Hausdorff operators on real Hardy spaces \(H^1\) over homogeneous spaces with local doubling property. (English) Zbl 1499.43010
Summary: We give conditions for boundedness of Hausdorff operators on real Hardy spaces \(H^1\) over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.
MSC:
| 43A85 | Harmonic analysis on homogeneous spaces |
| 47G10 | Integral operators |
| 22E30 | Analysis on real and complex Lie groups |
Keywords:
Hausdorff operator; locally compact group; homogeneous space; atomic Hardy space; hyperbolic plane; two-sphereReferences:
| [1] | Bourbaki, N., Éléments de mathématique. Topologie générale, Chapitres 5 à 10 (2007), Berlin-Heidelberg: Springer-Verlag, Berlin-Heidelberg · Zbl 0139.15904 · doi:10.1007/978-3-540-34486-5 |
| [2] | Bourbaki, N., Elements of Mathematics. Lie Groups and Lie Algebras, Chapters 1-3 (1989), Berlin-Heidelberg: Springer-Verlag, Berlin-Heidelberg · Zbl 0672.22001 |
| [3] | N. Bourbaki, Éléments de mathématique. Intégration, Chapters 1-9, 2nd ed., Hermann (Paris, 1965-1989). |
| [4] | N. Bourbaki, Elements de mathematique. Livre VI. Integration. Ch. 1-9. 2nd ed., Ch. 1-9. Hermann, Paris (1965-1969) |
| [5] | Carbonaro, A.; Mauceri, G.; Meda, S., H^1 and BMO for certain locally doubling metric measure spaces, Ann, Sci. Norm. Super. Pisa Cl. Sci., 5, 8, 543-582 (2009) · Zbl 1180.42008 |
| [6] | Chen, J.; Dai, J.; Fan, D.; Zhu, X., Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces, Science China Math., 61, 1647-1664 (2018) · Zbl 1401.42005 · doi:10.1007/s11425-017-9246-7 |
| [7] | Chen, J.; Fan, D.; Wang, S., Hausdorff operators on Euclidean space (a survey article), Appl. Math. J. Chinese Univ. Ser. B., 28, 548-564 (2013) · Zbl 1299.42078 · doi:10.1007/s11766-013-3228-1 |
| [8] | Chen, J.; Zhu, X., Boundedness of multidimensional Hausdorff operators on H^1(ℝ^n), J. Math. Anal. Appl., 409, 428-434 (2014) · Zbl 1309.42026 · doi:10.1016/j.jmaa.2013.07.042 |
| [9] | Chu, J. Y.; Fu, Z. W.; Wu, Q. Y., L^p and BMO bounds for weighted Hardy operators on the Heisenberg group, J. Inequal. Appl., 282, 1-12 (2016) · Zbl 1351.26034 |
| [10] | 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 · doi:10.1090/S0002-9904-1977-14325-5 |
| [11] | Delsarte, J., Hypergroupes et opérateurs de permutation et de transmutation, La théorie des équations aux dérivées partielles (Nancy, 9-15 avril 1956), Colloq. Internat. Centre Nat. Rech. Sci., LXXI, Paris-45 (1956) · Zbl 0075.27503 |
| [12] | Georgakis, C., The Hausdorff mean of a Fourier-Stieltjes transform, Proc. Amer. Math. Soc., 116, 465-471 (1992) · Zbl 0755.42005 · doi:10.1090/S0002-9939-1992-1096210-9 |
| [13] | Goldberg, R. R., Certain operators and Fourier transforms on L^2, Proc. Amer. Math. Soc., 10, 385-390 (1959) · Zbl 0092.11101 |
| [14] | Hardy, G. H., Divergent Series (1949), Oxford: Clarendon Press, Oxford · Zbl 0032.05801 |
| [15] | Jain, P.; Jain, S.; Stepanov, V. D., LCT based integral transforms and Hausdorff operators, Eurasian Math. J., 11, 57-71 (2020) · Zbl 1463.44008 · doi:10.32523/2077-9879-2020-11-1-57-71 |
