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In [[complex analysis]], the '''Hardy spaces''' (or '''Hardy classes''') ''H<sup>p</sup>'' are certain spaces of [[holomorphic function]]s on the [[unit disk]] or [[upper half plane]]. They were introduced by [[Frigyes Riesz]] {{harv|Riesz|1923}}, who named them after [[G. H. Hardy]], because of the paper {{harv|Hardy|1915}}. In [[real analysis]] '''Hardy spaces''' are certain spaces of [[Distribution (mathematics)|distributions]] on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the [[complex number|complex]] Hardy spaces, and are related to the [[Lp space|''L<sup>p</sup>'' spaces]] of [[functional analysis]]. For 1&nbsp;≤&nbsp;''p''&nbsp;≤&nbsp;∞ these real Hardy spaces ''H<sup>p</sup>'' are certain [[subset]]s of ''L<sup>p</sup>'', while for ''p'' < 1 the ''L<sup>p</sup>'' 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 domain]]s in the complex case,  or certain spaces of distributions on '''R'''<sup>''n''</sup> in the real case.

Hardy spaces have a number of applications in [[mathematical analysis]] itself, as well as in [[control theory]] (such as [[H infinity|''H''<sup>∞</sup> methods]]) and in [[scattering theory]].

==Hardy  spaces for the unit disk==
For  spaces of [[holomorphic]] functions on the open [[unit disk]], the Hardy space ''H''<sup>2</sup> consists of the functions ''f'' whose [[root mean square|mean square value]] on the circle of radius ''r'' remains bounded as ''r'' → 1 from below.

More generally, the Hardy space ''H<sup>p</sup>'' for 0 < ''p'' < ∞ is the class of holomorphic functions ''f'' on the open unit disk satisfying

:<math>\sup_{0<r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\theta}\right )\right|^p \; \mathrm{d}\theta\right)^\frac{1}{p}<\infty.</math>

This class ''H<sup>p</sup>'' is a vector space. The number on the left side of the above inequality is the Hardy space ''p''-norm for ''f'', denoted by <math>\|f\|_{H^p}.</math> It is a norm when ''p'' ≥ 1, but not when 0&nbsp;< ''p''&nbsp;<&nbsp;1.

The space ''H''<sup>∞</sup> is defined as the vector space of bounded holomorphic functions on the disk, with the norm

:<math>\|f\|_{H^\infty} = \sup_{|z|< 1} \left|f(z)\right|.</math>

For 0&nbsp;< p&nbsp;≤ q&nbsp;≤ ∞, the class ''H<sup>q</sup>'' is  a [[subset]] of ''H<sup>p</sup>'', and the ''H<sup>p</sup>''-norm is increasing with ''p'' (it is a consequence of [[Hölder's inequality]] that the ''L<sup>p</sup>''-norm is increasing for [[Probability space|probability measures]], i.e. [[Measure (mathematics)|measures]] with total mass 1).

==Hardy spaces on the unit circle==
The Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex [[Lp space|''L<sup>p</sup>'' spaces]] on the unit circle. This connection is provided by the following theorem {{harv|Katznelson|1976|loc=Thm 3.8}}:  Given ''f'' ∈ ''H<sup>p</sup>'', with ''p'' ≥ 0, the radial limit

:<math>\tilde f\left(e^{i\theta}\right) = \lim_{r\to 1} f\left(re^{i\theta}\right)</math>

exists for almost every θ. The function <math>\tilde f</math> belongs to the ''L<sup>p</sup>'' space for the unit circle, and one has that

:<math>\|\tilde f\|_{L^p} = \|f\|_{H^p}.</math>

Denoting the unit circle by '''T''', and by ''H<sup>p</sup>''('''T''') the vector subspace of ''L<sup>p</sup>''('''T''') consisting of all limit functions <math>\tilde f</math>, when ''f'' varies in ''H<sup>p</sup>'', one then has that for ''p''&nbsp;≥&nbsp;1,{{harv|Katznelson|1976}}

