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Hardy spaces associated with different homogeneities and boundedness of composition operators. (English) Zbl 1291.42018

The space under consideration is \(\mathbb R^n=\mathbb R^{n-1}\times\mathbb R\) with \(x=(x',x_n)\) where \(x'\in\mathbb R^{n-1}\) and \(x_n\in\mathbb R\). Two kinds of homogeneities are considered: one is the usual isotoropic one, and the other is one of nonisotropic type \((x',x_n)\to (\delta x', \delta^2 x_n)\). The metrics are the usual euclidean metric and nonisotropic metric \(|x|_h:=(|x'|^2+|x_n|)^{1/2}\). The authors consider two kinds of Calderón-Zygmund operators \(T_1\) and \(T_2\) associated with the usual isotropic homogeneity and the nonisotropic homogeneity, respectively. It is known that \(T_1\) and \(T_2\) are bounded on \(L^p(\mathbb R^n)\) \((1<p<\infty)\), and that they are bounded on the usual Hardy space \(H^p(\mathbb R^n)\) and the nonisotropic Hardy space \(H_h^p(\mathbb R^n)\) \((0<p\leq1)\), respectively. However, the composition operator \(T_1\circ T_2\) is not bounded on \(H^p(\mathbb R^n)\) nor \(H_h^p(\mathbb R^n)\). Using two good functions \(\psi^{(1)}, \psi^{(2)}\) characterizing \(H^p(\mathbb R^n)\) and \(H_h^p(\mathbb R^n)\), they construct a discrete Littlewood-Paley-Stein square function. Using this, they define the Hardy space \(H_{\text{comp}}^p(\mathbb R^n)\) associated with two different homogeneities. They show that the composition operator \(T_1\circ T_2\) is bounded on \(H_{\text{comp}}^p(\mathbb R^n)\). Also, they show, without using Journé’s coverring lemma, that for \(0<p\leq1\) it holds \(\|f\|_{L^p(\mathbb R^n))}\leq C\|f\|_{H_{\text{comp}}^p(\mathbb R^n)}\) for \(f\in L^2(\mathbb R^n)\cap H_{\text{comp}}^p(\mathbb R^n)\) and hence they prove that \(T_1\circ T_2\) is bounded from \(H_{\text{comp}}^p(\mathbb R^n)\) to \(L^p(\mathbb R^n)\). They give a Calderón-Zygmund decomposition for \(H_{\text{comp}}^p(\mathbb R^n)\), and an interpolation theorem on \(H_{\text{comp}}^p(\mathbb R^n)\). They comment that these results hold also in the cases of more general nonisotropic homogeneities.

MSC:

42B30 \(H^p\)-spaces
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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[1] Chang, S. Y. A. and Fefferman, R.: Some recent developments in Fourier anal- ysis and Hp-theory on product domains. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1-43. · Zbl 0557.42007 · doi:10.1090/S0273-0979-1985-15291-7
[2] Chang, S. Y. A. and Fefferman, R.: A continuous version of duality of H1 with BMO on the bidisc. Ann. of Math. (2) 112 (1980), no. 1, 179-201. · Zbl 0451.42014 · doi:10.2307/1971324
[3] Chang, S. Y. A. and Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Amer. J. Math. 104 (1982), no. 3, 455-468. · Zbl 0513.42019 · doi:10.2307/2374150
[4] Coifman, R. R. and Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), no. 44, 569-645. · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[5] Fefferman, C. and Stein, E. M.: Hp spaces of several variables. Acta. Math. 129 (1972), no. 3-4, 137-193. · Zbl 0257.46078 · doi:10.1007/BF02392215
[6] Fefferman, R.: The atomic decomposition of H1 in product spaces. Adv. Math. 55 (1985), 90-100. · Zbl 0606.42016 · doi:10.1016/0001-8708(85)90006-4
[7] Fefferman, R.: Harmonic analysis on product spaces. Ann. of Math. (2) 126 (1987), no. 1, 109-130. · Zbl 0644.42017 · doi:10.2307/1971346
[8] Ferguson, S. and Lacey, M.: A characterization of product BMO by commuta- tors. Acta Math. 189 (2002), no. 2, 143-160. · Zbl 1039.47022 · doi:10.1007/BF02392840
[9] Folland, G. B. and Stein, E. M.: Hardy spaces on homogeneous groups. Mathe- matical Notes 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. · Zbl 0508.42025
[10] Frazier, M. and Jawerth, B.: A discrete transform and decomposition of distri- bution. J. Func.Anal. 93 (1990), no. 1, 34-170. · Zbl 0716.46031 · doi:10.1016/0022-1236(90)90137-A
[11] Frazier, M., Jawerth, B. and Weiss, G.: Littlewood-Paley theory and the study of function spaces. CBMS Regional conference series in Mathematics 79, American Mathematical Society, Providence, RI, 1991. · Zbl 0757.42006
[12] García Cuerva, J. and Rubio de Francia, J. L.: Weighted norm inequalities and related topics. North-Holland Math. Studies 116, North-Holland, Amsterdam, 1985. · Zbl 0578.46046
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