Investigations of certain operators with respect to the Vilenkin system. (English) Zbl 0805.42019

The author examines Vilenkin analogues of Walsh series results of P. Simon for strong convergence [Acta Math. Hung. 49, 425-431 (1987; Zbl 0643.42020)] and for the Sunouchi operator [Acta Math. Hung. 46, 307-310 (1985; Zbl 0591.42019)]. He shows that if \(f\) belongs to the atomic Hardy space \(H^ 1(G_ m)\), for any Vilenkin group \(G_ m\), then \[ \lim_{n\to\infty} \log^{-1} n \sum^ n_{k=0} k^{-1} \| S_ k f\|_ 1= \| f\|_ 1, \] and if \(f\) belongs to the martingale Hardy space \(H(G_ m)\), for a Vilenkin group \(G_ m\) of bounded type, then \[ Tf:= \left(\sum^ \infty_{n=0} | S_{M_ n}f- \sigma_{M_ n} f|^ 2\right)^{{1\over 2}} \] is a bounded operator from \(H(G_ m)\) into \(L^ 1(G_ m)\). He shows this last result never holds if \(G_ m\) is of unbounded type. However, if \(H(G_ m)\) is replaced by \(H^ 1(G_ m)\), then \(T\) can be a bounded operator for some Vilenkin groups of unbounded type. In fact, the author proves that \(T\) is bounded from \(H^ 1(G_ m)\) to \(L^ 2(G_ m)\) if and only if \(m_ k':= M^{-1}_{k+1} \sum^{k-1}_{j= 0} M_{j+1}\log m_ j\) is bounded.


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Full Text: DOI


[1] G. H. Agaev, N. Ja. Vilenkin, G. M. Dzsafarli and A. I. Rubinstein,Multiplicative Systems of Functions and Harmonic Analysis on 0-Dimensional Groups, Izd. ELM (Baku, 1981) (in Russian).
[2] R. R. Coifman and G. L. Weiss, Extensions of Hardy spaces and their use in analysis,Bull. Amer. Math. Soc.,83 (1977), 569–645. · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[3] N. J. Fine, On the Walsh functions,TAMS,65 (1949), 372–414. · Zbl 0036.03604 · doi:10.1090/S0002-9947-1949-0032833-2
[4] S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system,Acta Math. Hungar.,45 (1985), 223–234. · Zbl 0577.42021 · doi:10.1007/BF01955039
[5] F. Schipp and P. Simon, On some (L, H)-type maximal inequalities with respect to the Walsh-Paley system,Colloquia Math. Soc. János Bolyai, 35. Functions, Series, Operators (Budapest, 1980), pp. 1039–1045.
[6] F. Schipp and P. Simon, Investigation of Haar and Franklin series in the Hardy spaces,Analysis Math.,8 (1982), 47–56. · Zbl 0496.42011 · doi:10.1007/BF02073771
[7] P. Simon and J. Pál, On a generalization of the concept of derivative,Acta Math. Acad. Sci. Hungar.,29 (1977), 155–164. · Zbl 0345.42011 · doi:10.1007/BF01896477
[8] P. Simon, Strong convergence of certain means with respect to the Walsh-Fourier series,Acta Math. Hungar.,49 (1987), 425–431. · Zbl 0643.42020 · doi:10.1007/BF01951006
[9] P. Simon, (L 1,H)-type estimations for some operators with respect to the Walsh-Paley system,Acta Math. Hungar.,46 (1985), 307–310. · Zbl 0591.42019 · doi:10.1007/BF01955744
[10] P. Simon, Investigations with respect to the Vilenkin system,Annales Univ. Sci. Budapest., Sectio Mathematica,27 (1982), 87–101. · Zbl 0586.43001
[11] B. Smith,A Strong Convergence Theorem for H 1(T), Lecture Notes in Math., 995, Springer (Berlin-New York, 1983), pp. 169–173.
[12] G. I. Sunouchi, On the Walsh-Kaczmarz series,Proc. Amer. Math. Soc.,2 (1951), 5–11. · Zbl 0044.07103 · doi:10.1090/S0002-9939-1951-0041259-1
[13] G. I. Sunouchi, Strong summability of Walsh-Fourier series,Tohoku Math. J.,16 (1964), 228–237. · Zbl 0146.08902 · doi:10.2748/tmj/1178243709
[14] N. Ja. Vilenkin, On a class of complete orthonormal systems,Izd. Akad. Nauk SSSR,11 (1947), 363–400 (in Russian). · Zbl 0036.35601
[15] A. Zygmund,Trigonometrical Series, Cambridge University Press (New York, 1959). · Zbl 0085.05601
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.