Project Euclid

References

• [1] H. Aoyama, Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J. 39 (2009), no. 2, 207–216.
  Zentralblatt MATH: 1190.46026
  Digital Object Identifier: doi:10.32917/hmj/1249046337
  Project Euclid: euclid.hmj/1249046337
  
• [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
  Zentralblatt MATH: 0306.60030
  Digital Object Identifier: doi:10.1214/aoms/1177699141
  Project Euclid: euclid.aoms/1177699141
  
• [3] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19–42.
  Zentralblatt MATH: 0301.60035
  Digital Object Identifier: doi:10.1214/aop/1176997023
  Project Euclid: euclid.aop/1176997023
  
• [4] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304.
  Zentralblatt MATH: 0223.60021
  Digital Object Identifier: doi:10.1007/BF02394573
  Project Euclid: euclid.acta/1485889655
  
• [5] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.
  Zentralblatt MATH: 0144.06402
  Digital Object Identifier: doi:10.1007/BF02392815
  Project Euclid: euclid.acta/1485889479
  
• [6] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013.
  Zentralblatt MATH: 1268.46002
  
• [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.
  Mathematical Reviews (MathSciNet): MR2210118
  Zentralblatt MATH: 1100.42012
  
• [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.
  Zentralblatt MATH: 1037.42023
  
• [9] D. Cruz-Uribe and L. D. Yang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  Zentralblatt MATH: 1311.42053
  Digital Object Identifier: doi:10.1512/iumj.2014.63.5232
  
• [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.
  Zentralblatt MATH: 1284.42084
  Digital Object Identifier: doi:10.1007/s10114-013-1766-3
  
• [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.
  Zentralblatt MATH: 1050.42018
  Digital Object Identifier: doi:10.1016/S0022-247X(03)00503-1
  
• [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.
  Zentralblatt MATH: 1354.42042
  Digital Object Identifier: doi:10.1215/17358787-3649326
  
• [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
  Zentralblatt MATH: 1341.42027
  Digital Object Identifier: doi:10.1515/fca-2015-0065
  
• [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].
  arXiv: 1809.07520v1
  
• [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.
  Zentralblatt MATH: 65.0266.01
  Digital Object Identifier: doi:10.4064/fm-32-1-122-132
  
• [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.
  Zentralblatt MATH: 1295.60052
  Digital Object Identifier: doi:10.1016/j.spl.2013.06.007
  
• [29] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  Zentralblatt MATH: 1244.42012
  Digital Object Identifier: doi:10.1016/j.jfa.2012.01.004
  
• [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.
  Zentralblatt MATH: 1293.42025
  Digital Object Identifier: doi:10.1007/s00020-013-2073-1
  
• [32] F. Schipp, W. R. Wade, and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  Zentralblatt MATH: 0727.42017
  
• [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.
  Zentralblatt MATH: 1143.42032
  Digital Object Identifier: doi:10.1016/j.jat.2007.05.004
  
• [35] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  Zentralblatt MATH: 0796.60049
  
• [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.
  Zentralblatt MATH: 1306.42003
  
• [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.
  Zentralblatt MATH: 06636005
  Digital Object Identifier: doi:10.1016/j.jfa.2016.07.006