On the maximal operators of Vilenkin-Fejéer means

The main aim of this paper is to prove that the maximal operator \overset{\sim}{s}\astf:=\underset{n \in P} \sup\frac{|s nf|}{\log2 (n+1)} is bounded from the Hardy space H1/2 to the space L1/2, where s nf are Fejér means of bounded Vilenkin-Fourier series.

Anahtar Kelimeler:

Vilenkin system, Fejéer means, martingale Hardy space

On the maximal operators of Vilenkin-Fejéer means

Keywords:

Vilenkin system, Fejéer means, martingale Hardy space,

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[1] Agaev, G. N., Vilenkin, N. Ya., Dzhafarly, G. M., Rubinshtein, A. I.: Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehim, 1981 (in Russian).
[2] Blahota, I., G´at, G., Goginava, U.: Maximal operators of Fejer means of double Vilenkin-Fourier series, Colloq. Math. 107, no. 2, 287–296, (2007).
[3] Blahota, I., G´at, G., Goginava, U.: Maximal operators of Fej´er means of Vilenkin-Fourier series, JIPAM. J. Inequal. Pure Appl. Math. 7, 1–7, (2006).
[4] Fujii, N. J.: A maximal inequality for H1 functions on generalized Walsh-Paley group, Proc. Amer. Math. Soc. 77, (1979), 111–116.
[5] G´at, G.: Ces`aro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory 124, no. 1, 25–43, (2003).
[6] Goginava, U.: Maximal operators of Fej´er-Walsh means, Acta Sci. Math. (Szeged) 74, no. 3-4, 615–624, (2008).
[7] Goginava, U.: The maximal operator of the Fej´er means of the character system of the p-series field in the Kaczmarz rearrangement, Publ. Math. Debrecen 71, no. 1-2, 43–55, (2007).
[8] Goginava, U.: Maximal operators of Fej´er means of double Walsh-Fourier series, Acta Math. Hungar. 115, no. 4, 333–340, (2007).
[9] Goginava, U., Nagy, K.: On the maximal operator of Walsh-Kaczmarz-Fejer means, Czechoslovak Math. J. (to appear).
[10] P´al, J., Simon, J.: On a generalization of the concept of derivate, Acta Math. Hung., 29, 155–164, (1977).
[11] Schipp, F.: Certain rearrangements of series in the Walsh series, Mat. Zametki, 18, 193–201, (1975).
[12] Simon, P.: Ces`aro summability wish respect to two-parameter Walsh systems, Monatsh. Math.,131, 321–334, (2000).
[13] Simon, P.: Investigations with respect to the Vilenkin system, Annales Univ. Sci. Budapest Eotv., Sect. Math., 28, 87–101, (1985).
[14] Vilenkin, N.Ya.: A class of complete orthonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat., 11, 363–400, (1947).
[15] Weisz, F.: Martingale Hardy spaces and their applications in Fourier Analysis, Springer, Berlin-Heidelberg-New York, 1994.
[16] Weisz, F.: Cesro summability of one and two-dimensional Fourier series, Anal. Math. 5 (1996), 353-367.re Appl. Math. 7, 1–7, (2006).
[17] Weisz, F.: Summability of multi-dimensional Fourier series and Hardy space, Kluwer Academic, Dordrecht, 2002.
[18] Weisz, F.: Weak type inequalities for the Walsh and bounded Ciesielski systems. Anal. Math. 30, no. 2, 147–160, (2004).
[19] Zygmund, A.: Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.