Generalized Frank characterizations of Muckenhoupt weights and homogeneous ball Banach Sobolev spaces

In this article, the authors first establish a new characterization of Muckenhoupt weights in terms of oscillations. As an application, the authors give a new characterization of homogeneous ball Banach Sobolev spaces, which extends the elegant characterization of Sobolev spaces obtained by R. L. Frank in 2024 and is a variant of the famous formula obtained by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung in 2024 with difference quotients replaced by oscillations. Moreover, the authors also obtain new representation formulae of gradients in terms of oscillations in ball Banach function spaces, which even include the critical case where Frank did not consider. Furthermore, via some counterexamples, we prove that all the main results are sharp. Applying these results, the authors further reveal the mutual equivalences among Muckenhoupt weights, the weighted upper estimate of the characterization of Frank, and the weighted upper estimate of the formula of Brezis et al.