:<math>g\in H^p\left(\mathbf{T}\right)\text{ if and only if } g\in L^p\left(\mathbf{T}\right)\text{ and } \hat{g}(n)=0 \text{ for all } n < 0,</math>

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

:<math>\forall n \in \mathbf{Z}, \ \ \ \hat{g}(n) = \frac{1}{2\pi}\int_0^{2\pi} g\left(e^{i\phi}\right) e^{-in\phi} \, \mathrm{d}\phi.</math>

The space ''H<sup>p</sup>''('''T''') is a closed subspace of ''L<sup>p</sup>''('''T'''). Since ''L<sup>p</sup>''('''T''') is a [[Banach space]] (for 1 ≤ ''p'' ≤ ∞), so is ''H<sup>p</sup>''('''T''').

The above can be turned around.  Given a function <math>\tilde f</math> ∈ ''L<sup>p</sup>''('''T'''), with ''p'' ≥ 1, one can regain a ([[Harmonic function|harmonic]]) function ''f'' on the unit disk by means of the [[Poisson kernel]] ''P<sub>r</sub>'':

:<math>f\left(re^{i\theta}\right)=\frac{1}{2\pi} \int_0^{2\pi} P_r(\theta-\phi) \tilde f\left(e^{i\phi}\right) \,\mathrm{d}\phi, \quad r < 1,</math>

and ''f'' belongs to ''H<sup>p</sup>'' exactly when <math>\tilde f</math> is in ''H<sup>p</sup>''('''T'''). Supposing that <math>\tilde f</math> is in ''H<sup>p</sup>''('''T'''). ''i.e.'' that <math>\tilde f</math> has Fourier coefficients (''a<sub>n</sub>'')<sub>''n''∈'''Z'''</sub> with ''a<sub>n</sub>'' = 0 for every ''n'' < 0,then the element ''f'' of the Hardy space ''H<sup>p</sup>'' associated to <math>\tilde f</math> is the holomorphic function

:<math>f(z)=\sum_{n=0}^\infty a_n z^n, \ \ \ |z| < 1.</math>

In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the [[causal]] solutions. Thus, the space ''H''<sup>2</sup> is seen to sit naturally inside ''L''<sup>2</sup> space, and is represented by [[infinite sequence]]s indexed by '''N'''; whereas ''L''<sup>2</sup> consists of [[bi-infinite sequence]]s indexed by '''Z'''.

=== Connection to real Hardy spaces on the circle===
When 1 ≤ ''p'' < ∞, the ''real Hardy spaces'' ''H<sup>p</sup>'' discussed further down in this article are easy to describe in the present context. A real function ''f'' on the unit circle belongs to the real Hardy space ''H<sup>p</sup>''('''T''') if it is the real part of a function in ''H<sup>p</sup>''('''T'''), and a complex function ''f'' belongs to the real Hardy space iff Re(''f'') and Im(''f'') belong to the space (see the section on real Hardy spaces below).

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

:<math> F(z) = \frac{1 + z}{1 - z}, \quad |z| < 1</math> 

for which 

:<math>f(e^{i\theta}):= \tilde F(e^{i\theta}) = i \, \cot(\tfrac{\theta}{2}).</math>

The function ''F'' is in ''H<sup>p</sup>'' for every ''p'' < 1, the radial limit ''f'' is in ''H<sup>p</sup>''('''T''') but Re(''f'') is 0 almost everywhere.  It is no longer possible to recover ''F'' from Re(''f''), and one cannot define real-''H<sup>p</sup>''('''T''') in the simple way above.

For the same function ''F'', let ''f<sub>r</sub>''(e<sup>iθ</sup>) = ''F''(''re''<sup>iθ</sup>). The limit when ''r'' → 1 of Re(''f<sub>r</sub>''), ''in the sense of'' [[Distribution (mathematics)|''distributions'']] on the circle, is a non-zero multiple of the [[Dirac distribution]] at ''z'' = 1. The Dirac distribution at any point of the unit circle belongs to real-''H<sup>p</sup>''('''T''') for every ''p''&nbsp;<&nbsp;1 (see below).

