Besov Spaces Associated to Monge-Ampere Equations(1/2)

Let ϕ : Rn 7→ R be a strictly convex and smooth function, and μ = detD2ϕ be theMonge-Amp`ere measure generated by ϕ. For x ∈ Rn and t > 0, let S(x, t) := {y ∈ Rn :ϕ(y) < ϕ(x)+∇ϕ(x) · (y−x)+t} denote the section. If μ satisfies the doubling property,Caffarelli and Guti´errez (Trans. AMS 348:1075–1092, 1996) provided a variant of theCalder´on-Zygmund decomposition and a John-Nirenberg-type inequality associated withsections. Under a stronger uniform continuity condition on μ, Caffarelli and Guti´errez(Amer. J. Math. 119:423–465, 1997) proved an invariant Harnack’s inequality for nonnegativesolutions of the Monge-Amp`ere equations with respect to sections. The purposeof this project is to establish a theory of Besov spaces associated with sections under onlythe doubling condition on μ and prove that Monge-Amp`ere singular integral operatorsare bounded on these spaces.