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

Abstract

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.

Keywords

• Vilenkin system
• Vilenkin group
• Vilenkin–Fourier series
• Nörlund means
• Martingale Hardy space
• W e a k − L p $Weak-L^{p}$ spaces