Anisotropic Smoothness Spaces via Level Sets

"Anisotropic Smoothness Spaces via Level Sets"

It has been understood for some time that the classical smoothness spaces, such as the Sobolev and Besov classes, are not satisfactory for certain problems in image processing and nonlinear PDEs. Their deficiency lies in their isotropy. Functions in these smoothness spaces must be simultaneously smooth in all directions. The anisotropic generalizations of these spaces also have the deficiency that they are biased in coordinate directions. While they allow different smoothness in certain directions, these directions must be aligned to the coordinate axes. In the application areas mentioned above, it would be desirable to measure smoothness in new ways which would allow one to have more local control over the smoothness directions. We introduce one possible approach to this problem based on defining smoothness via level sets. We present this approach in the case of functions defined on IRd. Our smoothness spaces depend on two smoothness indices (s1, s2). The first reflects the smoothness of the level sets of the function, while the second index reflects how smoothly the level sets themselves are changing. As a motivation, we start with d = 2 and investigate Besov smooth domains.