Weights for ergodic square functions (Q1090431)

The paper deals with weighted inequalities for a class of (nonlinear) operators associated to an invertible transformation which preserves a measure \(\mu\). This class includes ergodic square functions introduced by \textit{R. L. Jones} [Stud. Math. 60, 111-129 (1977; Zbl 0349.47007)]. In particular, it is shown that ergodic square functions are of weak type \((1,1)\) with respect to the measure \(wd\mu\) if and only if the weight function w satisfies an ergodic variant of the condition \(A_ 1\). Moreover, if w satisfies an ergodic variant of Muckenhoupt's \(A_ p\) condition, \(1<p<\infty,\) then ergodic square functions are of strong type \((p,p)\) with respect to the measure \(wd\mu.\) The converse of the last result is not true.