Analytic estimate for the torus cohomological equation ($\Vert f{\Vert }_{\sigma -\delta }\le C{\gamma }^{-1}{\delta }^{-(n+\tau )}\Vert g{\Vert }_{\sigma }$)

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.