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Sharp \((H_p, L_p)\) type inequalities of maximal operators of \(T\) means with respect to Vilenkin systems. (English) Zbl 1492.42034

Let \(f=(f^{(n)})_{n\in \mathbb{N}}\) be a martingale on a bounded type Vilenkin group and \(S_k f\) \((k=0,1,\dots)\) be the partial sums of the Vilenkin-Fourier series of \(f\). For a sequence of non-negative numbers \((q_k)\) with \(q_0>0\) the means \(T_n\) of the Vilenkin-Fourier series of \(f\) are defined as follows \[ T_n f=\frac{1}{Q_n}\sum_{k=0}^{n-1} q_k S_k f, \] where \(Q_n=q_0 +q_1 + \dots + q_{n-1}\).
Suppose \((q_k)\) is either a non-increasing sequence or a non-decreasing sequence satisfying the condition \(q_{n-1}/Q_n=O(1/n)\) \((n\rightarrow \infty)\). It is proved that for every \(p\in (0, 1/2]\) the following weighted maximal operator \[ M_pf=\sup_n\frac{|T_n f|}{(n+1)^{1/p -2} \log ^{2[1/2+p]} (n+1)} \] is bounded from the martingale Hardy space \(H_p\) to the space \(L_p\).
Suppose \((q_k)\) is a monotone sequence. For the parameters \(p\in(0,1/2)\) it is obtained the following estimate for \(L_p\)-norms of the means \(T_k f\) \[ \sum_{k=1}^{\infty}\frac{||T_k f||_p^{p}}{k^{2-2p}}\leq C_p||f||_{H_p}^{p}. \] For the endpoint case \(p=1/2\) under some additional restrictions on the sequence \((q_k)\) it is shown the estimate \[ \sum_{k=1}^{n}\frac{||T_k f||_{1/2}^{1/2}}{k \log n}\leq C||f||_{H_{1/2}}^{1/2} \;\;\;\;(n\in \mathbb{N}). \]

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
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References:

