Cohomological equation on a Diophantine torus (${\mathcal{D}}_{\omega }f=g$)

For a Diophantine frequency vector, the equation $\omega \cdot {\partial }_{x}f=g$ on a periodic torus has an analytic solution on each smaller complex strip exactly when $g$ has zero average. Its nonzero Fourier coefficients are $f_k=g_k/(ik\cdot \omega )$, and its zero coefficient is arbitrary. Thus the normalized zero-mean solution is unique.

For a sufficiently small analytic periodic vector field $w$, a near-identity angular diffeomorphism $\chi$ and a constant vector $a$ solve $D\chi \omega =\omega +w\circ \chi +a$. They satisfy $\chi -\mathrm{id}=O(w)$, $a=O(w)$; if $⟨w⟩=0$, then $a=O(w^2)$. For the defect $e={\mathcal{D}}_{\omega }\chi -\omega -w\circ \chi -a$, set $A=D\chi$, choose $\Delta a=⟨A^{-1}{⟩}^{-1}⟨A^{-1}e⟩$, and solve ${\mathcal{D}}_{\omega }v=A^{-1}(\Delta a-e)$ with zero mean. Updating $\chi$ by $Av$ leaves a quadratic defect $(De)v-[w(\chi +Av)-w(\chi )-Dw(\chi )Av]$. The torus small-divisor estimate and geometrically decreasing strip losses yield a convergent analytic iteration.

If $g$ is analytic on a complex strip of width $\sigma$ and has zero mean, the normalized solution satisfies $\Vert f{\Vert }_{\sigma -\delta }\le C_{n,\tau }{\gamma }^{-1}{\delta }^{-(n+\tau )}\Vert g{\Vert }_{\sigma }$. Fourier decay supplies $e^{-\delta |k{|}_1}$ after the strip loss, and lattice-shell counting bounds the remaining sum ${\sum }_{k\ne 0}|k{|}_1^{\tau }{e}^{-\delta |k{|}_1}$. This simple bound is sufficient for analytic conjugacy iterations even when sharper estimates are available.