This paper is a major contribution to the study of ${H}^{p}$ spaces, singular integrals, and harmonic analysis an ${\mathbb{R}}^{n}$. 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 ${L}^{p}\left(\mathbb{R}\right)$. More recently this definition has been generalised to ${\mathbb{R}}^{n}$ by introducing generalised conjugate harmonic functions in ${\mathbb{R}}_{+}^{n+1}=\{(x,t):x\in {\mathbb{R}}^{n}$, $t>0\}$. 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 ${L}^{1}\left({\mathbb{R}}^{n}\right)$ such that ${R}_{j}f\in {L}^{1}\left({\mathbb{R}}^{n}\right)$, $j=1,\cdots ,n$, where ${R}_{j}$ is the $j$-th Riesz transform. (In terms of Fourier transforms, ${\left({R}_{j}f\right)}^{\wedge}\left(y\right)={y}_{j}\widehat{f}\left(y\right)/\left|y\right|$. 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 ${\mathbb{R}}^{n}$, where ${f}_{Q}={\left|Q\right|}^{-1}{\int}_{Q}f$. 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 ${\mathbb{R}}^{n}$) 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 ${\mathbb{R}}_{+}^{n+1}$, 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

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

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