View source for Hardy spaces

{{MSC|42B30}}
[[Category:Harmonic analysis]]
{{TEX|done}}

''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 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.

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.

'''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|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]] $u$.

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}}.

====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 {{Cite|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|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====
{|
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|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}}
|-
|}