Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SUMMABILITY IN MUSIELAK-ORLICZ HARDY SPACES

  • Jun Liu (School of Mathematics China University of Mining and Technology) ;
  • Haonan Xia (School of Mathematics China University of Mining and Technology)
  • Received : 2022.12.30
  • Accepted : 2023.04.13
  • Published : 2023.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let 𝜑 : ℝn × [0, ∞) → [0, ∞) be a growth function and H𝜑(ℝn) the Musielak-Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called 𝜃-summability is considered for multi-dimensional Fourier transforms in H𝜑(ℝn). Precisely, with some assumptions on 𝜃, the authors first prove that the maximal operator of the 𝜃-means is bounded from H𝜑(ℝn) to L𝜑(ℝn). As consequences, some norm and almost everywhere convergence results of the 𝜃-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner-Riesz, Weierstrass and Picard-Bessel summations, are also presented.

Keywords

Acknowledgement

This project was financially supported by the National Natural Science Foundation of China (Grant No. 12001527), the Natural Science Foundation of Jiangsu Province (Grant No. BK20200647) and the Postdoctoral Science Foundation of China (Grant No. 2021M693422).

References

  1. A. Bonami, J. Cao, L. D. Ky, L. Liu, D. Yang, and W. Yuan, Multiplication between Hardy spaces and their dual spaces, J. Math. Pures Appl. (9) 131 (2019), 130-170. https://doi.org/10.1016/j.matpur.2019.05.003 
  2. A. Bonami, J. Feuto, and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat. 54 (2010), no. 2, 341-358. https://doi.org/10.5565/PUBLMAT_54210_03 
  3. A. Bonami, S. Grellier, and L. D. Ky, Paraproducts and products of functions in BMO(ℝn) and H1 (ℝn) through wavelets, J. Math. Pures Appl. (9) 97 (2012), no. 3, 230-241. https://doi.org/10.1016/j.matpur.2011.06.002 
  4. A. Bonami, L. D. Ky, Y. Liang, and D. Yang, Several remarks on Musielak-Orlicz Hardy spaces, Bull. Sci. Math. 181 (2022), Paper No. 103206, 35 pp. https://doi.org/10.1016/j.bulsci.2022.103206 
  5. P. L. Butzer and R. J. Nessel, Fourier analysis and approximation, Pure and Applied Mathematics, Vol. 40, Academic Press, New York, 1971. 
  6. L. A. E. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. https://doi.org/10.1007/BF02392815 
  7. X. Fan, J. He, B. Li, and D. Yang, Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces, Sci. China Math. 60 (2017), no. 11, 2093-2154. https://doi.org/10.1007/s11425-016-9024-2 
  8. C. L. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. https://doi.org/10.1007/BF02392215 
  9. H. G. Feichtinger and F. Weisz, Wiener amalgams and pointwise summability of Fourier transforms and Fourier series, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 3, 509-536. https://doi.org/10.1017/S0305004106009273 
  10. J. Garcia-Cuerva, Weighted Hp spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63 pp. 
  11. L. Grafakos, Classical Fourier Analysis, third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-1194-3 
  12. R. A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), 235-255, Southern Illinois Univ. Press, Carbondale, IL, 1968. 
  13. S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), no. 4, 959-982. http://projecteuclid.org/euclid.dmj/1077314345  1077314345
  14. L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931-2958. https://doi.org/10.1090/S0002-9947-2012-05727-8 
  15. L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), no. 1, 115-150. https://doi.org/10.1007/s00020-013-2111-z 
  16. L. D. Ky, Bilinear decompositions for the product space H1L × BMOL, Math. Nachr. 287 (2014), no. 11-12, 1288-1297. https://doi.org/10.1002/mana.201200101 
  17. M.-Y. Lee, Weighted norm inequalities of Bochner-Riesz means, J. Math. Anal. Appl. 324 (2006), no. 2, 1274-1281. https://doi.org/10.1016/j.jmaa.2005.07.085 
  18. B. Li, M. F. Liao, and B. D. Li, Some estimates for maximal Bochner-Riesz means on Musielak-Orlicz Hardy spaces, Math. Notes 107 (2020), no. 3-4, 618-627. https://doi.org/10.1134/s0001434620030293 
  19. Y. Liang, J. Z. Huang, and D. C. Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 413-428. https://doi.org/10.1016/j.jmaa.2012.05.049 
  20. J. Liu, D. D. Haroske, and D. C. Yang, New molecular characterizations of anisotropic Musielak-Orlicz Hardy spaces and their applications, J. Math. Anal. Appl. 475 (2019), no. 2, 1341-1366. https://doi.org/10.1016/j.jmaa.2019.03.017 
  21. J. Liu, F. Weisz, D. Yang, and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl. 25 (2019), no. 3, 874-922. https://doi.org/10.1007/s00041-018-9609-3 
  22. S. Sato, Weak type estimates for some maximal operators on the weighted Hardy spaces, Ark. Mat. 33 (1995), no. 2, 377-384. https://doi.org/10.1007/BF02559715 
  23. E. M. Stein, M. H. Taibleson, and G. L. Weiss, Weak type estimates for maximal operators on certain Hp classes, Rend. Circ. Mat. Palermo (2) 1981 (1981), no. suppl, suppl. 1, 81-97. 
  24. E. M. Stein and G. L. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton Univ. Press, Princeton, NJ, 1971. 
  25. J.-O. Stromberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), no. 3, 511-544. https://doi.org/10.1512/iumj.1979.28.28037 
  26. R. M. Trigub and E. S. Belinskii, Fourier Analysis and Approximation of Functions, Kluwer Acad. Publ., Dordrecht, 2004. https://doi.org/10.1007/978-1-4020-2876-2 
  27. F. Weisz, Summability of multi-dimensional Fourier series and Hardy spaces, Mathematics and its Applications, 541, Kluwer Acad. Publ., Dordrecht, 2002. https://doi.org/10.1007/978-94-017-3183-6 
  28. F. Weisz, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory 7 (2012), 1-179. 
  29. F. Weisz, Convergence and Summability of Fourier Transforms and Hardy Spaces, Birkhauser, Basel, 2017. 
  30. F. Weisz, Summability of Fourier transforms in variable Hardy and Hardy-Lorentz spaces, Jaen J. Approx. 10 (2018), no. 1-2, 101-131. 
  31. D. C. Yang, Y. Liang, and L. D. Ky, Real-variable theory of Musielak-Orlicz Hardy spaces, Lecture Notes in Mathematics, 2182, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-54361-1