Hardy space
Encyclopedia
In complex analysis
, the Hardy spaces (or Hardy classes) H^{p} are certain spaces of holomorphic function
s 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 certain spaces of distributions
on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex
Hardy spaces, and are related to the L^{p} spaces
of functional analysis
. For 1 ≤ p ≤ ∞ these real Hardy spaces H^{p} are certain subset
s of 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 domain
s 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
.
on the circle of radius r remains bounded as r → 1 from below.
More generally, the Hardy space H^{p} for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying
This class H^{p} is a vector space. The number on the left side of the above inequality is the Hardy space pnorm for f, denoted by It is a norm when p ≥ 1, but not when 0 < p < 1.
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 H^{p}, and the H^{p}norm is increasing with p (it is a consequence of Hölder's inequality
that the L^{p}norm is increasing for probability measures
, i.e. measures
with total mass 1).
on the unit circle. This connection is provided by the following theorem : Given f ∈ H^{p}, with p > 0, the radial limit
exists for almost every θ. The function belongs to the L^{p} space for the unit circle, and one has that
Denoting the unit circle by T, and by H^{p}(T) the vector subspace of L^{p}(T) consisting of all limit functions , when f varies in H^{p}, one then has that for p ≥ 1,
,
where the ĝ(n) are the Fourier coefficients of a function g integrable on the unit circle,
The space H^{p}(T) is a closed subspace of L^{p}(T). Since L^{p}(T) is a Banach space
(for 1 ≤ p ≤ ∞), so is H^{p}(T).
The above can be turned around. Given a function ∈ L^{p}(T), 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 H^{p}(T). Supposing that is in H^{p}(T). i.e. that has Fourier coefficients (a_{n})_{n ∈ Z} with a_{n} = 0 for every n < 0,
then the element f of the Hardy space H^{p} associated to is the holomorphic function
In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. 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 biinfinite sequences indexed by Z.
For p < 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid. For example, consider
The function F is in H^{p} for every p < 1, the radial limit f is in H^{p}(T) but its real part Re f is 0 almost everywhere. It is no longer possible to recover F from Re f, and one cannot define realH^{p}(T) in the simple way above.
For the same function F, let f_{r }(e^{i θ}) = F(r e^{i θ}). The limit when r → 1 of Re f_{r}, in the sense of distributions
on the circle, is a nonzero multiple of the Dirac distribution at z = 1. The Dirac distribution at any point of the unit circle belongs to realH^{p}(T) for every p < 1 (see below).
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 = 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^{1} because the above takes the form of the Poisson kernel
. This implies that
for almost every θ.
One says that h(z) is an inner (interior) function if and only if
h(z) ≤ 1 on the unit disk and the limit
exists for almost all
θ and its modulus
is equal to 1. 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 H^{p} if and only if the positive function φ belongs to L^{p}(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^{α}, α > 0, a family (G_{α}) of outer functions is obtained, with the properties:
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 H^{p} and a function in H^{q}. For example: every function in H^{1} is the product of two functions in H^{2}; every function in H^{p}, p < 1, can be expressed as product of several functions in some H^{q}, q > 1.
Let P_{r} 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 e^{i θ} → P_{r}(θ) on the circle. Namely, (f ∗ P_{r})(e^{i θ}) 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 H^{p}(T) consists of distributions f such that M f is in L^{p}(T).
The function F defined on the unit disk by F(r e^{i θ}) = (f ∗ P_{r})(e^{i θ}) is harmonic, and M f is the radial maximal function of F. When M f belongs to L^{p}(T) and p ≥ 1, the distribution f "is" a function in L^{p}(T), namely the boundary value of F. For p ≥ 1, the real Hardy space H^{p}(T) is a subset of L^{p}(T).
This mapping u → v extends to a bounded linear operator H on L^{p}(T), when 1 < p < ∞ (up to a scalar multiple, it is the Hilbert transform
on the unit circle), and H also maps L^{1}(T) to weakL^{1}(T). When 1 ≤ p < ∞, the following are equivalent for a real valued integrable function f on the unit circle:
When 1 < p < ∞, H(f) belongs to L^{p}(T) when f ∈ L^{p}(T), hence the real Hardy space H^{p}(T) coincides with L^{p}(T) in this case. For p = 1, the real Hardy space H^{1}(T) is a proper subspace of L^{1}(T).
