On weighted inequalities for martingale transform operators

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 .