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Some remarks on the dyadic Rademacher maximal function

Mikko Kemppainen (2013)

Colloquium Mathematicae

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) L p 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 L inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.

Strong topologies on vector-valued function spaces

Marian Nowak (2000)

Czechoslovak Mathematical Journal

Let ( X , · X ) be a real Banach space and let E be an ideal of L 0 over a σ -finite measure space ( Ø , Σ , μ ) . Let ( X ) be the space of all strongly Σ -measurable functions f Ø X such that the scalar function f ˜ , defined by f ˜ ( ø ) = f ( ø ) X for ø Ø , belongs to E . The paper deals with strong topologies on E ( X ) . In particular, the strong topology β ( E ( X ) , E ( X ) n ) ( E ( X ) n = the order continuous dual of E ( X ) ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

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