On small-denominators equations with large variable coefficients

Abstract.

On an n-torus, we study the first-order differential equation $-i(\nabla\cdot\omega)x(q) +Ex(q)+Bh(q)x(q) =b(q)$, where E,B are real parameters, h,b are analytic functions (the mean value of h vanishes) and $\omega$ is a diophantine n-vector incommensurable with E. Assuming that $E\ge1$ and $E^{\theta}\ge CB (0 <\theta <1)$, we derive an a priori estimate for analytic norms of the solution x, uniform in E and B.