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Pointwise Schauder estimates for optimal transport maps of rough densities

Arghya Rakshit

TL;DR

The paper proves a pointwise Schauder-type estimate for the potential $u$ of the optimal transport map between a rough source density $\rho_0$ and an almost-constant target, under the condition $p>n$ and small excess energy. By combining a variational framework with harmonic approximation and an iterative $\epsilon$-regularity scheme, the authors establish the existence of a quadratic polynomial $Q$ that approximates $u$ to leading order at small scales, with quantified $L^2$ and $L^\infty$ control. A key step is showing that the interpolation density is close to uniform in $L^1$ and that a harmonic function can efficiently approximate the velocity field, enabling decay of the excess energy across scales. The results yield a pointwise $C^{2,\frac{4\alpha}{4+n}}$ regularity in the bulk and are sharp in the $L^\infty$ sense, as demonstrated by a 2D counterexample and a detailed sharpness argument. This extends pointwise Schauder-type regularity to the optimal transport setting with rough densities, highlighting the role of $L^p$-control and energy decay in regularity theory for Monge–Ampère–type problems.

Abstract

We prove a pointwise $C^{2,\,α}$ estimate for the potential of the optimal transport map in the case that the densities are only close to constant in a certain $L^p$ sense.

Pointwise Schauder estimates for optimal transport maps of rough densities

TL;DR

The paper proves a pointwise Schauder-type estimate for the potential of the optimal transport map between a rough source density and an almost-constant target, under the condition and small excess energy. By combining a variational framework with harmonic approximation and an iterative -regularity scheme, the authors establish the existence of a quadratic polynomial that approximates to leading order at small scales, with quantified and control. A key step is showing that the interpolation density is close to uniform in and that a harmonic function can efficiently approximate the velocity field, enabling decay of the excess energy across scales. The results yield a pointwise regularity in the bulk and are sharp in the sense, as demonstrated by a 2D counterexample and a detailed sharpness argument. This extends pointwise Schauder-type regularity to the optimal transport setting with rough densities, highlighting the role of -control and energy decay in regularity theory for Monge–Ampère–type problems.

Abstract

We prove a pointwise estimate for the potential of the optimal transport map in the case that the densities are only close to constant in a certain sense.
Paper Structure (8 sections, 9 theorems, 144 equations, 3 figures)

This paper contains 8 sections, 9 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Counterexample when $\rho_0$ has $C^\alpha$ decay in dimension $n=2$
  3. Key Lemmas
  4. Harmonic Approximation
  5. one-step improvement
  6. $\epsilon$ Regularity
  7. Proof of Main Theorem
  8. Sharpness

Key Result

Theorem 1.1

Let $\rho_0$ and $\rho_1$ be two densities described as above, and let $T=\nabla u$ be the optimal transport map that takes $\rho_0$ to $\rho_1$. There exists a universal constant $\epsilon_1>0$ such that if $\rho_0$ satisfies the following measure-theoretic inequality: for all $0 < r < 1$, and then there exists a quadratic polynomial $Q$ such that, for all $0<r<\frac{1}{2}$, Moreover, for all

Figures (3)

  • Figure 1: Level sets for $\det D^2u(x,y)$
  • Figure 2: $u$ lies above $Q$
  • Figure 3: $u$ lies below $Q$

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 4.1
  • ...and 7 more