Hardy space

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

Incomplex analysis,theHardy spaces(orHardy classes)H^pare certainspacesofholomorphic functionson theunit diskorupper half plane.They were introduced byFrigyes Riesz(Riesz 1923), who named them afterG. H. Hardy,because of the paper (Hardy 1915). Inreal analysisHardy spacesare certain spaces ofdistributionson the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of thecomplexHardy spaces, and are related to theL^pspacesoffunctional analysis.For 1 ≤p< ∞ these real Hardy spacesH^pare certainsubsetsofL^p,while forp< 1 theL^pspaces have some undesirable properties, and the Hardy spaces are much better behaved.

There are also higher-dimensional generalizations, consisting of certain holomorphic functions ontube domainsin the complex case, or certain spaces of distributions onRⁿin the real case.

Hardy spaces have a number of applications inmathematical analysisitself, as well as incontrol theory(such asH^∞methods) and inscattering theory.

Hardy spaces for the unit disk

For spaces ofholomorphic functionson the openunit disk,theHardy spaceH²consists of the functionsfwhosemean square valueon the circle of radiusrremains bounded asr→ 1 from below.

More generally, the Hardy spaceH^pfor 0 <p< ∞ is the class of holomorphic functionsfon the open unit disk satisfying

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

This classH^pis a vector space. The number on the left side of the above inequality is the Hardy spacep-norm forf,denoted by$\|f\|_{H^{p}}.$It is a norm whenp≥ 1, but not when 0 <p< 1.

The spaceH^∞is defined as the vector space of bounded holomorphic functions on the disk, with the norm

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

For 0 < p ≤ q ≤ ∞, the classH^qis asubsetofH^p,and theH^p-norm is increasing withp(it is a consequence ofHölder's inequalitythat theL^p-norm is increasing forprobability measures,i.e.measureswith 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 complexL^pspaceson the unit circle. This connection is provided by the following theorem (Katznelson 1976,Thm 3.8): Givenf∈H^p,withp≥ 1, the radial limit

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

exists for almost every θ. The function${\tilde {f}}$belongs to theL^pspace for the unit circle,and one has that

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

Denoting the unit circle byT,and byH^p(T) the vector subspace ofL^p(T) consisting of all limit functions${\tilde {f}}$,whenfvaries inH^p,one then has that forp≥ 1,(Katznelson 1976)

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

where theĝ(n) are theFourier coefficientsof a functiongintegrable on the unit circle,

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

The spaceH^p(T) is a closed subspace ofL^p(T). SinceL^p(T) is aBanach space(for 1 ≤p≤ ∞), so isH^p(T).

The above can be turned around. Given a function${\tilde {f}}\in L^{p}(\mathbf {T} )$,withp≥ 1, one can regain a (harmonic) functionfon the unit disk by means of thePoisson kernelP_r:

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

andfbelongs toH^pexactly when${\tilde {f}}$is inH^p(T). Supposing that${\tilde {f}}$is inH^p(T),i.e.that${\tilde {f}}$has Fourier coefficients (a_n)_n∈Zwitha_n= 0 for everyn< 0, then the elementfof the Hardy spaceH^passociated to${\tilde {f}}$is the holomorphic function

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

In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as thecausalsolutions.Thus, the spaceH²is seen to sit naturally insideL²space, and is represented byinfinite sequencesindexed byN;whereasL²consists ofbi-infinite sequencesindexed byZ.

Connection to real Hardy spaces on the circle

When 1 ≤p< ∞, thereal Hardy spacesH^pdiscussed further downin this article are easy to describe in the present context. A real functionfon the unit circle belongs to the real Hardy spaceH^p(T) if it is the real part of a function inH^p(T), and a complex functionfbelongs 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

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

ThenFis inH^pfor every 0 <p< 1, and the radial limit

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

exists for a.e.θand is inH^p(T), but Re(f) is 0 almost everywhere, so it is no longer possible to recoverFfrom 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 functionF,letf_r(e^iθ) =F(re^iθ). The limit whenr→ 1 of Re(f_r),in the sense ofdistributionson the circle, is a non-zero multiple of theDirac distributionatz= 1. The Dirac distribution at a point of the unit circle belongs to real-H^p(T) for everyp< 1 (see below).

