{"history":[],"article":{"id":"240c1525-6f18-43a9-9840-d2404e0e4d5c","title":"Hardy space","slug":"hardy-space","content":":::summary\n:::\n\n# \nIn [complex analysis](https://bliptext.com/articles/complex-analysis), the **Hardy [spaces](https://bliptext.com/articles/space-mathematics)** (or **Hardy classes**) Hp are certain [spaces](https://bliptext.com/articles/space-mathematics) of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) on the [unit disk](https://bliptext.com/articles/unit-disk) or [upper half plane](https://bliptext.com/articles/upper-half-plane).\nThey were introduced by [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz), who named them after [G. H. Hardy](https://bliptext.com/articles/g-h-hardy), because of the paper.\nIn [real analysis](https://bliptext.com/articles/real-analysis) **Hardy spaces** are certain spaces of [distributions](https://bliptext.com/articles/distribution-mathematics) on the real line, which are (in the sense of [distributions](https://bliptext.com/articles/distribution-mathematics)) boundary values of the holomorphic functions of the [complex](https://bliptext.com/articles/complex-number) **Hardy spaces**, and are related to the Lp spaces of [functional analysis](https://bliptext.com/articles/functional-analysis).\nFor 1 ≤ p < ∞ these real Hardy spaces Hp are certain [subsets](https://bliptext.com/articles/subset) of Lp, while for p < 1 the Lp spaces have some undesirable properties, and the Hardy spaces are much better behaved.\nThere are also higher-dimensional generalizations, consisting of certain holomorphic functions on [tube domains](https://bliptext.com/articles/tube-domain) in the complex case, or certain spaces of distributions on **R**n in the real case.\nHardy spaces have a number of applications in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) itself, as well as in [control theory](https://bliptext.com/articles/control-theory) (such as H∞ methods) and in [scattering theory](https://bliptext.com/articles/scattering-theory).\n\n# Hardy spaces for the unit disk\nFor spaces of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) on the open [unit disk](https://bliptext.com/articles/unit-disk), the Hardy space H2 consists of the functions f whose [mean square value](https://bliptext.com/articles/root-mean-square) on the circle of radius r remains bounded as r → 1 from below.\nMore generally, the Hardy space Hp for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying\nThis class Hp is a vector space.\nThe number on the left side of the above inequality is the Hardy space p-norm for f, denoted by It is a norm when p ≥ 1, but not when 0 < p < 1.\nThe space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm\nFor 0 < p ≤ q ≤ ∞, the class Hq is a [subset](https://bliptext.com/articles/subset) of Hp, and the Hp-norm is increasing with p (it is a consequence of [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality) that the Lp-norm is increasing for [probability measures](https://bliptext.com/articles/probability-space), i.e. [measures](https://bliptext.com/articles/measure-mathematics) with total mass 1).\n\n# Hardy spaces on the unit circle\nThe Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex Lp spaces on the unit circle.\nThis connection is provided by the following theorem : Given f ∈ Hp, with p ≥ 1, the radial limit\nexists for almost every θ.\nThe function \\tilde f belongs to the Lp space for the unit circle, and one has that\nDenoting the unit circle by ******T******, and by Hp(******T******) the vector subspace of Lp(******T******) consisting of all limit functions \\tilde f, when f varies in Hp, one then has that for p ≥ 1,\nwhere the ĝ(n) are the [Fourier coefficients](https://bliptext.com/articles/fourier-coefficients) of a function g integrable on the unit circle,\n****T****he space Hp(****T****) is a closed subspace of Lp(****T****).\nSince Lp(****T****) is a [Banach space](https://bliptext.com/articles/banach-space) (for 1 ≤ p ≤ ∞), so is Hp(****T****).\nThe above can be turned around.\nGiven a function, with p ≥ 1, one can regain a ([harmonic](https://bliptext.com/articles/harmonic-function)) function f on the unit disk by means of the [Poisson kernel](https://bliptext.com/articles/poisson-kernel) Pr:\nand f belongs to Hp exactly when \\tilde f is in Hp(**T**).\nSupposing that \\tilde f is in Hp(**T**), i.e. that \\tilde f has Fourier coefficients (an)n∈Z with an = 0 for every n < 0, then the element f of the Hardy space Hp associated to \\tilde f is the holomorphic function\nIn applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the [causal](https://bliptext.com/articles/causal) solutions.\nThus, the space H2 is seen to sit naturally inside L2 space, and is represented by [infinite sequences](https://bliptext.com/articles/infinite-sequence) indexed by **N**; whereas L2 consists of bi-[infinite sequences](https://bliptext.com/articles/infinite-sequence) indexed by **Z**.