Noncommutative weak type estimate of Vilenkin derivatives

Let f∈L1(N), where N=L∞(Gm)⊗ˉM, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(Dn(f))n≥1∥Λ1,∞(N,ℓ∞)≤C∥f∥L1(N), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón–Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.