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Source: Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain 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 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 Lp spaces of functional analysis. For 1 ≤ p < ∞ these real Hardy spaces Hp are certain subsets of Lp, while for p < 1 the Lp 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 Rn 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

Hardy

Hardy spaces on the unit circle

= Connection to real Hardy spaces on the circle

Factorization into inner and outer functions (Beurling)

Hardy

space

Real-variable techniques on the unit circle

Hardy

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

e

i
φ

→

P

r

(
θ
−
φ
)
.

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

For 0 < p < ∞, the real Hardy space Hp(T) consists of distributions f such that M f is in Lp(T).
The function F defined on the unit disk by F(reiθ) = (f ∗ Pr)(eiθ) is harmonic, and M f is the radial maximal function of F. When M f belongs to Lp(T) and p ≥ 1, the distribution f "is" a function in Lp(T), namely the boundary value of F. For p ≥ 1, the real Hardy space Hp(T) is a subset of Lp(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,

u
(

e

i
θ

)
=

a

0

2

+

∑

k
⩾
1

a

k

cos
⁡
(
k
θ
)
+

b

k

sin
⁡
(
k
θ
)
⟶
v
(

e

i
θ

)
=

∑

k
⩾
1

a

k

sin
⁡
(
k
θ
)
−

b

k

cos
⁡
(
k
θ
)
.

{\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 Lp(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 ∈ Hp(T)
the function f and its conjugate H(f) belong to Lp(T)
the radial maximal function M f belongs to Lp(T).
When 1 < p < ∞, H(f) belongs to Lp(T) when f ∈ Lp(T), hence the real Hardy space Hp(T) coincides with Lp(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).

= Real Hardy spaces for 0 < p < 1

=
When 0 < p < 1, a function F in Hp cannot be reconstructed from the real part of its boundary limit function on the circle, because of the lack of convexity of Lp 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
)
=

∑

n
=
0

+
∞

c

n

z

n

,

|

z

|

<
1

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

is in Hp, it can be shown that cn = O(n1/p–1). It follows that the Fourier series

∑

n
=
0

+
∞

c

n

e

i
n
θ

{\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(reiθ) =(f ∗ Pr)(θ). The function F ∈ Hp can be reconstructed from the real distribution Re(f) on the circle, because the Taylor coefficients cn 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 Hp when 0 < N p < 1 (and N an integer ≥ 1).
A real distribution on the circle belongs to real-Hp(T) iff it is the boundary value of the real part of some F ∈ Hp. A Dirac distribution δx, at any point x of the unit circle, belongs to real-Hp(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 Hp(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

‖

H

p

=

sup

y
>
0

(

∫

−
∞

+
∞

|

f
(
x
+
i
y
)

|

p

d

x

)

1
p

.

{\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

‖
f

‖

H

∞

=

sup

z
∈

H

|

f
(
z
)

|

.

{\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 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

m
(
z
)
=
i
⋅

1
+
z

1
−
z

.

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

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

(
M
f
)
(
z
)
:=

π

1
−
z

f
(
m
(
z
)
)
.

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

is an isometric isomorphism of Hilbert spaces.

Real Hardy spaces for Rn

In analysis on the real vector space Rn, the Hardy space Hp (for 0 < p ≤ ∞) consists of tempered distributions f such that for some Schwartz function Φ with ∫Φ = 1, the maximal function

(

M

Φ

f
)
(
x
)
=

sup

t
>
0

|

(
f
∗

Φ

t

)
(
x
)

|

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

is in Lp(Rn), where ∗ is convolution and Φt (x) = t −nΦ(x / t). The Hp-quasinorm ||f ||Hp of a distribution f of Hp is defined to be the Lp norm of MΦf (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). The Hp-quasinorm is a norm when p ≥ 1, but not when p < 1.
If 1 < p < ∞, the Hardy space Hp is the same vector space as Lp, 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

f

k

(
x
)
=

1

[
0
,
1
]

(
x
−
k
)
−

1

[
0
,
1
]

(
x
+
k
)
,

k
>
0.

{\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 Hp has elements that are not functions, and its dual is the homogeneous Lipschitz space of order n(1/p − 1). When p < 1, the Hp-quasinorm is not a norm, as it is not subadditive. The pth power ||f ||Hpp is subadditive for p < 1 and so defines a metric on the Hardy space Hp, which defines the topology and makes Hp into a complete metric space.

