# Hardy classes

*, *

Classes of analytic functions in the disc for which

(*) |

where is the normalized Lebesgue measure on the circle ; this is equivalent to the condition that the subharmonic function has a harmonic majorant in . Among the Hardy classes one also reckons the class of bounded analytic functions in . The Hardy classes, which were introduced by F. Riesz in [1] and named by him in honour of G.H. Hardy, who first studied properties of -means under the condition (*), play an important role in various problems of boundary properties of functions, in harmonic analysis, in the theory of power series, linear operators, random processes, and in the theory of extremal and approximation problems.

For any the inclusions are strict, where is the Nevanlinna class of functions of bounded characteristic (cf. Function of bounded characteristic); in particular, the functions of the Hardy classes have almost-everywhere on angular boundary values (cf. Angular boundary value) , from which the original functions in can be uniquely recovered. If , then (the converse is not true for an arbitrary analytic function ), and

The classes , , are precisely the classes of analytic functions in that have boundary values and that can be recovered from them by means of the Cauchy integral. But functions that can be represented in by an integral of Cauchy or Cauchy–Stieltjes type belong, generally speaking, only to the classes , (the converse is not true). Univalent functions in belong to all the classes , . The condition is necessary and sufficient for an analytic function to be continuous in and absolutely continuous on . If maps the disc conformally onto a Jordan domain , then the condition is equivalent to the contour being rectifiable (see [2], [5]).

The existence of a one-to-one correspondence between the functions of Hardy classes and their boundary values makes it possible to regard, when this is convenient, functions as functions on , and then the classes become closed subspaces of the Banach spaces (these are complete linear metric spaces if ). For these subspaces coincide with the closures in of the polynomials in , and for with the collections of those functions in with vanishing Fourier coefficients of negative indices. The M. Riesz theorem asserts that the mapping that can be expressed in terms of Fourier series by

is a bounded projection of the Banach space onto for every , but not for or . This implies that the real spaces and , , are the same; for other values of these spaces are essentially distinct, both in their approximation characteristics, in the structure of the dual spaces and (for ) in relation to the properties of the Fourier coefficients (see [7], [9]).

The zero sets of non-trivial functions of the Hardy classes are completely characterized by the condition , which guarantees the uniform convergence on compacta inside of a canonical Blaschke product

For every function , , , there is an F. Riesz factorization , where is the Blaschke product constructed from the zeros of , and in . The function , in turn, decomposes into the product of the outer function

and the singular inner function

where , and is a non-negative singular measure on . The conditions

are equivalent, and almost-everywhere on . Functions of the form are called inner functions; they are completely characterized by the conditions in and almost-everywhere on . Frequently one uses the decomposition of an arbitrary function into the product of two functions in (see [4], [5]).

The class occupies a special place among the Hardy classes, since it is a Hilbert space with a reproducing kernel and has a simple description in terms of the Taylor coefficients:

The study of the operator of multiplication by , or the shift operator, in has played an important role; it turned out that all invariant subspaces of this operator are generated by inner functions , that is, are of the form (see [4]).

Under pointwise multiplication and the -norm the class is a Banach algebra and the space of maximal ideals and the Shilov boundary have a very complicated structure (see [4]); the problem of the density of the ideals , , in the space with the usual Gel'fand topology (the so-called Corona problem) was solved affirmatively on the basis of a description of the universal interpolation sequences, that is, sequences , , such that (see [5], [9]).

The Hardy classes , , of analytic functions in domains other than the disc can be defined (non-equivalently, in general) by starting out either from the condition that the functions have harmonic majorants in or from the condition of boundedness of the integrals over families of contours , , that in a certain sense approximate the boundary of . The first method also makes it possible to define Hardy classes on Riemann surfaces. The second method leads to classes that are better adapted for the solution of extremal and approximation problems; in the case of Jordan domains with a rectifiable boundary the latter classes are called Smirnov classes and are denoted by (see [2] and Smirnov class). For a half-plane, for example , the classes , , defined by the condition

are closely related in their properties to the Hardy classes for the disc, however, their applications in harmonic analysis are connected not with the theory of Fourier series, but with Fourier transforms.

