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Martingale Hardy–Orlicz-amalgam spaces

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Abstract

In this article, the authors first introduce a class of Orlicz-amalgam spaces, which defined on a probabilistic setting. Based on these Orlicz-amalgam spaces, the authors introduce a new kind of Hardy type spaces, namely martingale Hardy–Orlicz-amalgam spaces, which generalize the martingale Hardy-amalgam spaces very recently studied by Bansah and Sehba. Their characterizations via the atomic decompositions are also obtained. As applications of these characterizations, the authors construct the dual theorems in the new framework. Furthermore, the authors also present the boundedness of fractional integral operators \(I_\alpha \) on martingale Hardy–Orlicz-amalgam spaces.

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References

  1. Arai, R., Nakai, E., Sadasue, G.: Fractional integrals and their commutators on martingale Orlicz spaces. J. Math. Anal. Appl. 487(2), 123991 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bansah, J., Sehba, B.: Martingale hardy-amalgam spaces: atomic decompositions and duality, operator theory and harmonic analysis: OTHA 2020. In: Part I-New General Trends and Advances of the Theory 10, pp. 73–100. Springer International Publishing (2021)

  3. Birnbaum, Z., Orlicz, W.: Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen. Stud. Math. 3, 1–67 (1931)

    Article  Google Scholar 

  4. Burkholder, D.L.: Martingale transforms. Ann. Math. Statist. 37(6), 1494–1504 (1966)

    Article  MathSciNet  Google Scholar 

  5. Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970)

    Article  MathSciNet  Google Scholar 

  6. Chao, J.-A., Ombe, H.: Commutators on dyadic martingales. Proc. Jpn. Acad. Ser. A Math. Sci. 61(2), 35–38 (1985)

    Article  MathSciNet  Google Scholar 

  7. Doob, J.L.: Stochastic Processes. Chapman & Hall, London (1953)

    Google Scholar 

  8. Feichtinger, H.G.: Banach spaces of distributions of Wiener’s type and interpolation. In: Functional Analysis and Approximation (Oberwolfach, 1980), Internat. Ser. Number. Math. 60, pp. 153–165. Birkhäuser, Basel-Boston (1981)

  9. Feichtinger, H.G.: A characterization of minimal homogeneous Banach spaces. Proc. Am. Math. Soc. 81, 55–61 (1981)

    Article  MathSciNet  Google Scholar 

  10. Feichtinger, H.G.: Banach convolution algebras of Wiener type. In: Functions, Series, Operators, Vol. I, II (Budapest, 1980), Colloq. Math. Soc. János Bolyai, 35, pp. 509–524. North-Holland Publishing Co., Amsterdam (1983)

  11. Feichtinger, H.G.: Wiener amalgams over Euclidean spaces and some of their applications. In: K. Jarosz, editor, Function Spaces, Proc Conf, Edwardsville/IL (USA) 1990, Notes Pure Appl. Math., NuHAG; Classical, Marcel Dekker, 136, 123–137 (1992)

  12. Fournier, J.J., Stewart, J.: Amalgams of \(L^p\) and \(l^q\). Bull. Am. Math. Soc. 13, 1–21 (1985)

    Article  Google Scholar 

  13. Garsia, A.M.: Martingale inequalities: seminar notes on recent progress, Math. Benjamin, Lecture Note Series, New York (1973)

  14. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  15. Hao, Z., Jiao, Y.: Fractional integral on martingale Hardy spaces with variable exponents. Fract. Calc. Appl. Anal. 18(5), 1128–1145 (2015)

    Article  MathSciNet  Google Scholar 

  16. Hao, Z., Li, L.: Orlicz–Lorentz Hardy martingale spaces. J. Math. Anal. Appl. 482(1), 123520 (2020)

    Article  MathSciNet  Google Scholar 

  17. Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals II. Math. Z. 34(1), 403–439 (1932)

    Article  MathSciNet  Google Scholar 

  18. Heil, C.: An Introduction to Weighted Wiener Amalgams. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and their Applications (Chennai, January 2002), pp. 183–216. Allied Publishers, New Delhi (2003)

    Google Scholar 

  19. Ho, K.-P.: Martingale transforms and fractional integrals on rearrangement-invariant martingale Hardy spaces. Period. Math. Hungar. 81, 159–173 (2020)

    Article  MathSciNet  Google Scholar 

  20. Ho, K.-P.: Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces. J. Math. Anal. Appl. 503, 125279 (2021)

    Article  MathSciNet  Google Scholar 

  21. Ho, K.-P.: Singular integrals and sublinear operators on amalgam spaces and Hardy-amalgam spaces. Math. Scand. 127, 585–599 (2021)

    Article  MathSciNet  Google Scholar 

  22. Ho, K.-P.: Fractional integral operators on Orlicz slice Hardy spaces. Fract. Calc. Appl. Anal. 25, 12941305 (2022)

    Article  MathSciNet  Google Scholar 

  23. Holland, F.: Harmonic analysis on amalgams of \(L^p\) and \(l^q\). J. Lond. Math. Soc. (2) 10(3), 295–305 (1975)

    Article  MathSciNet  Google Scholar 

  24. Jiao, Y., Peng, L., Liu, P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Spaces Appl. 7, 153–166 (2009)

    Article  MathSciNet  Google Scholar 

  25. Jiao, Y., Weisz, F., Wu, L., Zhou, D.: Variable martingale Hardy spaces and their applications in Fourier analysis. Dissertationes Math. 550, 1–67 (2020)

