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''real-variable theory of, real-variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110090/h1100902.png" /> theory''
+
{{MSC|42B30}}
 +
[[Category:Harmonic analysis]]
 +
{{TEX|done}}
  
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.
+
''real-variable theory of, real-variable $\mathcal{H}^p$ theory''
  
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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====Definition====
 +
The real-variable Hardy spaces $\mathcal{H}^p = \mathcal{H}^p (\mathbb R^n)$ ($0<p<\infty$) are spaces of distributions on $\mathbb R^n$ (cf. [[Generalized functions, space of]]), originally defined as boundary values of [[Holomorphic function|holomorphic]] or [[Harmonic function|harmonic]] functions, which have assumed an important role in modern [[Harmonic analysis]]. They may be defined in terms of certain maximal functions.
  
<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/h11009013.png" /></td> </tr></table>
+
Specifically, suppose $\phi$ belongs to the Schwartz class $\mathscr{S} (\mathbb R^n)$ of rapidly decreasing smooth functions, and let $\phi_t (x) = t^{-n} \phi (x/t)$ for $t>0$. If $f\in \mathscr{S}' (\mathbb R^n)$, the space of tempered distributions, define the radial maximal function $m_\phi$ and the non-tangential maximal function $M_\phi$ by
 +
\begin{align}
 +
& m_\phi f (x) = \sup_{t>0} |f * \phi_t (x)|\\
 +
& M_{\phi} f (x) = \sup_{|y-x|<t} |f*\phi_t (y)|\, .
 +
\end{align}
 +
where $*$ denotes [[Convolution of functions|convolution of functions]]. C. Fefferman and E.M. Stein {{Cite|FS}} (see also {{Cite|St}}) proved the following theorem, also known as Fefferman-Stein theorem.
  
<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>
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'''Theorem 1'''
 +
For $f\in \mathscr{S}' (\mathbb R^n)$ and $0<p<\infty$, the following conditions are equivalent:
  
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) $m_\phi f \in L^p$ for some $\phi$ with $\int \phi \neq 0$;
  
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) $M_\phi f\in L^p$ for some $\phi$ with $\int \phi \neq 0$;
  
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) $M_\phi f\in L^p$ for $\phi \in \mathscr{S} (\mathbb R^n)$, and in fact $M_\phi f\in L^p$ uniformly for $\phi$ in a suitable bounded subset of $\mathscr{S} (\mathbb R^n)$.  
  
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.
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'''Definition 2'''
 +
$\mathcal{H}^p (\mathbb R^n)$ is the space of all $f\in \mathscr{S}' (\mathbb R^n)$ 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]]].
+
====Basic properties====
 +
For $p>1$, $\mathcal{H}^p$ coincides with $L^p$ and $\mathcal{H}^1$ is a proper subspace of $L^1$. For $p<1$ $\mathcal{H}^p$ contains distributions that are not functions. The connection with the complex-variable [[Hardy classes]] is given by the following characterization
  
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
+
'''Theorem 3'''
 +
Denote by $\mathbb R^2_+$ the upper half plane $\{(x,y): y>0\}$. A tempered distribution $f$ on $\mathbb R$ is in $\mathcal{H}^p$ if and only if it is the boundary value of a [[Harmonic function|harmonic function]] $u: \mathbb R^2_+\to \mathbb R$ such that
 +
\[
 +
\int_0^\infty \int_{-\infty}^\infty |(u + iv)(x+iy)|^p \, dx\, dy\; < \; \infty\, ,
 +
\]
 +
where $v$ is the [[Conjugate harmonic functions|harmonic conjugate of]] $u$.
  
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" />;
+
There is a similar characterization of $\mathcal{H}^p (\mathbb R^n)$ for $n>1$ in terms of systems of harmonic functions satisfying generalized [[Cauchy-Riemann equations]]; see {{Cite|FS}}.
  
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" />.
+
====Atomic decomposition====
 +
Another characterization of $\mathcal{H}^p$ for $p\leq 1$ is of great importance.  
  
<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
+
'''Definition 4'''
 +
A measurable function $\alpha$ is called a $p$-atom ($p\leq 1$) if
  
<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/h11009070.png" /></td> </tr></table>
+
i) $\alpha$ vanishes outside some ball of radius $r$ and $\sup_x |\alpha (x)| \leq r^{-n/p}$;
  
<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>
+
ii) The integral
 +
\[
 +
\int \alpha (x) P (x)\, dx
 +
\]
 +
vanishes for all polynomials $P$ of degree $\leq \frac{n-np}{p}$.
  
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 atomic decomposition theorem (see {{Cite|St}}) states that
  
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]]].
+
'''Theorem 5'''
 +
$f\in \mathcal{H}^p$ if and only if
 +
\[
 +
\sum_j c_j \alpha_j
 +
\]
 +
for some sequence $\alpha_j$ of $p$-atoms and some sequence of real numbers $\{c_j\}\in \ell^p$ (i.e. such that $\sum |c_j|^p < \infty$).
  
