Bourgain–Brezis inequalities and Besov spaces

Bourgain–Brezis inequalities are generalizations of the limiting Sobolev embedding $W_1^1 \to L_{d/(d-1)}$ to the case where the first gradient is replaced with a more involved vectorial differential operator. I will show how a passage to the Besov scale allows induction on scales arguments for such type inequalities. The described approach allows to significantly sharpen some Bourgain–Brezis inequalities and related estimates (e.g. non-linear generalizations of the aforementioned Sobolev embeddings suggested by Maz'ja) and also get rid of the differential structure. The core of the method is a construction of a special discrete model problem, which is interesting in itself. In particular, the martingale Besov spaces play a pivotal role there.