Malgrange division by quasianalytic functions

Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.