On the Coifman type inequality for the oscillation of one-sided averages

In this paper we study the Coifman type estimate for an oscillation operator related to the one-sided discrete square function S+. We prove that for any weight w, the Lp(w)-norm of this operator, and therefore the Lp(w)-norm of S+, is dominated by a constant times the Lp(w)-norm of the one-sided Hardy–Littlewood maximal function iterated two times. For the kth commutator with a BMO function we show that k+2 iterates of the one-sided Hardy–Littlewood maximal function are sufficient.