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Some new function spaces and their applications to harmonic analysis. (English) Zbl 0569.42016
This paper is devoted to the definition of a new family of function spaces and to the investigation of their fundamental properties. These spaces, called “tent spaces” are of functions on \(X\times {\mathbb{R}}_+\) where X is a Euclidean space and the spaces are so defined that the functions have “good” boundary values on the boundary \(X\) of this space. Such boundary values play a central role in harmonic analysis and the theory developed in this paper systemises a great deal of the earlier work. It is so rich in material that it is hardly possible in a short review to summarize the results in detail. To show the range of these methods the authors give a number of applications at the close of this paper, to maximal functions, to the Hilbert transform and to the theory of Hardy spaces.
Reviewer: S. J. Patterson

MSC:
42B25 Maximal functions, Littlewood-Paley theory
31B25 Boundary behavior of harmonic functions in higher dimensions
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[1] Bergh, J; Löfström, J, Interpolation spaces, (1976), Springer-Verlag New York/Berlin · Zbl 0344.46071
[2] Calderón, A.P, Intermediate spaces and interpolation, the complex method, Studia math., 24, 113-190, (1964) · Zbl 0204.13703
[3] Calderón, A.P; Torchinsky, A; Calderón, A.P; Torchinsky, A, Parabolic maximal functions associated with a dis-tribution, II, Adv. in math., Adv. in math., 24, 101-171, (1977) · Zbl 0355.46021
[4] Carleson, L, Interpolation of bounded analytic functions and the corona problem, Ann. of math. (2), 76, 547-599, (1962) · Zbl 0112.29702
[5] Chang, S.Y.A; Fefferman, R, A continuous version of duality of H1 and BMO, Ann. of math. (2), 112, 179-201, (1980)
[6] Coifman, R.R; Deng, D.G; Meyer, Y, Domaine de la racine carrée de certains opérateurs differentiels acrétifs, Ann. inst. Fourier (Grenoble), 33, 123-134, (1983) · Zbl 0497.35088
[7] Coifman, R.R; Meyer, Y; Stein, E.M, Un nouvel espace adapté a l’étude des opérateurs définis par des intégrales singulières, (), 1-15
[8] Coifman, R.R; Weiss, G, Extensions of Hardy spaces and their use in analysis, Bull. amer. math. soc., 83, 569-645, (1977) · Zbl 0358.30023
[9] \scD. G. Deng, unpublished manuscript.
[10] Johnson, R, Applications of Carleson measures to partial differential equations and Fourier multipliers, (), 16-72
[11] Fefferman, C; Stein, E.M, Hp spaces of several variables, Acta math., 129, 137-193, (1972) · Zbl 0257.46078
[12] Fefferman, C; Stein, E.M, Some maximal inequalities, Amer. J. math., 93, 107-115, (1971) · Zbl 0222.26019
[13] Jones, P, Interpolation between Hardy spaces, (), 437-451
[14] Nagel, A; Stein, E.M, On certain maximal functions and approach regions, Adv. in math., 54, 83-106, (1984) · Zbl 0546.42017
[15] Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton, N. J · Zbl 0207.13501
[16] Stein, E.M; Weiss, G, Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton Univ. Press Princeton, N. J · Zbl 0232.42007
[17] \scJ. M. Wilson, The atomic decomposition for Hardy spaces, preprint. · Zbl 0563.42012
[18] Wolff, T, A note on interpolation spaces, (), 199-204
[19] Yosida, K, Functional anlysis, (1980), Springer-Verlag New York/Berlin
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