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]]].
+
====Functional analytic structure====
 +
$\mathcal{H}^1$ is a [[Banach space]] (and obviously $\mathcal{H}^p$ is a Banach space for $p>1$, since it coincides with $L^p$). For $p<1$ $\mathcal{H}^p$ is a complete [[Topological vector space|topological vector space]]. In all these cases the topology is induced by any of the following equivalent quasi-norms
 +
\begin{align}
 +
&f\mapsto\;\; \int |m_\phi|^p\\
 +
&f\mapsto\;\; \int |M_\phi|^p\\
 +
&f\mapsto\;\; \inf \left\{ \sum |c_j|^p : \text{there is an atomic decomposition } f = \sum c_j \alpha_j\right\}\, .
 +
\end{align}
 +
A celebrated theorem of Fefferman {{Cite|FS}} (see also {{Cite|St}}), states that $\mathcal{H}^1$ is the dual of ${\rm BMO}$, the space  of functions with bounded mean oscillation (cf. also [[BMO-space]]). For $p<1$, the dual of $\mathcal{H}^p$ is the homogeneous Lipschitz space of order $\frac{n-np}{p}$; see {{Cite|FoS}}.
 +
 
 +
====Importance in harmonic analysis====
 +
The space ${\rm BMO}$ and the spaces $\mathcal{H}^p$ with $p\leq1$
 +
provide an extension to $p\in ]0, \infty]$ of the scale of classical $L^p$-spaces with $p\in ]1, \infty[$ that is in many respects more natural and useful than the corresponding $L^p$-extension. 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 $L^p$ only for $p>1$ are also well-behaved on $\mathcal{H}^p$ and ${\rm BMO}$. In addition, many important classes of singular distributions belong to $\mathcal{H}^p$, or are closely related to elements of $\mathcal{H}^p$, for suitable $p<1$. See {{Cite|FoS}}, {{Cite|St}}.
 +
 
 +
====Extensions====
 +
The real-variable $\mathcal{H}^p$ theory can be extended to spaces other than $\mathbb R^n$. A rather complete extension is available in the setting of homogeneous groups, i.e., simply-connected [[Lie group, nilpotent|nilpotent Lie groups]] with a one-parameter family of dilations; see {{Cite|FoS}}. (These groups include, in particular, $\mathbb R^n$ 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 {{Cite|CW}}}.
  
 
====References====
 
====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</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</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)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.M. Stein,  "Harmonic analysis" , Princeton Univ. Press  (1993)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|CW}}|| 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}}
 +
|-
 +
|valign="top"|{{Ref|FS}}|| C. Fefferman,  E.M. Stein,  "$\mathcal{H}^p$ spaces of several variables"  ''Acta Math.'' , '''129'''  (1972)  pp. 137–193   {{MR|0447953}} {{ZBL|0257.46078}}
 +
|-
 +
|valign="top"|{{Ref|FoS}}|| G.B. Folland,  E.M. Stein,  "Hardy spaces on homogeneous groups" , Princeton Univ. Press  (1982) {{MR|0657581}} {{ZBL|0508.42025}}
 +
|-
 +
|valign="top"|{{Ref|St}}|| E.M. Stein,  "Harmonic analysis" , Princeton Univ. Press  (1993) {{MR|1232192}} {{ZBL|0821.42001}}
 +
|-
 +
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Latest revision as of 13:36, 26 December 2013

2020 Mathematics Subject Classification: Primary: 42B30 [MSN][ZBL]

real-variable theory of, real-variable $\mathcal{H}^p$ theory

Definition

The real-variable Hardy spaces $\mathcal{H}^p = \mathcal{H}^p (\mathbb R^n)$ ($0<p<\infty$) are spaces of distributions on $\mathbb R^n$ (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 $\phi$ belongs to the Schwartz class $\mathscr{S} (\mathbb R^n)$ of rapidly decreasing smooth functions, and let $\phi_t (x) = t^{-n} \phi (x/t)$ for $t>0$. If $f\in \mathscr{S}' (\mathbb R^n)$, the space of tempered distributions, define the radial maximal function $m_\phi$ and the non-tangential maximal function $M_\phi$ by \begin{align} & m_\phi f (x) = \sup_{t>0} |f * \phi_t (x)|\\ & M_{\phi} f (x) = \sup_{|y-x|<t} |f*\phi_t (y)|\, . \end{align} where $*$ denotes convolution of functions. C. Fefferman and E.M. Stein [FS] (see also [St]) proved the following theorem, also known as Fefferman-Stein theorem.

Theorem 1 For $f\in \mathscr{S}' (\mathbb R^n)$ and $0<p<\infty$, the following conditions are equivalent:

1) $m_\phi f \in L^p$ for some $\phi$ with $\int \phi \neq 0$;

2) $M_\phi f\in L^p$ for some $\phi$ with $\int \phi \neq 0$;

3) $M_\phi f\in L^p$ for $\phi \in \mathscr{S} (\mathbb R^n)$, and in fact $M_\phi f\in L^p$ uniformly for $\phi$ in a suitable bounded subset of $\mathscr{S} (\mathbb R^n)$.

