The BMO Space, Function Smoothness, and the Curse of Dimensionality

In this talk, we introduce a modern perspective on the classical concept of function smoothness. Instead of focusing on pointwise analysis, we explore functions through their averages, offering a more comprehensive view of their behavior. Our approach presents a unifying theory connecting the celebrated bounded mean oscillation (BMO) space and the classical Sobolev space theory, both of which are fundamental in the study of elliptic partial differential equations. We will highlight key structures that were previously not fully understood, with the potential to unlock new results in both fields. Several open problems and conjectures accessible to a broad audience will also be discussed.