# Hardy spaces

*real-variable theory of, real-variable theory*

The real-variable Hardy spaces () are spaces of distributions on (cf. Generalized functions, space of), originally defined as boundary values of holomorphic or harmonic functions, which have assumed an important role in modern harmonic analysis. They may be defined in terms of certain maximal functions.

Specifically, suppose belongs to the Schwartz class of rapidly decreasing smooth functions, and let for . If , the space of tempered distributions, define the radial maximal function and the non-tangential maximal function by

where denotes convolution of functions. C. Fefferman and E.M. Stein [a2] (see also [a4]) proved that for and , the following conditions are equivalent (the Fefferman–Stein theorem):

1) for some with ;

2) for some with ;

3) for every , and in fact uniformly for in a suitable bounded subset of . is the space of all that satisfy these conditions.

For , coincides with , and is a proper subspace of . For , contains distributions that are not functions. A distribution on is in if and only if is the boundary value of a harmonic function on the upper half-plane such that , where is the harmonic conjugate of ; this is the connection with the complex-variable Hardy classes. There is a similar characterization of for in terms of systems of harmonic functions satisfying generalized Cauchy–Riemann equations; see [a2].

Another characterization of for is of great importance. A measurable function is called a -atom () if

i) vanishes outside some ball of radius and ;

ii) for all polynomials of degree . The atomic decomposition theorem (see [a4]) states that if and only if , where each is a -atom and .

is a complete topological vector space for , and a Banach space for , with topology defined by any of the quasi-norms , , or

By the celebrated Fefferman theorem [a2] (see also [a4]), the dual of is the space of functions of bounded mean oscillation (cf. also -space). For , the dual of is the homogeneous Lipschitz space of order ; see [a3].

The spaces () and provide an extension of the scale of -spaces () that is in many respects more natural and useful than the corresponding -spaces. Most importantly, many of the essential operations of harmonic analysis, e.g., singular integrals of Calderón–Zygmund type (cf. also Calderón–Zygmund operator; Singular integral), maximal operators and Littlewood–Paley functionals, that are well-behaved on only for are also well-behaved on and . In addition, many important classes of singular distributions belong to , or are closely related to elements of , for suitable . See [a3], [a4].

The real-variable theory can be extended to spaces other than . A rather complete extension is available in the setting of homogeneous groups, i.e., simply-connected nilpotent Lie groups (cf. also Lie group, nilpotent) with a one-parameter family of dilations; see [a3]. (These groups include, in particular, with non-isotropic dilations.) Parts of the theory have also been developed in the much more general setting of the Coifman–Weiss spaces of homogeneous type; see [a1].

#### References

[a1] | R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–645 |

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

[a3] | G.B. Folland, E.M. Stein, "Hardy spaces on homogeneous groups" , Princeton Univ. Press (1982) |

[a4] | E.M. Stein, "Harmonic analysis" , Princeton Univ. Press (1993) |

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Hardy spaces.

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