B-valued martingale Hardy-Lorentz-Karamata spaces. (English) Zbl 1422.60070
Summary: In this paper, we investigate the Hardy-Lorentz-Karamata spaces for Banach space-valued martingales. Relying on the geometrical properties of the underlying Banach spaces, we establish the atomic decompositions and characterize the dual spaces of these spaces. We also obtain some martingale inequalities in the setting of Hardy-Lorentz-Karamata spaces.
Keywords:
Banach-valued martingale; Hardy-Lorentz-Karamata space; atomic decomposition; duality; martingale inequalityReferences:
| [1] | Abu-Shammala, W., Torchinsky, A.: The Hardy-Lorentz spaces \[H^{p, q}({\mathbb{R}}^n)\] Hp,q(Rn). Studia Math. 182(3), 283-294 (2007) · Zbl 1129.42006 · doi:10.4064/sm182-3-7 |
| [2] | Agora, E., Antezana, J., Carro, M., Mara, J.: Weak-type boundedness of the Hardy-Littlewood maximal operator on weighted Lorentz spaces. J. Fourier Anal. Appl. 22(6), 1431-1439 (2016) · Zbl 1357.42014 · doi:10.1007/s00041-015-9456-4 |
| [3] | Ariño, M., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Amer. Math. Soc. 320(2), 727-735 (1990) · Zbl 0716.42016 |
| [4] | Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988) · Zbl 0647.46057 |
| [5] | Burkholder, D. L., Martingale transforms and the geometry of Banach spaces, 35-50 (1981), Berlin, Heidelberg · Zbl 0471.60012 |
| [6] | Burkholder, Donald L., Differential subordination of harmonic functions and martingales, 1-23 (1989), Berlin, Heidelberg · Zbl 0675.31003 |
| [7] | Chatterji, S.: Martingale convergence and the Radon-Nikodym theorem in Banach spaces. Math. Scand. 22, 21-41 (1968) · Zbl 0175.14503 · doi:10.7146/math.scand.a-10868 |
| [8] | Edmunds, David E.; Evans, W. Desmond, Hardy-type Operators, 11-61 (2004), Berlin, Heidelberg · Zbl 1099.46002 |
| [9] | Herz, C.: Bounded mean oscillation and regulated martingales. Trans. Amer. Math. Soc. 193, 199-215 (1974) · Zbl 0321.60041 · doi:10.1090/S0002-9947-1974-0353447-5 |
| [10] | Gogatishvilli, A., Opic, B., Trebels, W.: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278, 86-107 (2005) · Zbl 1080.46012 · doi:10.1002/mana.200310228 |
| [11] | Ho, K.: Fourier integrals and Sobolev embedding on rearrangement-invariant quasi-Banach function spaces, Ann. Acad. Sci. Fenn. Math. 41(2), 897-922 (2016) · Zbl 1355.42020 · doi:10.5186/aasfm.2016.4157 |
| [12] | Ho, K.: Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorentz-Karamata spaces. Q. J. Math. 65(3), 985-1009 (2014) · Zbl 1341.42039 · doi:10.1093/qmath/hat038 |
| [13] | Ho, K.: Littlewood-Paley spaces. Math. Scand. 108(1), 77-102 (2011) · Zbl 1216.46031 · doi:10.7146/math.scand.a-15161 |
| [14] | Ho, K.: Atomic decompositions of martingale Hardy-Morrey spaces. Acta Math. Hungar 149(1), 177-189 (2016) · Zbl 1399.60077 · doi:10.1007/s10474-016-0591-4 |
| [15] | Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab. 4(4), 587-599 (1976) · Zbl 0368.60022 · doi:10.1214/aop/1176996029 |
| [16] | Jiao, Y.: Carleson measures and vector-valued BMO martingales. Probab. Theory Relat. Fields 145(3-4), 421-434 (2009) · Zbl 1173.60319 · doi:10.1007/s00440-008-0173-7 |
| [17] | Jiao, Y., Peng, L., Liu, P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Spaces Appl. 7(2), 153-166 (2009) · Zbl 1173.60320 · doi:10.1155/2009/465079 |
| [18] | Jiao, Y., Wu, L., Popa, M.: Operator-valued martingale transforms in rearrangement invariant spaces and applications. Sci. China Math. 56(4), 831-844 (2013) · Zbl 1281.60041 · doi:10.1007/s11425-013-4570-8 |
| [19] | Jiao, Y., Xie, G., Zhou, D.: Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces. Q. J. Math. 66(2), 605-623 (2015) · Zbl 1317.42021 · doi:10.1093/qmath/hav003 |
| [20] | Jiao, Y., Wu, L., Peng, L.: Weak Orlicz-Hardy martingale spaces. Int. J. Math. 26(8), 1550062 (2015) · Zbl 1344.42019 · doi:10.1142/S0129167X15500627 |
| [21] | Jiao, Y., Wu, L., Yang, A., Yi, R.: The predual and John-Nirenberg inequalities on generalized BMO martingale spaces. Trans. Amer. Math. Soc. 369(1), 537-553 (2017) · Zbl 1353.60043 · doi:10.1090/tran/6657 |
