Sharp (Hp,Lp)(H_p,L_p) and (Hp,weak−Lp)(H_p,\text{weak}-L_p) type inequalities of weighted maximal operators of TT means with respect to Vilenkin systems

We discuss $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities of weighted maximal operators of $T$ means with respect to the Vilenkin systems with monotone coefficients, considered in \cite{tut4} and prove that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.Comment: arXiv admin note: substantial text overlap with arXiv:2101.09196, arXiv:1504.05974 by other author

On some weighted maximal operators of partial sums of Walsh-Fourier series in the space H1H_1

In the first part of this paper we describe the status of the art of this subject. In the second part we present and motivate some new results. Indeed, 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 $H_{1}(G)$ to the space $\text{weak}-L_{1}(G),$ but is not bounded from $H_{1}(G)$ to the space $L_{1}(G).

SHARP (Hp, Lp) AND (Hp, weak − Lp) TYPE INEQUALITIES OF WEIGHTED MAXIMAL OPERATORS OF T MEANS WITH RESPECT TO VILENKIN SYSTEMS

Source at https://rmi.tsu.ge/transactions/.We discuss (Hp, Lp) and (Hp, weak − Lp) type inequalities of weighted maximal operators of T means with respect to the Vilenkin systems with monotone coefficients [44] and prove that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out

a.e. Convergence of TT means with respect to Vilenkin systems of integrable functions

In this paper we derive converge of $T$ means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in $L_p$ norms of such $T$ means.Comment: arXiv admin note: text overlap with arXiv:2106.11836. substantial text overlap with arXiv:2101.09196 by other author

a.e. Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions

In this paper we derive converge of N\"orlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in $L_p$ norms of such N\"orlund means.Comment: arXiv admin note: substantial text overlap with arXiv:2107.0201

Some new (Hp – Lp) type inequalities for weighted maximal operators of partial sums of Walsh–Fourier series

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 Hp(G) to the Lebesgue space Lp(G), for 0<p<1. Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results

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

In this paper, we derive the maximal subspace of natural numbers nk: k≥ 0 , such that the restricted maximal operator, defined by supk∈N|σnkF| on this subspace of Fejér means of Walsh–Fourier series is bounded from the martingale Hardy space H1 / 2 to the Lebesgue space L1 / 2. The sharpness of this result is also proved.

Convergence of T means with respect to Vilenkin systems of integrable functions

In this paper, we derive the convergence of T means of Vilenkin–Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilenkin–Lebesgue points. Moreover, we discuss the pointwise and norm convergence in Lp norms of such T means

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

Abstract We prove and discuss some new weak type ( 1 , 1 ) $(1,1 )$ 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