On the WALA conjecture, Alberti representations and applications

Abstract

We prove the WALA conjecture of G. David and S. Semmes: every AD-regular Radon measure for which all real-valued Lipschitz functions satisfy the weak approximation by affine functions condition is uniformly rectifiable. In order to prove the conjecture, we identify the correct quantitative analogue of the decomposability bundle and establish several structural results for general AD-regular measures.

arXiv commentThis version of the paper has been extensively proofread and the exposition has been improved. Further sections, including applications developed jointly with Michele Villa and Mihalis Mourgoglou, will be added in the next version