Vector-valued partial sums on unbounded Vilenkin systems

Let $\text{Gm}={\prod }_{k\ge 0}{\mathbb{Z}}_{m_k}$ be a Vilenkin group that is not necessarily bounded, i.e., ${\sup }_k{m}_k=\mathrm{\infty }$. We prove that, for every UMD Banach space $X$ and every $1<p<\mathrm{\infty }$, the Vilenkin partial-sum operators are uniformly bounded on $L^p(\text{Gm};X)$, with a bound depending only on $p$ and the UMD constant of $X$, and not on $\mathbf{m}$. This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields $\mathcal{R}$-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.