==Factorization into inner and outer functions (Beurling)==
For 0&nbsp;<&nbsp;''p''&nbsp;≤&nbsp;∞, every non-zero function ''f'' in ''H<sup>p</sup>'' can be written as the product ''f'' = ''Gh'' where ''G'' is an ''outer function'' and ''h'' is an ''inner function'', as defined below {{harv|Rudin|1987|loc=Thm 17.17}}. This "[[Arne Beurling|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

:<math>G(z) = c\, \exp\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z} \log\!\left(\varphi\!\left(e^{i\theta} \right)\right)\, \mathrm{d}\theta \right)</math>

for some complex number ''c'' with |''c''| = 1, and some positive measurable function φ on the unit circle such that log(φ) is integrable on the circle.  In particular, when φ is integrable on the circle, ''G'' is in ''H''<sup>1</sup> because the above takes the form of the [[Poisson kernel]] {{harv|Rudin|1987|loc=Thm 17.16}}.  This implies that

:<math>\lim_{r\to 1^-}\left|G\left (re^{i\theta} \right)\right| = \varphi \left(e^{i\theta}\right )</math>

for almost every θ.

One says that ''h''(''z'') is an '''inner (interior) function''' if and only if |''h''(''z'')|&nbsp;≤&nbsp;1 on the unit disk and the limit

:<math>\lim_{r\to 1^-} h(re^{i\theta})</math>

exists for [[almost all]] θ and its [[Absolute value|modulus]] is equal to 1.  In particular, ''h'' is in ''H''<sup>∞</sup>. The inner function can be further factored into a form involving a [[Blaschke product]].

The function ''f'', decomposed as ''f'' = ''Gh'', is in ''H<sup>p</sup>'' if and only if the positive function φ belongs to ''L<sup>p</sup>''('''T'''), where φ is the 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 φ<sup>α</sup>, α > 0, a family (''G''<sub>α</sub>) of outer functions is obtained, with the properties:

:''G''<sub>1</sub>&nbsp;= ''G'', ''G''<sub>α+β</sub> = ''G''<sub>α</sub>&nbsp;''G''<sub>β</sub>&nbsp; and |''G''<sub>α</sub>| = |''G''|<sup>α</sup> 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<sup>r</sup>'' can be expressed as the product of a function in ''H<sup>p</sup>'' and a function in ''H<sup>q</sup>''. For example: every function in ''H''<sup>1</sup> is the product of two functions in ''H''<sup>2</sup>; every function in ''H<sup>p</sup>'', ''p'' < 1, can be expressed as product of several functions in some ''H<sup>q</sup>'', ''q''&nbsp;>&nbsp;1.

== Real-variable techniques on the unit circle ==
Real-variable techniques, mainly associated to the study of ''real Hardy spaces'' defined on '''R'''<sup>''n''</sup> (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<sub>r</sub>'' denote the Poisson kernel on the unit circle '''T'''. For a distribution ''f'' on the unit circle, set

:<math>(M f)(e^{i\theta})=\sup_{0<r<1} \left |(f * P_r) \left(e^{i\theta} \right)\right|,</math>

where the ''star'' indicates convolution between the distribution ''f'' and the function e<sup>iθ</sup> → ''P<sub>r</sub>''(θ) on the circle.  Namely, (''f'' ∗ ''P<sub>r</sub>'')(e<sup>iθ</sup>) is the result of the action of ''f'' on the ''C''<sup>∞</sup>-function defined on the unit circle by

:<math>e^{i\varphi} \rightarrow P_r(\theta - \varphi).</math>

For 0&nbsp;< ''p''&nbsp;< ∞, the ''real Hardy space'' ''H<sup>p</sup>''('''T''') consists of distributions ''f'' such that ''M&nbsp;f''&thinsp; is in ''L<sup>p</sup>''('''T''').

