Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series. (English) Zbl 1143.42032

Summary: The maximal operators for Cesàro or \((C,\alpha)\) and Riesz summability with respect to Walsh-Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from \(H_p\) to \(L_p\) for all \(1/(\alpha+1)<p<\infty\). In this work we prove that this boundedness result does not hold anymore if \(p\leq 1/(\alpha+1)\). However, for \(p=1/(\alpha+1)\) the maximal operators are bounded from \(H_{1/(\alpha+1)}\) to the weak \(L_{1/(\alpha+1)}\) space. To the proof some known estimations for the Cesàro and Riesz kernels have to be sharpened.


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
60G42 Martingales with discrete parameter
60G46 Martingales and classical analysis
46B42 Banach lattices
46E40 Spaces of vector- and operator-valued functions
Full Text: DOI


[1] Fine, N. J., On the Walsh functions, Trans. Amer. Math. Soc., 65, 372-414 (1949) · Zbl 0036.03604
[2] Fine, N. J., Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. USA, 41, 558-591 (1955) · Zbl 0065.05303
[3] Fujii, N., A maximal inequality for \(H^1\)-functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc., 77, 111-116 (1979) · Zbl 0415.43014
[4] Gát, G., On \((C, 1)\) summability of integrable functions with respect to the Walsh-Kaczmarz system, Studia Math., 130, 135-148 (1998) · Zbl 0905.42016
[5] 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, 43-45 (2007) · Zbl 1136.42024
[6] Schipp, F., Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest Sect. Math., 18, 189-195 (1975) · Zbl 0351.42012
[7] Schipp, F.; Simon, P., On some \((H, L_1)\)-type maximal inequalities with respect to the Walsh-Paley system, (Functions, Series, Operators, Proceedings Conference in Budapest, 1980, Coll. Math. Soc. J. Bolyai, vol. 35 (1981), North-Holland: North-Holland Amsterdam), 1039-1045 · Zbl 0535.42019
[8] Schipp, F.; Wade, W. R.; Simon, P.; Pál, J., Walsh Series: An Introduction to Dyadic Harmonic Analysis (1990), Adam Hilger: Adam Hilger Bristol, New York · Zbl 0727.42017
[9] Simon, P., Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest Sect. Math., 27, 87-101 (1985) · Zbl 0586.43001
[10] Simon, P., Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131, 321-334 (2000) · Zbl 0976.42014
[11] Simon, P., On the Cesáro summability with respect to the Walsh-Kaczmarz system, J. Approx. Theory, 106, 249-261 (2000) · Zbl 0987.42021
[12] Wade, W. R., Summability estimates of double Vilenkin-Fourier series, Math. Pannonica, 10, 67-75 (1999) · Zbl 0928.42012
[13] Weisz, F., The maximal Cesàro operator on Hardy spaces, Analysis, 18, 157-166 (1998) · Zbl 0944.42007
[14] Weisz, F., Riesz means of Fourier transforms and Fourier series on Hardy spaces, Studia Math., 131, 253-270 (1998) · Zbl 0934.42004
[15] Weisz, F., \( \theta \)-summation and Hardy spaces, J. Approx. Theory, 107, 121-142 (2000) · Zbl 0987.42012
[16] Weisz, F., \((C, \alpha)\) summability of Walsh-Fourier series, Anal. Math., 27, 141-155 (2001) · Zbl 0992.42016
[17] Weisz, F., Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and its Applications (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 1306.42003
[18] Weisz, F., Weak type inequalities for the Walsh and bounded Ciesielski systems, Anal. Math., 30, 147-160 (2004) · Zbl 1068.42026
[19] Zygmund, A., Trigonometric Series (2002), Cambridge Press: Cambridge Press London · Zbl 1084.42003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.