[Paper Review] A note on The maximal operators of the Nörlund logaritmic means of Vilenkin-Fourier series

George Tephnadze, G. Tutberidze|arXiv (Cornell University)|Jan 27, 2019

Advanced Harmonic Analysis Research4 citations

TL;DR

This paper investigates the boundedness of maximal operators associated with Nörlund logarithmic means of Vilenkin-Fourier series in martingale Hardy spaces $ H_p(G_m) $ for $ 0 < p < 1 $. It establishes sharp $ (H_p, L_p) $-type inequalities by proving that the operator $ \widetilde{L}_p^*f := \sup_n |L_n f| / \Theta(n+1) $ is bounded from $ H_p $ to $ L_p $ if and only if $ \Theta(n) \sim n^{1/p - 1} / \log n $, with the rate being optimal in the sense of sharpness.

ABSTRACT

The main aim of this paper is to investigate $\left(H_{p},L_{p} ight)$- type inequalities for the the maximal operators of Nörlund logaritmic means, for $0

Motivation & Objective

• To establish sharp $ (H_p, L_p) $-type inequalities for maximal operators of Nörlund logarithmic means in the context of Vilenkin-Fourier series.
• To determine the optimal growth rate of the weight function $ \Theta(n) $ such that the maximal operator $ \widetilde{L}_p^*f := \sup_n |L_n f| / \Theta(n+1) $ is bounded from $ H_p(G_m) $ to $ L_p(G_m) $.
• To prove the sharpness of the rate $ \Theta(n) \sim n^{1/p-1}/\log n $, showing that any slower-growing $ \varphi(n) $ leads to unboundedness.
• To extend known results on partial sum operators to the case of Nörlund logarithmic means in the critical range $ 0 < p < 1 $.

Proposed method

• Utilizes the maximal operator $ \widetilde{L}_p^*f := \sup_n |L_n f| / \Theta(n+1) $, where $ L_n f $ denotes the $ n $-th Nörlund logarithmic mean of the Vilenkin-Fourier series.
• Applies known sharp bounds for the maximal partial sum operator $ \widetilde{S}_p^*f := \sup_n |S_n f| / (n+1)^{1/p - 1} $, which is bounded from $ H_p $ to $ L_p $.
• Establishes a pointwise domination: $ |L_n f| / (n+1)^{1/p - 1} \leq \sup_k |S_k f| / (k+1)^{1/p - 1} $, enabling transfer of bounds from partial sums to logarithmic means.
• Constructs a specific martingale $ f_{n_k} = D_{M_{2n_k+1}} - D_{M_{2n_k}} $ with uniformly bounded $ H_p $-norm to test the sharpness of the weight function.
• Employs measure-theoretic estimates to show that $ \mu\{ |L_{M_{2n_k}+2} f_{n_k}| \geq 1 / (l_{M_{2n_k}+2} \varphi(M_{2n_k+2})) \} = 1 $, leading to divergence when $ \varphi(n) $ grows slower than $ \Theta(n) $.
• Combines the above estimates to prove that $ \| \widetilde{L}_p^* f \|_p \to \infty $ if $ \Theta(n)/\varphi(n) \to \infty $, establishing sharpness.

Experimental results

Research questions

• RQ1 What is the optimal growth rate of the weight function $ \Theta(n) $ such that the maximal operator $ \widetilde{L}_p^*f := \sup_n |L_n f| / \Theta(n+1) $ is bounded from $ H_p(G_m) $ to $ L_p(G_m) $ for $ 0 < p < 1 $?
• RQ2 Is the rate $ \Theta(n) \sim n^{1/p - 1} / \log n $ sharp in the sense that any slower-growing $ \varphi(n) $ leads to an unbounded maximal operator?
• RQ3 Can the sharpness of the logarithmic factor be proven via a constructive counterexample in the martingale Hardy space framework?
• RQ4 How do the properties of Nörlund logarithmic means compare to those of partial sums in terms of $ (H_p, L_p) $ boundedness for $ 0 < p < 1 $?
• RQ5 What is the relationship between the growth of $ \Theta(n) $ and the norm of the maximal operator in the $ H_p \to L_p $ setting?

Key findings

• The maximal operator $ \widetilde{L}_p^*f := \sup_n |L_n f| / (n+1)^{1/p - 1} $ is bounded from $ H_p(G_m) $ to $ L_p(G_m) $ for $ 0 < p < 1 $, extending known results for partial sums.
• The rate $ \Theta(n) \sim n^{1/p - 1} / \log n $ is sharp: if $ \varphi(n) $ grows slower than $ \Theta(n) $, then $ \sup_n |L_n f| / \varphi(n+1) $ is unbounded on $ H_p(G_m) $.
• A counterexample martingale $ f_{n_k} = D_{M_{2n_k+1}} - D_{M_{2n_k}} $ has uniformly bounded $ H_p $-norm but satisfies $ \| \widetilde{L}_p^* f_{n_k} \|_p \to \infty $ when $ \varphi(n) \ll \Theta(n) $.
• The sharpness is demonstrated by showing that $ \| \widetilde{L}_p^* f_{n_k} \|_p / \|f_{n_k}\|_{H_p} \to \infty $ as $ k \to \infty $, under the condition $ \Theta(n)/\varphi(n) \to \infty $.
• The paper proves that $ C_1 n^{1/p - 1} / \log n \leq \Theta(n) \leq C_2 n^{1/p - 1} $ for some absolute constants $ C_1, C_2 $, confirming the exact asymptotic order.
• The result confirms that the logarithmic factor is essential and cannot be removed or replaced by a slower-growing function in the denominator for boundedness.