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Summability of Fourier series in periodic Hardy spaces with variable exponent

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Abstract

Let \(p(\cdot ) \mathbb{T}\it ^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder condition and \(0<q \le \infty \). We introduce the periodic variable Hardy and Hardy–Lorentz spaces \(H_{p(\cdot )}(\mathbb{T}\it ^d)\) and \(H_{p(\cdot ),q}(\mathbb{T}\it ^d)\) and prove their atomic decompositions. A general summability method, the so called \(\theta \)-summability is considered for multi-dimensional Fourier series. Under some conditions on \(\theta \), it is proved that the maximal operator of the \(\theta \)-means is bounded from \(H_{p(\cdot )}(\mathbb{T}\it ^d)\) to \(L_{p(\cdot )}(\mathbb{T}\it ^d)\) and from \(H_{p(\cdot ),q}(\mathbb{T}\it ^d)\) to \(L_{p(\cdot ),q}(\mathbb{T}\it ^d)\). This implies some norm and almost everywhere convergence results for the summability means. The Riesz, Bochner–Riesz, Weierstrass, Picard and Bessel summations are investigated as special cases.

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References

  1. J. Bergh and J. Löfström, Interpolation Spaces, an Introduction, Springer (Berlin, 1976)

  2. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag (Basel, 1971)

  3. Cruz-Uribe, D., Fiorenza, A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^p\) Spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: The maximal function on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser/Springer (New York (2013)

    Book  Google Scholar 

  6. L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer (Berlin, 2011)

  7. R. E. Edwards, Fourier Series, a Modern Introduction, vol. 2, Springer (Berlin, 1982)

  8. Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–194 (1972)

    Article  MathSciNet  Google Scholar 

  9. Feichtinger, H.G., Weisz, F.: The Segal algebra \({ S}_0(^d)\) and norm summability of Fourier series and Fourier transforms. Monatsh. Math. 148, 333–349 (2006)

    Article  MathSciNet  Google Scholar 

  10. Feichtinger, H.G., Weisz, F.: Wiener amalgams and pointwise summability of Fourier transforms and Fourier series. Math. Proc. Cambridge Philos. Soc. 140, 509–536 (2006)

    Article  MathSciNet  Google Scholar 

  11. Fejér, L.: Untersuchungen über Fouriersche Reihen. Math. Ann. 58, 51–69 (1904)

    Article  Google Scholar 

  12. Gát, G.: Pointwise convergence of cone-like restricted two-dimensional \((C,1)\) means of trigonometric Fourier series. J. Approx. Theory 149, 74–102 (2007)

    Article  MathSciNet  Google Scholar 

  13. G. Gát, Almost everywhere convergence of sequences of Cesàro and Riesz means of integrable functions with respect to the multidimensional Walsh system, Acta Math. Sin., Engl. Ser., 30 (2014), 311–322

  14. Gát, G., Goginava, U., Nagy, K.: On the Marcinkiewicz-Fejér means of double Fourier series with respect to Walsh-Kaczmarz system. Studia Sci. Math. Hungar. 46, 399–421 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Goginava, U.: Marcinkiewicz-Fejér means of \(d\)-dimensional Walsh-Fourier series. J. Math. Anal. Appl. 307, 206–218 (2005)

    Article  MathSciNet  Google Scholar 

  16. Goginava, U.: Almost everywhere convergence of \((C, a)\)-means of cubical partial sums of d-dimensional Walsh-Fourier series. J. Approx. Theory 141, 8–28 (2006)

    Article  MathSciNet  Google Scholar 

  17. Goginava, U.: The maximal operator of the Marcinkiewicz-Fejér means of \(d\)-dimensional Walsh-Fourier series. East J. Approx. 12, 295–302 (2006)

    MathSciNet  Google Scholar 

  18. L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education (New Jersey, 2004)

  19. Gundy, R.F., Stein, E.M.: \(H^p\) theory for the poly-disc. Proc. Nat. Acad. Sci. USA 76, 1026–1029 (1979)

    Article  Google Scholar 

  20. Y. Jiao, F. Weisz, L. Wu, and D. Zhou, Variable martingale Hardy spaces and their applications in Fourier analysis, Dissertationes Math. (to appear)

  21. Jiao, Y., Zuo, Y., Zhou, D., Wu, L.: Variable Hardy-Lorentz spaces \(H^{p(\cdot ), q}(\mathbb{R}^n)\). Math. Nachr. 292, 309–349 (2019)

