Banach Journal of Mathematical Analysis

Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces

Kristóf Szarvas and Ferenc Weisz

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We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space Hp to the classical Lebesgue space Lp and from the variable dyadic martingale Hardy space Hp() to the variable Lebesgue space Lp(). Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from Hp() to Lp() and from the variable Hardy–Lorentz space Hp(),q to the variable Lorentz space Lp(),q. As a consequence, we can prove theorems about almost everywhere and norm convergence.

Article information

Banach J. Math. Anal., Volume 13, Number 3 (2019), 675-696.

Received: 27 June 2018
Accepted: 4 November 2018
First available in Project Euclid: 31 May 2019

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Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces
Secondary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 60G42: Martingales with discrete parameter 60G46: Martingales and classical analysis 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

variable Hardy spaces variable Hardy–Lorentz spaces Cesàro means Riesz means Cesàro and Riesz maximal operator boundedness


Szarvas, Kristóf; Weisz, Ferenc. Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces. Banach J. Math. Anal. 13 (2019), no. 3, 675--696. doi:10.1215/17358787-2018-0037.

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  • [1] H. Aoyama, Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J. 39 (2009), no. 2, 207–216.
  • [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
  • [3] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19–42.
  • [4] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304.
  • [5] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.
  • [6] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013.
  • [7] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264.
  • [8] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238.
  • [9] D. Cruz-Uribe and L. D. Yang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • [10] L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004) no. 2, 245–253.
  • [11] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
  • [12] N. Fujii, A maximal inequality for ${H}^{1}$-functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc. 77 (1979), no. 1, 111–116.
  • [13] G. Gát, On $(C,1)$ summability of integrable functions with respect to the Walsh-Kaczmarz system, Studia Math. 130 (1998), no. 2, 135–148.
  • [14] G. Gát, Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups, J. Approx. Theory 101 (1999), no. 1, 1–36.
  • [15] 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), no. 2, 311–322.
  • [16] G. Gát and U. Goginava, A weak type inequality for the maximal operator of $(C,\alpha)$-means of Fourier series with respect to the Walsh-Kaczmarz system, Acta Math. Hungar. 125 (2009), nos. 1–2, 65–83.
  • [17] U. Goginava, On some $(H_{p,q},L_{p,q})$-type maximal inequalities with respect to the Walsh-Paley system, Georgian Math. J. 7 (2000), no. 3, 475–488.
  • [18] U. Goginava, Almost everywhere summability of multiple Walsh-Fourier series, J. Math. Anal. Appl. 287 (2003), no. 1, 90–100.
  • [19] U. Goginava, Maximal operators of $(C,\alpha)$-means of cubic partial sums of d-dimensional Walsh-Fourier series, Anal. Math. 33 (2007), no. 4, 263–286.
  • [20] B. Golubov, A. Efimov, and V. Skvortsov, Walsh Series and Transforms, Math. Appl. (Soviet Ser.) 64, Kluwer Academic, Dordrecht, 1991.
  • [21] C. Herz, $H_{p}$-spaces of martingales, $0<p\leq 1$, Z. Wahrscheinlichkeitstheor. Verw. Geb. 28 (1974), 189–205.
  • [22] Y. Jiao, D. Zhou, Z. Hao, and W. Chen, Martingale Hardy spaces with variable exponents, Banach J. Math. Anal. 10 (2016), no. 4, 750–770.
  • [23] Y. Jiao, D. Zhou, F. Weisz, and Z. Hao, Fractional integral on martingale Hardy spaces with variable exponents, Fract. Calc. Appl. Anal. 18 (2015), no. 5, 1128–1145. Corrigendum, Fract. Calc. Appl. Anal. 20 (2017), no. 4, 1051–1052
  • [24] Y. Jiao, D. Zhou, F. Weisz, and L. Wu, Variable martingale Hardy spaces and their applications in Fourier analysis, preprint, arXiv:1809.07520v1 [math.PR].
  • [25] Y. Jiao, Y. Zuo, D. Zhou, and L. Wu, Variable Hardy-Lorentz spaces $H^{p(\cdot),q}(\mathbb{R}^{n})$, Math. Nachr. 292 (2019), no. 2, 309–349.
  • [26] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618.
  • [27] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122–132.
  • [28] E. Nakai and G. Sadasue, Maximal function on generalized martingale Lebesgue spaces with variable exponent, Statist. Probab. Lett. 83 (2013), no. 10, 2168–2171.
  • [29] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • [30] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R}^{n})$, Math. Inequal. Appl. 7 (2004), no. 2, 255–265.
  • [31] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148.
  • [32] F. Schipp, W. R. Wade, and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [33] P. Simon, Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math. 131 (2000), no. 4, 321–334.
  • [34] P. Simon and F. Weisz, Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series, J. Approx. Theory 151 (2008), no. 1, 1–19.
  • [35] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [36] F. Weisz, Cesàro summability of one- and two-dimensional Walsh-Fourier series, Anal. Math. 22 (1996), no. 3, 229–242.
  • [37] F. Weisz, Summability of Multi-Dimensional Fourier Series and Hardy Spaces, Math. Appl. 541, Kluwer Academic, Dordrecht, 2002.
  • [38] X. Yan, D. Yang, W. Yuan, and C. Zhou, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822–2887.