Factorization into inner and outer functions (Beurling)

For 0 <p≤ ∞, every non-zero functionfinH^pcan be written as the productf=GhwhereGis anouter functionandhis aninner function,as defined below (Rudin 1987,Thm 17.17). This "Beurlingfactorization "allows the Hardy space to be completely characterized by the spaces of inner and outer functions.

One says thatG(z)is anouter (exterior) functionif it takes the form

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

for some complex numbercwith |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,Gis inH¹because the above takes the form of thePoisson kernel(Rudin 1987,Thm 17.16). This implies that

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

for almost every θ.

One says thathis aninner (interior) functionif and only if |h| ≤ 1 on the unit disk and the limit

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

exists foralmost allθ and itsmodulusis equal to 1 a.e. In particular,his inH^∞.The inner function can be further factored into a form involving aBlaschke product.

The functionf,decomposed asf=Gh,is inH^pif and only if φ belongs toL^p(T), where φ is the positive function in the representation of the outer functionG.

LetGbe 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_αG_βand |G_α| = |G|^αalmost everywhere on the circle.

It follows that whenever 0 <p,q,r< ∞ and 1/r= 1/p+ 1/q,every functionfinH^rcan be expressed as the product of a function inH^pand a function inH^q.For example: every function inH¹is the product of two functions inH²;every function inH^p,p< 1, can be expressed as product of several functions in someH^q,q> 1.

Real-variable techniques on the unit circle

Real-variable techniques, mainly associated to the study ofreal Hardy spacesdefined onRⁿ(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.

LetP_rdenote the Poisson kernel on the unit circleT.For a distributionfon the unit circle, set

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

where thestarindicates convolution between the distributionfand the function e^iθ→P_r(θ) on the circle. Namely, (f∗P_r)(e^iθ) is the result of the action offon theC^∞-function defined on the unit circle by

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

For 0 <p< ∞, thereal Hardy spaceH^p(T) consists of distributionsfsuch thatM f  is inL^p(T).

The functionFdefined on the unit disk byF(re^iθ) = (f∗P_r)(e^iθ) is harmonic, andM f  is theradial maximal functionofF.WhenM f  belongs toL^p(T) andp≥ 1, the distributionf  "is"a function inL^p(T), namely the boundary value ofF.Forp≥ 1, thereal Hardy spaceH^p(T) is a subset ofL^p(T).

Conjugate function

To every real trigonometric polynomialuon the unit circle, one associates the realconjugate polynomialvsuch thatu+ ivextends to a holomorphic function in the unit disk,

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

This mappingu→vextends to a bounded linear operatorHonL^p(T), when 1 <p< ∞ (up to a scalar multiple, it is theHilbert transformon the unit circle), andHalso mapsL¹(T) toweak-L¹(T).When 1 ≤p< ∞, the following are equivalent for areal valuedintegrable functionfon the unit circle:

• the functionfis the real part of some functiong∈H^p(T)
• the functionfand its conjugateH(f)belong toL^p(T)
• the radial maximal functionM f  belongs toL^p(T).

When 1 <p< ∞,H(f)belongs toL^p(T) whenf∈L^p(T), hence the real Hardy spaceH^p(T) coincides withL^p(T) in this case. Forp= 1, the real Hardy spaceH¹(T) is a proper subspace ofL¹(T).

The case ofp= ∞ was excluded from the definition of real Hardy spaces, because the maximal functionM f  of anL^∞function is always bounded, and because it is not desirable that real-H^∞be equal toL^∞.However, the two following properties are equivalent for a real valued functionf

• the functionf  is the real part of some functiong∈H^∞(T)
• the functionf  and its conjugateH(f)belong toL^∞(T).