The function ''F'' defined on the unit disk by ''F''(''re''<sup>iθ</sup>) = (''f'' ∗ ''P<sub>r</sub>'')(e<sup>iθ</sup>) is harmonic, and ''M&nbsp;f''&thinsp; is the ''radial maximal function'' of ''F''. When ''M&nbsp;f''&thinsp; belongs to ''L<sup>p</sup>''('''T''') and ''p''&nbsp;&ge;&nbsp;1, the distribution ''f''&thinsp; "''is''" a function in ''L<sup>p</sup>''('''T'''), namely the boundary value of ''F''. For ''p''&nbsp;&ge; 1, the ''real Hardy space'' ''H<sup>p</sup>''('''T''') is a subset of ''L<sup>p</sup>''('''T''').

=== Conjugate function ===
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,

:<math> u(e^{i\theta}) = \frac{a_0}{2} + \sum_{k \ge 1} a_k \cos(k \theta) + b_k \sin(k \theta) \longrightarrow v(e^{i\theta}) = \sum_{k \ge 1} a_k \sin(k \theta) - b_k \cos(k \theta). </math>

This mapping ''u'' → ''v'' extends to a bounded linear operator ''H'' on ''L<sup>p</sup>''('''T'''), when 1&nbsp;< ''p''&nbsp;< ∞ (up to a scalar multiple, it is the [[Hilbert transform]] on the unit circle), and ''H'' also maps ''L''<sup>1</sup>('''T''') to [[Lp space#Weak Lp|weak-''L''<sup>1</sup>('''T''')]]. When 1&nbsp;&le; ''p''&nbsp;< ∞, 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'' ∈ ''H<sup>p</sup>''('''T''')
* the function ''f'' and its conjugate ''H(f)'' belong to ''L<sup>p</sup>''('''T''')
* the radial maximal function ''M f''&thinsp;  belongs to ''L<sup>p</sup>''('''T''').

When 1 < ''p'' < ∞, ''H(f)'' belongs to ''L<sup>p</sup>''('''T''') when ''f'' ∈ ''L<sup>p</sup>''('''T'''), hence the real Hardy space ''H<sup>p</sup>''('''T''') coincides with ''L<sup>p</sup>''('''T''') in this case. For ''p'' = 1,  the real Hardy space ''H''<sup>1</sup>('''T''') is a proper subspace of ''L''<sup>1</sup>('''T''').

The case of ''p'' = ∞ was excluded from the definition of real Hardy spaces, because the maximal function ''M&nbsp;f''&thinsp; of an ''L''<sup>∞</sup> function is always bounded, and because it is not desirable that real-''H''<sup>∞</sup> be equal to ''L''<sup>∞</sup>. However, the two following properties are equivalent for a real valued function ''f''
* the function ''f''&thinsp; is the real part of some function ''g'' ∈ ''H''<sup>∞</sup>('''T''')
* the function ''f''&thinsp; and its conjugate ''H(f)'' belong to ''L''<sup>∞</sup>('''T''').

=== Real Hardy spaces for 0 < ''p'' < 1 ===
When 0 < ''p'' < 1, a function ''F'' in ''H<sup>p</sup>'' cannot be reconstructed from the real part of his boundary limit ''function'' on the circle, because of the lack of convexity of ''L<sup>p</sup>'' in this case.  Convexity fails but a kind of "''complex convexity''" remains, namely the fact that ''z'' → |''z''|<sup>''q''</sup> is [[Subharmonic function#Subharmonic functions in the complex plane|subharmonic]] for every ''q'' > 0. As a consequence, if

:<math> F(z) = \sum_{n=0}^{+\infty} c_n z^n, \quad |z| < 1</math>

is in ''H<sup>p</sup>'', it can be shown that ''c<sub>n</sub>'' = O(''n''<sup>1/''p''–1</sup>). It follows that the Fourier series

:<math> \sum_{n=0}^{+\infty} c_n e^{in \theta}</math>

converges in the sense of distributions to a distribution ''f'' on the unit circle, and ''F''(''re''<sup>iθ</sup>)&nbsp;=(''f''&nbsp;∗&nbsp;''P<sub>r</sub>'')(θ).  The function ''F'' ∈ ''H<sup>p</sup>'' can be reconstructed from the real distribution Re(''f'') on the circle, because the Taylor coefficients ''c<sub>n</sub>'' 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''&nbsp;<&nbsp;1. Distributions do appear, as it is seen with functions ''F''(''z'') = (1−''z'')<sup>–''N''</sup> (for |''z''| < 1), that belong to ''H<sup>p</sup>'' when 0&nbsp;< ''N''&nbsp;''p''&nbsp;< 1 (and ''N'' an integer ≥&nbsp;1).

