Mappings of BMO–bounded distortion Kari Astala Tadeusz Iwaniec Pekka Koskela Gaven Martin ∗ 1 Introduction This paper can be viewed as a sequel to the work [9] where the theory of map- pings of BMO–bounded distortion is developed, largely in even dimensions, using singular integral operators and recent developments in the theory of higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions. The planar case was studied earlier by G. David [4]. In particular he obtained existence theorems, modulus of continuity estimates and bounds on area distortion for mappings of BMO–distortion (in fact, in slightly more generality). We obtain similar results in all even dimensions. One of our main new results here is the extension of the classical theorem of Painlev´ e concerning removable singularties for bounded analytic functions to the class of mappings of BMO bounded distortion. The setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem, or “the measurable Riemann mapping theorem”, which is not available in higher dimensions. We give a construction to show our results are qualitatively optimal. Another surprising fact is that there are domains which support no bounded quasiregular mappings, but admit * Research of all authors supported in part by grants from the N.Z. Marsden Fund. Also the U.S. National Science Foundation (TI), DMS–9706611 and the Academy of Finland (KA+PK), SA–34082. 1