Some remarks on interior elliptic regularity estimates

With the help of certain inequalities which we have established for multiplication in Sobolev-Slobodeckij spaces, we will give a simple proof of a version of elliptic estimate for elliptic operators that belong to a certain general class of operators denoted by D-m(e,q). This version of elliptic estimate is not restricted to second order operators and the coefficients are not assumed to be smooth. Our proof is tailored to accommodate fractional Sobolev spaces and our methods are elementary in nature in the sense that they do not directly appeal to the theory of pseudo-differential operators or Littlewood-Paley theory. The elliptic estimate whose proof is presented in this paper has applications in the existence theory of several important PDEs that appear in the study of Einstein constraint equations in general relativity.