Boundedness of Monge-Ampere singular integral operators on Besov spaces

Abstract

Let $\phi: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $\mu= \text{det}\,D^2 \phi$ be the Monge-Amp\`ere measure generated by $\phi.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n: \phi(y)<\phi(x)+\nabla \phi(x)\cdot(y-x)+t\}$ denote the section. If $\mu$ satisfies the doubling property, Caffarelli and Guti\'errez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calder\'on-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-Amp\`ere 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-Amp\`ere singular integral operators are bounded on these spaces.