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Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales

Year 2025, Volume: 8 Issue: Special Issue: ICCMA, 18 - 38, 16.12.2025
https://doi.org/10.33205/cma.1818318

Abstract

Applying the martingale transform and $K$-method of interpolation spaces, we investigate the interchanging relations between Hardy-Lorentz-Karamata spaces of predictable martingales. More precisely, let $0

References

  • N. H. Abel: Untersuchungen uber die Reihe 1 + m/1 x + m(m+1)/1·2 x2 + · · · , J. Reine Angew. Math., 1 (1826), 311–339.
  • W. Abu-Shammala, A. Torchinsky: The Hardy-Lorentz spaces Hp,q(Rn), Studia Math., 182 (2007), 283–294.
  • C. Bennett, R. Sharpley: Interpolation of Operators, Academic Press, New York (1988).
  • J. Bergh, J. Löfström: Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York (1976).
  • N. H. Bingham, C.M. Goldie and J.L. Teugels: Regular Variation, Cambridge Univ. Press, Cambridge (1987).
  • D. L. Burkholder: Martingale transforms, Ann. Math. Stat., 37 (1966), 1494–1504.
  • D. L. Burkholder: Distribution function inequalities for martingales, Ann. Probab., 1 (1) (1973), 19–42.
  • D. L. Burkholder, B. Davis, R. F. Gundy: Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkley Symp., 2 (1972).
  • D. L. Burkholder, R. F. Gundy: Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math., 124 (1970), 249–304.
  • M. J. Carro, J. A. Raposo and J. Soria: Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities, Mem. Amer. Math. Soc., 187 (2007).
  • D. Chamorro, P. G. Lemari´e-Rieusset: Real interpolation method, Lorentz spaces and refined Sobolev inequalities, J. Funct. Anal., 265 (12) (2013), 3219–3232.
  • J. A. Chao, R. L. Long: Martingale transforms and Hardy spaces, Probab. Th. Rel. Fields, 91 (1992), 399–404.
  • M. Ciesielski, G. Lewicki: Sequence Lorentz spaces and their geometric structure, J. Geom. Anal., 29 (3) (2019), 1929–1952.
  • J. Doob: Stochastic Process, New York: Wiley (1953).
  • D. E. Edmunds, W. D. Evans: Hardy Operators, Function Spaces and Embedding, Springer-Verlag, Berlin (2004).
  • D. E. Edmunds, R. Kerman and L. Pick: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307–355.
  • D. E. Edmunds, B. Opic: Alternative characterisations of Lorentz-Karamata spaces, Czech. Math. J., 58 (2008), 517–540.
  • P. Fern´andz-Mart´inez, T. M. Signes: An application of interpolation theory to renorming of Lorentz-Karamata type spaces, Ann. Acad. Sci. Fenn. Math., 39 (1) (2014), 97–107.
  • A. M. Garsia: Martingale inequalities: Seminar notes on recent progress,W. A. Benjamin, Inc., Reading, Mass. London-Amsterdam (1973).
  • A. Gogatishvili, M. Kˇrepela, L. Pick and F. Soudsk´y: Embeddings of Lorentz-type spaces involving weighted integral means, J. Funct. Anal., 273 (9) (2017), 2939–2980.
  • A. Gogatishvili, B. Opic and W. Trebels: Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr., 278 (1-2) (2005), 86–107.
  • L. Grafakos: Classical fourier analysis, New York, Springer (2008).
  • J. Gustavsson: A function parameter in connection with interpolation of Banach spaces, Math. Scand., 42 (1978), 289–305.
  • Z. Hao, X. Ding, L. Li and F. Weisz: Real interpolation for variable martingale Hardy-Lorentz-Karamata spaces, Anal. Appl., 22 (8) (2024), 1389–1416.
  • Z. Hao, L. Li: New Doob’s maximal inequalities for martingales, Acta Math. Sci., 43 (2) (2023), 531–538.
  • Z. Hao, L. Li, L. Long and F. Weisz: Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators, Fract. Calc. Appl. Anal., 27 (2) (2024), 554–615.
  • Z. Hao, L. Li and F. Weisz: Real interpolation for martingale Orlicz-Lorentz-Karamata Hardy spaces, Bull. Sci. Math., 205 (2) (2025), No. 103690.
  • K.-P. Ho: Littlewood-Paley Spaces, Math. Scand., 108 (2011), 77–102.
  • K.-P. Ho: Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorentz-Karamata spaces, Q. J. Math., 65 (3) (2014), 985–1009.
  • K.-P. Ho: Martingale inequalities on rearrangement-invariant quasi-Banach function spaces, Acta Sci. Math. (Szeged), 83 (3-4) (2017), 619–627.
  • N. Ioku: Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation, J. Funct. Anal., 266 (5) (2014), 2944–2958.
  • Y. Jiao: Lp,q-norm estimates associated with Burkholder’s inequalities, Sci. China Math., 54 (2011), 2713–2721.
  • Y. Jiao, T. Ma and P. Liu: Embeddings of Lorentz spaces of vector-valued martingales, Funct. Anal. Appl., 44 (2010), 237–240.
  • Y. Jiao, L. Peng and P. Liu: Atomic decompositions of Lorentz martingale spaces and applications, J. Funct. Spaces Appl., 7 (2009), 153-166.
  • Y. Jiao, G. Xie and D. Zhou: Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces, Q. J. Math., 66 (2) (2015), 605–623.
  • L. Li, K. Liu and Y.Wang: Martingale inequalities in Orlicz-Karamata modular spaces, Banach J. Math. Anal., 18 (2024), Article ID: 54.
  • W. Li, L. Wu and Y. Jiao: Real interpolation of martingale Hardy-Lorentz-Karamata spaces, Sci. China Math., 50 (12) (2020), 1793–1806.
  • R. L. Long: Martingale Spaces and Inequalities, Peking University Press, Beijing (1993).
  • J. S. Neves: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings, Dissertationes Math., 405 (2002), 1–46.
  • D. Peˇsa: Lorentz Karamata spaces, arXiv: 2006.14455v4 (2023).
  • F. Weisz: Martingale Hardy spaces for 0 < p ≤ 1, Probab. Th. Rel. Fields, 84 (3) (1990), 361–376.
  • F. Weisz: Hardy spaces of predictable martingales, Anal. Math., 20 (3) (1994), 225–233.
  • F. Weisz: Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer (2006).
  • L.Wu, D. Zhou and Y. Jiao: Modular inequalities in martingale Orlicz-Karamata spaces, Math. Nachr., 291 (8-9) (2018), 1450–1462.
  • L. Yu, M. He: Interchange between Hardy-Lorentz spaces of predictable martingales, J. Math. Anal. Appl., 472 (2019), 1277–1291.
  • D. Zhou, L. Wu and Y. Jiao: Martingale weak Orlicz-Karamata-Hardy spaces associated with concave functions, J. Math. Anal. Appl., 456 (1) (2017), 543–562.
There are 46 citations in total.

