Investigations with respect to the Vilenkin system. (English) Zbl 0586.43001

Let \(m=(m_ 0,m_ 1,...,m_ k,...)\) be a sequence of natural numbers with \(m_ k\geq 2\) and denote by \({\mathbb{Z}}_{m_ k}\) the \(m_ k\)-th discrete cyclic group. Let \(G_ m\) be the direct product of \({\mathbb{Z}}_{m_ k}\) and \(\{\psi_ n\}\) the character group of \(G_ m\) that is called the Vilenkin system. In this paper, a Bernstein type inequality, the (C,1)-summation in the Hardy space and the conjugation are investigated. For example, the author proves that the maximal operator \(\sigma^*: H^ 1(G_ m)\to L^ 1(G_ m)\) is bounded if and only if \(\sup_{n}m_ n<\infty\). The part ”if” is known [cf. N. Fujii, Proc. Am. Math. Soc. 77, 111-116 (1979; Zbl 0415.43014)]. In the case of \(\sup_{n}m_ n=\infty\), the \(H^ 1\)-space is not atomic, but - modifying the concept of the ”atoms” - the author proves the part of ”only if”.
Reviewer: T.Nakazi


43A17 Analysis on ordered groups, \(H^p\)-theory
26D15 Inequalities for sums, series and integrals
43A55 Summability methods on groups, semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)


Zbl 0415.43014