Hardy spaces associated to the sections

Abstract

In this paper we define the Hardy space H 1 F (R n ) associated with a family F of sections and a doubling measure µ ,w hereF is closely related to the Monge-Ampere equation. Furthermore, we show that the dual space of H 1 F (R n) is just the space BMOF (Rn), which was first defined by Caffarelli and Gu tierrez. We also prove that the Monge-Ampere singular integral operator is bounded from H 1 F (R n ) to L 1 (R n ,dµ ). 1. Introduction. In 1996, Caffarelli and Gutierrez (CG1) studied real variable theory related to the Monge-Ampere equation. They gave a Besicovitch type covering lemma for a family F of convex sets in Euclidean n-space R n ,w hereF ={ S(x, t); x ∈ R n and t> 0} and S(x, t) is called a section (see the definition below) satisfying certain axioms of affine invariance. In terms of the sections, Caffarelli and Gutierrez set up a variant of the Calderon- Zygmund decomposition by applying this covering lemma and the doubling condition of a Borel measure µ. The decomposition plays an important role in the study of the linearized Monge-Ampere equation (CG2). As an application of the above decomposition, Caffarelli and Gutierrez defined the Hardy-Littlewood maximal operator M and BMOF (R n ) space as- sociated to a family F of sections and the doubling measure µ, and obtained the weak type (1,1) boundedness of M and the John-Nirenberg inequality for BMOF (R n ) in (CG1).