Zoll magnetic systems on the two-torus: a Nash-Moser construction

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional $S$, and then look for its zeros by means of a Nash-Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator $\mathrm{d} S$ in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator $\mathrm{d} S\circ \mathrm{d} S^*$ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.