Mappings of BMO-distortion and Beltrami-type operators (Q1411689)

In this interesting paper the authors are concerned with mappings with a distortion function bounded above by a BMO function of small BMO-norm. Since quasiregular mappings are assumed a priori to have bounded distortion and since the BMO-norm of a constant function is zero, it follows that the class of mappings considered by the authors contains the quasiregular mappings and can be viewed as a natural extension of them. It is shown that to a large extent the regularity properties of mappings with a distortion function in BMO are controlled by the invertibility properties of a certain family of singular integral operators, closely related to the Beurling-Ahlfors transform. These operators are called Beltrami-type operators. The authors establish, in even dimensions, improved integrability and regularity beyond the a priori assumptions, and study the removable sets for bounded mappings in their class. In the final part of the paper, the authors make a comment about the odd-dimensional case.