Embeddings of B-valued Lorentz-Karamata spaces associated with martingale transforms

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Abstract

In this article, the embeddings of Banach space valued (briefly by B-valued) Lorentz-Karamata martingale spaces are established. Different from the previous approach, we employ the technique of martingale transforms defined by some operator sequences, which can often supply a clear and succinct method in the research of martingale theory and Harmonic analysis. It’s worth mentioning that the results obtained here are connected closely with the geometrical properties of the Banach spaces in which the martingales take values, such as p-uniform smoothness and q-uniform convexity.

Keywords

martingale transform / Lorentz-Karamata space / embedding / B-valued martingale / uniform convexity / uniform smoothness

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References

[1] G. Pisier, Martingales with values in uniformly convex spaces, Isr. J. Math., 20 (1975), 326-350. https://doi.org/10.1007/BF02760337

[2] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequence are unconditional, Ann. Probab., 9 (1981), 997-1011. https://doi.org/10.1214/aop/1176994270

[3] P. D. Liu, Martingale Spaces and the Geometric Property of Banach Spaces (in Chinese), Science Press, Beijing, 2007.

[4] G. Pisier, Martingales in Banach Spaces, Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316480588

[5] W. A. Woyczynski, Geometry and martingales in Banach spaces, in Probability-Winter School, (2006), 229-275. https://doi.org/10.1007/BFb0081956

[6] Y. Jiao, L. Wu, A. M. Yang, R. Yi, The predual and John-Nirenberg inequalities on generalized BMO martingale spaces, Trans. Amer. Math. Soc., 369 (2017), 537-553. https://doi.org/10.1090/tran/6657

[7] Y. Jiao, G. H. Xie, D. J. Zhou, Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces, Q. J. Math., 66 (2015), 605-623. https://doi.org/10.1093/qmath/hav003

[8] K. Ho, Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorentz-Karamata spaces, Q. J. Math., 65 (2014), 985-1009. https://doi.org/10.1093/qmath/hat038

[9] Z. W. Hao, L. B. Li, L. Long, F. Weisz, Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators, Fract. Calc. Appl. Anal., 27 (2024), 554-615. https://doi.org/10.1007/s13540-024-00259-3

[10] Z. W. Hao, L. B. Li, F. Weisz, Real interpolation for martingale Orlicz-Lorentz-Karamata Hardy spaces, Bull. Sci. Math., 205 (2025), 103690. https://doi.org/10.1016/j.bulsci.2025.103690

[11] L. Yu, Duals of Hardy spaces of B-valued martingales for 0 < r ≤ 1, Acta Math. Sin., 30 (2014), 1365-1380. https://doi.org/10.1007/s10114-014-3611-8

[12] L. Yu, Duality theorem for B-valued martingale Orlicz-Hardy spaces associated with concave functions, Math. Nachr., 289 (2016), 756-774. https://doi.org/10.1002/mana.201400174

[13] L. Yu, Dual spaces of weak Orlicz-Hardy spaces for vector-valued martingales, J. Math. Anal. Appl., 437 (2016), 71-89. https://doi.org/10.1016/j.jmaa.2015.12.048

[14] K. T. Liu, W. W. Li, T. Yue, B-valued martingale Hardy-Lorentz-Karamata spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), 2395-2422. https://doi.org/10.1007/s40840-018-0607-z

[15] Z. W. Hao, L. Wang, F. Weisz, Two-parameter B-valued martingale Hardy-Lorentz-Karamata spaces: Inequalities and dualities, Math. Nachr., 298 (2025), 677-708. https://doi.org/10.1002/mana.202400204

[16] D. L. Burkholder, Martingale transform, Ann. Math. Stat., 37 (1966), 1494-1504. https://doi.org/10.1214/aoms/1177699141

[17] D. L. Burkholder, R. F. Gundy, Extrapolation and interpolation of quasilinear operators on martingales, Acta Math., 124 (1970), 249-304. https://doi.org/10.1007/bf02394573

[18] R. L. Long, Martingale Spaces and Inequalities, Peking University Press, Beijing, 1993. https://doi.org/10.1007/978-3-322-99266-6

[19] L. Yu, Orlicz norm inequalities for operator-valued martingale transforms, Stat. Probab. Lett., 78 (2008), 2664-2670. https://doi.org/10.1016/j.spl.2008.03.025

[20] T. Martínez, J. L. Torrea, Operator-valued martingale transforms, Tohoku Math. J., 52 (2000), 449-474. https://doi.org/10.2748/tmj/1178207823

[21] Y. Jiao, T. Ma, P. D. Liu, Embeddings of Lorentz spaces of vector-valued martingales, Funct. Anal. Appl., 44 (2010), 237-240. https://doi.org/10.1007/s10688-010-0033-y

[22] D. Edmunds, W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer (Berlin), 2004. https://doi.org/10.1007/978-3-662-07731-3

[23] J. Neves, Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings, Dissertationes Math., 405 (2002), 46. https://doi.org/10.4064/dm405-0-1

[24] Y. Jiao, L. Wu, M. Popa, Operator-valued martingale transforms in rearrangement invariant spaces and applications, Sci. China Math., 56 (2013), 831-844. https://doi.org/10.1007/s11425-013-4570-8

[25] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer-Verlerg, Berlin, 1994. https://doi.org/10.1007/BFb0073448