    Article  MathSciNet  Google Scholar 

  26. Jiao, Y., Weisz, F., Xie, G., Yang, D.: Martingale Musielak–Orlicz–Lorentz hardy spaces with applications to dyadic fourier analysis. J. Geom. Anal. 31, 11002–11050 (2021)

    Article  MathSciNet  Google Scholar 

  27. Jiao, Y., Wu, L., Yang, A., Yi, R.: The predual and John–Nirenberg inequalities on generalized BMO martingale spaces. Trans. Am. Math. Soc. 369(1), 537–553 (2017)

    Article  MathSciNet  Google Scholar 

  28. Jiao, Y., Wu, L., Zhou, D.: Noncommutative martingale Hardy–Orlicz spaces: dualities and inequalities. Sci. China Math. 66, 2081–2104 (2023)

    Article  MathSciNet  Google Scholar 

  29. Kamińnska, A.: Indices, convexity and concavity in Musielak–Orlicz spaces. Funct. Approx. 26, 67–84 (1998)

    MathSciNet  Google Scholar 

  30. Kikuchi, M.: New martingale inequalities in rearrangement function spaces. Proc. Edinb. Math. Soc. 47(2), 633–657 (2004)

    Article  MathSciNet  Google Scholar 

  31. Kikuchi, M.: On some inequalities for Doob decompositions in Banach function spaces. Math. Z. 265(4), 865–887 (2010)

    Article  MathSciNet  Google Scholar 

  32. Li, L., Wang, Y., Yang, A.: Atomic decompositions of martingale Hardy Lorentz Amalgam spaces and applications. Acta Math. Hungar (to appear)

  33. Long, R.L.: Martingale Spaces and Inequalities. Peking University Press, Beijing (1993)

    Book  Google Scholar 

  34. Maligranda, L.: Orlicz spaces and Interpolation, Seminars in Math., vol. 5. Campinas: Univ. Estadual, de Campinas (1989)

  35. Miyamoto, T., Nakai, E., Sadasue, G.: Martingale Orlicz Hardy spaces. Math. Nachr. 285, 670–686 (2012)

    Article  MathSciNet  Google Scholar 

  36. Nakai, E., Sadasue, G.: Characterizations of boundedness for generalized fractional integrals on martingale Morrey spaces. Math. Inequal. Appl. 20, 929–947 (2017)

    MathSciNet  Google Scholar 

  37. Nakai, E., Sawano, Y.: Orlicz Hardy spaces and their duals. Sci. China Math. 57(5), 903–962 (2014)

  38. Rao, M.M.: Linear functional on Orlicz spaces: general theory. Pacific J. Math. 25, 553–585 (1968)

    Article  MathSciNet  Google Scholar 

  39. Sadasue, G.: Fractional integrals on martingale Hardy spaces for \(0<p\le 1\). Mem. Osaka Kyoiku Univ. Ser. III 60(1), 1–7 (2011)

    MathSciNet  Google Scholar 

  40. Watari, C.: Multipliers for Walsh Fourier series. Tohoku Math. J. 16(3), 239–251 (1964)

    Article  MathSciNet  Google Scholar 

  41. Weisz, F.: Martingale Hardy spaces for \(0<p\le 1\). Probab. Theory Relat. Fields 84(3), 361–376 (1990)

    Article  MathSciNet  Google Scholar 

  42. Weisz, F.: Martingale Hardy spaces and their Applications in Fourier Analysis. Springer, Berlin (1994)

    Book  Google Scholar 

  43. Weisz, F., Xie, G., Yang, D.: Dual spaces for martingale Musielak–Orlicz Lorentz Hardy spaces. Bull. Sci. Math. 179, 103154 (2022)

    Article  MathSciNet  Google Scholar 

  44. Xie, G., Weisz, F., Yang, D., Jiao, Y.: New martingale inequalities and applications to Fourier analysis. Nonlinear Anal. 182, 143–192 (2019)

    Article  MathSciNet  Google Scholar 

  45. Zhang, Y., Yang, D., Yuan, W., Wang, S.: Real-variable characterizations of Orlicz-slice Hardy spaces. Anal. Appl. 17, 597–664 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for giving valuable comments and suggestions which helped to improve the final version of this paper. Libo Li is supported by the NSFC (No. 12101223) and Hunan Provincial Natural Science Foundation (No. 2022JJ40146). Kaituo Liu is supported by the Doctoral Scientific Research Foundation of Hubei University of Automotive Technology (No. BK201805).

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Libo Li & Yao Wang

  2. School of Mathematics, Physics and Optoelectronic Engineering, Hubei University of Automotive Technology, Shiyan, 442002, Hubei, China

    Kaituo Liu

Authors
  1. Libo Li
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  2. Kaituo Liu
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  3. Yao Wang
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Correspondence to Kaituo Liu.

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Communicated by Yong Jiao.

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Li, L., Liu, K. & Wang, Y. Martingale Hardy–Orlicz-amalgam spaces. Ann. Funct. Anal. 15, 37 (2024). https://doi.org/10.1007/s43034-024-00338-9

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