Hardy spaces

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