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H${}^{p}$ spaces of several variables. (English) Zbl 0257.46078

This paper is a major contribution to the study of ${H}^{p}$ spaces, singular integrals, and harmonic analysis an ${ℝ}^{n}$. 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 ${L}^{p}\left(ℝ\right)$. More recently this definition has been generalised to ${ℝ}^{n}$ by introducing generalised conjugate harmonic functions in ${ℝ}_{+}^{n+1}=\left\{\left(x,t\right):x\in {ℝ}^{n}$, $t>0\right\}$. 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 ${L}^{1}\left({ℝ}^{n}\right)$ such that ${R}_{j}f\in {L}^{1}\left({ℝ}^{n}\right)$, $j=1,\cdots ,n$, where ${R}_{j}$ is the $j$-th Riesz transform. (In terms of Fourier transforms, ${\left({R}_{j}f\right)}^{\wedge }\left(y\right)={y}_{j}\stackrel{^}{f}\left(y\right)/|y|$. When $n=1$ the definition says that the Hilbert transform of $f$ is in ${L}^{1}$, or equivalently $\int f\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx=0$ and $f=g+\overline{h}$ where $g$ and $h$ are in the classical “analytic” ${H}^{1}$). The authors prove that the dual of ${H}^{\text{'}}$ is the space of all functionals of the form $\phi \to \int f\phi$ (suitably interpreted if $f\phi \notin {L}^{1}$), where $\phi$ is a function of bounded mean oscillation (BMO), which means that there is a constant $C>0$ (depending an $\phi$) such that ${\int }_{Q}|f-{f}_{Q}|\le C|Q|$ for any cube $Q$ in ${ℝ}^{n}$, 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 ${ℝ}^{n}$) 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. 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 $p$ $\left(0. They define these first as spaces of harmonic functions an ${ℝ}_{+}^{n+1}$, without reference to boundary values. Specifically, a harmonic function ${u}_{0}$ is in ${H}^{p}$ if there exist harmonic functions ${u}_{1},\cdots ,{u}_{n}$ satisfying

$\partial {u}_{j}/\partial {x}_{i}=0,\phantom{\rule{1.em}{0ex}}\sum _{i=0}^{n}\partial {u}_{i}/\partial {x}_{i}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\underset{t>0}{sup}{\int }_{{ℝ}^{n}}{|u\left(x,t\right)|}^{p}\phantom{\rule{0.166667em}{0ex}}dx<\infty ,$

where ${|u|}^{2}={\sum }_{i=0}^{n}{|{u}_{i}|}^{2}$. This definition is appropriate if $p>\left(n-1\right)/n$. In general a more elaborate version (here omitted) is needed (the point is that ${|u|}^{p}$ is subharmonic only if $p\ge \left(n-1\right)/n$). The main result is as follows: let $u$ be harmonic in ${ℝ}_{+}^{n+1}$ and define ${u}^{*}\left(x\right)={sup}_{t}|u\left(x,t\right)|$. 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 $u\left(x,t\right)\to f\left(x\right)$ as $t\to 0$, in the distribution sense, where $f$ is a tempered distribution on ${ℝ}^{n}$. 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 $\phi$ on ${ℝ}^{n}$, decreasing rapidly at $\infty$, with $\int \phi =1$. Put ${\phi }_{t}\left(x\right)={t}^{-n}\phi \left(x/t\right)$, and for any tempered disribution $f$ write ${f}^{*}\left(x\right)={sup}_{t>0}|{\phi }_{t}*f\left(x\right)|$. Then (for $0) $f\in {H}^{p}$ 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 ${ℝ}^{n}$. The paper concludes with a proof that certain singular integral operators map ${H}^{p}$ to itself.

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, ${H}^{p}$-spaces 46E30 Spaces of measurable functions 42B25 Maximal functions, Littlewood-Paley theory 30D55 H (sup p)-classes (MSC2000) 42A50 Conjugate functions, conjugate series, singular integrals, one variable 44A35 Convolution (integral transforms) 46F10 Operations with distributions (generalized functions)
##### References:
 [1] Burkholder, D. L. & Gundy, R. F., Distribution function inequalities for the area integral. To appear in Studia Math. [2] Burkholder, D. L., Gundy, R. F. &Silverstein, M. L., A maximal function characterization of the classH p .Trans. Amer. Math. Soc., 157 (1971), 137–153. [3] Calderón, A. P., Commutators of singular integral operators.Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092–1099. · Zbl 0151.16901 · doi:10.1073/pnas.53.5.1092 [4] –, Intermediate spaces and interpolation, the complex method.Studia Math., 24 (1964), 113–190. [5] Calderón, A. P. &Zygmund, A., On the existence of certain singular integrals.Acta Math., 88 (1952), 85–139. · Zbl 0047.10201 · doi:10.1007/BF02392130 [6] –, On higher gradients of harmonic functions.Studia Math., 24 (1964), 211–226. [7] Duren, P. L., Romberg, B. W. &Shields, A. L., Linear functionals onH p spaces with 0