Anisotropic Variant of the BMO--type seminorms

The purpose of this talk is to present the relation between certain BMO--type seminorms and the total variation of SBV functions

Recently, Bourgain, Brezis and Mironescu introduced a new BMO--type space $B\subset L^1(Q)$ on the unit cube $Q\subset {\mathbb{R}}^n$, by mean of the seminorm $$\Vert f{\Vert }_B=\underset{0<\epsilon <1}{sup}[f{]}_{\epsilon }$$ where $[f{]}_{\epsilon }$ is defined with a suitable maximization procedure. The space $B$ contains BMO and the space BV of functions of bounded variation.

Later Ambrosio, Bourgain, Brezis and Figalli give a new characterization of the perimeter of sets in terms of this seminorm considering characteristic functions of sets, was studied. Further results characterizing total variation of SBV functions and norm of Sobolev functions, independent of theory of distributions, were given by Fusco, Moscariello, Sbordone.

Using a different approach, by considering in $\text{(???)}$, instead of cubes, covering families by translations of a given open bounded set with Lipschitz boundary, we give a representation formula of the total variation of SBV function.