On a class of complete orthonormal systems

In [14] we investigated some Vilenkin-Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak -L 1/(1+α) , (0 < α 1). In this paper we construct a martingale in the space H 1/(1+α) , which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin-Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α) . In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

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“…The Vilenkin system is orthonormal and complete in L 2 (G m ) (see [17]). The norm (or quasi-norm) of the space…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

Memić

Persson

Tephnadze

2016

Studia Scientiarum Mathematicarum Hungarica

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“…Let oj^ y ■ denote the element of this partition which contains x. We have uN,v,j2 Ç wJv>,/i»"(wiv>,/2) = *(wJv>,/2)' and' frorn (12) and (9) l5fe,,/j)^^)-5fe,,/2)^^1 <l5fe.,/j)^yW-«iV>I,,/j)/Ar'V)l + IS*«'. /^'"(X) -S5'y .…”

Section: '»=V0mentioning

confidence: 99%

On Rearrangements of Vilenkin-Fourier Series which Preserve almost Everywhere Convergence

Gosselin¹,

Young²

1975

Transactions of the American Mathematical Society

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ABSTRACT. It is known that the partial sums of Vilenkin-Fourier series of Lq functions (q > 1) converge a.e. In this paper we establish the L result for a class of rearrangements of the Vilenkin-Fourier series, and the Lq result (1 < q < 2) for a subclass of rearrangements.In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. 2Balashov. The L result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.

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“…Each ψ n is a character of G m , and all the characters of G m are of this form [9]. Define the m-adic addition as…”

Section: Introductionmentioning

confidence: 99%

Integrals of weighted maximal logarithmic kernels on bounded Vilenkin groups

Mező

Nagy

2012

Period Math Hung

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On a class of complete orthonormal systems

Cited by 90 publications

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A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

On Rearrangements of Vilenkin-Fourier Series which Preserve almost Everywhere Convergence

Integrals of weighted maximal logarithmic kernels on bounded Vilenkin groups

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