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

Project Details


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.
Effective start/end date1/08/1531/07/16

UN Sustainable Development Goals

In 2015, UN member states agreed to 17 global Sustainable Development Goals (SDGs) to end poverty, protect the planet and ensure prosperity for all. This project contributes towards the following SDG(s):

  • SDG 1 - No Poverty
  • SDG 10 - Reduced Inequalities
  • SDG 17 - Partnerships for the Goals


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