On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi

Masato Kikuchi

Colloquium Mathematicae, Tome 131 (2013), p. 13-26 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f = {(f ₙ)}_{n \in ℤ ₊} \in ℳ$ , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${| | M g | |}_{X} {\leq | | M f | |}_{X}$ , then ${| | S g | |}_{X} {\leq C | | S f | |}_{X}$ , where C is a constant independent of f and g.

Publié le : 2013-01-01

Zbl 1281.60042

EUDML-ID : urn:eudml:doc:286638

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     author = {Masato Kikuchi},
     title = {On a relation between norms of the maximal function and the square function of a martingale},
     journal = {Colloquium Mathematicae},
     volume = {131},
     year = {2013},
     pages = {13-26},
     zbl = {1281.60042},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-2}
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Masato Kikuchi. On a relation between norms of the maximal function and the square function of a martingale. Colloquium Mathematicae, Tome 131 (2013) pp. 13-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-2/