Fourier transform of a compactly supported distribution

For $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$, its Fourier transform is the smooth functionIt has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

Let a distribution have support in a compact convex set $K$. Fixed neighborhood cutoffs initially give an exponential bound larger than the support function $H_K$. Cutoffs of width $\epsilon$ cost powers of ${\epsilon }^{-1}$ in derivative estimates but enlarge the exponential by only $e^{C\epsilon |\mathrm{Im}z|}$. Choosing $\epsilon =(1+|z|{)}^{-1}$ absorbs the cutoff cost into a polynomial and retains the exact exponential $e^{H_K(\mathrm{Im}z)}$.

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.