A real distribution on the circle belongs to real-''H<sup>p</sup>''('''T''') iff it is the boundary value of the real part of some ''F'' ∈ ''H<sup>p</sup>''. A Dirac distribution δ<sub>''x''</sub>, at any point ''x'' of the unit circle, belongs to real-''H<sup>p</sup>''('''T''') for every ''p'' < 1; derivatives δ′<sub>''x''</sub> belong when ''p'' < 1/2, second derivatives δ′′<sub>''x''</sub> when ''p'' < 1/3, and so on.

==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.

The Hardy space ''H<sup>p</sup>''('''H''') on the [[upper half-plane]] '''H''' is defined to be the space of holomorphic functions ''f'' on '''H''' with bounded (quasi-)norm, the norm being given by

:<math>\|f\|_{H^p} = \sup_{y>0} \left ( \int|f(x+ iy)|^p\, \mathrm{d}x \right)^{\frac{1}{p}}.</math>

The corresponding ''H''<sup>∞</sup>('''H''') is defined as functions of bounded norm, with the norm given by

:<math>\|f\|_{H^\infty} = \sup_{z\in\mathbf{H}}|f(z)|.</math>

Although the [[unit disk]] '''D''' and the upper half-plane '''H''' can be mapped to one another by means of [[Möbius transformation]]s, they are not interchangeable 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 square|''H''<sup>2</sup>]], one may still state the following theorem: Given the Möbius transformation ''m'' : '''D''' → '''H''' with

:<math>m(z)= i \cdot \frac{1+z}{1-z}</math>

then there is an [[Isometry|isometric]] [[isomorphism]] ''M'' : ''H''<sup>2</sup>('''H''') → ''H''<sup>2</sup>('''D''') with

:<math>(Mf)(z)=\frac{\sqrt{\pi}}{1-z} f(m(z)).</math>

==Real Hardy spaces for '''R'''<sup>''n''</sup>==
In analysis on the real vector space '''R'''<sup>''n''</sup>, the Hardy space ''H<sup>p</sup>'' (for 0&nbsp;<&nbsp;''p''&nbsp;≤&nbsp;∞) consists of [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distributions]] ''f'' such that for some [[Schwartz function]] Φ with ∫Φ&nbsp;=&nbsp;1, the [[maximal function]]

:<math>(M_\Phi f)(x)=\sup_{t>0}|(f*\Phi_t)(x)|</math>

is in ''L<sup>p</sup>''('''R'''<sup>''n''</sup>), where ∗ is convolution and {{nowrap|Φ<sub>''t''&thinsp;</sub>(''x'') {{=}}}} {{nowrap|''t''<sup>&thinsp;&minus;''n''</sup>Φ(''x''&thinsp;/&thinsp;''t'')}}. The ''H<sup>p</sup>''-[[quasinorm]] ||''f''&thinsp;||<sub>''Hp''</sub> of a distribution ''f'' of ''H<sup>p</sup>'' is defined to be the ''L<sup>p</sup>'' norm of ''M''<sub>Φ</sub>''f'' (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms).  The ''H<sup>p</sup>''-quasinorm is a norm when ''p'' ≥ 1, but not when ''p''&nbsp;<&nbsp;1.

