Operator-valued martingale transforms
Let \(B_1\) and \(B_2\) be two Banach spaces, \((\Omega,{\mathcal F}, \mathbb{F}= \{{\mathcal F}_n\}_{n\geq 1}, P)\) be a stochastic basis, \(f=\{f_n\}_{n\geq 1}\) be a \(B_1\)-valued martingale and \(\{v_n\}_{n\geq 1}\) be an \({\mathcal L}(B_1,B_2)\) valued \(\mathbb{F} \)-predictable uniformly bounded sequence. Denoting \[ \|f\|_{L^p_{B_1}} = \sup_{n\geq 1} \|f_n\|_{L^p}, \quad (Tf)_n: =\sum^n_{k=2} v_k(f_k-f_{k-1}) +v_1f_1, \] \[ f^* =\sup_{n\geq 1} |f_n|_{B_1}, \quad (Tf)^*=\sup_{n \geq 1} \bigl|(Tf)_n\bigr |_{B_2} \] and assuming that \(\sup_{n\geq 1}\|v_n \|_{{\mathcal L}(B_1,B_2)}\leq 1\), it is proved that the following statements are equivalent: (i) there exists \(C>0\) such that for any \(\lambda>0\), \(\lambda P\{(Tf)^* >\lambda\}\leq C\|f\|_{L^1_{B_1}}\); (ii) there exists \(C>0\) such that for any \(\lambda>0\), \(\lambda P\{(Tf)^* >\lambda\}\leq C\|f^* \|_{L^1}\); (iii) given any \(p\), \(1\leq p<\infty\), there exists \(C_p>0\) such that \(\|(Tf^*) \|_{L^p}\leq C_p\|f^*\|_{L^p}\); (iv) given any \(p\), \(1<p< \infty\), there exists \(C_p>0\) such that \(\|(Tf^*) \|_{L^p} \leq C_p\|f\|^p_{L_{B_1}}\); (v) there exist \(p_0\), \(1<p_0 <\infty\), and a constant \(C_0\) such that \(\|(Tf^*) \|_{L^{p_0}} \leq C_0\|f \|_{L^{p_0}_{B_1}}\). If \(B_2\) has the Radon-Nikodym property, then any of these conditions implies \[ \|f\|_{L^1_{B_1}} <\infty \Rightarrow Tf \text{ converges a.s.} \] Several important applications of this result are given.
scientific article; zbMATH DE number 1558410
scientific article