The p = ∞ case 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 realH^{∞} be equal to L^{∞}. However, the two following properties are equivalent for a real valued function f
is in H^{p}, it can be shown that c_{n} = O(n^{1/p–1}). It follows that the Fourier series
converges in the sense of distributions to a distribution f on the unit circle, and F(r e^{i θ}) = (f ∗ P_{r})(θ). The function F ∈ H^{p} can be reconstructed from the real distribution Re f on the circle, because the Taylor coefficients c_{n} 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 do appear, as it is seen with functions F(z) = (1 – z)^{–N} (for z < 1),
that belong to H^{p} when 0 < N p < 1 (and N an integer ≥ 1).
A real distribution on the circle belongs to realH^{p}(T) iff it is the boundary value of the real part of some F ∈ H^{p}. A Dirac distribution δ_{x}, at any point x of the unit circle, belongs to realH^{p}(T) for every p < 1; derivatives δ ’_{x} belong when p < 1/2, second derivatives δ ’’_{x} when p < 1/3, and so on.
The Hardy space on the upper halfplane is defined to be the space of holomorphic functions f on with bounded (quasi)norm, the norm being given by
The corresponding is defined as functions of bounded norm, with the norm given by
Although the unit disk and the upper halfplane can be mapped to oneanother by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (onedimensional) Lebesgue measure
while the real line does not. However, for H^{2}
, one may still state the following theorem: Given the Möbius transformation with
then there is an isometric
isomorphism
with
is in L^{p}(R^{n}), where ∗ is convolution and Φ_{t}(x) = t^{−n}Φ(x/t).
The H^{p}quasinorm ƒ_{Hp} of a distribution ƒ of H^{p} is defined to be the L^{p} norm of M_{Φ}ƒ (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). The H^{p}quasinorm is a norm when p ≥ 1, but not when p < 1.
If 1 < p < ∞, the Hardy space H^{p} is the same vector space as L^{p}, with equivalent norm. When p = 1, the Hardy space H^{1} is a proper subspace of L^{1}. One can find sequences in H^{1} that are bounded in L^{1} but unbounded in H^{1}, for example on the line
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 n(1/p − 1).
When p < 1, the H^{p}quasinorm is not a norm, as it is not subadditive. 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.
whose order i_{1} + ··· + i_{n} is at most n(1/p − 1) vanish. For example, the integral of f must vanish in order that f ∈ H^{p}, 0 < p ≤ 1, and as long as p > n/(n+1) this is also sufficient.
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 (here B denotes the Euclidean volume of B in R^{n}). The H^{p}quasinorm of an arbitrary H^{p}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^{p} has an atomic decomposition as a convergent infinite combination of H^{p}atoms,
where the a_{j} are H^{p}atoms and the c_{j} are scalars.
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 < p < 1 (on the circle, the corresponding representation is valid for 0 < p < 1, but on the line, Haar functions do not belong to H^{p} when p ≤ 1/2 because their maximal function is equivalent at infinity to a x^{–2} for some a ≠ 0).
on some probability space (Ω, Σ, P), with respect to an increasing sequence of σfields (Σ_{n})_{n ≥ 0}. Assume for simplicity that Σ is equal to the σfield generated by the sequence (Σ_{n})_{n ≥ 0}. The maximal function of the martingale is defined by
Let 1 ≤ p < ∞. The martingale (M_{n})_{n ≥ 0} belongs to martingaleH^{p} when M^{∗} ∈ L^{p}.
If M^{∗} ∈ L^{p}, the martingale (M_{n})_{n ≥ 0} is bounded in L^{p}, hence it converges almost surely to some function f by the martingale convergence theorem
. Moreover, M_{n} converges to f in L^{p}norm by the dominated convergence theorem
, hence M_{n} can be expressed as conditional expectation of f on Σ_{n}. It is thus possible to identify martingaleH^{p} with the subspace of L^{p}(Ω, Σ, P) consisting of those f such that the martingale
belongs to martingaleH^{p}.