| [16] | Kawazoe, T., Atomic Hardy spaces on semisimple Lie groups, Japan. J. Math., 11, 293-343 (1985) · Zbl 0626.43009 · doi:10.4099/math1924.11.293 |
| [17] | Kawazoe, T.; Liu, J., On a weak L^1 property of maximal operators on non-compact semisimple Lie groups, Tokyo J. Math., 25, 165-180 (2002) · Zbl 1010.22014 · doi:10.3836/tjm/1244208943 |
| [18] | Lancaster, P.; Tismenetsky, M., The Theory of Matrices (1985), Orlando, FL: Academic Press, Inc., Orlando, FL · Zbl 0558.15001 |
| [19] | Lang, S., SL_2(ℝ) (1975), Mass.-London-Amsterdam: Addison-Wesley (Reading, Mass.-London-Amsterdam · Zbl 0311.22001 |
| [20] | Lerner, A.; Liflyand, E., Multidimensional Hausdorff operators on the real Hardy space, J. Austr. Math. Soc., 83, 79-86 (2007) · Zbl 1143.47023 · doi:10.1017/S1446788700036399 |
| [21] | B.M. Levitan, The Theory of Generalized Displacement Operators, Nauka (Moscow) (In Russian): English translation: Generalized Translation Operators and Some of their Applications, Don Goelman Israel Program for Scientific Translations, Daniel Davey & Co., Inc. (Jerusalem, 1964). · Zbl 0192.48902 |
| [22] | Liflyand, E., Hausdorff operators on Hardy spaces, Eurasian Math. J., 4, 101-141 (2013) · Zbl 1328.47039 |
| [23] | Liflyand, E.; Móricz, F., The Hausdorff operator is bounded on the real Hardy space H1(R), Proc. Amer. Math. Soc., 128, 1391-1396 (2000) · Zbl 0951.47038 · doi:10.1090/S0002-9939-99-05159-X |
| [24] | A. R. Mirotin, On the boundedness of Hausdorff operators on real Hardy spaces H^1 over homogeneous spaces of groups with local doubling property, arXiv:2007.10836. · Zbl 1431.43006 |
| [25] | Mirotin, A. R., On the general form of linear functionals on the Hardy spaces H^1 over compact Abelian groups and some of its applications, Indag. Math., 28, 451-462 (2017) · Zbl 1377.42027 · doi:10.1016/j.indag.2016.11.023 |
| [26] | Mirotin, A. R., Boundedness of Hausdorff operators on Hardy spaces H^1 over locally compact groups, J. Math. Anal. Appl., 473, 519-533 (2019) · Zbl 1431.43006 · doi:10.1016/j.jmaa.2018.12.065 |
| [27] | Mirotin, A. R., Hausdorff operators on homogeneous spaces of locally compact groups, J. Beloruss. Gos. Univ. Math. Inform., 2, 28-35 (2020) |
| [28] | Ruan, J.; Fan, D.; Wu, Q., Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group, Math. Inequal. Appl., 22, 307-329 (2019) · Zbl 1439.42033 |
| [29] | Stahl, S., Poincare Half-plane. Gateway to Modern Geometry (1993), (Boston-London: Jones and Bartlett Publishers, (Boston-London · Zbl 0771.51001 |
| [30] | Stylogiannis, G., Hausdorff operators on Bergman spaces of the upper half plane, Concr. Oper., 7, 69-80 (2020) · Zbl 1505.47035 · doi:10.1515/conop-2020-0005 |
| [31] | Volosivets, S. S., Hausdorff operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces, Math. Notes, 93, 382-391 (2013) · Zbl 1270.42020 · doi:10.1134/S0001434613030048 |
| [32] | Wu, Q.; Fan, D., Hardy space estimates of Hausdorff operators on the Heisenberg group, Nonlinear Anal., 164, 135-154 (2017) · Zbl 1515.47058 · doi:10.1016/j.na.2017.09.001 |
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.