= Atomic decomposition

=
When 0 < p ≤ 1, a bounded measurable function f of compact support is in the Hardy space Hp if and only if all its moments

∫

R

n

f
(
x
)

x

1

i

1

…

x

n

i

n

d

x
,

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

whose order i1+ ... +in is at most n(1/p − 1), vanish. For example, the integral of f must vanish in order that f ∈ Hp, 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 Hp-atom (here |B| denotes the Euclidean volume of B in Rn). The Hp-quasinorm of an arbitrary Hp-atom is bounded by a constant depending only on p and on the Schwartz function Φ.
When 0 < p ≤ 1, any element f of Hp has an atomic decomposition as a convergent infinite combination of Hp-atoms,

f
=
∑

c

j

a

j

,

∑

|

c

j

|

p

<
∞

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

where the aj are Hp-atoms and the cj 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 Hp-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 Hp when p ≤ 1/2 because their maximal function is equivalent at infinity to a x−2 for some a ≠ 0).

Martingale Hp

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

M

∗

=

sup

n
≥
0

|

M

n

|

.

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

Let 1 ≤ p < ∞. The martingale (Mn)n≥0 belongs to martingale-Hp when M* ∈ Lp.
If M* ∈ Lp, the martingale (Mn)n≥0 is bounded in Lp; hence it converges almost surely to some function f by the martingale convergence theorem. Moreover, Mn converges to f in Lp-norm by the dominated convergence theorem; hence Mn can be expressed as conditional expectation of f on Σn. It is thus possible to identify martingale-Hp with the subspace of Lp(Ω, Σ, P) consisting of those f such that the martingale

M

n

=
E
⁡

(

f

|

Σ

n

)

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

belongs to martingale-Hp.
Doob's maximal inequality implies that martingale-Hp coincides with Lp(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 Lp-norm of the maximal function to that of the square function of the martingale

S
(
f
)
=

(

|

M

0

|

2

+

∑

n
=
0

∞

|

M

n
+
1

−

M

n

|

2

)

1
2

.

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

Martingale-Hp can be defined by saying that S(f)∈ Lp (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 (Bt) 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

t
∧
τ

)

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

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

= Example: dyadic martingale-H1

=
In this example, Ω = [0, 1] and Σn 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 (hk)

f
=
∑

c

k

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

∫

0

1

(

∑

|

c

k

h

k

(
x
)

|

2

)

1
2

d

x
.

{\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).

Notes

References

Burkholder, Donald L.; Gundy, Richard F.; Silverstein, Martin L. (1971), "A maximal function characterization of the class Hp", Transactions of the American Mathematical Society, 157: 137–153, doi:10.2307/1995838, JSTOR 1995838, MR 0274767, S2CID 53996980.
Cima, Joseph A.; Ross, William T. (2000), The Backward Shift on the Hardy Space, American Mathematical Society, ISBN 978-0-8218-2083-4
Colwell, Peter (1985), Blaschke Products - Bounded Analytic Functions, Ann Arbor: University of Michigan Press, ISBN 978-0-472-10065-1
Duren, P. (1970), Theory of Hp-Spaces, Academic Press
Fefferman, Charles; Stein, Elias M. (1972), "Hp spaces of several variables", Acta Mathematica, 129 (3–4): 137–193, doi:10.1007/BF02392215, MR 0447953.
Folland, G.B. (2001) [1994], "Hardy spaces", Encyclopedia of Mathematics, EMS Press
Garsia, Adriano M. (1973), Martingale Inequalities: Seminar notes on recent progress, Mathematics Lecture Notes Series, W. A. Benjamin MR0448538
Hardy, G. H. (1915), "On the mean value of the modulus of an analytic function", Proceedings of the London Mathematical Society, 14: 269–277, doi:10.1112/plms/s2_14.1.269, JFM 45.1331.03
Hoffman, Kenneth (1988), Banach Spaces of Analytic Functions, Dover Publications, ISBN 978-0-486-65785-1
Katznelson, Yitzhak (1976), An Introduction to Harmonic Analysis, Dover Publications, ISBN 978-0-486-63331-2
Koosis, P. (1998), Introduction to Hp Spaces (Second ed.), Cambridge University Press
Mashreghi, J. (2009), Representation Theorems in Hardy Spaces, Cambridge University Press, ISBN 9780521517683
Müller, Paul F. X. (2005), Isomorphisms Between H1 spaces, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), Basel: Birkhäuser, ISBN 978-3-7643-2431-5, MR 2157745
Riesz, F. (1923), "Über die Randwerte einer analytischen Funktion", Mathematische Zeitschrift, 18: 87–95, doi:10.1007/BF01192397, S2CID 121306447
Rudin, Walter (1987), Real and Complex Analysis, McGraw-Hill, ISBN 978-0-07-100276-9
Shvedenko, S.V. (2001) [1994], "Hardy classes", Encyclopedia of Mathematics, EMS Press

Hardy spaces for the unit disk

Hardy spaces on the unit circle

= Connection to real Hardy spaces on the circle

Factorization into inner and outer functions (Beurling)

Real-variable techniques on the unit circle

= Conjugate function

= Real Hardy spaces for 0 < p < 1

Hardy spaces for the upper half plane

Real Hardy spaces for Rn

= Atomic decomposition

Martingale Hp

= Example: dyadic martingale-H1

Notes

References

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