On the maximal operator of Walsh-Kaczmarz-Fejér means

Ushangi Goginava; Károly Nagy

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 673-686
  • ISSN: 0011-4642

Abstract

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In this paper we prove that the maximal operator σ ˜ κ , * f : = sup n ∈ ℙ | σ n κ f | log 2 ( n + 1 ) , where σ n κ f is the n -th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space H 1 / 2 ( G ) to the space L 1 / 2 ( G ) .

How to cite

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Goginava, Ushangi, and Nagy, Károly. "On the maximal operator of Walsh-Kaczmarz-Fejér means." Czechoslovak Mathematical Journal 61.3 (2011): 673-686. <http://eudml.org/doc/197059>.

@article{Goginava2011,
abstract = {In this paper we prove that the maximal operator \[\tilde\{\sigma \}^\{\kappa ,*\}f:=\sup \_\{n\in \{\mathbb \{P\}\}\}\frac\{|\{\sigma \}\_n^\kappa f|\}\{\log ^\{2\}(n+1)\},\] where $\{\sigma \}_n^\kappa f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_\{1/2\}( G) $ to the space $L_\{1/2\}( G).$},
author = {Goginava, Ushangi, Nagy, Károly},
journal = {Czechoslovak Mathematical Journal},
keywords = {Walsh-Kaczmarz system; Fejér means; maximal operator; Walsh-Kaczmarz system; Fejér mean; maximal operator},
language = {eng},
number = {3},
pages = {673-686},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the maximal operator of Walsh-Kaczmarz-Fejér means},
url = {http://eudml.org/doc/197059},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Goginava, Ushangi
AU - Nagy, Károly
TI - On the maximal operator of Walsh-Kaczmarz-Fejér means
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 673
EP - 686
AB - In this paper we prove that the maximal operator \[\tilde{\sigma }^{\kappa ,*}f:=\sup _{n\in {\mathbb {P}}}\frac{|{\sigma }_n^\kappa f|}{\log ^{2}(n+1)},\] where ${\sigma }_n^\kappa f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1/2}( G) $ to the space $L_{1/2}( G).$
LA - eng
KW - Walsh-Kaczmarz system; Fejér means; maximal operator; Walsh-Kaczmarz system; Fejér mean; maximal operator
UR - http://eudml.org/doc/197059
ER -

References

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  1. Agaev, G. N., Vilenkin, N. Ya., Dzhafarli, G. M., Rubinshtein, A. I., Multiplicative systems of functions and harmonic analysis on 0-dimensional groups, ``ELM'' Baku 180 p (1981), Russian. (1981) 
  2. Fine, J., 10.1073/pnas.41.8.588, Proc. Nat. Acad. Sci. USA 41 (1955), 558-591. (1955) Zbl0065.05303MR0070757DOI10.1073/pnas.41.8.588
  3. Fujii, N. J., Cesàro summability of Walsh-Fourier series, Proc. Amer. Math. Soc. 77 (1979), 111-116. (1979) 
  4. Gát, G., 10.4064/sm-130-2-135-148, Studia Math. 130 (1998), 135-148. (1998) Zbl0905.42016MR1623340DOI10.4064/sm-130-2-135-148
  5. Gát, G., Goginava, U., Nagy, K., On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system, Studia Sci. Math. Hungarica 46 (2009), 399-421. (2009) MR2657025
  6. Goginava, U., The maximal operator of the Fejér means of the character system of the p -series field in the Kaczmarz rearrangement, Publ. Math. Debrecen 71 (2007), 43-55. (2007) Zbl1136.42024MR2340033
  7. Goginava, U., 10.1007/s10474-007-5268-6, Acta Math. Hungar. 115 (2007), 333-340. (2007) Zbl1174.42336MR2327986DOI10.1007/s10474-007-5268-6
  8. Goginava, U., Maximal operators of Fejér-Walsh means, Acta Sci. Math. (Szeged) 74 (2008), 615-624. (2008) Zbl1199.42127MR2487936
  9. Goginava, U., The maximal operator of the Marcinkiewicz-Fejér means of the d -dimensional Walsh-Fourier series, East J. Approx. 12 (2006), 295-302. (2006) MR2252557
  10. Schipp, F., Wade, W. R., Simon, P., Pál, J., Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol-New York (1990). (1990) MR1117682
  11. Schipp, F., Certain rearrengements of series in the Walsh series, Mat. Zametki 18 (1975), 193-201. (1975) MR0390633
  12. Schipp, F., 10.1007/BF02079908, Anal. Math. 2 (1976), 65-76. (1976) Zbl0343.42009MR0415190DOI10.1007/BF02079908
  13. Simon, P., 10.1007/s006050070004, Monatsh. Math. 131 (2000), 321-334. (2000) MR1813992DOI10.1007/s006050070004
  14. Simon, P., 10.1006/jath.2000.3488, J. Approx. Theory 106 (2000), 249-261. (2000) Zbl0987.42021MR1788275DOI10.1006/jath.2000.3488
  15. Skvortsov, V. A., 10.1007/BF02350811, Analysis Math. 7 (1981), 141-150. (1981) Zbl0472.42014MR0633073DOI10.1007/BF02350811
  16. Šneider, A. A., On series with respect to the Walsh functions with monotone coefficients, Izv. Akad. Nauk SSSR Ser. Math. 12 (1948), 179-192. (1948) MR0025605
  17. Yano, S. H., 10.2748/tmj/1178245527, Tohoku Math. J. 3 (1951), 223-242. (1951) MR0045236DOI10.2748/tmj/1178245527
  18. Young, W. S., 10.1090/S0002-9939-1974-0350310-6, Proc. Amer. Math. Soc. 44 (1974), 353-358. (1974) MR0350310DOI10.1090/S0002-9939-1974-0350310-6
  19. Weisz, F., Martingale Hardy spaces and their applications in Fourier analysis, Springer-Verlang, Berlin (1994). (1994) Zbl0796.60049MR1320508
  20. Weisz, F., Summability of multi-dimensional Fourier series and Hardy space, Kluwer Academic, Dordrecht (2002). (2002) MR2009144
  21. Weisz, F., 10.1007/BF02205221, Anal. Math. 22 (1996), 229-242. (1996) Zbl0866.42020MR1627638DOI10.1007/BF02205221
  22. Weisz, F., 10.1023/B:AMHU.0000028241.87331.c5, Acta Math. Hungar. 103 (2004), 139-176. (2004) Zbl1060.42021MR2047878DOI10.1023/B:AMHU.0000028241.87331.c5

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