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Now showing items 1-10 of 10

#### Some inequalities related to strong convergence of Riesz logarithmic means

(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-03-23)

In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series.

#### On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces

(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-03)

In this paper we characterize subsequences of Fejér means with respect to Vilenkin systems, which are bounded from the Hardy space <i>H<sub>p</sub></i> to the Lebesgue space <i>L<sub>p</sub></i>, for all 0 < p < 1/2. The result is in a sense sharp.

#### A sharp boundedness result for restricted maximal operators of Vilenkin-Fourier series on martingale Hardy spaces

(Journal article; Peer reviewed, 2018-09-20)

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space
H
p
to the Lebesgue space
L
p
for all
0<p≤1
. We also prove that the result is sharp in a particular sense.

#### Vilenkin–Lebesgue Points and Almost Everywhere Convergence for Some Classical Summability Methods

(Journal article; Tidsskriftartikkel; Peer reviewed, 2022-09-17)

The concept of Vilenkin–Lebesgue points was introduced in
[12], where the almost everywhere convergence of Fejer means of
Vilenkin–Fourier series was proved. In this paper, we present a different
(and simpler) approach to prove a similar result, which can be used to
prove that the corresponding result holds also in a more general context,
namely for regular Norlund and T-means.

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

(Peer reviewed; Journal article; Tidsskriftsartikkel, 2016)

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 H1/(1+α) to the space weak-L1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H1/(1+α), which satisfies the conditions considered in [14], and so ...

#### Some weak type inequalities and almost everywhere convergence of Vilenkin–Nörlund means

(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-05-04)

We prove and discuss some new weak type (1, 1) inequalities of maximal operators of
Vilenkin–Nörlund means generated by monotone coefficients. Moreover, we use
these results to prove a.e. convergence of such Vilenkin–Nörlund means. As
applications, both some well-known and new inequalities are pointed out.

#### (Hp− Lp -Type inequalities for subsequences of Nörlund means of Walsh–Fourier series

(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-04-07)

#### Some New Weak (Hp- Lp)-Type Inequality for Weighted Maximal Operators of Partial Sums of Walsh–Fourier Series

(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-08-18)

In this paper, we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space H<sub>p</sub>(G)
to the Lebesgue space weak−L<sub>p</sub>(G),
for 0<p<1.
Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known ...

#### Some new restricted maximal operators of Fejér means of Walsh–Fourier series

(Journal article; Tidsskriftartikkel, 2023-09-12)

In this paper, we derive the maximal subspace of natural numbers { <i>n<sub>k</sub></i> : <i>k</i> ≥ 0 }, such that the restricted maximal operator, defined by sup<sub><i>k</i>∈ℕ</sub> | σ<i><sub>n<sub>k</sub></sub>F</i> | on this subspace of Fejér means of Walsh–Fourier series is bounded from the martingale Hardy space <i>H</i><sub>1/2</sub> to the Lebesgue space ...

#### Sharpness of some Hardy-type inequalities

(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-12-04)

The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure
. There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also the involved function spaces are ...