OAI.WeakHessian.every_geodesic

The theorem states that, on a complete separable metric space X with Borel σ-algebra and a Borel measure m with full support (positive measure on every nonempty open set) and finite on bounded sets, satisfying the defined RCD(K,N) condition for real K and N>1, the following holds. Here RCD(K,N) means the defined curvature-dimension condition CD(K,N), phrased through optimal dynamical plans of geodesics with distortion coefficients and Rényi entropy integrals, together with quadratic Cheeger energy (the parallelogram identity for Cheeger energy on finite-energy functions). Let F be bounded and globally Lipschitz and let G be bounded and continuous, and suppose (F,G) satisfies the weak Hessian upper bound: for every compactly supported test function g (bounded, globally Lipschitz, in the Laplacian domain with Sobolev Laplacian) and every nonnegative, compactly supported, globally Lipschitz h, the weak Hessian of F in direction g tested against h is at most the integral of h·G·Γ(g,g) with respect to m, where Γ is the polarized minimal weak gradient pairing. Then for every constant-speed minimizing geodesic σ:[0,1]→X, including constant curves, the second distributional derivative of F∘σ is bounded by G∘σ in the following sense. For every smooth, nonnegative, compactly supported φ whose support lies in (0,1), the integral over [0,1] of F(σ(t))φ''(t) is at most d(σ(0),σ(1))² times the integral over [0,1] of G(σ(t))φ(t).