Paley wiener and microlocal analysis

This pair develops only the foundational Fourier calculus on tempered distributions. Two major continuations are deliberately not recorded here as proved results.

Paley-Wiener theory. The compact-support calculation on the A page proves only that the real-frequency Fourier transform of a compactly supported distribution is smooth and polynomially bounded. Paley-Wiener theory goes much further: it studies holomorphic continuation to complex frequency and relates quantitative growth there to the support of the original distribution. Neither direction of that characterization is proved in this pair, so it must not be used as a dependency supplied here.

Microlocal analysis. The examples here distinguish global support only when a compact-support hypothesis makes a convolution or Fourier argument legal. Microlocal analysis refines singular support by retaining cotangent directions of nonsmoothness and studies how those directions propagate under differential and pseudodifferential operators. Wavefront sets, pseudodifferential calculus, and propagation theorems require substantial new definitions and estimates and are outside this pair.

The cited notes state the Paley-Wiener theorem in §11.2.5 and identify the pseudodifferential framework as part of microlocal analysis in §14.3. Those source statements are orientation only here; this remark is not a supplier for either theory.