Real Hardy spaces for 0 <p< 1

When 0 <p< 1, a functionFinH^pcannot be reconstructed from the real part of its boundary limitfunctionon the circle, because of the lack of convexity ofL^pin this case. Convexity fails but a kind of "complex convexity"remains, namely the fact thatz→ |z|^qissubharmonicfor everyq> 0. As a consequence, if

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

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

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

converges in the sense of distributions to a distributionfon the unit circle, andF(re^iθ) =(f∗P_r)(θ). The functionF∈H^pcan be reconstructed from the real distribution Re(f) on the circle, because the Taylor coefficientsc_nofFcan be computed from the Fourier coefficients of Re(f).

Distributions on the circle are general enough for handling Hardy spaces whenp< 1. Distributions that are not functions do occur, as is seen with functionsF(z) = (1−z)^−N(for |z| < 1), that belong toH^pwhen 0 <Np< 1 (andNan 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 someF∈H^p.A Dirac distribution δ_x,at any pointxof the unit circle, belongs to real-H^p(T) for everyp< 1; derivatives δ′_xbelong whenp< 1/2, second derivatives δ′′_xwhenp< 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 spaceH^p(H) on theupper half-planeHis defined to be the space of holomorphic functionsfonHwith bounded norm, the norm being given by

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

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

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

Although theunit diskDand the upper half-planeHcan be mapped to one another by means ofMöbius transformations,they are not interchangeableas domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional)Lebesgue measurewhile the real line does not. However, forH²,one has the following theorem: ifm:D→Hdenotes the Möbius transformation

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

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

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

is anisometricisomorphismof Hilbert spaces.

Real Hardy spaces forRⁿ

In analysis on the real vector spaceRⁿ,the Hardy spaceH^p(for 0 <p≤ ∞) consists oftempered distributionsfsuch that for someSchwartz functionΦ with ∫Φ = 1, themaximal function

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

is inL^p(Rⁿ), where ∗ is convolution andΦ_t(x) =t⁻ⁿΦ(x / t).TheH^p-quasinorm||f ||_Hpof a distributionfofH^pis defined to be theL^pnorm ofM_Φf(this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). TheH^p-quasinorm is a norm whenp≥ 1, but not whenp< 1.

If 1 <p< ∞, the Hardy spaceH^pis the same vector space asL^p,with equivalent norm. Whenp= 1, the Hardy spaceH¹is a proper subspace ofL¹.One can find sequences inH¹that are bounded inL¹but unbounded inH¹,for example on the line

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

TheL¹andH¹norms are not equivalent onH¹,andH¹is not closed inL¹.The dual ofH¹is the spaceBMOof functions ofbounded mean oscillation.The spaceBMOcontains unbounded functions (proving again thatH¹is not closed inL¹).

Ifp< 1 then the Hardy spaceH^phas elements that are not functions, and its dual is the homogeneous Lipschitz space of ordern(1/p− 1). Whenp< 1, theH^p-quasinorm is not a norm, as it is not subadditive. Thepth power ||f ||_Hp^pis subadditive forp< 1 and so defines a metric on the Hardy spaceH^p,which defines the topology and makesH^pinto a complete metric space.

Atomic decomposition

When 0 <p≤ 1, a bounded measurable functionfof compact support is in the Hardy spaceH^pif and only if all its moments

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

whose orderi₁+... +i_nis at mostn(1/p− 1), vanish. For example, the integral offmust vanish in order thatf∈H^p,0 <p≤ 1, and as long asp>n / (n+1)this is also sufficient.

If in additionfhas support in some ballBand is bounded by |B|^−1/pthenfis called anH^p-atom(here |B| denotes the Euclidean volume ofBinRⁿ). TheH^p-quasinorm of an arbitraryH^p-atom is bounded by a constant depending only onpand on the Schwartz function Φ.

When 0 <p≤ 1, any elementfofH^phas anatomic decompositionas a convergent infinite combination ofH^p-atoms,

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

where thea_jareH^p-atoms and thec_jare scalars.

On the line for example, the difference of Dirac distributionsf= δ₁−δ₀can be represented as a series ofHaar functions,convergent inH^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 toH^pwhenp≤ 1/2 because their maximal function is equivalent at infinity toax⁻²for somea≠ 0).

