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The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class Cm. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class Cm or Cc. Such a property is characteristic for the solutions f: Rn -X R of the equations k (*) z ai(x, t)f(Oki(x, t)) = F(x, f(X1(x)), . . . , f(XS(x))) + b(x, t), i=l where x E Rn, t E Rr n > 1, r > 1 and where the functions O: Rn+r Rn, A>: Rn -Rn, ai: Rn+r -R, b: Rn+r +R, F: Rn+s * R satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating (*) and fixing t is of constant strength, hypoelliptic at a point x0. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of (*), is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.

The regularity of the locally integrable and continuous solutions of nonlinear functional equations