View source for Hardy spaces

''real-variable theory of, real-variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100902.png" /> theory''

The real-variable Hardy spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100903.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100904.png" />) are spaces of distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100905.png" /> (cf. [[Generalized functions, space of|Generalized functions, space of]]), originally defined as boundary values of holomorphic or harmonic functions, which have assumed an important role in modern [[Harmonic analysis|harmonic analysis]]. They may be defined in terms of certain maximal functions.

Specifically, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100906.png" /> belongs to the Schwartz class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100907.png" /> of rapidly decreasing smooth functions, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100908.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100909.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009010.png" />, the space of tempered distributions, define the radial maximal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009011.png" /> and the non-tangential maximal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009012.png" /> by

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009015.png" /> denotes [[Convolution of functions|convolution of functions]]. C. Fefferman and E.M. Stein [[#References|[a2]]] (see also [[#References|[a4]]]) proved that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009017.png" />, the following conditions are equivalent (the Fefferman–Stein theorem):

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009018.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009019.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009020.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009021.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009023.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009024.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009025.png" />, and in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009026.png" /> uniformly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009027.png" /> in a suitable bounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009028.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009029.png" /> is the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009030.png" /> that satisfy these conditions.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009032.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009033.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009034.png" /> is a proper subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009035.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009037.png" /> contains distributions that are not functions. A distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009039.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009040.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009041.png" /> is the boundary value of a [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009042.png" /> on the upper half-plane such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009044.png" /> is the harmonic conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009045.png" />; this is the connection with the complex-variable [[Hardy classes|Hardy classes]]. There is a similar characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009046.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009047.png" /> in terms of systems of harmonic functions satisfying generalized [[Cauchy–Riemann equations|Cauchy–Riemann equations]]; see [[#References|[a2]]].

Another characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009049.png" /> is of great importance. A measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009050.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009052.png" />-atom (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009053.png" />) if

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009054.png" /> vanishes outside some ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009056.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009057.png" /> for all polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009058.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009059.png" />. The atomic decomposition theorem (see [[#References|[a4]]]) states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009060.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009061.png" />, where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009062.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009063.png" />-atom and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009064.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009065.png" /> is a complete [[Topological vector space|topological vector space]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009066.png" />, and a [[Banach space|Banach space]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009067.png" />, with topology defined by any of the quasi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009069.png" />, or

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009071.png" /></td> </tr></table>

By the celebrated Fefferman theorem [[#References|[a2]]] (see also [[#References|[a4]]]), the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009072.png" /> is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009073.png" /> of functions of bounded mean oscillation (cf. also [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009074.png" />-space]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009075.png" />, the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009076.png" /> is the homogeneous Lipschitz space of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009077.png" />; see [[#References|[a3]]].

The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009078.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009079.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009080.png" /> provide an extension of the scale of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009081.png" />-spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009082.png" />) that is in many respects more natural and useful than the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009083.png" />-spaces. Most importantly, many of the essential operations of harmonic analysis, e.g., singular integrals of Calderón–Zygmund type (cf. also [[Calderón–Zygmund operator|Calderón–Zygmund operator]]; [[Singular integral|Singular integral]]), maximal operators and Littlewood–Paley functionals, that are well-behaved on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009084.png" /> only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009085.png" /> are also well-behaved on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009087.png" />. In addition, many important classes of singular distributions belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009088.png" />, or are closely related to elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009089.png" />, for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009090.png" />. See [[#References|[a3]]], [[#References|[a4]]].

The real-variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009091.png" /> theory can be extended to spaces other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009092.png" />. A rather complete extension is available in the setting of homogeneous groups, i.e., simply-connected nilpotent Lie groups (cf. also [[Lie group, nilpotent|Lie group, nilpotent]]) with a one-parameter family of dilations; see [[#References|[a3]]]. (These groups include, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009093.png" /> 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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.R. Coifman,   G. Weiss,   "Extensions of Hardy spaces and their use in analysis"  ''Bull. Amer. Math. Soc.'' , '''83'''  (1977)  pp. 569–645  {{MR|0447954}} {{ZBL|0358.30023}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Fefferman,   E.M. Stein,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h11009094.png" /> spaces of several variables"  ''Acta Math.'' , '''129'''  (1972)  pp. 137–193  {{MR|0447953}} {{ZBL|0257.46078}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.B. Folland,   E.M. Stein,   "Hardy spaces on homogeneous groups" , Princeton Univ. Press  (1982)  {{MR|0657581}} {{ZBL|0508.42025}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.M. Stein,   "Harmonic analysis" , Princeton Univ. Press  (1993)  {{MR|1232192}} {{ZBL|0821.42001}} </TD></TR></table>