MartingaleH^p

Let (M_n)_n≥0be amartingaleon 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.Themaximal functionof the martingale is defined by

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

Let 1 ≤p< ∞. The martingale (M_n)_n≥0belongs tomartingale-H^pwhenM*∈L^p.

IfM*∈L^p,the martingale (M_n)_n≥0is bounded inL^p;hence it converges almost surely to some functionfby themartingale convergence theorem.Moreover,M_nconverges tofinL^p-norm by thedominated convergence theorem;henceM_ncan be expressed as conditional expectation offon Σ_n.It is thus possible to identify martingale-H^pwith the subspace ofL^p(Ω, Σ,P) consisting of thosefsuch that the martingale

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

belongs to martingale-H^p.

Doob's maximal inequalityimplies that martingale-H^pcoincides withL^p(Ω, Σ,P) when 1 <p< ∞. The interesting space is martingale-H¹,whose dual is martingale-BMO (Garsia 1973).

The Burkholder–Gundy inequalities (whenp> 1) and the Burgess Davis inequality (whenp= 1) relate theL^p-norm of the maximal function to that of thesquare functionof the martingale

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

Martingale-H^pcan be defined by saying thatS(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 complexBrownian motion(B_t) in the complex plane, starting from the pointz= 0 at timet= 0. Let τ denote the hitting time of the unit circle. For every holomorphic functionFin the unit disk,

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

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

Example: dyadic martingale-H¹

In this example, Ω = [0, 1] and Σ_nis the finite field generated by the dyadic partition of [0, 1] into 2ⁿintervals of length 2⁻ⁿ,for everyn≥ 0. If a functionfon [0, 1] is represented by its expansion on theHaar system(h_k)

$f=\sum c_{k}h_{k},$

then the martingale-H¹norm offcan be defined by theL¹norm of the square function

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

This space, sometimes denoted byH¹(δ), is isomorphic to the classical realH¹space on the circle (Müller 2005). The Haar system is anunconditional basisforH¹(δ).

References

• Burkholder, Donald L.; Gundy, Richard F.; Silverstein, Martin L. (1971), "A maximal function characterization of the classH^p",Transactions of the American Mathematical Society,157:137–153,doi:10.2307/1995838,JSTOR1995838,MR0274767,S2CID53996980.
• Cima, Joseph A.; Ross, William T. (2000),The Backward Shift on the Hardy Space,American Mathematical Society,ISBN978-0-8218-2083-4
• Colwell, Peter (1985),Blaschke Products - Bounded Analytic Functions,Ann Arbor:University of Michigan Press,ISBN978-0-472-10065-1
• Duren, P.(1970),Theory of H^p-Spaces,Academic Press
• Fefferman, Charles;Stein, Elias M.(1972), "H^pspaces of several variables ",Acta Mathematica,129(3–4): 137–193,doi:10.1007/BF02392215,MR0447953.
• 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. BenjaminMR0448538
• 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,JFM45.1331.03
• Hoffman, Kenneth (1988),Banach Spaces of Analytic Functions,Dover Publications,ISBN978-0-486-65785-1
• Katznelson, Yitzhak(1976),An Introduction to Harmonic Analysis,Dover Publications,ISBN978-0-486-63331-2
• Koosis, P. (1998),Introduction toH_pSpaces(Second ed.),Cambridge University Press
• Mashreghi, J.(2009),Representation Theorems in Hardy Spaces,Cambridge University Press,ISBN9780521517683
• Müller, Paul F. X. (2005),Isomorphisms BetweenH¹spaces,Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), Basel:Birkhäuser,ISBN978-3-7643-2431-5,MR2157745
• Riesz, F.(1923), "Über die Randwerte einer analytischen Funktion",Mathematische Zeitschrift,18:87–95,doi:10.1007/BF01192397,S2CID121306447
• Rudin, Walter(1987),Real and Complex Analysis,McGraw-Hill,ISBN978-0-07-100276-9
• Shvedenko, S.V. (2001) [1994],"Hardy classes",Encyclopedia of Mathematics,EMS Press