Weighted norm inequalities for Vilenkin-Fourier series

Abstract

Let S n f {S_n}f be the n n th partial sum of the Vilenkin-Fourier series of f ∈ L 1 f \in {L^1} . For 1 > p > ∞ 1 > p > \infty , we characterize all weight functions w w such that if f ∈ L p ( w ) f \in {L^p}(w) , S n f {S_n}f converges to f f in L p ( w ) {L^p}(w) . We also determine all weight functions w w such that { S n } \{ {S_n}\} is uniformly of weak type ( 1 , 1 ) (1,1) with respect to w w .