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2 Kemppainen M.

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On the Rademacher maximal function

100%

Kemppainen M.

nr 1

1-31

This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^{p}$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^{p}$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for $L^{p}$-boundedness and also to provide a characterization by concave functions.

Some remarks on the dyadic Rademacher maximal function

100%

Kemppainen M.

tom 131

nr 1

113-128

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.