If 1&nbsp;< ''p''&nbsp;< ∞, the Hardy space ''H<sup>p</sup>'' is the same vector space as ''L<sup>p</sup>'', with equivalent norm. When ''p''&nbsp;= 1, the Hardy space ''H''<sup>1</sup> is a proper subspace of ''L''<sup>1</sup>.  One can find sequences in ''H''<sup>1</sup> that are bounded in ''L''<sup>1</sup> but unbounded in ''H''<sup>1</sup>, for example on the line

:<math> f_k(x) = \mathbf{1}_{[0, 1]}(x - k) -  \mathbf{1}_{[0, 1]}(x + k), \ \ \ k > 0.</math>

The ''L''<sup>1</sup> and ''H''<sup>1</sup> norms are not equivalent on ''H''<sup>1</sup>, and ''H''<sup>1</sup> is not closed in ''L''<sup>1</sup>.  The dual of ''H''<sup>1</sup> is the space ''BMO'' of functions of [[bounded mean oscillation]].  The space ''BMO'' contains unbounded functions (proving again that ''H''<sup>1</sup> is not closed in ''L''<sup>1</sup>).

If ''p''&nbsp;<&nbsp;1 then the Hardy space ''H<sup>p</sup>'' has elements that are not functions, and its dual is the homogeneous Lipschitz space of order ''n''(1/''p''&nbsp;−&nbsp;1). When ''p'' < 1, the ''H<sup>p</sup>''-quasinorm is not  a norm, as it is not subadditive. The ''p''th power ||''f''&thinsp;||<sub>''Hp''</sub><sup>''p''</sup> is subadditive for ''p''&nbsp;<&nbsp;1 and so defines a metric on the Hardy space ''H<sup>p</sup>'', which defines the topology and makes ''H<sup>p</sup>'' into a complete metric space.

=== Atomic decomposition ===
When 0 < ''p'' ≤ 1, a bounded measurable function ''f'' of compact support is in the Hardy space ''H<sup>p</sup>'' if and only if all its moments

:<math>\int_{\mathbf{R}^n} f(x)x_1^{i_1}\ldots x_n^{i_n}\, \mathrm{d}x, </math>

whose order ''i''<sub>1</sub>+ ... +''i<sub>n</sub>'' is at most ''n''(1/''p''&nbsp;−&nbsp;1), vanish. For example, the integral of ''f'' must vanish in order that ''f'' ∈ ''H<sup>p</sup>'', 0&nbsp;< ''p'' ≤ 1, and as long as ''p''&nbsp;> {{nowrap|''n''&thinsp;/&thinsp;(''n''+1)}} this is also sufficient.

If in addition ''f'' has support in some ball ''B'' and is bounded by |''B''|<sup>−1/''p''</sup> then ''f'' is called an '''''H<sup>p</sup>''-atom''' (here |''B''| denotes the Euclidean volume of ''B'' in '''R'''<sup>''n''</sup>).  The ''H<sup>p</sup>''-quasinorm of an arbitrary ''H<sup>p</sup>''-atom is bounded by a constant depending only on ''p'' and on the Schwartz function Φ.

When 0 < ''p'' ≤ 1, any element ''f'' of ''H<sup>p</sup>'' has an '''atomic decomposition''' as a convergent infinite combination of ''H<sup>p</sup>''-atoms,

:<math>f = \sum c_j a_j, \ \ \ \sum |c_j|^p < \infty</math>

where the ''a<sub>j</sub>'' are ''H<sup>p</sup>''-atoms and the ''c<sub>j</sub>'' are scalars.

On the line for example, the difference of Dirac distributions ''f''&nbsp;= δ<sub>1</sub>−δ<sub>0</sub> can be represented as a series of [[Haar wavelet|Haar functions]], convergent in ''H<sup>p</sup>''-quasinorm when 1/2&nbsp;< ''p''&nbsp;<&nbsp;1 (on the circle, the corresponding representation is valid for 0&nbsp;< ''p''&nbsp;< 1, but on the line, Haar functions do not belong to ''H<sup>p</sup>'' when ''p'' ≤ 1/2 because their maximal function is equivalent at infinity to ''a''&nbsp;''x''<sup>–2</sup> for some ''a''&nbsp;≠&nbsp;0).