Definition 2 $\mathcal{H}^p (\mathbb R^n)$ is the space of all $f\in \mathscr{S}' (\mathbb R^n)$ that satisfy these conditions.

Basic properties

For $p>1$, $\mathcal{H}^p$ coincides with $L^p$ and $\mathcal{H}^1$ is a proper subspace of $L^1$. For $p<1$ $\mathcal{H}^p$ contains distributions that are not functions. The connection with the complex-variable Hardy classes is given by the following characterization

Theorem 3 Denote by $\mathbb R^2_+$ the upper half plane $\{(x,y): y>0\}$. A tempered distribution $f$ on $\mathbb R$ is in $\mathcal{H}^p$ if and only if it is the boundary value of a harmonic function $u: \mathbb R^2_+\to \mathbb R$ such that \[ \int_0^\infty \int_{-\infty}^\infty |(u + iv)(x+iy)|^p \, dx\, dy\; < \; \infty\, , \] where $v$ is the harmonic conjugate of $u$.

There is a similar characterization of $\mathcal{H}^p (\mathbb R^n)$ for $n>1$ in terms of systems of harmonic functions satisfying generalized Cauchy-Riemann equations; see [FS].

Atomic decomposition

Another characterization of $\mathcal{H}^p$ for $p\leq 1$ is of great importance.

Definition 4 A measurable function $\alpha$ is called a $p$-atom ($p\leq 1$) if

i) $\alpha$ vanishes outside some ball of radius $r$ and $\sup_x |\alpha (x)| \leq r^{-n/p}$;

ii) The integral \[ \int \alpha (x) P (x)\, dx \] vanishes for all polynomials $P$ of degree $\leq \frac{n-np}{p}$.

The atomic decomposition theorem (see [St]) states that

Theorem 5 $f\in \mathcal{H}^p$ if and only if \[ \sum_j c_j \alpha_j \] for some sequence $\alpha_j$ of $p$-atoms and some sequence of real numbers $\{c_j\}\in \ell^p$ (i.e. such that $\sum |c_j|^p < \infty$).

Functional analytic structure

$\mathcal{H}^1$ is a Banach space (and obviously $\mathcal{H}^p$ is a Banach space for $p>1$, since it coincides with $L^p$). For $p<1$ $\mathcal{H}^p$ is a complete topological vector space. In all these cases the topology is induced by any of the following equivalent quasi-norms \begin{align} &f\mapsto\;\; \int |m_\phi|^p\\ &f\mapsto\;\; \int |M_\phi|^p\\ &f\mapsto\;\; \inf \left\{ \sum |c_j|^p : \text{there is an atomic decomposition } f = \sum c_j \alpha_j\right\}\, . \end{align} A celebrated theorem of Fefferman [FS] (see also [St]), states that $\mathcal{H}^1$ is the dual of ${\rm BMO}$, the space of functions with bounded mean oscillation (cf. also BMO-space). For $p<1$, the dual of $\mathcal{H}^p$ is the homogeneous Lipschitz space of order $\frac{n-np}{p}$; see [FoS].

Importance in harmonic analysis

The space ${\rm BMO}$ and the spaces $\mathcal{H}^p$ with $p\leq1$ provide an extension to $p\in ]0, \infty]$ of the scale of classical $L^p$-spaces with $p\in ]1, \infty[$ that is in many respects more natural and useful than the corresponding $L^p$-extension. 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 $L^p$ only for $p>1$ are also well-behaved on $\mathcal{H}^p$ and ${\rm BMO}$. In addition, many important classes of singular distributions belong to $\mathcal{H}^p$, or are closely related to elements of $\mathcal{H}^p$, for suitable $p<1$. See [FoS], [St].

Extensions

The real-variable $\mathcal{H}^p$ theory can be extended to spaces other than $\mathbb R^n$. A rather complete extension is available in the setting of homogeneous groups, i.e., simply-connected nilpotent Lie groups with a one-parameter family of dilations; see [FoS]. (These groups include, in particular, $\mathbb R^n$ 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 [CW]}.

References

[CW] R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–645 MR0447954 Zbl 0358.30023
[FS] C. Fefferman, E.M. Stein, "$\mathcal{H}^p$ spaces of several variables" Acta Math. , 129 (1972) pp. 137–193 MR0447953 Zbl 0257.46078
[FoS] G.B. Folland, E.M. Stein, "Hardy spaces on homogeneous groups" , Princeton Univ. Press (1982) MR0657581 Zbl 0508.42025
[St] E.M. Stein, "Harmonic analysis" , Princeton Univ. Press (1993) MR1232192 Zbl 0821.42001
How to Cite This Entry:
Hardy spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_spaces&oldid=13621
This article was adapted from an original article by G.B. Folland (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article