Microlocal Analysis

Job: localize singularities of distributions in phase space \(T^*X\) — where and in which directions they oscillate — and propagate them under PDE and FIOs.

Core

Classical local analysis asks where a function is singular (\(\operatorname{sing\,supp}\)). Microlocal analysis refines this to the wavefront set \(\mathrm{WF}(u)\subset T^*X\setminus 0\), enabling symbolic calculus of pseudodifferential operators and Hörmander’s propagation of singularities along bicharacteristics.

Page Focus
Foundations distributions, wavefront set, \(\Psi\)DO symbols, microlocal elliptic regularity, propagation (Hörmander)
Fourier integral operators FIO sketch, canonical relations, PDE applications

Theorem spine

  1. \(\mathrm{WF}(u)=\emptyset\Leftrightarrow u\) smooth; \(\pi(\mathrm{WF}(u))=\operatorname{sing\,supp}u\).
  2. \(\Psi\)DOs act microlocally: \(\mathrm{WF}(Au)\subset\mathrm{WF}(u)\); elliptic \(A\) preserves WF.
  3. Elliptic regularity (microlocal): \(Au=f\) and \(A\) elliptic at \((x,\xi)\) \(\Rightarrow\) \((x,\xi)\notin\mathrm{WF}(u)\setminus\mathrm{WF}(f)\).
  4. Propagation: for real principal type \(P\), \(\mathrm{WF}(u)\setminus\mathrm{WF}(Pu)\) is invariant under the Hamilton flow of \(p=\sigma(P)\).
  5. FIOs quantize canonical relations; parametrices for hyperbolic Cauchy problems are FIOs.

Interfaces

See it

Wave equation. \(P=\partial_t^2-\Delta\), principal symbol \(p=\tau^2-|\xi|^2\). Singularities of solutions travel at speed \(1\) along light rays — bicharacteristic strips of \(p=0\) in \(T^*(\mathbb{R}_t\times\mathbb{R}^n_x)\). Microlocal elliptic regularity is vacuous on the characteristic set; propagation fills in the dynamics.

Check

If \(u\) is a compactly supported distribution and \(\hat u\) is rapidly decreasing in a cone \(\Gamma\), what can you say about \(\mathrm{WF}(u)\)?

No point \((x,\xi)\) with \(\xi\) in the dual cone directions of \(\Gamma\) (more precisely: if a conic neighborhood of \(\xi_0\) has rapid decay of the localized Fourier transform, then \((x,\xi_0)\notin\mathrm{WF}(u)\) for \(x\in\operatorname{supp}\varphi\) of the cutoff). Rapid decay in all directions \(\Rightarrow\) \(u\) smooth on \(\operatorname{supp}\varphi\).

Why do elliptic operators not propagate singularities along nontrivial bicharacteristics?

Ellipticity means \(p(x,\xi)\neq 0\) on \(T^*X\setminus 0\), so the characteristic set is empty; there is no Hamilton flow of interest on \(p=0\). Elliptic regularity removes singularities entirely where \(Pu\) is smooth.

Pitfalls

  • Wavefront set is conic in \(\xi\): \((x,\xi)\in\mathrm{WF}\Rightarrow(x,\lambda\xi)\in\mathrm{WF}\) for \(\lambda>0\).
  • Characteristic set of \(P\) is where \(\sigma_{\mathrm{prin}}(P)=0\) — do not confuse with \(\mathrm{WF}(u)\).
  • \(\Psi\)DO calculus needs symbol classes (\(S^m_{\rho,\delta}\)); formal symbols without remainder estimates are not operators.

Refs

  • Hörmander, The Analysis of Linear Partial Differential Operators I–IV
  • Grigis–Sjöstrand, Microlocal Analysis for Differential Operators
  • Zworski, Semiclassical Analysis
  • Duistermaat, Fourier Integral Operators

Next

Foundations · Fourier integral operators · PDE · Functional analysis · Fourier analysis