Fefferman, Charles Louis; Stein, Elias M. \(H^p\) spaces of several variables. (English) Zbl 0257.46078 Acta Math. 129, 137-193 (1972). 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. Reviewer: Alexander M. Davie Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 53 ReviewsCited in 1458 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Burkholder, D. L. & Gundy, R. F., Distribution function inequalities for the area integral. To appear in Studia Math. · Zbl 0219.31009 [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. · Zbl 0223.30048 [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. · Zbl 0204.13703 [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. · Zbl 0168.37002 [7] Duren, P. L., Romberg, B. W. &Shields, A. L., Linear functionals onH p spaces with 0<p<1.J. Reine Angew. Math., 238 (1969), 32–60. · Zbl 0176.43102 [8] Eymard, P., AlgèbresA p et convoluteurs deL p . Sém. Bourbaki, 367, 1969/70. [9] Fefferman, C. L., Inequalities for strongly singular convolution operators.Acta Math., 124 (1970), 9–36. · Zbl 0188.42601 · doi:10.1007/BF02394567 [10] –, Characterizations of bounded mean oscillation.Bull. Amer. Math. Soc., 77 (1971), 587–588. · Zbl 0229.46051 · doi:10.1090/S0002-9904-1971-12763-5 [11] Fefferman, C. L., to appear. [12] Flett, T. M., On the rate of growth of mean values of holomorphic and harmonic functions.Proc. London Math. Soc. 20 (1970), 748–768. · Zbl 0211.39203 [13] Hardy, G. H. &Littlewood, J. E., Some properties of conjugate functions.J. Reine Angew. Math., 167 (1932), 405–423. · Zbl 0003.20203 [14] John, F. &Nirenberg, L., On functions of bounded mean oscillation.Comm. Pure Appl. Math., 14 (1961), 415–426. · Zbl 0102.04302 · doi:10.1002/cpa.3160140317 [15] Hirschman, I. I., Jr., Multiplier transformations I.Duke Math. J., 26 (1956), 222–242. [16] Hörmander, L.,L p estimates for (pluri-)subharmonic functions.Math. Scand., 20 (1967), 65–78. · Zbl 0156.12201 [17] Segovia, C., On the area function of Lusin.Studia Math., 33 (1969), 312–343. · Zbl 0203.10502 [18] Spanne, S., Sur l’interpolation entre les espacesL k p{\(\Phi\)} .Ann. Scuola Norm. Sup. Pisa, 20 (1966), 625–648. · Zbl 0203.12403 [19] Stein, E. M., A maximal function with applications to Fourier series.Ann. of Math., 68 (1958), 584–603. · Zbl 0085.05602 · doi:10.2307/1970157 [20] –, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz.Trans. Amer. Math. Soc., 88 (1958), 430–466. · Zbl 0105.05104 · doi:10.1090/S0002-9947-1958-0112932-2 [21] –, Singular integrals, harmonic functions and differentiability properties of functions of several variables.Proc. Symp. in Pure Math., 10 (1967), 316–335. [22] Stein, E. M.,Singular integrals and differentiability properties of functions. Princeton, 1970. · Zbl 0207.13501 [23] –,L p boundedness of certain convolution operators.Bull. Amer. Math. Soc., 77 (1971), 404–405. · Zbl 0217.44503 · doi:10.1090/S0002-9904-1971-12716-7 [24] Stein, E. M. &Weiss, G., On the interpolation of analytic families of operators acting onH p spaces.Tôhoku Math. J., 9 (1957), 318–339. · Zbl 0083.34201 · doi:10.2748/tmj/1178244785 [25] –, On the theory of harmonic functions of several variables I. The theory ofH p spaces.Acta Math., 103 (1960), 25–62. · Zbl 0097.28501 · doi:10.1007/BF02546524 [26] Stein, E. M. & Weiss, G.,Introduction to Fourier analysis on Euclidean spaces. Princeton, 1971. · Zbl 0232.42007 [27] –, Generalization of the Cauchy-Riemann equations and representations of the rotation group.Amer. J. Math., 90 (1968), 163–196. · Zbl 0157.18303 · doi:10.2307/2373431 [28] Wainger, S., Special trigonometric series ink dimensions.Mem. Amer. Math. Soc., 59 (1965). · Zbl 0136.36601 [29] Zygmund, A.,Trigonometric series. Cambridge, 1959. · Zbl 0085.05601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.