Endpoint gradient bounds for centered convex maximal operators and probability kernels

We establish the endpoint gradient estimate for centered homothetic maximal averages in every dimension d≥2 over each fixed convex body with C1,θ boundary, 0<θ≤1. In particular, the centered ball maximal operator satisfies the W1,1 gradient bound in every dimension. Neither strict convexity nor a lower bound on the support gap is required. The geometric argument combines concentration in gradient values with nonhomogeneous calibrated rows and a summable radial correction. It extends the scalar refinement, direction-history construction and global assembly of the planar centered-disk proof of OpenAI. A separate transfer theorem gives sharp singular-derivative identities for bounded compact probability kernels once local BV output is known, and gradient continuity under an additional spatial-tail condition. Quantitative contact and translation estimates yield further profile and kernel examples. The estimates retain the fixed body's regularity data; the endpoint for arbitrary C¹ and nonsmooth convex bodies remains open here.