Lemma 3.20 . [043B]

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Lemma 3.20.

(Exponential decay of higher Fourier modes) In the situation of Lemma 3.19, the higher Fourier modes of $\nabla_{g_{a}}^{2}\Delta_{a}^{-1}f$ admit estimate in the region $\{\ell>A^{1/2}\}\subset\mathcal{B}^{+}_{\nu}$ ,

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}(f-\bar{f})|_{g_{a}}\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.