Details

Primary Language English
Subjects Lie Groups, Harmonic and Fourier Analysis
Journal Section Research Article
Authors

Zhiwei Hao 0000-0001-5553-3398

Mei Li 0009-0004-3829-745X

Yao Wang 0009-0001-9707-4035

Ferenc Weisz 0000-0002-7766-2745

Submission Date November 10, 2025
Acceptance Date November 28, 2025
Early Pub Date December 16, 2025
Publication Date December 16, 2025
Published in Issue Year 2025 Volume: 8 Issue: Special Issue: ICCMA

Cite

APA Hao, Z., Li, M., Wang, Y., Weisz, F. (2025). Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. Constructive Mathematical Analysis, 8(Special Issue: ICCMA), 18-38. https://doi.org/10.33205/cma.1818318
AMA Hao Z, Li M, Wang Y, Weisz F. Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. CMA. December 2025;8(Special Issue: ICCMA):18-38. doi:10.33205/cma.1818318
Chicago Hao, Zhiwei, Mei Li, Yao Wang, and Ferenc Weisz. “Interchange Between Hardy-Lorentz-Karamata Spaces of Predictable Martingales”. Constructive Mathematical Analysis 8, no. Special Issue: ICCMA (December 2025): 18-38. https://doi.org/10.33205/cma.1818318.
EndNote Hao Z, Li M, Wang Y, Weisz F (December 1, 2025) Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. Constructive Mathematical Analysis 8 Special Issue: ICCMA 18–38.
IEEE Z. Hao, M. Li, Y. Wang, and F. Weisz, “Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales”, CMA, vol. 8, no. Special Issue: ICCMA, pp. 18–38, 2025, doi: 10.33205/cma.1818318.
ISNAD Hao, Zhiwei et al. “Interchange Between Hardy-Lorentz-Karamata Spaces of Predictable Martingales”. Constructive Mathematical Analysis 8/Special Issue: ICCMA (December2025), 18-38. https://doi.org/10.33205/cma.1818318.
JAMA Hao Z, Li M, Wang Y, Weisz F. Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. CMA. 2025;8:18–38.
MLA Hao, Zhiwei et al. “Interchange Between Hardy-Lorentz-Karamata Spaces of Predictable Martingales”. Constructive Mathematical Analysis, vol. 8, no. Special Issue: ICCMA, 2025, pp. 18-38, doi:10.33205/cma.1818318.
Vancouver Hao Z, Li M, Wang Y, Weisz F. Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. CMA. 2025;8(Special Issue: ICCMA):18-3.