Hausdorff operators on Hardy spaces

E. Liflyand

Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 101-141.

Résumé

Hausdorff operators (Hausdorff summability methods) appeared long ago aiming to solve certain classical problems in analysis. Modern theory of Hausdorff operators started with the work of Siskakis in complex analysis setting and with the work of Liflyand–Móricz in the Fourier transform setting. In this paper a survey of the main results on Hausdorff operators in various settings is given. Many open problems in the subject are formulated.

Bibliographie

[1] C. R. Adams, “Hausdorff transformations for double sequences”, Bull. Amer. Math. Soc., 39 (1933), 303–312 | DOI | MR | Zbl

[2] L. A. Aizenberg, “On Temlyakov integrals and boundary properties of analytic functions of two complex variables”, Doklady Akad. Nauk SSSR, 120 (1958), 935– 938 (in Russian) | MR

[3] L. A. Aizenberg, “On Temlyakov integrals and boundary properties of analytic functions of many complex variables”, Uchen. Zapiski MOPI, 77, 1959, 13–35 (in Russian)

[4] L. Aizenberg, E. Liflyand, “Hardy spaces in Reinhardt domains, and Hausdorff operators”, Illinois J. Math., 53 (2009), 1033–1049 | MR | Zbl

[5] L. Aizenberg, E. Liflyand, A. Vidras, Hausdorff operators in Hardy spaces on Cartan type domains in $\mathbb C^n$, submitted

[6] T. Alberico, R. Corporente, C. Sbordone, “Explicit bounds for composition operators preserving $BMO(\mathbb R)$”, Georgian Math. J., 14 (2007), 21–32 | MR | Zbl

[7] K. F. Andersen, “Boundedness of Hausdorff operators on $L^p(\mathbb R^n)$, $H^1(\mathbb R^n)$ and $BMO(\mathbb R^n)$”, Acta Sci. Math. (Szeged), 69 (2003), 409–418 | MR | Zbl

[8] A. G. Arvanitidis, A. G. Siskakis, Cesàro operators on the Hardy spaces of the half-plane, Preprint, 2010, arXiv: 1006.1520 | MR

[9] S. Baron, Introduction in the theory of the summability of series, Valgus, Tallinn, 1977 (in Russian) | MR | Zbl

[10] G. Bourdaud, M. Lanza de Cristoforis, W. Sickel, “Functional calculus on BMO and related spaces”, J. Func. Anal., 189 (2002), 515–538 | DOI | MR | Zbl

[11] G. Brown, F. Móricz, “The Hausdorff and the quasi Hausdorff operators on the spaces $L^p$, $1\le p\infty$”, Math. Inequal. Appl., 3 (2000), 105–115 | MR | Zbl

[12] G. Brown, F. Móricz, “Multivariate Hausdorff operators on the spaces $L^p(\mathbb R^n)$”, J. Math. Anal. Appl., 271 (2002), 443–454 | DOI | MR | Zbl

[13] G. Gao, H. Jia, “Boundedness of commutators of high-dimensional Hausdorff operators”, J. Funct. Spaces Appl., 2012 (2012), Article ID 541205, 12 pp. | DOI | MR

[14] D.-C. Chang, R. Gilbert, S. Stević, “Hausdorff operator on the unit polydisk in $\mathbb C^n$”, Complex Variab. Ellipt. Eq., 51 (2006), 329–345 | DOI | MR | Zbl

[15] J. C. Chen, D. S. Fan, C. J. Zhang, “Multilinear Hausdorff operators and their best constants”, Acta Math. Sinica, 28 (2012), 1521–1530 | DOI | MR | Zbl

[16] J. C. Chen, D. S. Fan, C. J. Zhang, “Boundedness of Hausdorff operators on some product Hardy type spaces”, Appl. Math. J. Chinese Univ., 27:1 (2012), 114–126 | DOI | MR | Zbl

[17] J. C. Chen, D. S. Fan, J. Li, “Hausdorff operators on function spaces”, Chin. Ann. Math. Ser. B, 33:4 (2012), 537–556 | DOI | MR | Zbl

[18] R. R. Coifman, G. Weiss, “Extensions of Hardy spaces and their use in analysis”, Bull. Amer. Math. Soc., 83 (1977), 569–645 | DOI | MR | Zbl

[19] G. Dafni, “Local VMO and weak convergence in $h^1$”, Canad. Math. Bull., 45 (2002), 46–59 | DOI | MR | Zbl

[20] P. Duren, Theory of $H^p$ spaces, Academic Press, New York–London, 1970 | MR | Zbl

[21] C. Fefferman, E. M. Stein, “$H^p$ spaces of several variables”, Acta Math., 129 (1972), 137–193 | DOI | MR | Zbl

[22] R. Fefferman, “Some recent developments in Fourier analysis and $H^p$ theory and product domains. II”, Function spaces and applications, Proc. US-Swed. Semin., Lect. Notes Math., 1302, Lund/Swed., 1988, 44–51 | DOI | MR | Zbl

