##
**\(H^p\) 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 \(\mathbb R^n\). 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 \(L^p(\mathbb R)\). 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 \(L^1(\mathbb R^n)\) such that \(R_jf\in L^1(\mathbb R^n)\), \(j=1,\dots,n\), 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 \(\int_Q |f-f_Q|\leq C|Q|\) for any cube \(Q\) in \(\mathbb R^n\), 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 \(\mathbb R^n\)) 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 \(u_1,\dots,u_n\) satisfying

\[ \partial u_j/\partial x_i=0, \quad \sum_{i=0}^n \partial u_i/\partial x_i \quad\text{and}\quad \sup_{t>0} \int_{\mathbb R^n}|u(x,t)|^p\,dx<\infty, \]

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 \(p\geq (n-1)/n\)). The main result is as follows: let \(u\) be harmonic in \(\mathbb R_+^{n+1}\) 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 \(\mathbb R^n\). 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 \(\varphi\) on \(\mathbb R^n\), decreasing rapidly at \(\infty\), with \(\int\varphi=1\). Put \(\varphi_t(x)= t^{-n} \varphi(x/t)\), and for any tempered distribution \(f\) 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 \(\mathbb R^n\).

The paper concludes with a proof that certain singular integral operators map \(H^p\) to itself.

The first main result is the description of the dual of \(H^1\). \(H^1\) is the Banach space of all functions \(f\) in \(L^1(\mathbb R^n)\) such that \(R_jf\in L^1(\mathbb R^n)\), \(j=1,\dots,n\), 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 \(\int_Q |f-f_Q|\leq C|Q|\) for any cube \(Q\) in \(\mathbb R^n\), 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 \(\mathbb R^n\)) 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 \(u_1,\dots,u_n\) satisfying

\[ \partial u_j/\partial x_i=0, \quad \sum_{i=0}^n \partial u_i/\partial x_i \quad\text{and}\quad \sup_{t>0} \int_{\mathbb R^n}|u(x,t)|^p\,dx<\infty, \]

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 \(p\geq (n-1)/n\)). The main result is as follows: let \(u\) be harmonic in \(\mathbb R_+^{n+1}\) 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 \(\mathbb R^n\). 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 \(\varphi\) on \(\mathbb R^n\), decreasing rapidly at \(\infty\), with \(\int\varphi=1\). Put \(\varphi_t(x)= t^{-n} \varphi(x/t)\), and for any tempered distribution \(f\) 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 \(\mathbb R^n\).

The paper concludes with a proof that certain singular integral operators map \(H^p\) to itself.

Reviewer: Alexander M. Davie

### MSC:

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

42B25 | Maximal functions, Littlewood-Paley theory |

30H10 | Hardy spaces |

42A50 | Conjugate functions, conjugate series, singular integrals |

44A35 | Convolution as an integral transform |

46F10 | Operations with distributions and generalized functions |

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\textit{C. L. Fefferman} and \textit{E. M. Stein}, Acta Math. 129, 137--193 (1972; Zbl 0257.46078)

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### References:

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