[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)) · Zbl 0588.43001
[2] Blahota, I., On a norm inequality with respect to Vilenkin-like systems, Acta Math. Hungar., 89, 1-2, 15-27 (2000) · Zbl 0973.42020
[3] Blahota, I.; Tephnadze, G., Strong convergence theorem for Vilenkin-Fejér means, Publ. Math. Debrecen, 85, 1-2, 181-196 (2014) · Zbl 1340.42065
[4] Gát, G., Investigations of certain operators with respect to the Vilenkin system, Acta Math. Hung., 61, 131-149 (1993) · Zbl 0805.42019
[5] Gát, G., Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory, 124, 1, 25-43 (2003) · Zbl 1032.43003
[6] Goginava, U., The maximal operator of Marcinkiewicz-Fejé r means of the d-dimensional Walsh-Fourier series, (English summary) East, J. Approx., 12, 3, 295-302 (2006) · Zbl 1487.42067
[7] Goginava, U.; Nagy, K., On the maximal operator of Walsh-Kaczmarz-Fejer means, Czechoslovak Math. J., 61, 3, 673-686 (2011) · Zbl 1249.42011
[8] Fujii, N., A maximal inequality for \(H^1\)-functions on a generalized Walsh-Paley group, Proc. Am. Math. Soc., 77, 1, 111-116 (1979) · Zbl 0415.43014
[9] Lukkassen, D.; Persson, LE; Tephnadze, G.; Tutberidze, G., Some inequalities related to strong convergence of Riesz logarithmic means of Vilenkin-Fourier series, J. Inequal. Appl. (2020) · Zbl 1503.26064
[10] Moore, CN, Summable Series and Convergence Factors (1966), New York: Dover Publications Inc, New York · Zbl 0142.30704
[11] Móricz, F.; Siddiqi, A., Approximation by Nörlund means of Walsh-Fourier series, (English summary), J. Approx. Theory, 70, 3, 375-389 (1992) · Zbl 0757.42009
[12] Nagy, K., Approximation by Nörlund means of Walsh-Kaczmarz-Fourier series, Georgian Math. J., 18, 1, 147-162 (2011) · Zbl 1210.42043
[13] Nagy, K., Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series, Anal. Math., 36, 4, 299-319 (2010) · Zbl 1240.42133
[14] Nagy, K., Approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions, Math. Inequal. Appl., 15, 2, 301-322 (2012) · Zbl 1243.42038
[15] Pál, J.; Simon, P., On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar, 29, 1-2, 155-164 (1977) · Zbl 0345.42011
[16] Persson, LE; Tephnadze, G.; Wall, P., Maximal operators of Vilenkin-Nörlund means, J. Fourier Anal. Appl., 21, 1, 76-94 (2015) · Zbl 1311.42071
[17] Persson, LE; Tephnadze, G.; Wall, P., On an approximation of 2-dimensional Walsh-Fourier series in the martingale Hardy spaces, Ann. Funct. Anal., 9, 1, 137-150 (2018) · Zbl 1382.42017
[18] Persson, LE; Tephnadze, G.; Wall, P., Some new \((H_p, L_p)\) type inequalities of maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients, J. Math. Inequal., 9, 4, 1055-1069 (2015) · Zbl 1329.42030
[19] Persson, LE; Tephnadze, G.; Tutberidze, G., On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces, Oper Matrices, 14, 1, 283-294 (2020) · Zbl 1465.42031
[20] Persson, L. E., Tephnadze, G., Tutberidze, G., Wall, P.: Strong summability result of Vilenkin-Fejér means on bounded Vilenkin groups. Ukr. Math. J., 73(4), (2021), 544-555. · Zbl 1479.42075
[21] Schipp, F., Certain rearrangements of series in the Walsh system, Mat. Zametki, 18, 2, 193-201 (1975) · Zbl 0349.42013
[22] Simon, P., Cesáro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131, 4, 321-334 (2000) · Zbl 0976.42014
[23] Simon, P., Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest. Eõtvõs Sect. Math., 27, 87-101 (1984) · Zbl 0586.43001
[24] Simon, P., Strong convergence theorem for Vilenkin-Fourier series, J. Math. Anal. Appl., 245, 52-68 (2000) · Zbl 0987.42022
[25] Tephnadze, G., Fejér means of Vilenkin-Fourier series, Studia Sci. Math. Hungar., 49, 1, 79-90 (2012) · Zbl 1265.42099
[26] Tephnadze, G., On the maximal operators of Vilenkin-Fejér means, Turk. J. Math., 37, 2, 308-318 (2013) · Zbl 1278.42037
[27] Tephnadze, G., On the maximal operators of Vilenkin-Fejér means on Hardy spaces, Math. Inequal. Appl., 16, 2, 301-312 (2013) · Zbl 1263.42008
[28] Tephnadze, G., On the maximal operators of Walsh-Kaczmarz-Fejér means, Period. Math. Hung., 67, 1, 33-45 (2013) · Zbl 1299.42097
[29] Tephnadze, G., Approximation by Walsh-Kaczmarz-Fejér means on the Hardy space, Acta Math. Sci., 34, 5, 1593-1602 (2014) · Zbl 1324.42043
[30] Tephnadze, G., On the maximal operators of Riesz logarithmic means of Vilenkin-Fourier series, Stud. Sci. Math. Hung., 51, 1, 105-120 (2014) · Zbl 1299.42098
[31] Tephnadze, G., On the partial sums of Vilenkin-Fourier series, J. Contemp. Math. Anal., 49, 1, 23-32 (2014) · Zbl 1345.43002
[32] Tephnadze, G., On the partial sums of Walsh-Fourier series, Colloq. Math., 141, 2, 227-242 (2015) · Zbl 1338.42040
[33] Tutberidze, G., A note on the strong convergence of partial sums with respect to Vilenkin system, J. Cont. Math. Anal., 54, 6, 319-324 (2019) · Zbl 1435.20050
[34] Tutberidze, G.: Modulus of continuity and boundedness of subsequences of Vilenkin- Fejér means in the martingale Hardy spaces. Geo. Math. J., (2021), doi:10.1515/gmj-2021-2106 · Zbl 1482.42072
[35] Tutberidze, G., Maximal operators of \(T\) means with respect to the Vilenkin system, Nonlinear Stud., 27, 4, 1-11 (2020) · Zbl 1478.42028
[36] Vilenkin, N. Ya., On a class of complete orthonormal systems, Am. Math. Soc. Transl. (2), 28, 1-35 (1963) · Zbl 0125.34304
[37] Weisz, F., Martingale Hardy Spaces and their Applications in Fourier Analysis. Lecture Notes in Mathematics (1994), Berlin: Springer, Berlin · Zbl 0796.60049
[38] Weisz, F., Cesáro summability of one and two-dimensional Fourier series, Anal. Math. Stud., 5, 353-367 (1996) · Zbl 0866.42019
[39] Weisz, F.: Hardy spaces and Cesáro means of two-dimensional Fourier series. In approximation Theory and Function Series, Budapest (Hungary), 1995, Volume 5 of Bolyai Soc. Math. Studia, pages 353-367, 1996. · Zbl 0866.42019
[40] Weisz, F., \(Q\)-summability of Fourier series, Acta Math. Hung., 103, 1-2, 139-175 (2004) · Zbl 1060.42021
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