Doob's maximal inequality
implies that martingaleH^{p} coincides with L^{p}(Ω, Σ, P) when 1 < p < ∞. The interesting space is martingaleH^{1}, whose dual is martingaleBMO .
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 square function of the martingale
MartingaleH^{p} can be defined by saying that S(f) ∈ L^{p} .
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 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 martingaleH^{p} iff F ∈ H^{p} .
(h_{k})
then the martingaleH^{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 . The Haar system is an unconditional basis
for H^{1}(δ).
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the Hardy spaces (or Hardy classes) H^{p} are certain spaces of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complexvalued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on the unit disk or upper half plane. They were introduced by Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
, who named them after G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, because of the paper . In real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
Hardy spaces are certain spaces of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
Hardy spaces, and are related to the L^{p} spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the pnorm for finitedimensional vector spaces...
of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. For 1 ≤ p ≤ ∞ these real Hardy spaces H^{p} are certain subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of 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 domain
Tube domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip in the complex plane to several complex variables...
s 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
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
itself, as well as in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
(such as H^{∞} methods) and in scattering theory
Scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
.
Hardy spaces for the unit disk
For spaces of holomorphic functions on the open unit disk, the Hardy space H^{2} consists of the functions ƒ whose mean square valueRoot mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...
on the circle of radius r remains bounded as r → 1 from below.
More generally, the Hardy space H^{p} for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying
This class H^{p} is a vector space. The number on the left side of the above inequality is the Hardy space pnorm for f, denoted by It is a norm when p ≥ 1, but not when 0 < p < 1.
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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of H^{p}, and the H^{p}norm is increasing with p (it is a consequence of Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
that the L^{p}norm is increasing for probability measures
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a realworld process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
, i.e. measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
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 L^{p} spacesLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the pnorm for finitedimensional vector spaces...
on the unit circle. This connection is provided by the following theorem : Given f ∈ H^{p}, with p > 0, the radial limit
exists for almost every θ. The function belongs to the L^{p} space for the unit circle, and one has that
Denoting the unit circle by T, and by H^{p}(T) the vector subspace of L^{p}(T) consisting of all limit functions , when f varies in H^{p}, one then has that for p ≥ 1,
,
where the ĝ(n) are the Fourier coefficients of a function g integrable on the unit circle,
The space H^{p}(T) is a closed subspace of L^{p}(T). Since L^{p}(T) is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
(for 1 ≤ p ≤ ∞), so is H^{p}(T).
The above can be turned around. Given a function ∈ L^{p}(T), with p ≥ 1, one can regain a (harmonic
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
) function f on the unit disk by means of the Poisson kernel
Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the twodimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...
P_{r}:
and f belongs to H^{p} exactly when is in H^{p}(T). Supposing that is in H^{p}(T). i.e. that has Fourier coefficients (a_{n})_{n ∈ Z} with a_{n} = 0 for every n < 0,
then the element f of the Hardy space H^{p} associated to is the holomorphic function
In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. 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 biinfinite sequences indexed by Z.
Connection to real Hardy spaces on the circle
When 1 ≤ p < ∞, the real Hardy spaces H^{p} 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^{p}(T) if it is the real part of a function in H^{p}(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
The function F is in H^{p} for every p < 1, the radial limit f is in H^{p}(T) but its real part Re f is 0 almost everywhere. It is no longer possible to recover F from Re f, and one cannot define realH^{p}(T) in the simple way above.
For the same function F, let f_{r }(e^{i θ}) = F(r e^{i θ}). The limit when r → 1 of Re f_{r}, in the sense of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
on the circle, is a nonzero multiple of the Dirac distribution at z = 1. The Dirac distribution at any point of the unit circle belongs to realH^{p}(T) for every p < 1 (see below).
Factorization into inner and outer functions (Beurling)
For 0 < p ≤ ∞, every nonzero function ƒ 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 . This "BeurlingArne Beurling
Arne CarlAugust Beurling was a Swedish mathematician and professor of mathematics at Uppsala University and later at the Institute for Advanced Study in Princeton, New Jersey....
