BILIPSCHITZ EMBEDDINGS OF SPHERESINTO JET SPACE CARNOT GROUPS NOT ADMITTING LIPSCHITZ EXTENSIONS

摘要

For all k,n > 1, we construct a biLipschitz embedding of Sn into the jet space Carnot group Jk(Rn) that does not admit a Lipschitz extension to Bn+1. Let f : Bn R be a smooth, positive function with kth-order derivatives that are approximately linear near Bn.The embedding is given by taking the jet of f on the upper hemisphere and the jet of -f on the lower hemisphere, where we view Sn as two copies Bn. To prove the lack of a Lipschitz extension, we apply a factorization result of Wenger and Young for n = 1 and modify an argument of Rigot and Wenger for n > 2.