A decomposition technique for John domains

We develop a method to decompose functions with mean value zero that are defined on a (possibly unbounded) John domain into a countable sum of functions with mean value zero and support in cubes or balls.This method enables us to generalize results known for simple domains to the class of John domains and domains satisfying a certain chain condition.As applications we present the solvability of the divergence equation div u = f , the negative norm theorem, Korn's inequality, Poincaré's inequality and a localized version of the Fefferman-Stein inequality.We present the results for weighted Lebesgue spaces and Orlicz spaces.