SOBOLEV REGULARITY OF THE BEURLING TRANSFORM ON PLANAR DOMAINS

Consider a Lipschitz domain Omega and the Beurling transform of its characteristic function B chi Omega(z )= -p.v.1/pi z(2) *chi Omega O(z). It is shown that if the outward unit normal vector N of the boundary of the domain is in the trace space of W-n,W-p (Omega) (i.e., the Besov space B-p,p(n-1/p) (partial derivative Omega)) then B chi Omega is an element of W-n,W-p(Omega). Moreover, when p > 2 the boundedness of the Beurling transform on W-n,W-p(Omega) follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.