A variant of Caffarelli’s contraction theorem for probability distributions of negative powers

Abstract

Let μ be a probability measure on Rd with barycenter at the origin, which is supported on an open, bounded and convex set K having smooth boundary and positive Gauss curvature. Suppose that dμx=ϱx−αdx, with α>d+1 and ϱ:K→0,+∞ is C∞-smooth, satisfies ɛ0-convex condition for some ɛ0>0 such that ϱ and its first derivatives are bounded in K. In this paper, we prove that for any convex solution ψ:Rd→0,+∞ to the following Monge–Ampère equation (1)[ϱ∇ψx]−αdet(∇2ψx)=(ψx)−α,x∈Rdthe function x↦logψx has second derivatives to be bounded by a constant Cɛ0 depending on ɛ0, α, ψ(x0) and values of first, second and fourth derivatives of ψ at a fixed point x0∈Rd. This estimate is as a variant case of the Caffarelli contraction theorem by replacing target densities of e−W by densities of ϱ−α that Cauchy distributions (i.e., ϱx−α=1+|x|2−α) are typical examples.