## Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces.(English)Zbl 1418.42037

Summary: 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 $$H_p$$ to the classical Lebesgue space $$L_p$$ and from the variable dyadic martingale Hardy space $$H_{p(\cdot)}$$ to the variable Lebesgue space $$L_{p(\cdot)}$$. Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from $$H_{p(\cdot)}$$ to $$L_{p(\cdot)}$$ and from the variable Hardy-Lorentz space $$H_{p(\cdot),q}$$ to the variable Lorentz space $$L_{p(\cdot),q}$$. As a consequence, we can prove theorems about almost everywhere and norm convergence.

### MSC:

 42B30 $$H^p$$-spaces 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 ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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### References:

 [1] H. Aoyama, Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J. 39 (2009), no. 2, 207-216. · Zbl 1190.46026 [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. · Zbl 0306.60030 [3] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42. · Zbl 0301.60035 [4] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. · Zbl 0223.60021 [5] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. · Zbl 0144.06402 [6] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013. · Zbl 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. · Zbl 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. · Zbl 1037.42023 [9] D. Cruz-Uribe and L. D. Yang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447-493. · Zbl 1311.42053 [10] L. Diening, Maximal function on generalized Lebesgue spaces $$L^{p(\cdot)}$$, Math. Inequal. Appl. 7 (2004) no. 2, 245-253. · Zbl 1071.42014 [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. · Zbl 1222.46002 [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. · Zbl 0415.43014 [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. · Zbl 0905.42016 [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. · Zbl 1284.42084 [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. · Zbl 0979.42014 [18] U. Goginava, Almost everywhere summability of multiple Walsh-Fourier series, J. Math. Anal. Appl. 287 (2003), no. 1, 90-100. · Zbl 1050.42018 [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
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