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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 CalderÃ³n-Zygmund type. A more refined version of this result, involving CalderÃ³n'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)

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