Tent spaces at endpoints

Abstract

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, (Tqp(R+n+1))*=Tq'p'(R+n+1) for 1<p,q<∞, and (T∞1(R+n+1))*=C(R+n+1), (Tq1(R+n+1))*=Tq'∞(R+n+1) for 1<q<∞, where C(R+n+1) denotes the Carleson measure space on R+n+1. We prove that (Cv(R+n+1))*=T∞1(R+n+1), which we introduced recently, where Cv(R+n+1) is the vanishing Carleson measure space on R+n+1. We also give the characterizations of Tq∞(R+n+1) by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on Lq(Rn) of the paraproduct πF associated with the tent space Tq∞(R+n+1), and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces T2p(R+n+1) (1≤p≤∞) and Lp(Rn), Hp(Rn) and BMO(Rn) spaces.