Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups

In the theory of orthonormal systems, the study of summability methods plays a central role. For the bounded Vilenkin system, a fundamental theorem asserts that the Lₚ-norm convergence of certain operator sequences follows from a mild γ-type condition on the coefficients, without requiring any positivity or monotonicity assumptions. In this paper, we extend this classical result (Theorem~V) to the setting of almost everywhere convergence. In particular, we show that if the coefficients ,n\ satisfy \[ (tk,n)ₖ₌₀ⁿ_{γ} = O\!(n-1/γ'), 1 < γ ≤ 2, \] then the corresponding sequence converges almost everywhere with respect to the bounded Vilenkin system. This work continues the line of research initiated by Blahota and Gát (J. Geom. Anal., 2025). Moreover, we address an open problem posed by Blahota and Gát, providing both affirmative and negative results. Finally, we present several applications of the main theorem to classical and generalized summability methods, thereby illustrating its broad applicability within the context of Fourier analysis.