\n\n## Connection to real Hardy spaces on the circle\nWhen 1 ≤ p < ∞, the real Hardy spaces Hp discussed further down in this article are easy to describe in the present context.\nA real function f on the unit circle belongs to the real Hardy space Hp(****T****) if it is the real part of a function in Hp(****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).\nThus 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.\nFor 0 < p < 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid.\nFor example, consider the function\nThen F is in Hp for every 0 < p < 1, and the radial limit\nexists for a.e. θ and is in Hp(**T**), but Re(f) is 0 almost everywhere, so it is no longer possible to recover F from Re(f).\nAs a consequence of this example, one sees that for 0 < p < 1, one cannot characterize the real-Hp(**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.\nFor the same function F, let fr(eiθ) = F(reiθ).\nThe limit when r → 1 of Re(fr), in the sense of distributions on the circle, is a non-zero multiple of the [Dirac distribution](https://bliptext.com/articles/dirac-distribution) at z = 1.\n**T**he Dirac distribution at a point of the unit circle belongs to real-Hp(**T**) for every p < 1 (see below).\n\n# Factorization into inner and outer functions (Beurling)\nFor 0 < p ≤ ∞, every non-zero function f in Hp can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below.\nThis \"[Beurling](https://bliptext.com/articles/arne-beurling) factorization\" allows the Hardy space to be completely characterized by the spaces of inner and outer functions.\nOne says that G(z) is an **outer (exterior) function** if it takes the form\nfor some complex number c with |c| = 1, and some positive measurable function \\varphi on the unit circle such that is integrable on the circle.\nIn particular, when \\varphi is integrable on the circle, G is in H1 because the above takes the form of the [Poisson kernel](https://bliptext.com/articles/poisson-kernel).\nThis implies that\nfor almost every θ.\nOne says that h is an **inner (interior) function** if and only if |h| ≤ 1 on the unit disk and the limit\nexists for [almost all](https://bliptext.com/articles/almost-all) θ and its [modulus](https://bliptext.com/articles/absolute-value) is equal to 1 a.e. In particular, h is in H∞.\nThe inner function can be further factored into a form involving a [Blaschke product](https://bliptext.com/articles/blaschke-product).\n**T**he function f, decomposed as f = Gh, is in Hp if and only if φ belongs to Lp(**T**), where φ is the positive function in the representation of the outer function G.\nLet G be an outer function represented as above from a function φ on the circle.\nReplacing φ by φα, α > 0, a family (Gα) of outer functions is obtained, with the properties:\nIt follows that whenever 0 < p, q, r < ∞ and 1/r = 1/p + 1/q, every function f in Hr can be expressed as the product of a function in Hp and a function in Hq.\nFor example: every function in H1 is the product of two functions in H2; every function in Hp, p < 1, can be expressed as product of several functions in some Hq, q > 1.\n\n# Real-variable techniques on the unit circle\n**R**eal-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.\nIt is a common practice to allow for complex functions (or distributions) in these \"real\" spaces.\nThe definition that follows does not distinguish between real or complex case.\nLet Pr denote the Poisson kernel on the unit circle **T**.\nFor a distribution f on the unit circle, set\nwhere the star indicates convolution between the distribution f and the function eiθ → Pr(θ) on the circle.\nNamely, (f ∗ Pr)(eiθ) is the result of the action of f on the C∞-function defined on the unit circle by\nFor 0 < p < ∞, the real Hardy space Hp(****T****) consists of distributions f such that M f is in Lp(****T****).\nThe 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.\nWhen M f belongs to Lp(****T****) and p ≥ 1, the distribution f \"is\" a function in Lp(****T****), namely the boundary value of F.\nFor p ≥ 1, the real Hardy space Hp(****T****) is a subset of Lp(****T****).\n\n## Conjugate function\nTo 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,\n******T******his 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](https://bliptext.com/articles/hilbert-transform) on the unit circle), and H also maps L1(******T******) to weak-L1(******T******).\nWhen 1 ≤ p < ∞, the following are equivalent for a real valued integrable function f on the unit circle:\nWhen 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.\nFor p = 1, the real Hardy space H1(****T****) is a proper subspace of L1(****T****).\nThe 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∞.\nHowever, the two following properties are equivalent for a real valued function f\n\n## Real Hardy spaces for 0 < p < 1\nWhen 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.\nConvexity fails but a kind of \"complex convexity\" remains, namely the fact that z → |z|q is [subharmonic](https://bliptext.com/articles/subharmonic-function) for every q > 0.\nAs a consequence, if\nis in Hp, it can be shown that cn = O(n1/p–1).