On multipliers into martingale $$SL^\infty $$ spaces for arbitrary filtrations

Abstract

In this paper we study the following problem: for a given bounded positive function f on a filtered probability space can we find another function (a multiplier) m, \(0\le m\le 1\), such that the function mf is not “too small” but its square function is bounded? We explicitly show how to construct such multipliers for the usual martingale square function and for so-called conditional square function. Besides that, we show that for the usual square function more general statement can be obtained by application of a non-constructive abstract correction theorem by Kislyakov.