Weak type estimate for the partial sums of Noncommutative Vilenkin-Fourier series

Abstract

Let N = L ∞ ( G m ) ⊗ ¯ M \mathcal {N}=L_{\infty }(G_{\mathbf {m}})\bar {\otimes }\mathcal {M} where M \mathcal {M} is a semifinite von Neumann algebra and G m G_{\mathbf {m}} is a bounded Vilenkin group. Considering the partial sums S n ( f ) S_n(f) of the Vilenkin-Fourier series of f ∈ L 1 ( N ) f\in L_1(\mathcal {N}) , we prove that there is a universal constant c c independent of f f , n n and M \mathcal {M} such that ‖ S n ( f ) ‖ L 1 , ∞ ( N ) ≤ c ‖ f ‖ L 1 ( N ) . \begin{equation*} \|S_n(f)\|_{L_{1,\infty }(\mathcal {N})}\leq c\|f\|_{L_1(\mathcal {N})}. \end{equation*} Consequently, by transference argument, we obtain Scheckter and Sukochev’s result in the noncommutative bounded Vilenkin system setting. We also show the H 1 H_1 - L 1 L_1 boundedness for modified partial sums.