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On weighted inequalities for martingale transform operators

✍ Scribed by Teresa Martínez

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
224 KB
Volume
251
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

For 1 < p < ∞, the almost surely finiteness of $ E \left(v ^{- {p^{\prime} \over p}} \vert {\cal F}_{1} \right) $ is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|ℱ~n~)} with f ∈ L^p^(v dP). This condition is also equivalent to have weighted inequalities from L^p^(v dP) into L^p^(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|ℱ~1~) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.

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