\nIt follows that the Fourier series\nconverges in the sense of distributions to a distribution f on the unit circle, and F(reiθ) =(f ∗ Pr)(θ).\nThe 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).\nDistributions on the circle are general enough for handling Hardy spaces when p < 1.\nDistributions 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).\nA real distribution on the circle belongs to real-Hp(**T**) iff it is the boundary value of the real part of some F ∈ Hp.\nA 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.\n\n# Hardy spaces for the upper half plane\nIt 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.\nThe ******H******ardy space ******H******p(******H******) on the [upper half-plane](https://bliptext.com/articles/upper-half-plane) ******H****** is defined to be the space of holomorphic functions f on ******H****** with bounded norm, the norm being given by\nThe corresponding **H**∞(**H**) is defined as functions of bounded norm, with the norm given by\nAlthough the [unit disk](https://bliptext.com/articles/unit-disk) **D** and the upper half-plane **H** can be mapped to one another by means of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation), they are not interchangeable as domains for **H**ardy spaces.\nContributing to this difference is the fact that the unit circle has finite (one-dimensional) [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) while the real line does not.\n**H**owever, for **H**2, one has the following theorem: if m : **D** → **H** denotes the Möbius transformation\nThen the linear operator M : **H**2(**H**) → **H**2(**D**) defined by\nis an [isometric](https://bliptext.com/articles/isometry) [isomorphism](https://bliptext.com/articles/isomorphism) of Hilbert spaces.\n\n# Real Hardy spaces for Rn\nIn analysis on the real vector space **R**n, the Hardy space Hp (for 0 < p ≤ ∞) consists of [tempered distributions](https://bliptext.com/articles/distribution-mathematics) f such that for some [Schwartz function](https://bliptext.com/articles/schwartz-function) Φ with ∫Φ = 1, the [maximal function](https://bliptext.com/articles/maximal-function)\nis in Lp(**R**n), where ∗ is convolution and Φt (x) = t −nΦ(x / t).\nThe Hp-[quasinorm](https://bliptext.com/articles/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).\nThe Hp-quasinorm is a norm when p ≥ 1, but not when p < 1.\nIf 1 < p < ∞, the Hardy space Hp is the same vector space as Lp, with equivalent norm.\nWhen p = 1, the Hardy space H1 is a proper subspace of L1.\nOne can find sequences in H1 that are bounded in L1 but unbounded in H1, for example on the line\nThe L1 and H1 norms are not equivalent on H1, and H1 is not closed in L1.\nThe dual of H1 is the space BMO of functions of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation).\nThe space BMO contains unbounded functions (proving again that H1 is not closed in L1).\nIf 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).\nWhen p < 1, the Hp-quasinorm is not a norm, as it is not subadditive.\nThe 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.\n\n## Atomic decomposition\nWhen 0 < p ≤ 1, a bounded measurable function f of compact support is in the Hardy space Hp if and only if all its moments\nwhose order i1+ ... +in is at most n(1/p − 1), vanish.\nFor 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.\nIf 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 **R**n).\nThe Hp-quasinorm of an arbitrary Hp-atom is bounded by a constant depending only on p and on the Schwartz function Φ.\nWhen 0 < p ≤ 1, any element f of Hp has an **atomic decomposition** as a convergent infinite combination of Hp-atoms,\nwhere the aj are Hp-atoms and the cj are scalars.\nOn the line for example, the difference of Dirac distributions f = δ1−δ0 can be represented as a series of [Haar functions](https://bliptext.com/articles/haar-wavelet), 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](https://bliptext.com/articles/haar-wavelet) 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).\n\n# Martingale Hp\nLet (Mn)n≥0 be a [martingale](https://bliptext.com/articles/martingale-probability-theory) on some probability space (Ω, Σ, P), with respect to an increasing sequence of σ-fields (Σn)n≥0.\nAssume for simplicity that Σ is equal to the σ-field generated by the sequence (Σn)n≥0.\nThe maximal function of the martingale is defined by\nLet 1 ≤ p < ∞.\nThe martingale (Mn)n≥0 belongs to martingale-Hp when M* ∈ Lp.\nIf 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](https://bliptext.com/articles/doob-s-martingale-convergence-theorems).\nMoreover, Mn converges to f in Lp-norm by the [dominated convergence theorem](https://bliptext.com/articles/dominated-convergence-theorem); hence Mn can be expressed as conditional expectation of f on Σn.\nIt is thus possible to identify martingale-Hp with the subspace of Lp(Ω, Σ, P) consisting of those f such that the martingale\nbelongs to martingale-Hp.