*H*(^{p}**T**) - the function
*f*and its conjugate*H(f)*belong to*L*(^{p}**T**) - the radial maximal function
*M f*belongs to*L*(^{p}**T**).

When 1 < *p* < ∞, *H(f)* belongs to *L ^{p}*(

The case of *p* = ∞ was excluded from the definition of real Hardy spaces, because the maximal function *M f* 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**).

When 0 < *p* < 1, a function *F* in *H ^{p}* cannot be reconstructed from the real part of its boundary limit

is in *H ^{p}*, it can be shown that

converges in the sense of distributions to a distribution *f* on the unit circle, and *F*(*re*^{iθ}) =(*f* ∗ *P _{r}*)(θ). The function

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 *H ^{p}* when 0 <

A real distribution on the circle belongs to real-*H ^{p}*(

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 *H ^{p}*(

The corresponding *H*^{∞}(**H**) is defined as functions of bounded norm, with the norm given by

Although the unit disk **D** and the upper half-plane **H** can be mapped to one another by means of Möbius transformations, they are not interchangeable^{[clarification needed]} 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 *H*^{2}, one has the following theorem: if *m* : **D** → **H** denotes the Möbius transformation

Then the linear operator *M* : *H*^{2}(**H**) → *H*^{2}(**D**) defined by

is an isometric isomorphism of Hilbert spaces.

In analysis on the real vector space **R**^{n}, the Hardy space *H ^{p}* (for 0 <

is in *L ^{p}*(

If 1 < *p* < ∞, the Hardy space *H ^{p}* is the same vector space as

The *L*^{1} and *H*^{1} norms are not equivalent on *H*^{1}, and *H*^{1} is not closed in *L*^{1}. The dual of *H*^{1} is the space *BMO* of functions of bounded mean oscillation. The space *BMO* contains unbounded functions (proving again that *H*^{1} is not closed in *L*^{1}).

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

When 0 < *p* ≤ 1, a bounded measurable function *f* of compact support is in the Hardy space *H ^{p}* if and only if all its moments

whose order *i*_{1}+ ... +*i _{n}* is at most

If in addition *f* has support in some ball *B* and is bounded by |*B*|^{−1/p} then *f* is called an ** H^{p}-atom** (here |

When 0 < *p* ≤ 1, any element *f* of *H ^{p}* has an

where the *a _{j}* are

On the line for example, the difference of Dirac distributions *f* = δ_{1}−δ_{0} can be represented as a series of Haar functions, convergent in *H ^{p}*-quasinorm when 1/2 <

Let (*M _{n}*)

Let 1 ≤ *p* < ∞. The martingale (*M _{n}*)

If *M** ∈ *L ^{p}*, the martingale (

belongs to martingale-*H ^{p}*.

Doob's maximal inequality implies that martingale-*H ^{p}* coincides with

The Burkholder–Gundy inequalities (when *p* > 1) and the Burgess Davis inequality (when *p* = 1) relate the *L ^{p}*-norm of the maximal function to that of the

Martingale-*H ^{p}* can be defined by saying that

Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex Brownian motion (*B _{t}*) in the complex plane, starting from the point

is a martingale, that belongs to martingale-*H ^{p}* iff

In this example, Ω = [0, 1] and Σ_{n} is the finite field generated by the dyadic partition of [0, 1] into 2^{n} 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 (*h _{k}*)

then the martingale-*H*^{1} norm of *f* can be defined by the *L*^{1} norm of the square function

This space, sometimes denoted by *H*^{1}(δ), is isomorphic to the classical real *H*^{1} space on the circle (Müller 2005). The Haar system is an unconditional basis for *H*^{1}(δ).

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