The Boundedness Of Operator With Respect To Vilenkin-Like System

Denote m:=(m₀, m₁,…) a sequence of positive integers such that m_k≥2, k∈N.Vilenkin space G_m=∏_k=0^∞Z_mk is generated by sequence m: =(m_k, k∈N), where Z_mk={0, 1,·m_k-1}. G_m is called bounded Vilenkin space if the sequence m=(m₀, m₁,…) isL^∞bounded, otherwise we call G_m unbounded Vilenkin space. For Vilenkin space G_m, wewill introduce a new orthonormal system (ψ_n, n∈N), whereψ_n =∏_k=0^∞r_k^nk and r_k^nk isgeneralized Rademacher function. Actually, it is the generalization of the known Walsh-Paleysystem and Vilenkin system. For this new orthonormal system, we also can define L_p spaceand H_p space where0＜p≤∞. we know the atomic decomposition is a useful tool to dealwith the martingale spaces with small index and can be used in both one-parameter and two-parameter case. With atomic decomposition, we will discuss the boundedness of weightedaverage maximal operator H^*f: =sup_n|H_nf|, partial maximal operatorσ^*f=sup_n|σ_Mnf|and convergence of sequence {H_n f, n∈N}, {σ_Mn f, n∈N}, in bounded case and unboundedcase respectively, where H_n f is the weighted means of partial sums of function f,σ_Mn f is theCesaro means of function f.The framework of this thesis consists of the following parts:Chapter 1 is a brief introduction about background and the main result of this thesis.In Chapter 2, we introduces some prearrangement about Vilenkin-like system and listsome examples and character. We study the relation among Vilenkin-like system, Walsh-Paley system and Vilenkin system. So we have atomic decomposition in martingale space with small index generated by Vilenkin-like system.In Chapter 3, for bounded Vilenkin-like system i. e. the Vilenkin space G_m is bounded.we give the proof the weighted average of Dirichlet kernels V_n=1/W_n∑_k=0^n-1ω_k D_k are uniformlyL₁ bounded in bounded Vilenkin space, where W_n ==∑_k=0^n-1ω_k,(ω_k, k∈N) is sequence ofpositive integer. In one dimensional Vilenkin space, We will prove the the operator H^*=sup_n |H_n f| is bounded from Hardy space H^*(G_m) to space L_p(G_m), where 1/2＜p≤∞.From the boundedness of operator of H^*, we have the sequence H_n f converges in L_p normand almost everywhere, where 1≤p＜∞. From Doob inequality, we have the operatorS^*=sup_n |S_n f| is of type (p, p). For two-dimensional Vilenkin space we prove the operatorTf=sup_n=(n₁, N₂∈p²,β^-1≤n₁/n₂≤β)|H_n f| is of weak (1,1) and type (p,p), (H₁, L₁) for 1＜p≤∞, where H_n f is the convolution of function f and weighted average of Dirichler kernel in two-parameter. In this case the character of generalized Rademacher function i. e. |r_kⁿ|=1((?)k, n∈N) For bounded Vilenkin-like we prove that the Young inequality holds. As a consequence ofYoung inequality, we prove the following inequality is true (sum form k=1 to∞k^P-2‖S_k f‖_p^p)^1/p≤C‖f‖H_p(0＜p＜1).For Vilenkin system, we give the estimation about (C,α) kernels i. e. |K_n^α(t)|≤C·1/A_n^α·1/t^1+α, t∈(0, 1). and proves the operatorσ^α* f=sup_n|f*K_n^α| is of weak type (1,1) and type (p,p)(H₁,L₁) and the sequence {σ_n^αf, n∈N} converges a. e..In Chapter 4, the weighted average of Dirichlet kernels are not uniformly L₁ boundedwith respect to unbounded Vilenkin-like system. We will find a new method to prove theoperatorσ^* f=sup_n |f* K_Mn| is of weak type (1,1)and the sequence {σ_n^αf, n∈N} convergesa. e.. There are several main innovation made in this thesis.At first, we introduce a new orthornormal system which is the generalization of Walsh-Paley system and Vilenkin system. We establish a more generalized operator named weightedaverage operator and give the proof of boundedness of weighted average maximal operatorand convergence of weighted average of sequence by using the martingale theory and atomicdecomposition.Secondly, we not only resolve the boundedness of operator with respect to boundedVilenkin-like system and also get some new result with respect to Vilenkin-like system whichis that operatorσ^* f=sup_n|f*K_Mn|is of weak type (1,1)and the sequence {σ_n^αf, n∈N}converges a. e..Thirdly, applying the atomic decomposition, we study the some question about theVilenkin system and establish the estimation about (C,α)kernels with respect to Vilenkinsystem and find a new method in discussing the boundedness of operatorσ^α* and convergenceof sequence {σ_n^αf, n∈N}.