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Zbl 0257.46078
Fefferman, Charles Louis; Stein, Elias M.
H$^p$ spaces of several variables. (English)
[J] Acta Math. 129, 137-193 (1972). ISSN 0001-5962; ISSN 1871-2509/e

This paper is a major contribution to the study of $H^p$ spaces, singular integrals, and harmonic analysis an $\Bbb R^n$. Classically the theory of $H^p$ spaces arose from analytic function theory. $H^p$ was defined as the space of analytic functions in the upper half plane with boundary values in $L^p(\Bbb R)$. More recently this definition has been generalised to $\Bbb R^n$ by introducing generalised conjugate harmonic functions in $\Bbb R_+^{n+1}=\{(x,t): x\in\Bbb R^n$, $t>0\}$. The authors present several intrinsic descriptions of $H^p$, of a real variable nature, not involving conjugate functions. These results greatly clarify the meaning of $H^p$, as well as throwing new light on the behaviour of convolution operators an $L^p$. The following is a summary of some of the main results. The first main result is the description of the dual of $H^1$. $H^1$ is the Banach space of all functions $f$ in $L^1(\Bbb R^n)$ such that $R_jf\in L^1(\Bbb R^n)$, $j=1,\dots,n$, where $R_j$ is the $j$-th Riesz transform. (In terms of Fourier transforms, $(R_jf)^\wedge(y)= y_j\hat f(y)/|y|$. When $n=1$ the definition says that the Hilbert transform of $f$ is in $L^1$, or equivalently $\int f(x)\,dx=0$ and $f=g+\bar h$ where $g$ and $h$ are in the classical ``analytic" $H^1$). The authors prove that the dual of $H'$ is the space of all functionals of the form $\varphi\to\int f\varphi$ (suitably interpreted if $f\varphi\not\in L^1$), where $\varphi$ is a function of bounded mean oscillation (BMO), which means that there is a constant $C>0$ (depending an $\varphi$) such that $\int_Q |f-f_Q|\le C|Q|$ for any cube $Q$ in $\Bbb R^n$, where $f_Q= |Q|^{-1}\int_Q f$. The proof of this remarkable and deep result depends an an inequality of Carleson and some clever manipulations of Littlewood Paley functions. The hard part is to show that a BMO function defines a bounded functional on $H^1$. The essential difficulties are already present in the case $n=1$. This duality leads to a new approach to convolution operators which brings out the usefulness of BMO as a substitute for $L^\infty$. If $T$ is a convolution operator (i.e. $Tf= K*f$ for some distribution $K$ an $\Bbb R^n$) mapping $L^\infty$ into BMO, then the authors show, using the duality and a description of $L^p$ related to BMO, that $T$ maps $H^1$ into $H^1$, BMO into BMO, and $L^p$ into $L^p$ for $1<p<\infty$. The condition that $T$ map $L^\infty$ into BMO is relatively easy to verify for singular integral operators of Calderón-Zygmund type. A more refined version of this result, involving Calderón's complex method of interpolation, enables the authors to prove new results on $L^p$ multipliers. The authors then turn to $H^p$ spaces for general $p$ $(0<p<\infty)$. They define these first as spaces of harmonic functions an $\Bbb R_+^{n+1}$, without reference to boundary values. Specifically, a harmonic function $u_0$ is in $H^p$ if there exist harmonic functions $u_1,\dots,u_n$ satisfying $$\partial u_j/\partial x_i=0, \quad \sum_{i=0}^n \partial u_i/\partial x_i \quad\text{and}\quad \sup_{t>0} \int_{\Bbb R^n}|u(x,t)|^p\,dx<\infty,$$ where $|u|^2= \sum_{i=0}^n |u_i|^2$. This definition is appropriate if $p>(n-1)/n$. In general a more elaborate version (here omitted) is needed (the point is that $|u|^p$ is subharmonic only if $p\ge (n-1)/n$). The main result is as follows: let $u$ be harmonic in $\Bbb R_+^{n+1}$ and define $u^*(x)= \sup_t |u(x,t)|$. Then $u\in H^p$ if and only if $u^*\in L^p$. (For $n=1$ this was proved by Burkholder, Gundy and Silverstein in 1971). Finally the authors consider boundary values. If $u\in H^p$ then $u(x,t)\to f(x)$ as $t\to0$, in the distribution sense, where $f$ is a tempered distribution on $\Bbb R^n$. Denote the set of such $f$ also by $H^p$. Then $H^p= L^p$ for $p>1$ and for $p=1$ this definition is consistent with the earlier one. The last result above characterises $H^p$ in terms of Poisson integrals (as $u$ is the Poisson integral of $f$). The authors show that the Poisson kernel can be replaced by any smooth approximate identity -- more precisely, fix a smooth function $\varphi$ on $\Bbb R^n$, decreasing rapidly at $\infty$, with $\int\varphi=1$. Put $\varphi_t(x)= t^{-n} \varphi(x/t)$, and for any tempered disribution $f$ write $f^*(x)= \sup_{t>0} |\varphi_t*f(x)|$. Then (for $0<p<\infty$) $f\in H^p$ if and only if $f^*\in L^p$. (``Non-tangential" versions of this and the preceding result are also given). This result implies for example that one could define $H^p$ in terms of the wave equation rather than Laplace's and get the same space of functions on $\Bbb R^n$. The paper concludes with a proof that certain singular integral operators map $H^p$ to itself.

Display scanned Zentralblatt-MATH page with this review.
[Alexander M. Davie]

MSC 2000:: *46J15 Banach algebras of differentiable functions
46E30 Spaces of measurable functions
42B25 Maximal functions
30D55 H (sup p)-classes
42A50 Singular integrals, one variable
44A35 Convolution
46F10 Operations with distributions (generalized functions)

Cited in: Zbl 1240.42031 Zbl 1220.42013 Zbl 1191.35078 Zbl 1189.46030 Zbl 1128.42011 Zbl 1102.42012 Zbl 1051.43004 Zbl 1036.42020 Zbl 1032.42023 Zbl 1013.42011 Zbl 0979.31002 Zbl 1033.42009 Zbl 0952.47029 Zbl 0883.32003 Zbl 0881.43002 Zbl 0812.43001 Zbl 0716.42017 Zbl 0701.60063 Zbl 0677.42021 Zbl 0677.30030 Zbl 0679.42009 Zbl 0639.42019 Zbl 0601.31005 Zbl 0562.42019 Zbl 0544.30036 Zbl 0503.46036 Zbl 0536.42022 Zbl 0522.42016 Zbl 0508.46037 Zbl 0495.46035 Zbl 0464.47029 Zbl 0495.47024 Zbl 0503.46020 Zbl 0473.42013 Zbl 0437.35042 Zbl 0433.42019 Zbl 0456.46063 Zbl 0446.58015 Zbl 0432.42013 Zbl 0431.46019 Zbl 0429.35076 Zbl 0421.46026 Zbl 0418.32006 Zbl 0403.43004 Zbl 0459.30034

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