[23] P. Galanopoulos, A. G. Siskakis, “Hausdorff matrices and composition operators”, Ill. J. Math., 45 (2001), 757–773 | MR | Zbl

[24] P. Galanopoulos, M. Papadimitrakis, “Hausdorff and quasi-Hausdorff matrices on spaces of analytic functions”, Canad. J. Math., 58 (2006), 548–579 | DOI | MR | Zbl

[25] H. L. Garabedian, “Hausdorff matrices”, Amer. Math. Monthly, 46 (1939), 390–410 | DOI | MR | Zbl

[26] I. M. Gel'fand, Lectures on Linear Algebra, Interscience Publishers, 1978

[27] C. Georgakis, “The Hausdorff mean of a Fourier–Stieltjes transform”, Proc. Am. Math. Soc., 116 (1992), 465–471 | DOI | MR | Zbl

[28] D. V. Giang, F. Móricz, “The Cesàro operator is bounded on the Hardy space $H^1$”, Acta Sci. Math., 61 (1995), 535–544 | MR | Zbl

[29] D. V. Giang, F. Móricz, “Hardy spaces on the plane and double Fourier transform”, J. Fourier Anal. Appl., 2 (1996), 487–505 | DOI | MR | Zbl

[30] D. V. Giang, F. Móricz, “The two dimensional Cesàro operator is bounded on the multi-parameter Hardy space $\mathcal H^1(\mathbb R\times\mathbb R)$”, Acta Sci. Math., 63 (1997), 279–288 | MR | Zbl

[31] R. R. Goldberg, “Averages of Fourier coefficients”, Pacific J. Math., 9 (1959), 695–699 | DOI | Zbl

[32] R. R. Goldberg, “Convolutions and general transforms on $L^p$”, Duke Math. J., 27 (1960), 251–259 | DOI | MR | Zbl

[33] D. Goldberg, “A local version of real Hardy spaces”, Duke Math. J., 46 (1979), 27–42 | DOI | MR | Zbl

[34] M. Goldberg, “Equivalence constants for $l_p$ norms of matrices”, Linear and Multilinear Algebra, 21 (1987), 173–179 | DOI | MR | Zbl

[35] Russian Acad. Sci. Sb. Math., 188:7 (1998), 1041–1054 | DOI | DOI | MR | Zbl

[36] B. I. Golubov, “On boundedness of the Hardy and Bellman operators in the spaces $H$ and $BMO$”, Numer. Funct. Anal. Optimiz., 21 (2000), 145–158 | DOI | MR | Zbl

[37] G. M. Goluzin, Geometric theory of Functions of a Complex Variable, AMS, Providence, 1969 | Zbl

[38] Y. Gotoh, “On composition operators which preserve $BMO$”, Pacific J. Math., 201 (2001), 289–307 | DOI | MR | Zbl

[39] G. H. Hardy, J. E. Littlewood, “Some properties of fractional integrals. II”, Math. Z., 34 (1932), 403–439 | DOI

[40] G. H. Hardy, “An inequality for Hausdorff means”, J. London Math. Soc., 18 (1943), 46–50 | DOI | MR | Zbl

[41] G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949 | MR | Zbl

[42] F. Hausdorff, “Summationsmethoden und Momentfolgen. I”, Math. Z., 9 (1921), 74–109 | DOI | MR | Zbl

[43] W. K. Hayman, P. B. Kennedy, Subharmonic functions, v. I, Academic Press, London, 1976 | Zbl

[44] H. Helson, Harmonic analysis, Addison-Wesley, 1983 | MR | Zbl

[45] E. Hille, J. D. Tamarkin, “On the summability of Fourier series, III”, Math. Ann., 108 (1933), 525–577 | DOI | Zbl

[46] R. A. Horn, Ch. R. Johnson, Matrix analysis, Cambridge Univ. Press, Cambridge, 1985 | Zbl

[47] W. A. Hurwitz, L. L. Silverman, “The consistency and equivalence of certain definitions of summability”, Trans. Amer. Math. Soc., 18 (1917), 1–20 | DOI | MR | Zbl

[48] A. Jakimovski, B. E. Rhoades, J. Tzimbalario, “Hausdorff matrices as bounded operators over $\ell^p$”, Math. Z., 138 (1974), 173–181 | DOI | MR | Zbl

[49] P. W. Jones, “Homeomorphisms of the line which preserve $BMO$”, Ark. Mat., 21 (1983), 229–231 | DOI | MR | Zbl

[50] Y. Kanjin, “The Hausdorff operators on the real Hardy spaces $H^p(\mathbb R)$”, Studia Math., 148 (2001), 37–45 | DOI | MR | Zbl

[51] G. M. Khenkin, E. M. Chirka, “Boundary properties of holomorphic functions of several complex variables”, Modern Prob. Math., 4 (1975), 13–142 (in Russian) | MR

[52] H. Kober, “A note on Hilbert's operator”, Bull. Amer. Math. Soc., 48:1 (1942), 421–426 | DOI | MR

