Hardy space

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 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 subsets 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 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.

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 ''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 < ''p'' < ∞ is the class of holomorphic functions ''f'' on the open unit disk satisfying

:$\sup_\left(\frac \int_0^\left, f \left (re^\right )\^p \; \mathrm\theta\right)^\frac<\infty.$

This class ''H^p'' is a vector space. The number on the left side of the above inequality is the Hardy space ''p''-norm for ''f'', denoted by $\, f\, _.$ 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

:$\, f\, _ = \sup_ \left, f(z)\.$

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).

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'' spaces on the unit circle. This connection is provided by the following theorem : Given ''f'' ∈ ''H^p'', with ''p'' ≥ 1, the radial limit

:$\tilde f\left(e^\right) = \lim_ f\left(re^\right)$

exists for almost every θ. The function $\tilde f$ belongs to the ''L^p'' space for the unit circle, and one has that

:$\, \tilde f\, _ = \, f\, _.$

Denoting the unit circle by T, and by ''H^p''(T) the vector subspace of ''L^p''(T) consisting of all limit functions $\tilde f$, when ''f'' varies in ''H^p'', one then has that for ''p'' ≥ 1,

:$g\in H^p\left(\mathbf\right)\text g\in L^p\left(\mathbf\right)\text \hat(n)=0 \text n < 0,$

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

:$\forall n \in \mathbf, \ \ \ \hat(n) = \frac\int_0^ g\left(e^\right) e^ \, \mathrm\phi.$

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 $\tilde f \in L^p (\mathbf T)$, with ''p'' ≥ 1, one can regain a (harmonic) function ''f'' on the unit disk by means of the Poisson kernel ''P_r'':

:$f\left(re^\right)=\frac \int_0^ P_r(\theta-\phi) \tilde f\left(e^\right) \,\mathrm\phi, \quad r < 1,$

and ''f'' belongs to ''H^p'' exactly when $\tilde f$ is in ''H^p''(T). Supposing that $\tilde f$ is in ''H^p''(T), ''i.e.'' that $\tilde f$ 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 $\tilde f$ is the holomorphic function

:$f(z)=\sum_^\infty a_n z^n, \ \ \ , z, < 1.$

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 bi-infinite 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). Thus for 1 ≤ ''p'' < ∞, the real Hardy space contains the Hardy space, but is much bigger, since no relationship is imposed between the real and imaginary part of the function.

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

:$F(z) = \frac, \quad , z, < 1.$

Then ''F'' is in ''H^p'' for every 0 < ''p'' < 1, and the radial limit

:$f(e^):= \lim_ F(r e^) = i \, \cot(\tfrac).$

exists for a.e. ''θ'' and is in ''H^p''(T), but Re(''f'') is 0 almost everywhere, so it is no longer possible to recover ''F'' from Re(''f''). As a consequence of this example, one sees that for 0 < ''p'' < 1, one cannot characterize the real-''H^p''(T) (defined below) in the simple way given above, but must use the actual definition using maximal functions, which is given further along somewhere below.

For the same function ''F'', let ''f_r''(e^iθ) = ''F''(''re''^iθ). The limit when ''r'' → 1 of Re(''f_r''), ''in the sense of'' ''distributions'' on the circle, is a non-zero multiple of the Dirac distribution at ''z'' = 1. The Dirac distribution at a point of the unit circle belongs to real-''H^p''(T) for every ''p'' < 1 (see below).

Factorization into inner and outer functions (Beurling)

For 0 < ''p'' ≤ ∞, every non-zero function ''f'' in ''H^p'' 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

:$G(z) = c\, \exp\left(\frac\int_^\frac \log\!\left(\varphi\!\left(e^ \right)\right)\, \mathrm\theta \right)$

for some complex number ''c'' with , ''c'', = 1, and some positive measurable function $\varphi$ on the unit circle such that $\log(\varphi)$ is integrable on the circle. In particular, when $\varphi$ is integrable on the circle, ''G'' is in ''H''¹ because the above takes the form of the Poisson kernel . This implies that

:$\lim_\left, G\left (re^ \right)\ = \varphi \left(e^\right )$

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

:$\lim_ h(re^)$

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 ''H^p'' if and only if φ belongs to ''L^p''(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 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.

