Hardy space

For spaces of holomorphic functions on the open unit disk, the Hardy space H² 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

${\displaystyle \sup _{0\leqslant r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .}$ 

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 ${\displaystyle \|f\|_{H^{p}}.}$  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

${\displaystyle {\|f\|}_{H^{\infty }}=\sup _{|z|<1}\left|f(z)\right|.}$ 

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).

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 f ∈ H^p, with p ≥ 1, the radial limit

${\displaystyle {\tilde {f}}\left(e^{i\theta }\right)=\lim _{r\to 1}f\left(re^{i\theta }\right)}$ 

exists for almost every θ. The function ${\displaystyle {\tilde {f}}}$  belongs to the L^p space for the unit circle,^{[clarification needed]} and one has that

${\displaystyle {\|{\tilde {f}}\|}_{L^{p}}={\|f\|}_{H^{p}}.}$ 

Denoting the unit circle by T, and by H^p(T) the vector subspace of L^p(T) consisting of all limit functions ${\displaystyle {\tilde {f}}}$ , when f varies in H^p, one then has that for p ≥ 1,(Katznelson 1976)

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

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

${\displaystyle \forall n\in \mathbf {Z} ,\ \ \ {\hat {g}}(n)={\frac {1}{2\pi }}\int _{0}^{2\pi }g\left(e^{i\phi }\right)e^{-in\phi }\,\mathrm {d} \phi .}$ 

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 ${\displaystyle {\tilde {f}}\in L^{p}(\mathbf {T} )}$ , with p ≥ 1, one can regain a (harmonic) function f on the unit disk by means of the Poisson kernel P_r:

${\displaystyle f\left(re^{i\theta }\right)={\frac {1}{2\pi }}\int _{0}^{2\pi }P_{r}(\theta -\phi ){\tilde {f}}\left(e^{i\phi }\right)\,\mathrm {d} \phi ,\quad r<1,}$ 

and f belongs to H^p exactly when ${\displaystyle {\tilde {f}}}$  is in H^p(T). Supposing that ${\displaystyle {\tilde {f}}}$  is in H^p(T), i.e. that ${\displaystyle {\tilde {f}}}$  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 ${\displaystyle {\tilde {f}}}$  is the holomorphic function

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},\ \ \ |z|<1.}$ 

In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions.^{[clarification needed]} Thus, the space H² is seen to sit naturally inside L² space, and is represented by infinite sequences indexed by N; whereas L² consists of bi-infinite sequences indexed by Z.

When 1 ≤ p < ∞, the real Hardy spaces H^p discussed further down^{[clarification needed]} 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

${\displaystyle F(z)={\frac {1+z}{1-z}},\quad |z|<1.}$ 

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

${\displaystyle f(e^{i\theta }):=\lim _{r\to 1}F(re^{i\theta })=i\,\cot({\tfrac {\theta }{2}}).}$ 

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,^{[clarification needed]} 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).

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 (Rudin 1987, Thm 17.17). This "Beurling factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions.^[1][2]

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

${\displaystyle G(z)=c\,\exp \left({\frac {1}{2\pi }}\int _{-\pi }^{\pi }{\frac {e^{i\theta }+z}{e^{i\theta }-z}}\log \!\left(\varphi \!\left(e^{i\theta }\right)\right)\,\mathrm {d} \theta \right)}$ 

for some complex number c with |c| = 1, and some positive measurable function ${\displaystyle \varphi }$  on the unit circle such that ${\displaystyle \log(\varphi )}$  is integrable on the circle. In particular, when ${\displaystyle \varphi }$  is integrable on the circle, G is in H¹ because the above takes the form of the Poisson kernel (Rudin 1987, Thm 17.16). This implies that

${\displaystyle \lim _{r\to 1^{-}}\left|G\left(re^{i\theta }\right)\right|=\varphi \left(e^{i\theta }\right)}$ 

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

${\displaystyle \lim _{r\to 1^{-}}h(re^{i\theta })}$ 

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 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:

G₁ = 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 H¹ is the product of two functions in H²; every function in H^p, p < 1, can be expressed as product of several functions in some H^q, q > 1.

Real-variable techniques, mainly associated to the study of real Hardy spaces defined on Rⁿ (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

${\displaystyle (Mf)(e^{i\theta })=\sup _{0<r<1}\left|(f*P_{r})\left(e^{i\theta }\right)\right|,}$ 

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

${\displaystyle e^{i\varphi }\rightarrow P_{r}(\theta -\varphi ).}$ 

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).