    Article  MathSciNet  Google Scholar 

  22. Kempka, H., Vybíral, J.: Lorentz spaces with variable exponents. Math. Nachr. 287, 938–954 (2014)

    Article  MathSciNet  Google Scholar 

  23. Latter, R.H.: A characterization of \({H}^p{({ R}}^n)\) in terms of atoms. Studia Math. 62, 92–101 (1978)

    Article  MathSciNet  Google Scholar 

  24. Lebesgue, H.: Recherches sur la convergence des séries de Fourier. Math. Ann. 61, 251–280 (1905)

    Article  MathSciNet  Google Scholar 

  25. Liu, J., Weisz, F., Yang, D., Yuan, W.: Variable anisotropic Hardy spaces and their applications. Taiwanese J. Math. 22, 1173–1216 (2018)

    Article  MathSciNet  Google Scholar 

  26. Liu, J., Weisz, F., Yang, D., Yuan, W.: Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications. J. Fourier Anal. Appl. 25, 874–922 (2019)

    Article  MathSciNet  Google Scholar 

  27. Lorentz, G.: Some new functional spaces. Ann. of Math. 51, 37–55 (1950)

    Article  MathSciNet  Google Scholar 

  28. S. Lu, Four Lectures on Real \({H}^p\) Spaces, World Scientific (Singapore, 1995)

  29. Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)

    Article  MathSciNet  Google Scholar 

  30. Nekvinda, A.: Hardy-Littlewood maximal operator on \(L^{p(x)}(\mathbb{R})\). Math. Inequal. Appl. 7, 255–265 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Persson, L.E., Tephnadze, G., Wall, P.: Maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21, 76–94 (2015)

    Article  MathSciNet  Google Scholar 

  32. Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equ. Oper. Theory 77, 123–148 (2013)

    Article  MathSciNet  Google Scholar 

  33. Simon, P.: Cesàro summability with respect to two-parameter Walsh systems. Monatsh. Math. 131, 321–334 (2000)

    Article  MathSciNet  Google Scholar 

  34. Simon, P.: \((C,\alpha )\) summability of Walsh-Kaczmarz-Fourier series. J. Approx. Theory 127, 39–60 (2004)

    Article  MathSciNet  Google Scholar 

  35. P. Simon, On a theorem of Feichtinger and Weisz, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput., 39 (2013), 391–403

  36. Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ, Press (Princeton, NJ (1993)

    MATH  Google Scholar 

  37. E. M. Stein, M. H. Taibleson, and G. Weiss, Weak type estimates for maximal operators on certain \({H}^p\) classes, Rend. Circ. Mat. Palermo, Suppl., 1 (1981), 81–97

  38. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ, Press (Princeton, N.J. (1971)

    MATH  Google Scholar 

  39. Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers. Boston, London, Dordrecht (2004)

    Book  Google Scholar 

  40. Uchiyama, A.: Hardy Spaces on the Euclidean Space. Springer Monographs in Mathematics, Springer (Berlin (2001)

    Book  Google Scholar 

  41. Weisz, F.: Summability of Multi-dimensional Fourier Series and Hardy Spaces. Kluwer Academic Publishers (Dordrecht, Boston, London, Mathematics and Its Applications (2002)

    Book  Google Scholar 

  42. Weisz, F.: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory 7, 1–179 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Weisz, F.: Convergence and Summability of Fourier Transforms and Hardy Spaces. Applied and Numerical Harmonic Analysis, Springer, Birkhäuser (Basel (2017)

    Book  Google Scholar 

  44. Weisz, F.: Summability of Fourier transforms in variable Hardy and Hardy-Lorentz spaces. Jaen J. Approx. 10, 101–131 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Yan, X., Yang, D., Yuan, W., Zhuo, C.: Variable weak Hardy spaces and their applications. J. Funct. Anal. 271, 2822–2887 (2016)

    Article  MathSciNet  Google Scholar 

  46. A. Zygmund, Trigonometric Series, 3rd ed., Cambridge University Press (London, 2002)

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Authors and Affiliations

  1. Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C, H-1117, Budapest, Hungary

    F. Weisz

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  1. F. Weisz
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Correspondence to F. Weisz.

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This research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH, K115804 and KH130426.

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Weisz, F. Summability of Fourier series in periodic Hardy spaces with variable exponent. Acta Math. Hungar. 162, 557–583 (2020). https://doi.org/10.1007/s10474-020-01056-z

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