ON THE REGULARITY OF SETS IN RIEMANNIAN MANIFOLDS

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

WAVELET CHARACTERIZATIONS OF MULTI-DIRECTIONAL REGULARITY

The study of d dimensional traces of functions of m several variables leads to directional behaviors. The purpose of this paper is two-fold. Firstly, we extend the notion of one direction pointwise Hölder regularity introduced by Jaffard to multi-directions. Secondly, we characterize multi-directional pointwise regularity by Triebel anisotropic wavelet coefficients (resp. leaders), and also by Calderón anisotropic continuous wavelet transform.

SHEARLET TRANSFORMS AND DIRECTIONAL REGULARITIES

In an effort to characterize uniform and pointwise Hölder regularities, we obtain necessary decay rates and sufficient decay rates of continuous and discrete shearlet transform across scales. They are the same rates as those of the Hart Smith and continuous curvelet transforms. We then consider the situation where the regularity on a line in a non-parallel direction is significantly lower than the directional regularity along the line in a neighborhood of the line. Similar to that of the Hart Smith and continuous curvelet transforms, a set of necessary conditions for this direction of singularity is that the continuous shearlet transform decays half an order faster in directions "away" from the direction of the line and that the decay rate in directions "near" the line depends also on the horizontal distance from the line to the parallel line containing the center of the shearlet function. These similarities among the three transforms call for a general theory of transforms "with parabolic scaling."