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 ![]() |
[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 ![]() |
[10] | W. Rudin, "Function theory in the unit ball in ![]() |
[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, "![]() |
Hardy classes. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_classes&oldid=15564