A decomposition of smooth functions relative to a quasi-homogeneous polynomial

Let F be a quasi-homogeneous polynomial in R-n which is the sum of monomials +/-x(i)(pi), p(i) is an integer greater than or equal to 2, 1 less than or equal to i less than or equal to n, n greater than or equal to 2. Every C-infinity function phi has a decomposition phi = A phi + B phi, where A phi(x) is a sum of terms x(lambda)phi lambda[F(x)] and B phi(x) a sum of terms D(ij)psi(ij)(x), with D-ij = partial derivative(i)F partial derivative(j) - partial derivative(j)F partial derivative(i), 1 less than or equal to i < j less than or equal to n. phi lambda and psi(ij) are C-infinity functions which depends on phi in a linear continuous manner. An application to the study of distributions invariant with respect to F is given when F is positive.