  A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is
established. Boundary regularity of domains is replaced with information on
boundary traces of trial functions and of their derivatives up to some explicit
minimal order. The relevant Sobolev inequalities involve constants independent
of the geometry of the domain, and exhibit the same critical exponents as in
the classical inequalities on regular domains. Our approach relies upon new
representation formulas for Sobolev functions, and on ensuing pointwise
estimates which hold in any open set.
