Power substitution in quasianalytic Carleman classes

Consider an equation of the form f(x)=g(xk), where k>1 and f(x) is a function in a given Carleman class of smooth functions. For each k, we construct a Carleman-type class which contains all the smooth solutions g(x) to such equations. We prove, under regularity assumptions, that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions