Derivatives of Sub-Riemannian Geodesics are $L_p$-H\"older Continuous

TL;DR

This paper proves that derivatives of sub-Riemannian geodesics are always $L_p$-H"older continuous, leading to new insights into their smoothness, approximation, and related inequalities.

Contribution

It establishes the $L_p$-H"older continuity of derivatives of sub-Riemannian geodesics, a long-standing open problem in the field.

Findings

Decay of Fourier coefficients on abnormal controls

Approximation rate of geodesics by smooth functions

Generalization of the Poincaré inequality

Abstract

This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always $L_p$-H\"older continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincar\'e inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials.