Fourier decay in a bounded complex strip

For a test function $\phi$ supported in a fixed ball of radius $R$ and a fixed height bound $H$, repeated integration by parts givesApply $(1-\Delta {)}^L$ to the compactly supported smooth function $e^{x\cdot \eta }\phi (x)$ 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.