[53] P. Koosis, Introduction to $H_p$ Spaces, Cambridge Univ. Press, Cambridge, 1980 | MR | Zbl

[54] A. M. Kytmanov, The Bochner–Martinelli integral and its applications, Birkhäuser, Basel–Boston–Berlin, 1995 | MR | Zbl

[55] G. Leibowitz, “Discrete Hausdorff transformations”, Proc. Amer. Math. Soc., 38 (1973), 541–544 | DOI | MR | Zbl

[56] A. Lerner, E. Liflyand, “Multidimensional Hausdorff operator on the real Hardy space”, J. Austr. Math. Soc., 83 (2007), 65–72 | DOI | MR

[57] E. Liflyand, “Open problems on Hausdorff operators”, Complex Analysis and Potential Theory, Proc. Conf. Satellite to ICM 2006 (Gebze, Turkey, 8–14 Sept. 2006), eds. T. Aliyev Azeroglu, P. M. Tamrazov, World Sci., 2007, 280–285 | MR | Zbl

[58] E. Liflyand, “Boundedness of multidimensional Hausdorff operators on $H^1(\mathbb R^n)$”, Acta Sci. Math. (Szeged), 74 (2008), 845–851 | MR | Zbl

[59] E. Liflyand, A. Miyachi, “Boundedness of the Hausdorff operators in $H^p$ spaces, $0

1$”, Studia Math., 194:3 (2009), 279–292 | DOI | MR | Zbl

[60] E. Liflyand, F. Móricz, “The Hausdorff operator is bounded on the real Hardy space $H^1(\mathbb R)$”, Proc. Am. Math. Soc., 128 (2000), 1391–1396 | DOI | MR | Zbl

[61] E. Liflyand, F. Móricz, “The multi-parameter Hausdorff operator is bounded on the product Hardy space $H^{11}(\mathbb R\times\mathbb R)$”, Analysis, 21 (2001), 107–118 | DOI | Zbl

[62] E. Liflyand, F. Móricz, “Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces”, Acta Math. Hungar., 97:1–2 (2002), 133–143 | DOI | MR | Zbl

[63] L. Lorch, D. J. Newman, “The Lebesgue constants for regular Hausdorff methods”, Canad. J. Math., 13 (1961), 283–298 | DOI | MR | Zbl

[64] A. Miyachi, “Weak factorization of distributions in $H^p$ spaces”, Pacific J. Math., 115 (1984), 165–175 | DOI | MR | Zbl

[65] A. Miyachi, “Boundedness of the Cesàro Operator in Hardy spaces”, J. Fourier Anal. Appl., 10 (2004), 83–92 | DOI | MR | Zbl

[66] F. Móricz, “Multivariate Hausdorff operators on the spaces $H^1(\mathbb R^n)$ and $BMO(\mathbb R^n)$”, Analysis Math., 31 (2005), 31–41 | DOI | MR | Zbl

[67] R. E. Powell, S. M. Shah, Summability theory and applications, Van Nostrand Reinhold Co., London, 1972 | Zbl

[68] I. I. Privalov, Randeigenschaften analytischer Functionen, Deutscher Verein der Vissenschaftlern, Berlin, 1956 | Zbl

[69] B. E. Rhoades, “Generalized Hausdorff matrices bounded on $\ell^p$ and $c$”, Acta Sci. Math. (Szeged), 43 (1981), 333–345 | MR | Zbl

[70] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York–Amsterdam, 1969 | MR | Zbl

[71] O. Rudolf, Hausdorff-Operatoren auf BK-Räumen und Halbgruppen linearer Operatoren, Mitt. Math. Sem. Giessen, 241, 2000 | MR | Zbl

[72] S. Saks, Theory of the integral, 1939

[73] A. G. Siskakis, “Composition operators and the Cesàro operator on $H^p$”, J. London Math. Soc. (2), 36 (1987), 153–164 | DOI | MR | Zbl

[74] A. G. Siskakis, “The Cesàro operator is bounded on $H^1$”, Proc. Am. Math. Soc., 110 (1990), 461–462 | MR | Zbl

[75] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970 | MR

[76] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton Univ. Press, Princeton, 1972 | MR | Zbl

[77] E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, N.J., 1993 | MR | Zbl

[78] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971 | MR | Zbl

[79] E. L. Stout, “$H^p$ functions on stricly pseudoconvex domains”, Amer. J. Math., 98 (1976), 821–852 | DOI | MR | Zbl

[80] F. Ustina, “Lebesgue constants for double Hausdorff means”, Bull. Austral. Math. Soc., 31 (1986), 17–22 | MR

[81] F. Weisz, “Singular integrals on product domains”, Arch. Math., 77 (2001), 328–336 | DOI | MR | Zbl

[82] F. Weisz, “The boundedness of the Hausdorff operator on multi-dimensional Hardy spaces”, Analysis, 24 (2004), 183–195 | DOI | MR | Zbl

[83] D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, N.J., 1946 | MR

[84] A. Zygmund, Trigonometric series, v. II, Cambridge Univ. Press, Cambridge, 1959 | MR | Zbl