| [22] | Jiao, Y., Zuo, Y., Zhou, D., Wu, L.: Variable Hardy-Lorentz spaces \[H^{p(\cdot ),q}({\mathbb{R}}^n)\] Hp(·),q(Rn), submitted, (2017) |
| [23] | Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer, Berlin (1979) · Zbl 0403.46022 |
| [24] | Liu, K., Zhou, D., Jiao, Y.: Hardy-Lorentz spaces for B-valued martingales. J. Math. Anal. Appl. 450(2), 1401-1420 (2017) · Zbl 1383.60035 · doi:10.1016/j.jmaa.2017.01.093 |
| [25] | Liu, K., Zhou, D.: Dual spaces of weak martingale Hardy-Lorentz-Karamata spaces. Acta Math. Hungar 151(1), 50-68 (2017) · Zbl 1399.60079 · doi:10.1007/s10474-016-0670-6 |
| [26] | Liu, P.: Martingale inequalities and convexity and smoothness of Banach spaces. Acta Math. Sinica. 32(6), 765-775 (1989) · Zbl 0693.46008 |
| [27] | Liu, P.: Martingale spaces and geometrical properties of Banach spaces. Sci. China Ser. A 34(5), 513-527 (1991) · Zbl 0735.60047 |
| [28] | Liu, P.: Martingale and Geomertry of Banach Spaces. Science Press, Beijing (2007) |
| [29] | Liu, P., Hou, Y.: Atomic decompositions of Banach-space-valued martingales. Sci. China Ser. A 42(1), 38-47 (1999) · Zbl 0928.46020 · doi:10.1007/BF02872048 |
| [30] | Liu, P., Yu, L.: \[B B\]-valued martingale spaces with small index and atomic decompositions. Sci. China Ser. A 44(11), 1361-1372 (2001) · Zbl 1002.60036 · doi:10.1007/BF02877064 |
| [31] | Lorentz, G.: Some new functional spaces. Ann. Math. 51, 37-55 (1950) · Zbl 0035.35602 · doi:10.2307/1969496 |
| [32] | Long, R.: Martingale Spaces and Inequalities. Peking University Press, Beijing (1993) · Zbl 0783.60047 · doi:10.1007/978-3-322-99266-6 |
| [33] | Neves, J.: Lorentz-Karamata spaces. Bessel and Riesz potentials and embeddings, Dissertationes Math. 405, 46 (2002) · Zbl 1017.46021 · doi:10.4064/dm405-0-1 |
| [34] | Pisier, G.: Martingales with values in uniformly convex spaces. Israel J. Math. 20(3-4), 326-350 (1975) · Zbl 0344.46030 · doi:10.1007/BF02760337 |
| [35] | Pisier, G.: Martingales in Banach spaces. Cambridge University Press, Cambridge (2016) · Zbl 1382.46002 · doi:10.1017/CBO9781316480588 |
| [36] | Shimura, T.: Decomposition of nondecreasing slowly varying functions and the domain of attraction of Gaussian distributions. J. Math. Soc. Jpn. 43(4), 775-793 (1991) · Zbl 0744.60026 · doi:10.2969/jmsj/04340775 |
| [37] | Weisz, F.: Martingale Hardy spaces for \[0<p\le 1 0\]<p≤1. Probab. Theory Relat. Fields 84(3), 361-376 (1990) · Zbl 0687.60046 · doi:10.1007/BF01197890 |
| [38] | Weisz, Ferenc, Real interpolation, 164-182 (1994), Berlin, Heidelberg · Zbl 0796.60049 · doi:10.1007/BFb0073453 |
| [39] | Weisz, F.: Dual spaces of multi-parameter martingale Hardy spaces. Q. J. Math. 67(1), 137-145 (2016) · Zbl 1334.42049 · doi:10.1093/qmath/hav040 |
| [40] | Wu, L., Zhou, D., Jiao, Y.: Modular inequalities in the martingale Orlicz-Karamata spaces, submitted · Zbl 1394.60040 |
| [41] | Yang, A.: Atomic decompositions of vector-valued weak Musielak-Orlicz martingale spaces and applications. J. Math. Anal. Appl. 449(1), 670-681 (2017) · Zbl 1382.46020 · doi:10.1016/j.jmaa.2016.12.025 |
| [42] | Yu, L.: Duals of Banach-space-valued martingale Hardy spaces. Kyungpook Math. J. 41(2), 259-275 (2001) · Zbl 1002.60038 |
| [43] | Yu, L.: Duals of Hardy spaces of \[{\bf B} B\]-valued martingales for \[0<r\le 1 0\]<r≤1. Acta Math. Sin. 30(8), 1365-1380 (2014) · Zbl 1307.60046 · doi:10.1007/s10114-014-3611-8 |
| [44] | Yu, L.: Dual spaces of weak Orlicz-Hardy spaces for vector-valued martingales. J. Math. Anal. Appl. 437(1), 71-89 (2016) · Zbl 1347.42042 · doi:10.1016/j.jmaa.2015.12.048 |
| [45] | Yu, L.: Duality theorem for B-valued martingale Orlicz-Hardy spaces associated with concave functions. Math. Nachr. 289(5-6), 756-774 (2016) · Zbl 1343.42033 · doi:10.1002/mana.201400174 |
| [46] | Zhou, D., Wu, L., Jiao, Y.: Martingale weak Orlicz-Karamata-Hardy spaces associated with concave functions. J. Math. Anal. Appl. 456(1), 543-562 (2017) · Zbl 1385.46021 · doi:10.1016/j.jmaa.2017.07.022 |
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.