==Martingale ''H<sup>p</sup>''==
Let (''M<sub>n</sub>'')<sub>''n''≥0</sub> be a [[Martingale (probability theory)|martingale]] on some probability space (Ω,&nbsp;Σ,&nbsp;''P''), with respect to an increasing sequence of σ-fields (Σ<sub>''n''</sub>'')<sub>''n''≥0</sub>. Assume for simplicity that Σ is equal to the σ-field generated by the sequence (Σ<sub>''n''</sub>)<sub>''n''≥0</sub>. The ''maximal function'' of the martingale is defined by

:<math> M^* = \sup_{n \ge 0} \, |M_n|.</math>

Let 1 ≤ ''p'' < ∞. The martingale (''M<sub>n</sub>'')<sub>''n''≥0</sub> belongs to ''martingale''-''H<sup>p</sup>'' when ''M*'' ∈ ''L<sup>p</sup>''.

If ''M*'' ∈ ''L<sup>p</sup>'', the martingale (''M<sub>n</sub>'')<sub>''n''≥0</sub> is bounded in ''L<sup>p</sup>'', hence it converges almost surely to some function ''f'' by the [[Doob's martingale convergence theorems|martingale convergence theorem]]. Moreover, ''M<sub>n</sub>'' converges to ''f'' in ''L<sup>p</sup>''-norm by the [[dominated convergence theorem]], hence ''M<sub>n</sub>'' can be expressed as conditional expectation of ''f'' on Σ<sub>''n''</sub>. It is thus possible to identify martingale-''H<sup>p</sup>'' with the subspace of ''L<sup>p</sup>''(Ω,&nbsp;Σ,&nbsp;''P'') consisting of those ''f'' such that the martingale

:<math>M_n = E \bigl( f | \Sigma_n \bigr)</math>

belongs to martingale-''H<sup>p</sup>''.

[[Doob's martingale inequality|Doob's maximal inequality]] implies that martingale-''H<sup>p</sup>'' coincides with ''L<sup>p</sup>''(Ω,&nbsp;Σ,&nbsp;''P'') when 1 < ''p'' < ∞.  The interesting space is martingale-''H''<sup>1</sup>, whose dual is martingale-''BMO'' {{harv|Garsia|1973}}.

The Burkholder–Gundy inequalities (when ''p''&nbsp;>&nbsp;1) and the Burgess Davis inequality (when ''p'' = 1) relate the ''L<sup>p</sup>''-norm of the maximal function to that of the ''square function'' of the martingale

:<math> S(f) = \left( |M_0|^2 + \sum_{n=0}^{\infty} |M_{n+1} - M_n|^2 \right)^{\frac{1}{2}}. </math>

Martingale-''H<sup>p</sup>'' can be defined by saying that ''S''(''f'')∈ ''L<sup>p</sup>''.{{harv|Garsia|1973}}.

Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex [[Wiener process|Brownian motion]] (''B<sub>t</sub>'') 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,

:<math> M_t = F(B_{t \wedge \tau}) </math>

is a martingale, that belongs to martingale-''H<sup>p</sup>'' iff ''F''&nbsp;∈&nbsp;''H<sup>p</sup>'' {{Harv |Burkholder|Gundy|Silverstein|1971}}.

=== Example: dyadic martingale-''H''<sup>1</sup> ===
In this example, Ω = [0, 1] and Σ<sub>''n''</sub> is the finite field generated by the dyadic partition of [0, 1] into 2<sup>''n''</sup> intervals of length 2<sup>−''n''</sup>,  for every ''n'' ≥ 0. If a function ''f'' on [0, 1] is represented by its expansion on the [[Haar wavelet|Haar system]] (''h<sub>k</sub>'')

:<math> f = \sum c_k h_k,</math>

then the martingale-''H''<sup>1</sup> norm of ''f'' can be defined by the ''L''<sup>1</sup> norm of the square function

:<math> \int_0^1 \Bigl( \sum |c_k h_k(x)|^2 \Bigr)^{\frac{1}{2}} \, \mathrm{d}x.</math>

This space, sometimes denoted by ''H''<sup>1</sup>(δ), is isomorphic to the classical real ''H''<sup>1</sup> space on the circle {{harv |Müller|2005}}.  The Haar system is an [[Schauder basis|unconditional basis]] for ''H''<sup>1</sup>(δ).

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{{DEFAULTSORT:Hardy Space}}
[[Category:Hardy spaces]]
[[Category:Operator theory]]