Embeddings of Lorentz Spaces of Vector-Valued Martingales

We obtain new embedding theorems for Lorentz spaces of vector-valued martingales, thus generalizing the classical martingale inequalities. In contrast to earlier methods, we use martingale transformations defined by sequences of operators and identify the operator $S^{(p)}(f)$ for a martingale $f$ ranging in a Banach space $X$ with the maximal operator for some $\ell^p(X)$-valued martingale transform. The obtained inequalities are closely related to geometric properties of the Banach space in question.