Bourgain—Brezis inequalities and related topics

Bourgain—Brezis inequalities is an informal name for a class of estimates that resemble the Sobolev embedding theorem and include the norms in the limit spaces $L_1$ or $L_{\infty}$. Those specific norms, one the one hand, make the inequalities harder to prove, and on the other hand, link them to problems in geometric measure theory. The phenomenon may be traced back to the work of Gagliardo and Nirenberg in late 50-s. It got an intensive development in the 2000-s motivated by a fresh insight by Bourgain and Brezis. Nowadays, the field gets its inspiration from connections to geometric measure theory. I will survey the field, draw the said connections, and, if time permits, outline the recent "probabilistic’’ approach suggested by Ayoush, Wojciechowski, and me.