SYSTEMATIC MATHEMATICS

Microlocal Analysis & Pseudodifferential Operators

How can a distribution be singular at one point in one covector direction but smooth there in another, why does an elliptic operator reveal exactly the same wavefront as its forcing, and how do wave packets, boundary data, tomographic lines, or scattering states move singular information? This Stage 5 doctoral-specialization core answers those questions in seventy-six visible bilingual chapters across twelve content-sized units of lengths 5, 7, 6, 8, 7, 5, 6, 8, 6, 5, 7, and 6. It begins with cotangent frequency, cutoffs, covectors, nonstationary phase, stationary phase, singular support, localized Fourier decay, wavefront sets, Dirac and conormal examples, pseudolocality, and the product/pullback criteria. It then builds symbol classes and seminorms, classical homogeneous expansions, asymptotic summation, ellipticity, parameter symbols, Kohn–Nirenberg and Weyl quantization, proper support, amplitude reduction, composition, adjoints, coordinate invariance, diagonal kernels, Sobolev mapping, Calderón–Vaillancourt, compactness, Fredholm theory, sharp Gårding, parametrices, elliptic estimates, systems, and the boundary-symbol limitation. The second half develops Hamilton fields, bicharacteristics, positive commutators, real-principal-type propagation, wave rays, glancing and radial boundaries, nondegenerate phases, Lagrangian distributions, Fourier integral operators, canonical graphs, Egorov, clean composition, wave propagators, caustics, semiclassical symbols, coherent states, semiclassical wavefront, WKB, transmission, Calderón and Boutet de Monvel structures, b-analysis, subellipticity, Weyl and heat asymptotics, meromorphic resolvents, conductivity linearization, Radon visibility, scattering relations, numerical audits, and six research-reconstruction dossiers. Every chapter defines its spaces, phase variables, covector signs, symbol order, quantization, support, mapping scale, remainder, and first failure. The course supplies doctoral research preparation in microlocal PDE and operator analysis; it does not claim complete coverage of Hörmander symbol types, paradifferential and rough calculi, global index theory, analytic microlocal analysis, hyperfunctions, paired or intersecting Lagrangians, corner/edge calculi, nonlinear inverse problems, resonance theory, quantum chaos, or any universal postdoctoral frontier.

Before this course: Completed Sobolev Spaces, Distributions & Weak PDE, Harmonic Analysis & Wavelets, Differential Forms/de Rham/Hodge Theory, Differential Geometry & Manifolds, Functional Analysis, and the supporting Real/Complex Analysis and PDE courses. Learners must already be able to use distributional Fourier transforms, Sobolev mapping and compactness, cotangent bundles and symplectic forms, weak elliptic estimates, spectral decomposition, and coordinate changes. No prior complete course in wavefront sets, pseudodifferential operators, Fourier integral operators, propagation of singularities, boundary calculi, or semiclassical analysis is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Why position alone cannot describe a singularity

Objective: Why is the pair (x_0,ξ_0) more informative than singular support?

At one point, two distributions may be singular in different directions. A vertical jump edge and a horizontal jump edge share a crossing point but react differently to rapidly oscillating probes. Microlocal analysis separates position x from nonzero covector direction ξ. Before using a formula, identify the open set or manifold, the distribution or Sobolev space, the Fourier convention, the conic set, the operator order, and whether a conclusion is local, microlocal, or global.

Read the notation from this chapter's definition rather than treating symbols as decoration: For x∈R^n and ξ∈R^n∖{0}, the pair (x,ξ) records a base position and a nonzero frequency covector. Scaling ξ by t>0 keeps its direction, so microlocal singular sets are conic in ξ. The exact result established here is: Multiplying by a cutoff localizes position, while decay of the Fourier transform of that localized distribution tests directional smoothness; neither operation alone records both pieces of information.

The proof has a checkable route: 1. Choose χ∈C_c^∞ equal to one near the base point x_0, so χu forgets behavior far from x_0. 2. Compute or estimate 𝓕(χu)(ξ). Rapid decay in an open cone Γ around ξ_0 means repeated integration by parts is available in those directions. 3. Changing χ while retaining χ=1 near x_0 changes only a term supported away from x_0; convolution estimates preserve rapid decay in a slightly smaller cone. Every estimate must state which constants are uniform, which variables are large, and which lower-order or smoothing remainder is permitted.

Reconstruct the concrete calculation “Let u(x_1,x_2)=H(x_1), choose χ(x)=χ_1(x_1)χ_2(x_2) near the origin, and identify which frequency component can fail to decay rapidly.” before generalizing it. Then test the nearest failure: The zero covector is excluded because it contains no direction and compact-frequency behavior does not detect differentiability. A large value of one Fourier coefficient also does not by itself prove singularity; failure of rapid decay throughout every suitable cone is required. This separates a theorem from a formal manipulation.

Multiplying by a cutoff localizes position, while decay of the Fourier transform of that localized distribution tests directional smoothness; neither operation alone records both pieces of information.

Choose χ∈C_c^∞ equal to one near the base point x_0, so χu forgets behavior far from x_0.

Compute or estimate 𝓕(χu)(ξ). Rapid decay in an open cone Γ around ξ_0 means repeated integration by parts is available in those directions.

Changing χ while retaining χ=1 near x_0 changes only a term supported away from x_0; convolution estimates preserve rapid decay in a slightly smaller cone.

Let u(x_1,x_2)=H(x_1), choose χ(x)=χ_1(x_1)χ_2(x_2) near the origin, and identify which frequency component can fail to decay rapidly.

  1. The localized transform factors as 𝓕(χ_1H)(ξ_1)𝓕(χ_2)(ξ_2).
  2. The smooth compactly supported factor χ_2 has a rapidly decreasing transform in ξ_2.
  3. The jump in x_1 produces only algebraic decay in ξ_1, so singular covectors are normal to the line x_1=0 and have ξ_2=0.

Result: The possible singular directions at the origin are nonzero multiples of dx_1, not arbitrary nonzero covectors.