Hardy space

"Hardy class" redirects here. For the warships, see Hardy class destroyer.

In complex analysis, the Hardy spaces (or Hardy classes) ${\displaystyle H^{p}}$ are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz ( Riesz 1923 ), who named them after G. H. Hardy, because of the paper ( Hardy 1915 ). In real analysis Hardy spaces are spaces of distributions on the real n-space ${\displaystyle \mathbb {R} ^{n}}$, defined (in the sense of distributions) as boundary values of the holomorphic functions. are related to the L^p spaces. ^[1] For ${\displaystyle 1\leq p<\infty }$ these Hardy spaces are subsets of ${\displaystyle L^{p}}$ spaces, while for ${\displaystyle 0<p<1}$ the ${\displaystyle L^{p}}$ spaces have some undesirable properties, and the Hardy spaces are much better behaved. Hence, ${\displaystyle H^{p}}$ spaces can be considered extensions of ${\displaystyle L^{p}}$ spaces. ^[2]

Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g. ${\displaystyle H^{\infty }}$ methods) and scattering theory.

Definition

On the unit disk

The Hardy space ${\displaystyle H^{p}}$ for ${\displaystyle 0<p<\infty }$ is the class of holomorphic functions ${\displaystyle f}$ on the open unit disk ${\displaystyle \mathbb {D} =\{z\in \mathbb {C}$ :|z|<1\}} satisfying $${\displaystyle \sup _{0\leqslant r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .}$$ If ${\displaystyle p\geq 1}$ then the equation coincides with the definition of the Hardy space p-norm, denoted by ${\displaystyle \|f\|_{H^{p}}.}$

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) ( Rudin 1987 , Def 17.7).

On the unit circle

The Hardy spaces can also be viewed as closed vector subspaces of the complex L^p spaces on the unit circle ${\displaystyle \mathbb {T} =\{z\in \mathbb {C}$ :|z|=1\}}. This connection is provided by the following theorem ( Katznelson 1976 , Thm 3.8): Given ${\displaystyle f\in H^{p}}$ with ${\displaystyle p\geq 1}$, then 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 ${\displaystyle \theta }$ and ${\displaystyle {\tilde {f}}\in L^{p}(\mathbb {T} )}$ such that ^{[ clarification needed ]}$${\displaystyle {\|{\tilde {f}}\|}_{L^{p}}={\|f\|}_{H^{p}}.}$$ Denote 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 )

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

where the ${\displaystyle {\hat {g}}_{n}}$ are the Fourier coefficients defined as $${\displaystyle {\hat {g}}_{n}={\frac {1}{2\pi }}\int _{0}^{2\pi }g\left(e^{i\phi }\right)e^{-in\phi }\,\mathrm {d} \phi ,\quad \forall n\in \mathbb {Z} .}$$ 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., ${\displaystyle {\tilde {f}}}$ has Fourier coefficients (a_n)_n∈Z with a_n = 0 for every n < 0, then the associated holomorphic function f of H^p is given by $${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},\ \ \ |z|<1.}$$ In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. For example, the Hardy space H² consists of functions whose mean square value remains bounded as ${\displaystyle r\to 1}$ from below. 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.

On the upper half plane

The Hardy space on the upper half-plane ${\displaystyle \mathbb {H} =\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}}$ is defined to be the space of holomorphic functions ${\displaystyle f}$ on ${\displaystyle \mathbb {H} }$ with bounded norm, 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 ${\displaystyle H^{\infty }(\mathbb {H} )}$ is defined as functions of bounded norm, with the norm given by $${\displaystyle \|f\|_{H^{\infty }}=\sup _{z\in \mathbb {H} }|f(z)|.}$$ The unit disk is isomorphic to the upper half-plane by means of a Möbius transformation. For example, let ${\displaystyle m:\mathbb {D} \rightarrow \mathbb {H} }$ denote the Möbius transformation $${\displaystyle m(z)=i{\frac {1+z}{1-z}}.}$$ Then the linear operator ${\displaystyle M:H^{2}(\mathbb {H} )\rightarrow H^{2}(\mathbb {D} )}$ defined by $${\displaystyle (Mf)(z):={\frac {\sqrt {\pi }}{1-z}}f(m(z)),}$$ is an isometric isomorphism of Hardy spaces.

A similair approach applies to, e.g., the right half-plane.

On the real vector space

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.

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

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

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).

Link between real- and complex-variable Hardy spaces

Real-variable techniques, mainly associated to the study of real Hardy spaces defined on Rⁿ, 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 spaceH^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 spaceH^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 polynomialv 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).

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

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

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.

Beurling factorization

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. ^{[3] [4]}

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.

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

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

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 ).

Example

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).

See also

• H²
• H^∞ methods
• Paley-Wiener theorem

Notes

1. Folland 2001.
2. Stein & Murphy 1993, p. 88.
3. Beurling, Arne (1948). "On two problems concerning linear transformations in Hilbert space". Acta Mathematica . 81: 239–255. doi: 10.1007/BF02395019 .
4. Voichick, Michael; Zalcman, Lawrence (1965). "Inner and outer functions on Riemann surfaces". Proceedings of the American Mathematical Society . 16 (6): 1200–1204. doi: 10.1090/S0002-9939-1965-0183883-1 .

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