Boundedness of Monge-Ampere singular integral operators on Besov spaces

Abstract

Let $\varphi :{\mathbb{R}}^n\mapsto \mathbb{R}$ be a strictly convex and smooth function, and $\mu =\text{det}{D}^2\varphi$ be the Monge-Ampère measure generated by $\varphi .$ For $x\in {\mathbb{R}}^n$ and $t>0$, let $S(x,t):=\{y\in {\mathbb{R}}^n:\varphi (y)<\varphi (x)+\nabla \varphi (x)\cdot (y-x)+t\}$ denote the section. If $\mu$ satisfies the doubling property, Caffarelli and Gutiérrez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calderón-Zygmund decomposition and a John-Nirenberg-type inequality associated with sections. Under a stronger uniform continuity condition on $\mu$, they also (Amer. J. Math. 119:423--465, 1997) proved an invariant Harnack's inequality for nonnegative solutions of the Monge-Ampère equations with respect to sections. The purpose of this paper is to establish a theory of Besov spaces associated with sections under only the doubling condition on $\mu$ and prove that Monge-Ampère singular integral operators are bounded on these spaces.