An inequality for the predictable projection of an adapted process

Let ( fn)n 1 be a stochastic process adapted to the filtration (~’n)n o. Denoting by. (gn)Nn=1 the predictable projection of this process, i.e., gn = we show that the inequality [E( | gn |q)p/q]1/p 2 [E( I fn Iq)pI9 ] lip or, in more abstract terms ~(gn)Nn=1~Lp(lqN) ~ 2~(fn)Nn=1~Lp(lyN) holds true (with the obvious interpretation in the case of p = oo or g = oo). Several similar results, pertaining also to the case p > q, are known in the literature. The present result may have some interest in view of the following reasons: (1) the case p = 1 and 2 q oo seems to be new; (2) we obtain 2 as a uniform constant which is sharp in the case p == 1, q = oo and (3) the proof is very easy. We denote by filtration on a probability space (5~,.~, P) and let En be the conditional expectation with respect to 7n. Let ( f,~)n 1 be a stochastic process which will be assumed in most of this note to be adapted to (~n)~ 1 and to satisfy the appropriate integrability conditions so that the subsequent statements make sense; we denote by the predictable projection of ( f n~n 1, i.e., gn = For 1 p, q oo we define