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

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