On the WALA conjecture, Alberti representations and applications
arXiv:2605.29804
The paper proves the WALA conjecture, showing that any AD‑regular Radon measure whose Lipschitz functions satisfy a weak affine approximation condition is uniformly rectifiable, and develops quantitative tools like a decomposability bundle for such measures.
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.
This 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