On Weak Convergence in H 1 ( R d )

Abstract

We prove that the a.e. convergence of a sequence of functions bounded in HI (Rd) to a function in LI (Rd) implies weak convergence. Let HI (Rd) denote the Hardy space of functions on Rd whose Poisson maximal function lies in L1, and let BMO(Rd) denote the dual space (2) of functions of bounded mean oscillation. The space VMO(Rd), the closure of the Schwartz space 52 in BMO(Rd), is the predual of H1(Rd). We answer a question of Lions and Meyer by proving the following result. The theorem is also valid for Martingale H1, and the proof we give carries over directly to that setting. Theorem. Suppose {fn} is a sequence of HI (Rd) functions such that fllf IIH < 1 for all n and such that fn(x) -* f(x) for almost every x E Rd. Then f E HI(Rd), IIf IIH < 1, and J n fp dx I fQ dx Rd R