SavedAdd your journal

Journals / Constructive Mathematical Analysis / 2025 / Volume: 8 Issue: Special Issue: ICCMA Special Issue

Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales

Zhiwei Hao(Hunan University of Science and Technology)0000-0001-5553-3398
Mei Li(Hunan University of Science and Technology)0009-0004-3829-745X
Yao Wang(Hunan University of Science and Technology)0009-0001-9707-4035
Ferenc Weisz(Eötvös Loránd University)0000-0002-7766-2745
Pages
—
Publication Date
2025-12-16
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

More precisely, let $0

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Applying the martingale transform and $K$-method of interpolation spaces, we investigate the interchanging relations between Hardy-Lorentz-Karamata spaces of predictable martingales.
📊
Key Findings & Data:Applying the martingale transform and $K$-method of interpolation spaces, we investigate the interchanging relations between Hardy-Lorentz-Karamata spaces of predictable martingales.
💡
Conclusion & Impact:More precisely, let $0
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Martingale transform · Interpolation space · Hardy-Lorentz-Karamata space

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

🔖

Cite This Article

Hao, Z., Li, M., Wang, Y., Weisz, F. (2025). Interchange between Hardy-Lorentz-Karamata spaces of predictable martingales. Constructive Mathematical Analysis, Volume: 8 Issue: Special Issue: ICCMA Special Issue. https://doi.org/10.33205/cma.1818318

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (47)

  1. 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.
  2. W. Abu-Shammala, A. Torchinsky: The Hardy-Lorentz spaces Hp,q(Rn), Studia Math., 182 (2007), 283–294.
  3. C. Bennett, R. Sharpley: Interpolation of Operators, Academic Press, New York (1988).
  4. J. Bergh, J. Löfström: Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York (1976).
  5. N. H. Bingham, C.M. Goldie and J.L. Teugels: Regular Variation, Cambridge Univ. Press, Cambridge (1987).
  6. D. L. Burkholder: Martingale transforms, Ann. Math. Stat., 37 (1966), 1494–1504.
  7. D. L. Burkholder: Distribution function inequalities for martingales, Ann. Probab., 1 (1) (1973), 19–42.
  8. D. L. Burkholder, B. Davis, R. F. Gundy: Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkley Symp., 2 (1972).
  9. D. L. Burkholder, R. F. Gundy: Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math., 124 (1970), 249–304.
  10. 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).
  11. D. Chamorro, P. G. Lemari´e-Rieusset: Real interpolation method, Lorentz spaces and refined Sobolev inequalities, J. Funct. Anal., 265 (12) (2013), 3219–3232.
  12. J. A. Chao, R. L. Long: Martingale transforms and Hardy spaces, Probab. Th. Rel. Fields, 91 (1992), 399–404.
  13. M. Ciesielski, G. Lewicki: Sequence Lorentz spaces and their geometric structure, J. Geom. Anal., 29 (3) (2019), 1929–1952.
  14. J. Doob: Stochastic Process, New York: Wiley (1953).
  15. D. E. Edmunds, W. D. Evans: Hardy Operators, Function Spaces and Embedding, Springer-Verlag, Berlin (2004).
  16. D. E. Edmunds, R. Kerman and L. Pick: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307–355.
  17. D. E. Edmunds, B. Opic: Alternative characterisations of Lorentz-Karamata spaces, Czech. Math. J., 58 (2008), 517–540.
  18. 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.
  19. A. M. Garsia: Martingale inequalities: Seminar notes on recent progress,W. A. Benjamin, Inc., Reading, Mass. London-Amsterdam (1973).
  20. 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.
  21. A. Gogatishvili, B. Opic and W. Trebels: Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr., 278 (1-2) (2005), 86–107.
  22. L. Grafakos: Classical fourier analysis, New York, Springer (2008).
  23. J. Gustavsson: A function parameter in connection with interpolation of Banach spaces, Math. Scand., 42 (1978), 289–305.
  24. 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.
  25. Z. Hao, L. Li: New Doob’s maximal inequalities for martingales, Acta Math. Sci., 43 (2) (2023), 531–538.
  26. 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.
  27. Z. Hao, L. Li and F. Weisz: Real interpolation for martingale Orlicz-Lorentz-Karamata Hardy spaces, Bull. Sci. Math., 205 (2) (2025), No. 103690.
  28. K.-P. Ho: Littlewood-Paley Spaces, Math. Scand., 108 (2011), 77–102.
  29. K.-P. Ho: Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorentz-Karamata spaces, Q. J. Math., 65 (3) (2014), 985–1009.
  30. K.-P. Ho: Martingale inequalities on rearrangement-invariant quasi-Banach function spaces, Acta Sci. Math. (Szeged), 83 (3-4) (2017), 619–627.
  31. N. Ioku: Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation, J. Funct. Anal., 266 (5) (2014), 2944–2958.
  32. Y. Jiao: Lp,q-norm estimates associated with Burkholder’s inequalities, Sci. China Math., 54 (2011), 2713–2721.
  33. Y. Jiao, T. Ma and P. Liu: Embeddings of Lorentz spaces of vector-valued martingales, Funct. Anal. Appl., 44 (2010), 237–240.
  34. Y. Jiao, L. Peng and P. Liu: Atomic decompositions of Lorentz martingale spaces and applications, J. Funct. Spaces Appl., 7 (2009), 153-166.
  35. 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.
  36. L. Li, K. Liu and Y.Wang: Martingale inequalities in Orlicz-Karamata modular spaces, Banach J. Math. Anal., 18 (2024), Article ID: 54.
  37. W. Li, L. Wu and Y. Jiao: Real interpolation of martingale Hardy-Lorentz-Karamata spaces, Sci. China Math., 50 (12) (2020), 1793–1806.
  38. R. L. Long: Martingale Spaces and Inequalities, Peking University Press, Beijing (1993).
  39. J. S. Neves: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings, Dissertationes Math., 405 (2002), 1–46.
  40. D. Peˇsa: Lorentz Karamata spaces, arXiv: 2006.14455v4 (2023).
  41. F. Weisz: Martingale Hardy spaces for 0 < p ≤ 1, Probab. Th. Rel. Fields, 84 (3) (1990), 361–376.
  42. F. Weisz: Hardy spaces of predictable martingales, Anal. Math., 20 (3) (1994), 225–233.
  43. F. Weisz: Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer (2006).
  44. L.Wu, D. Zhou and Y. Jiao: Modular inequalities in martingale Orlicz-Karamata spaces, Math. Nachr., 291 (8-9) (2018), 1450–1462.
  45. L. Yu, M. He: Interchange between Hardy-Lorentz spaces of predictable martingales, J. Math. Anal. Appl., 472 (2019), 1277–1291.
  46. 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.
  47. 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.