Inequalities for scalar-valued linear operators that extend to their vector-valued analogues

A theorem of Marcinkiewicz and Zygmund asserts that a linear operator satisfying a strong type (L^r, L^q) inequality with norm M automatically extends to a vector-valued operator satisfying a strong type (L^r(l^v), L^q(l^v)) inequality with norm not exceeding C_{r, q}^(γ)M. In this paper, this theorem is proved in a more general context by replacing the L^q metric with a more general class of metrics. In doing so, the theorem of Marcinkiewicz and Zygmund is not only extended to more general contexts, but improvements of that theorem are also realized. In particular, our results show that operators satisfying weak type inequalities automatically extend to their vector-valued analogues; also the constant C_{r, q}^(γ) may be taken as one in the theorem of Marcinkiewicz and Zygmund whenever q ? r, and this includes most cases of interest.