On the growth and stability of real-analytic functions

This paper addresses the issue of whether integrals of real-analytic functi ons remain finite under small deformations. An approach based on uniform es timates for certain classes of one-dimensional integrals is introduced. It is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of Karpushkin, as well as produc e new results in higher dimensions. In dimension three, the new stability r esults are sharp, as shown by the well-known example of Varchenko.