The behaviour of square functions from ergodic theory in l^∞

In this paper, we analyze carefully the behaviour in L^∞(ℝ) of the square functions S and S_I, originating from ergodic theory. First, we show that we can find some function f ∈ L^∞(ℝ), such that Sf equals infinity on a nonzero measurable set. Second, we can find compact supported function f ∈ L^∞(ℝ) and I such that SIf does not belong to BMO space. Finally, we show that S is bounded from L^∞ c , the space of compactly supported L^∞(ℝ) functions, to BMO space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl (2000). That is, S_I are uniformly bounded in L^p(ℝ) with respect to I for 2 < p < ∞.