Some properties of the vector-valued Banach ideal space E(X) derived from those of E and X.
Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on ℝⁿ. In addition, to compensate for the lack of an inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.
Let be a real Banach space and let be an ideal of over a -finite measure space . Let be the space of all strongly -measurable functions such that the scalar function , defined by for , belongs to . The paper deals with strong topologies on . In particular, the strong topology ( the order continuous dual of ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.