= Fourier decay in a bounded complex strip
For a <test function> $\varphi$ supported in a fixed ball of radius $R$ and a fixed height bound $H$, repeated <integration by parts> gives
$$
|\widehat\varphi(\xi+i\eta)|\leq C_{R,H,L}
\max_{|\alpha|\leq2L}\|\partial^\alpha\varphi\|_\infty(1+|\xi|^2)^{-L},
\qquad |\eta|\leq H.
$$
Apply $(1-\Delta)^L$ to the compactly supported smooth function $e^{x\cdot\eta}\varphi(x)$ inside the real-frequency Fourier integral. The bounded $\eta$ absorbs its derivative and exponential factors into the constant. This estimate makes the numerator of a <Hörmander staircase> integral absolutely integrable and makes truncated contour endpoints vanish.
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