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,

${\displaystyle u(e^{i\theta })={\frac {a_{0}}{2}}+\sum _{k\geqslant 1}a_{k}\cos(k\theta )+b_{k}\sin(k\theta )\longrightarrow v(e^{i\theta })=\sum _{k\geqslant 1}a_{k}\sin(k\theta )-b_{k}\cos(k\theta ).}$ 

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 L¹(T) to weak-L¹(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 H¹(T) is a proper subspace of L¹(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).

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

${\displaystyle F(z)=\sum _{n=0}^{+\infty }c_{n}z^{n},\quad |z|<1}$ 

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

${\displaystyle \sum _{n=0}^{+\infty }c_{n}e^{in\theta }}$ 

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.

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

${\displaystyle \|f\|_{H^{p}}=\sup _{y>0}\left(\int _{-\infty }^{+\infty }|f(x+iy)|^{p}\,\mathrm {d} x\right)^{\frac {1}{p}}.}$ 

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

${\displaystyle \|f\|_{H^{\infty }}=\sup _{z\in \mathbf {H} }|f(z)|.}$ 

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^{[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 H², one has the following theorem: if m : D → H denotes the Möbius transformation

${\displaystyle m(z)=i\cdot {\frac {1+z}{1-z}}.}$ 

Then the linear operator M : H²(H) → H²(D) defined by

${\displaystyle (Mf)(z):={\frac {\sqrt {\pi }}{1-z}}f(m(z)).}$ 

is an isometric isomorphism of Hilbert spaces.

In analysis on the real vector space Rⁿ, 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

${\displaystyle (M_{\Phi }f)(x)=\sup _{t>0}|(f*\Phi _{t})(x)|}$ 

is in L^p(Rⁿ), where ∗ is convolution and Φ_t(x) = t⁻ⁿΦ(x / t). The H^p-quasinorm ||f ||_Hp of a distribution f of H^p is defined to be the L^p norm of M_Φf (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 H¹ is a proper subspace of L¹. One can find sequences in H¹ that are bounded in L¹ but unbounded in H¹, for example on the line

${\displaystyle f_{k}(x)=\mathbf {1} _{[0,1]}(x-k)-\mathbf {1} _{[0,1]}(x+k),\ \ \ k>0.}$ 

The L¹ and H¹ norms are not equivalent on H¹, and H¹ is not closed in L¹. The dual of H¹ is the space BMO of functions of bounded mean oscillation. The space BMO contains unbounded functions (proving again that H¹ is not closed in L¹).

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 ||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.

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

${\displaystyle \int _{\mathbf {R} ^{n}}f(x)x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\,\mathrm {d} x,}$ 

whose order i₁+ ... +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 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 Rⁿ). 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,

${\displaystyle f=\sum c_{j}a_{j},\ \ \ \sum |c_{j}|^{p}<\infty }$ 

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).

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

${\displaystyle M^{*}=\sup _{n\geq 0}\,|M_{n}|.}$ 

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

${\displaystyle M_{n}=\operatorname {E} {\bigl (}f|\Sigma _{n}{\bigr )}}$ 

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-H¹, whose dual is martingale-BMO (Garsia 1973).

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

${\displaystyle S(f)=\left(|M_{0}|^{2}+\sum _{n=0}^{\infty }|M_{n+1}-M_{n}|^{2}\right)^{\frac {1}{2}}.}$ 

Martingale-H^p can be defined by saying that S(f)∈ L^p (Garsia 1973).

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,

${\displaystyle M_{t}=F(B_{t\wedge \tau })}$ 

is a martingale, that belongs to martingale-H^p iff F ∈ H^p (Burkholder, Gundy & Silverstein 1971).

In this example, Ω = [0, 1] and Σ_n is the finite field generated by the dyadic partition of [0, 1] into 2ⁿ 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)

${\displaystyle f=\sum c_{k}h_{k},}$ 

then the martingale-H¹ norm of f can be defined by the L¹ norm of the square function

${\displaystyle \int _{0}^{1}{\Bigl (}\sum |c_{k}h_{k}(x)|^{2}{\Bigr )}^{\frac {1}{2}}\,\mathrm {d} x.}$ 

This space, sometimes denoted by H¹(δ), is isomorphic to the classical real H¹ space on the circle (Müller 2005). The Haar system is an unconditional basis for H¹(δ).

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