New martingale inequalities in rearrangement-invariant function spaces

We establish various martingale inequalities in a rearrangement-invariant (RI) Banach function space. If X is an RI space that is not too small, we associate with it RI spaces H-P(X) (1 less than or equal to p < infinity) and K(X), and discuss martingale inequalities in these spaces. One of our results is as follows. Let I less than or equal to p < infinity, let f = (f(n)) be an L-P-bounded martingale, and let \f\(P) = g + h be the Doob decomposition of the submartingale \f\(P) = (\f\(P)) into a martingale g = (g(n)) and a predictable non-decreasing process h = (h(n)) with h(0) = 0. Then, in the case where 1 < p < infinity, we obtain the inequalities \\h(infinity)(1/P)\\ less than or equal to 2\\f(infinity)\\H-P(x) and \\sup(n)\g(n)\(1/P)\\(x) less than or equal to 4\\f(infinity)\\H-P(x), and, in the case where p = 1, we obtain the inequalities [GRAPHICS] For some specific choices of X, we can give explicit expressions for H-P(X) and K(X). For example, H-1(L-1) = Llog L, H-P(Lp,infinity) = L-P,L-1, and soon. Furthermore, if the Boyd indicesof X satisfy 0 < alpha(x) less than or equal to betax < 1/P (respectively, 0 < alpha(x)), then H-P(X) = X (respectively, K(X) = X). In any case, 'Hp(X) is embedded in K(X), and K(X) is embedded in X.