Let \((\Omega,\Phi,\mu)\) be a probability space with filtration \(\{\Phi_j,j= 0,1,2,\dots\}\) consisting of a nondecreasing sequence of subfields of \(\Phi\), and let \((X,\|.\|_X)\), \((Y,\|.\|_Y)\), \((Z,\|.\|_Z)\) be Banach spaces with norms \(\|.\|_X\), \(\|.\|_Y\), \(\|.\|_Z\). The space of bounded linear operators from \(X\) into \(Y\) is denoted by \({\mathcal B}(X,Y)\). The sequence \(\{f_j\), \(j= 0, 1,2,\dots\}\) of \(X\)-valued functions on \(\Omega\) is a martingale with respect to \(\{\Phi_j\), \(j= 0,1,2,\dots\}\), and the sequence \(\{d_j\), \(j= 0,1,2,\dots\}\) is a martingale difference with respect to \(\{\Phi_j\), \( j= 0,1,2,\dots\}\) if \(f_j\in L^1((\Omega,\Phi,\mu), X)\), \(E(f_{j+1}\mid \Phi_j)= f_j\), \(d_j\in L^1((\Omega,\Phi,\mu), X)\), \(E(d_{j+1}\mid\Phi_j)= 0\) for each ‘admissible’ \(j\), \(f_k= \sum^k_{j=0} d_j\), \(d_k= f_k- f_{k-1}\), \(f_0= 0\). The set of \(http://matrix:8080/math/input/X\)-valued martingales with respect to \(\{\Phi_j, j= 0,1,2,\dots\}\) is denoted by \({\mathcal M}(\{\Phi_j\}, X)\), and the set of \(X\)-valued martingale differences with respect to sub-filtrations of \(\{\Phi_j, j= 0,1,2,\dots\}\) is denoted by \({\mathcal D}(\{\Phi_j\}, X)\).
If \(v_j:\Omega\to Z\) and \(v_j\) is \(\Phi_{j-1}\)-measurable for \(j= 1,2,3,\dots\), then \(\{v_j, j= 0,1,2,\dots\}\) is said to be predictable, and if \(Z={\mathcal B}(X, Y)\) with norm \(\| v_j\|_{X\to Y}\leq 1\), \(j= 0,1,2,\dots\), then \(\{v_j,j= 0,1,2,\dots\}\) is said to be a \(\{\Phi_j\}\)-multiplier sequence. The operator \(T_v\) is defined by \(T_v(f)_k= \sum^k_{j=1} v_j d_j\), \(f= (f_0,f_1,f_2,\dots)\). In the main results, the authors show that if \(1\leq p<\infty\), \[ \mathbb{R}(\{v_j(\omega), j= 0,1,2,\dots\}\leq C_p\text{ for almost all }\omega\text{ in }\Omega,\tag{1} \] where for \(\tau\subseteq B(X, Y)\), \(\mathbb{R}(\tau)= \text{inf}\{b> 0:\| \sum^n_{j=0} r_j(.)Q_j(x_j)\|_p\leq b\| \sum^n_{j=0} r_j(.) x_j\|_p\}\), \(n\) is a positive integer, \(\{Q_j, j= 1,2,\dots, n\}\subseteq\tau\), \(\{x_j, j= 1,2,\dots\}\subseteq X\), \(r_j\), \(j= 0,1,2,\dots\) are Rademacher functions defined on \((0,1)\}\), then \[ \|\sum^m_{j=0} v_j d_j\|_p\leq \alpha_p \beta_p C_p\| f_m\|_p\tag{2} \] for pre-defined constants \(\alpha_p\), \(\beta_p\), \(\{d_j\}\in {\mathcal D}(\{\Phi_j\}, X)\), with similar estimate to some extension \((\Omega^\rho, \Phi^\rho, \mu^\rho)\) of \((\Omega,\Phi,\mu)\). Further results indicate that if \(\{\Phi_j\}\) is an atomic filtration, then an initial condition of the form (2) leads to an estimate of the form (1).