The Hardy classes of analytic functions in the unit ball and the unit polydisc of the space are defined by the condition (*), where the circle is replaced by the sphere or the distinguished boundary of the polydisc. The specific nature of the higher-dimensional case becomes manifest, first of all, in the absence of a simple characterization of the zero sets and of a factorization of functions in the Hardy classes (see [6], [10]). Hardy classes can also be defined in various ways for other domains in (see [10]).

The higher-dimensional analogues of the Hardy classes (see [3]) are the so-called Hardy spaces, that is, spaces , , of Riesz systems: real-valued vector functions , , , satisfying the generalized Cauchy–Riemann conditions

for which

The definition of these spaces can also be given in terms of only the "real parts" of the systems by requiring that the function be harmonic and that its maximal function

There are other characterizations of real-variable spaces. spaces can also be defined on homogeneous groups, i.e. Lie groups with underlying manifold and dilations (see [11]).

For the transition from the function to its boundary values yields an identification of the spaces and ; therefore, of interest is only the case . It was just within the framework of the spaces that fundamental results of the theory of Hardy classes such as the realization of the dual space as the space of functions of bounded mean oscillation (see [8], [9]) and the atomic decomposition of the classes , (see [7]), were first established. The characterization of Hardy classes in terms of the maximal function requires in a number of cases a recourse to probability concepts connected with Brownian motion (see [8]).

Abstract Hardy classes arise in the theory of uniform algebras (cf. Uniform algebra) and are not directly connected with analytic functions. Let a closed algebra of continuous functions on a compact set and a certain homomorphism be fixed; there exists a positive measure on representing : , . By definition, the classes , , are the closures (the weak closure for ) of the algebra in the spaces ; the study of the classes makes it possible to obtain additional information on (see [12]).

#### References

[1] | F. Riesz, "Ueber die Randwerte einer analytischen Funktion" Math. Z. , 18 (1923) pp. 87–95 |

[2] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

[3] | E. Stein, G. Weiss, "On the theory of harmonic functions of several variables" Acta Math. , 103 (1960) pp. 25–62 |

[4] | K. Hoffman, "Banach spaces of analytic functions" , Prentice-Hall (1962) |

[5] | P.L. Duren, "Theory of spaces" , Acad. Press (1970) |

[6] | W. Rudin, "Function theory in polydiscs" , Benjamin (1969) |

[7] | R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 : 4 (1977) pp. 569–645 |

[8] | K.E. Petersen, "Brownian motion, Hardy spaces, and bounded mean oscillation" , Cambridge Univ. Press (1977) |

[9] | P. Koosis, "Introduction to -spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press (1980) |

[10] | W. Rudin, "Function theory in the unit ball in " , Springer (1980) |

[11] | G.B. Folland, E.M. Stein, "Hardy spaces on homogeneous groups" , Princeton Univ. Press (1982) |

[12] | T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969) |

#### Comments

The result concerning the shift operator is commonly known as Beurling's theorem. The indicated solution of the corona problem in this article is due to L. Carleson. There is another, more recent, proof by T. Wolff based on the existence of good solutions of the inhomogeneous Cauchy–Riemann equations. The result about the dual of being BMO, the functions of bounded mean oscillation, is due to C. Fefferman. It is usually stated for , because otherwise one has to introduce an unusual complex multiplication on real BMO, see [a1].

A measurable on is called a BMO-function, or function of class BMO, if is locally integrable (i.e. is integrable over any compact subset) and if, putting

(the average of over the bounded interval ), one has

where the supremum is over all bounded intervals .

There are BMO-spaces on other domains, e.g. the unit circle ( an arc, integration with respect to Lebesgue (or Haar) measure).

An important subclass is the class of VMO-functions, the class of functions of vanishing mean oscillation. Let , be as above. Write, for ,

Then if and as . (See [a1].)

For spaces of several variables see also [a2].

In addition to the numerous application areas already mentioned, the Hardy classes, especially , are important in control theory, cf. control theory.

#### References

[a1] | J.-B. Garnett, "Bounded analytic functions" , Acad. Press (1981) |

[a2] | C. Fefferman, E.M. Stein, " spaces of several variables" Acta Math. , 129 (1972) pp. 137–193 |

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Hardy classes.

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