Some inequalities related to strong convergence of Riesz logarithmic means

Abstract

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.

Bounded operators on Martingale Hardy spaces

“…In this thesis (see also [80]) we also proved that if 0 < p < 1/2 and f ∈ H p (G m ), there exists an absolute constant c p , depending only on p, such that the inequality Móricz and Siddiqi [86] investigate the approximation properties of some special Nörlund means of Walsh-Fourier series of L p functions in norm. The case when {q k = 1/k : k ∈ N} was excluded, since the methods of Móricz and Siddiqi are not applicable to Nörlund logarithmic means.…”

“…Moreover, in [149] it was proved that the maximal operator of Riesz means is bounded from the Hardy space H p to the Lebesgue space L p for p > 1/2 but not when 0 < p ≤ 1/2. Strong convergence theorems and boundedness of weighted maximal operators of Riesz logarithmic means was considered in Lukkassen, Persson, Tutberidze, Tephnadze [80] and Tephnadze [149].…”

Some new results for subsequences of Nörlund logarithmic means of Walsh-Fourier series

“…In Theorem 2 of this paper we give a negative answer to this question. In particular, we further develop some methods considered in [2,11] and prove that for any 0 < p < 1, there exists a martingale f ∈ H p such that sup n∈N L 2 n f weak−Lp = ∞. Moreover, in our Theorem 1 we generalize the result of Goginava [9] and prove that for any f ∈ L 1 (G) and for any Lebesgue point x,…”

Sharp $\left( H_{p},L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients

“…are bounded from H p to the space L p , for 0 < p ≤ 1/2 and the rate of weights are sharp. Moreover, in [9] was also proved that if 0 < p < 1/2 and f ∈ H p (G m ), then there exists an absolute constant c p , depending only on p, such that the inequality holds:…”