Accessible Unlicensed Requires Authentication Published online by De Gruyter October 21, 2021

Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

Giorgi Tutberidze

Abstract

In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space Hp to the Lebesgue space Lp for all 0<p<12.

MSC 2010: 42C10; 42B25

Funding source: Shota Rustaveli National Science Foundation

Award Identifier / Grant number: PHDF-18-476

Funding statement: The research was supported by Shota Rustaveli National Science Foundation grant PHDF-18-476.

Acknowledgements

The author would like to thank the referee for helpful suggestions which improved the final version of the paper.

References

[1] G. N. Agaev, N. Y. Vilenkin, G. M. Dzhafarli and A. I. Rubinshteĭn, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups (in Russian), “Èlm”, Baku, 1981. Search in Google Scholar

[2] I. Blahota, G. Gát and U. Goginava, Maximal operators of Fejér means of Vilenkin–Fourier series, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 4, Article ID 149. Search in Google Scholar

[3] I. Blahota, G. Gát and U. Goginava, Maximal operators of Fejér means of double Vilenkin–Fourier series, Colloq. Math. 107 (2007), no. 2, 287–296. Search in Google Scholar

[4] I. Blahota and G. Tephnadze, Strong convergence theorem for Vilenkin–Fejér means, Publ. Math. Debrecen 85 (2014), no. 1–2, 181–196. Search in Google Scholar

[5] N. Fujii, A maximal inequality for H1-functions on a generalized Walsh–Paley group, Proc. Amer. Math. Soc. 77 (1979), no. 1, 111–116. Search in Google Scholar

[6] G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory 124 (2003), no. 1, 25–43. Search in Google Scholar

[7] U. Goginava, Maximal operators of Fejér means of double Walsh–Fourier series, Acta Math. Hungar. 115 (2007), no. 4, 333–340. Search in Google Scholar

[8] U. Goginava and K. Nagy, On the maximal operator of Walsh–Kaczmarz–Fejér means, Czechoslovak Math. J. 61(136) (2011), no. 3, 673–686. Search in Google Scholar

[9] J. Pál and P. Simon, On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1–2, 155–164. Search in Google Scholar

[10] L.-E. Persson and G. Tephnadze, A sharp boundedness result concerning some maximal operators of Vilenkin–Fejér means, Mediterr. J. Math. 13 (2016), no. 4, 1841–1853. Search in Google Scholar

[11] L.-E. Persson, G. Tephnadze and G. Tutberidze, On the boundedness of subsequences of Vilenkin–Fejér means on the martingale Hardy spaces, Oper. Matrices 14 (2020), no. 1, 283–294. Search in Google Scholar

[12] L.-E. Persson, G. Tephnadze, G. Tutberidze and P. Wall, Some new results concerning strong convergence of Fejér means with respect to Vilenkin systems, Ukraïn. Mat. Zh. 73 (2021), no. 4, 544–555. Search in Google Scholar

[13] L.-E. Persson, G. Tephnadze and P. Wall, Maximal operators of Vilenkin–Nörlund means, J. Fourier Anal. Appl. 21 (2015), no. 1, 76–94. Search in Google Scholar

[14] L. E. Persson, G. Tephnadze and P. Wall, On an approximation of 2-dimensional Walsh–Fourier series in martingale Hardy spaces, Ann. Funct. Anal. 9 (2018), no. 1, 137–150. Search in Google Scholar

[15] F. Schipp, W. R. Wade and P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. Search in Google Scholar

[16] P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984), 87–101. Search in Google Scholar

[17] P. Simon, Cesaro summability with respect to two-parameter Walsh systems, Monatsh. Math. 131 (2000), no. 4, 321–334. Search in Google Scholar

[18] F. Šipp, Certain rearrangements of series in the Walsh system (in Russian), Mat. Zametki 18 (1975), no. 2, 193–201. Search in Google Scholar

[19] G. Tephnadze, On the maximal operators of Vilenkin–Fejér means, Turkish J. Math. 37 (2013), no. 2, 308–318. Search in Google Scholar

[20] G. Tephnadze, On the maximal operators of Vilenkin–Fejér means on Hardy spaces, Math. Inequal. Appl. 16 (2013), no. 1, 301–312. Search in Google Scholar

[21] G. Tephnadze, On the maximal operators of Walsh–Kaczmarz–Fejér means, Period. Math. Hungar. 67 (2013), no. 1, 33–45. Search in Google Scholar

[22] G. Tephnadze, On the Vilenkin–Fourier coefficients, Georgian Math. J. 20 (2013), no. 1, 169–177. Search in Google Scholar

[23] G. Tephnadze, Approximation by Walsh–Kaczmarz–Fejér means on the Hardy space, Acta Math. Sci. Ser. B (Engl. Ed.) 34 (2014), no. 5, 1593–1602. Search in Google Scholar

[24] G. Tephnadze, On the partial sums of Walsh–Fourier series, Colloq. Math. 141 (2015), no. 2, 227–242. Search in Google Scholar

[25] G. Tephnadze, On the convergence of Fejér means of Walsh–Fourier series in the space Hp, Izv. Nats. Akad. Nauk Armenii Mat. 51 (2016), no. 2, 54–70. Search in Google Scholar

[26] G. Tephnadze, On the convergence of partial sums with respect to Vilenkin system on the Martingale Hardy spaces, Izv. Nats. Akad. Nauk Armenii Mat. 53 (2018), no. 5, 77–94. Search in Google Scholar

[27] G. Tutberidze, A note on the strong convergence of partial sums with respect to Vilenkin system, Izv. Nats. Akad. Nauk Armenii Mat. 54 (2019), no. 6, 81–87. Search in Google Scholar

[28] G. Tutberidze, Maximal operators of T means with respect to the Vilenkin system, Nonlinear Stud. 27 (2020), no. 4, 1157–1167. Search in Google Scholar

[29] N. Vilenkin, On a class of complete orthonormal systems, Bull. Acad. Sci. URSS. Sér. Math. 11 (1947), 363–400. Search in Google Scholar

[30] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. Search in Google Scholar

[31] F. Weisz, Cesàro summability of one- and two-dimensional Walsh–Fourier series, Anal. Math. 22 (1996), no. 3, 229–242. Search in Google Scholar

[32] F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series, Approximation Theory and Function Series (Budapest 1995), Bolyai Soc. Math. Stud. 5, János Bolyai Mathematical Society, Budapest (1996), 353–367. Search in Google Scholar

[33] F. Weisz, Weak type inequalities for the Walsh and bounded Ciesielski systems, Anal. Math. 30 (2004), no. 2, 147–160. Search in Google Scholar

Received: 2019-02-13
Revised: 2020-12-25
Accepted: 2021-02-07
Published Online: 2021-10-21

© 2021 Walter de Gruyter GmbH, Berlin/Boston