Real-variable techniques on the unit circle

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 T. For a distribution ''f'' on the unit circle, set

:$(M f)(e^)=\sup_ \left , (f * P_r) \left(e^ \right)\,$

where the ''star'' indicates convolution between the distribution ''f'' and the function e^iθ → ''P_r''(θ) on the circle. Namely, (''f'' ∗ ''P_r'')(e^iθ) is the result of the action of ''f'' on the ''C''^∞-function defined on the unit circle by

:$e^ \rightarrow P_r(\theta - \varphi).$

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''(''re''^iθ) = (''f'' ∗ ''P_r'')(e^iθ) 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'' + i''v'' extends to a holomorphic function in the unit disk,

:$u(e^) = \frac + \sum_ a_k \cos(k \theta) + b_k \sin(k \theta) \longrightarrow v(e^) = \sum_ a_k \sin(k \theta) - b_k \cos(k \theta).$

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 weak-''L''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 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).

Real Hardy spaces for 0 < ''p'' < 1

When 0 < ''p'' < 1, a function ''F'' in ''H^p'' cannot be reconstructed from the real part of its 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, if

:$F(z) = \sum_^ c_n z^n, \quad , z, < 1$

is in ''H^p'', it can be shown that ''c_n'' = O(''n''^1/''p''–1). It follows that the Fourier series

:$\sum_^ c_n e^$

converges in the sense of distributions to a distribution ''f'' on the unit circle, and ''F''(''re''^iθ) =(''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 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 < ''N'' ''p'' < 1 (and ''N'' an integer ≥ 1).

A real distribution on the circle belongs to real-''H^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 real-''H^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 half-plane (usually the right half-plane or upper half-plane) are used.

The Hardy space ''H^p''(H) on the upper half-plane H is defined to be the space of holomorphic functions ''f'' on H with bounded norm, the norm being given by

:$\, f\, _ = \sup_ \left ( \int_^, f(x+ iy), ^p\, \mathrmx \right)^.$

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

:$\, f\, _ = \sup_, f(z), .$

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 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

:$m(z)= i \cdot \frac.$

Then the linear operator ''M'' : ''H''²(H) → ''H''²(D) defined by

:$(Mf)(z):=\frac f(m(z)).$

is an isometric isomorphism of Hilbert spaces.

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 ''f'' such that for some Schwartz function Φ with ∫Φ = 1, the maximal function

:$(M_\Phi f)(x)=\sup_, (f*\Phi_t)(x),$

is in ''L^p''(R^''n''), where ∗ is convolution and . The ''H^p''- quasinorm , , ''f'' , , _''Hp'' of a distribution ''f'' of ''H^p'' is defined to be the ''L^p'' norm of ''M''_Φ''f'' (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

:$f_k(x) = \mathbf_(x - k) - \mathbf_(x + k), \ \ \ k > 0.$

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 ''p''th power , , ''f'' , , _''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 ''f'' of compact support is in the Hardy space ''H^p'' if and only if all its moments

:$\int_ f(x)x_1^\ldots x_n^\, \mathrmx,$

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'' > this is also sufficient.

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 , ''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,

:$f = \sum c_j a_j, \ \ \ \sum , c_j, ^p < \infty$

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'' = δ₁−δ₀ 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).

Martingale ''H^p''

Let (''M_n'')_''n''≥0 be a martingale on some probability space (Ω, Σ, ''P''), with respect to an increasing sequence of σ-fields (Σ_''n'')_''n''≥0. Assume for simplicity that Σ is equal to the σ-field generated by the sequence (Σ_''n'')_''n''≥0. The ''maximal function'' of the martingale is defined by

:$M^* = \sup_ \, , M_n, .$

Let 1 ≤ ''p'' < ∞. The martingale (''M_n'')_''n''≥0 belongs to ''martingale''-''H^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 martingale-''H^p'' with the subspace of ''L^p''(Ω, Σ, ''P'') consisting of those ''f'' such that the martingale

:$M_n = \operatorname E \bigl( f , \Sigma_n \bigr)$

belongs to martingale-''H^p''.

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

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

:$S(f) = \left( , M_0, ^2 + \sum_^ , M_ - M_n, ^2 \right)^.$

Martingale-''H^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,

:$M_t = F(B_)$

is a martingale, that belongs to martingale-''H^p'' iff ''F'' ∈ ''H^p'' .

Example: dyadic martingale-''H''¹

In this example, Ω = , 1and Σ_''n'' is the finite field generated by the dyadic partition of , 1into 2^''n'' intervals of length 2^−''n'', for every ''n'' ≥ 0. If a function ''f'' on , 1is represented by its expansion on the Haar system (''h_k'')

:$f = \sum c_k h_k,$

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

:$\int_0^1 \Bigl( \sum , c_k h_k(x), ^2 \Bigr)^ \, \mathrmx.$

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).

Notes

References

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{{DEFAULTSORT:Hardy Space

Complex analysis
Operator theory