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 = 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^{1} because the above takes the form of the Poisson kernel
Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the twodimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...
. This implies that
for almost every θ.
One says that h(z) is an inner (interior) function if and only if
h(z) ≤ 1 on the unit disk and the limit
exists for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
θ and its modulus
Absolute value
In mathematics, the absolute value a of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of 3 is also 3...
is equal to 1. In particular, h is in H^{∞}. The inner function can be further factored into a form involving a Blaschke product
Blaschke product
In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a sequence of prescribed complex numbersinside the unit disc.Blaschke products were introduced by...
.
The function f, decomposed as f = Gh, is in H^{p} if and only if the positive function φ belongs to L^{p}(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^{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 H^{p} and a function in H^{q}. For example: every function in H^{1} is the product of two functions in H^{2}; every function in H^{p}, p < 1, can be expressed as product of several functions in some H^{q}, q > 1.
Realvariable techniques on the unit circle
Realvariable 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 T. For a distribution f on the unit circle, set
where the star indicates convolution between the distribution f and the function e^{i θ} → P_{r}(θ) on the circle. Namely, (f ∗ P_{r})(e^{i θ}) 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 H^{p}(T) consists of distributions f such that M f is in L^{p}(T).
The function F defined on the unit disk by F(r e^{i θ}) = (f ∗ P_{r})(e^{i θ}) is harmonic, and M f is the radial maximal function of F. When M f belongs to L^{p}(T) and p ≥ 1, the distribution f "is" a function in L^{p}(T), namely the boundary value of F. For p ≥ 1, the real Hardy space H^{p}(T) is a subset of L^{p}(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 L^{p}(T), when 1 < p < ∞ (up to a scalar multiple, it is the Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
on the unit circle), and H also maps L^{1}(T) to weakL^{1}(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 ∈ 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}(T) when f ∈ L^{p}(T), hence the real Hardy space H^{p}(T) coincides with L^{p}(T) in this case. For p = 1, the real Hardy space H^{1}(T) is a proper subspace of L^{1}(T).
The p = ∞ case 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 realH^{∞} 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).
Real Hardy spaces when 0 < p < 1
When 0 < p < 1, a function F in H^{p} cannot be reconstructed from the real part of his boundary limit function on the circle, because of the lack of convexity of L^{p} 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, ifis in H^{p}, it can be shown that c_{n} = O(n^{1/p–1}). It follows that the Fourier series
converges in the sense of distributions to a distribution f on the unit circle, and F(r e^{i θ}) = (f ∗ P_{r})(θ). The function F ∈ H^{p} can be reconstructed from the real distribution Re f on the circle, because the Taylor coefficients c_{n} 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 do appear, as it is seen with functions F(z) = (1 – z)^{–N} (for z < 1),
that belong to H^{p} when 0 < N p < 1 (and N an integer ≥ 1).
A real distribution on the circle belongs to realH^{p}(T) iff it is the boundary value of the real part of some F ∈ H^{p}. A Dirac distribution δ_{x}, at any point x of the unit circle, belongs to realH^{p}(T) for every p < 1; derivatives δ ’_{x} belong when p < 1/2, second derivatives δ ’’_{x} 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 halfplane (usually the right halfplane or upper halfplane) are used.The Hardy space on the upper halfplane is defined to be the space of holomorphic functions f on with bounded (quasi)norm, the norm being given by
The corresponding is defined as functions of bounded norm, with the norm given by
Although the unit disk and the upper halfplane can be mapped to oneanother by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (onedimensional) Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ndimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
while the real line does not. However, for H^{2}
H square
In mathematics and control theory, H2, or Hsquare is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space...
, one may still state the following theorem: Given the Möbius transformation with
then there is an isometric
Isometry
In mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
with
Real Hardy spaces for R^{n}
In analysis on the real vector space R^{n}, the Hardy space H^{p} (for 0 < p ≤ ∞) consists of tempered distributions ƒ such that for some Schwartz function Φ with ∫Φ = 1, the maximal functionMaximal function
Maximal functions appear in many forms in harmonic analysis . One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations...
is in L^{p}(R^{n}), where ∗ is convolution and Φ_{t}(x) = t^{−n}Φ(x/t).