\n[Doob's maximal inequality](https://bliptext.com/articles/doob-s-martingale-inequality) implies that martingale-Hp coincides with Lp(Ω, Σ, P) when 1 < p < ∞.\nThe interesting space is martingale-H1, whose dual is martingale-BMO.\nThe 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\nMartingale-Hp can be defined by saying that S(f)∈ Lp.\nMartingales with continuous time parameter can also be considered.\nA direct link with the classical theory is obtained via the complex [Brownian motion](https://bliptext.com/articles/wiener-process) (Bt) in the complex plane, starting from the point z = 0 at time t = 0.\nLet τ denote the hitting time of the unit circle.\nFor every holomorphic function F in the unit disk,\nis a martingale, that belongs to martingale-Hp iff F ∈ Hp.\n\n## Example: dyadic martingale-H1\nIn 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.\nIf a function f on [0, 1] is represented by its expansion on the [Haar system](https://bliptext.com/articles/haar-wavelet) (hk)\nthen the martingale-H1 norm of f can be defined by the L1 norm of the square function\nThis space, sometimes denoted by H1(δ), is isomorphic to the classical real H1 space on the circle.\nThe Haar system is an [unconditional basis](https://bliptext.com/articles/schauder-basis) for H1(δ).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:08:31.749Z","updated_at":"2025-01-25T19:08:31.749Z","search_vector":"'hardi':1 'space':2","search_content":" \n \n\n# \nIn [complex analysis](https://bliptext.com/articles/complex-analysis), the **Hardy [spaces](https://bliptext.com/articles/space-mathematics)** (or **Hardy classes**) Hp are certain [spaces](https://bliptext.com/articles/space-mathematics) of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) on the [unit disk](https://bliptext.com/articles/unit-disk) or [upper half plane](https://bliptext.com/articles/upper-half-plane).\nThey were introduced by [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz), who named them after [G. H. Hardy](https://bliptext.com/articles/g-h-hardy), because of the paper.\nIn [real analysis](https://bliptext.com/articles/real-analysis) **Hardy spaces** are certain spaces of [distributions](https://bliptext.com/articles/distribution-mathematics) on the real line, which are (in the sense of [distributions](https://bliptext.com/articles/distribution-mathematics)) boundary values of the holomorphic functions of the [complex](https://bliptext.com/articles/complex-number) **Hardy spaces**, and are related to the Lp spaces of [functional analysis](https://bliptext.com/articles/functional-analysis).\nFor 1 ≤ p 0, a family (Gα) of outer functions is obtained, with the properties:\nIt follows that whenever 0 1.\n\n# Real-variable techniques on the unit circle\n**R**eal-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.\nIt is a common practice to allow for complex functions (or distributions) in these \"real\" spaces.\nThe definition that follows does not distinguish between real or complex case.\nLet Pr denote the Poisson kernel on the unit circle **T**.\nFor a distribution f on the unit circle, set\nwhere the star indicates convolution between the distribution f and the function eiθ → Pr(θ) on the circle.\nNamely, (f ∗ Pr)(eiθ) is the result of the action of f on the C∞-function defined on the unit circle by\nFor 0 0.\nAs a consequence, if\nis in Hp, it can be shown that cn = O(n1/p–1).\nIt follows that the Fourier series\nconverges in the sense of distributions to a distribution f on the unit circle, and F(reiθ) =(f ∗ Pr)(θ).\nThe 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).\nDistributions on the circle are general enough for handling Hardy spaces when p n / (n+1) this is also sufficient.\nIf 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 **R**n).\nThe Hp-quasinorm of an arbitrary Hp-atom is bounded by a constant depending only on p and on the Schwartz function Φ.\nWhen 0 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\nMartingale-Hp can be defined by saying that S(f)∈ Lp.\nMartingales with continuous time parameter can also be considered.\nA direct link with the classical theory is obtained via the complex [Brownian motion](https://bliptext.com/articles/wiener-process) (Bt) in the complex plane, starting from the point z = 0 at time t = 0.\nLet τ denote the hitting time of the unit circle.\nFor every holomorphic function F in the unit disk,\nis a martingale, that belongs to martingale-Hp iff F ∈ Hp.\n\n## Example: dyadic martingale-H1\nIn 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.\nIf a function f on [0, 1] is represented by its expansion on the [Haar system](https://bliptext.com/articles/haar-wavelet) (hk)\nthen the martingale-H1 norm of f can be defined by the L1 norm of the square function\nThis space, sometimes denoted by H1(δ), is isomorphic to the classical real H1 space on the circle.\nThe Haar system is an [unconditional basis](https://bliptext.com/articles/schauder-basis) for H1(δ).","revisionCount":0},"hasMore":false,"nextCursor":null}

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