The H^{p}quasinorm ƒ_{Hp} of a distribution ƒ of H^{p} is defined to be the L^{p} norm of M_{Φ}ƒ (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). The H^{p}quasinorm is a norm when p ≥ 1, but not when p < 1.
If 1 < p < ∞, the Hardy space H^{p} is the same vector space as L^{p}, with equivalent norm. When p = 1, the Hardy space H^{1} is a proper subspace of L^{1}. One can find sequences in H^{1} that are bounded in L^{1} but unbounded in H^{1}, for example on the line
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
Bounded mean oscillation
In harmonic analysis, a function of bounded mean oscillation, also known as a BMO function, is a realvalued function whose mean oscillation is bounded...
. 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 n(1/p − 1).
When p < 1, the H^{p}quasinorm is not a norm, as it is not subadditive. 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.
Atomic decomposition
When 0 < p ≤ 1, a bounded measurable function ƒ of compact support is in the Hardy space H^{p} if and only if all its momentswhose order i_{1} + ··· + i_{n} is at most n(1/p − 1) vanish. For example, the integral of f must vanish in order that f ∈ H^{p}, 0 < p ≤ 1, and as long as p > n/(n+1) this is also sufficient.
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 (here B denotes the Euclidean volume of B in R^{n}). The H^{p}quasinorm of an arbitrary H^{p}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^{p} has an atomic decomposition as a convergent infinite combination of H^{p}atoms,
where the a_{j} are H^{p}atoms and the c_{j} are scalars.
On the line for example, the difference of Dirac distributions f = δ_{1} − δ_{0} can be represented as a series of Haar functions
Haar wavelet
In mathematics, the Haar wavelet is a certain sequence of rescaled "squareshaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis...
, convergent in H^{p}quasinorm when 1/2 < p < 1 (on the circle, the corresponding representation is valid for 0 < p < 1, but on the line, Haar functions do not belong to H^{p} when p ≤ 1/2 because their maximal function is equivalent at infinity to a x^{–2} for some a ≠ 0).
Martingale H^{p}
Let (M_{n})_{n ≥ 0} be a martingaleMartingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
on some probability space (Ω, Σ, P), with respect to an increasing sequence of σfields (Σ_{n})_{n ≥ 0}. Assume for simplicity that Σ is equal to the σfield generated by the sequence (Σ_{n})_{n ≥ 0}. The maximal function of the martingale is defined by
Let 1 ≤ p < ∞. The martingale (M_{n})_{n ≥ 0} belongs to martingaleH^{p} when M^{∗} ∈ L^{p}.
If M^{∗} ∈ L^{p}, the martingale (M_{n})_{n ≥ 0} is bounded in L^{p}, hence it converges almost surely to some function f by the martingale convergence theorem
Doob's martingale convergence theorems
In mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the longtime limits of supermartingales, named after the American mathematician Joseph Leo Doob....
. Moreover, M_{n} converges to f in L^{p}norm by the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
, hence M_{n} can be expressed as conditional expectation of f on Σ_{n}. It is thus possible to identify martingaleH^{p} with the subspace of L^{p}(Ω, Σ, P) consisting of those f such that the martingale
belongs to martingaleH^{p}.
Doob's maximal inequality
Doob's martingale inequality
In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time...
implies that martingaleH^{p} coincides with L^{p}(Ω, Σ, P) when 1 < p < ∞. The interesting space is martingaleH^{1}, whose dual is martingaleBMO .
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 square function of the martingale
MartingaleH^{p} can be defined by saying that S(f) ∈ L^{p} .
Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex Brownian motion
Wiener process
In mathematics, the Wiener process is a continuoustime stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
(B_{t}) 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 martingaleH^{p} iff F ∈ H^{p} .
Example: dyadic martingaleH^{1}
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 systemHaar wavelet
In mathematics, the Haar wavelet is a certain sequence of rescaled "squareshaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis...
(h_{k})
then the martingaleH^{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 . The Haar system is an unconditional basis
Schauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums...
for H^{1}(δ).