Reference texts:
- Distributions/Fourier transform/functional analysis
- - [H1] The Analysis of Linear Partial Differential Operators, Vol I, by Hörmander
- - [St] A Guide to Distribution Theory and Fourier Transforms by Strichartz
- - [RS] Functional Analysis by Reed–Simon
- - The Theory of Distributions by Richards–Youn
- Microlocal analysis (the homogeneous theory)
- - [H3] The Analysis of Linear Partial Differential Operators, Vol III, by Hörmander
- - [GS] Microlocal Analysis for Differential Operators by Grigis–Sjöstrand
- - Harmonic Analysis in Phase Space by Folland
- - Notes by Melrose, Taylor, Hintz, etc.
- Semiclassical analysis (microlocal with small parameter $h$)
- - [Zw] Semiclassical Analysis by Zworski
- - Spectral Asymptotics in the Semi-Classical Limit by Dimassi–Sjöstrand
1/23
Intro to PsiDOs and microlocalization
1/27
Crash course on function spaces
[RS] is a comprehensive reference for functional analysis. In particular, § III discusses Banach space, § V.2 Fréchet spaces, and § V.3 the spaces of Schwartz functions and of tempered distributions.
1/30
Fourier transform on Schwartz space
[St, § 3] is a gentle introduction to the theory of Fourier transform. I am roughly following [Zw, § 3], which is an abridged version of [H1, § 7].
2/1
Proof of ``key lemma''
See [Zw, Lemma 3.3 on p. 31]. Some additional argument is needed to justify the assertions in the last paragraph.
2/3
Interlude: calculus and linear algebra (i.e., FT of a Gaussian, carefully)
Stein and Shakarchi's Complex Analysis computes using Cauchy's theorem the FT of a one-dimensional Gaussian on p. 43. Higher dimensional analogues are computed in [Z, p. 29].
2/6
FT on $L^p$ spaces (Riemann–Lebesgue, Plancherel, Riesz–Thorin)
Excellent place to start is [RS, Chapter on Fourier Transform].
2/8
Some abstract harmonic analysis on LCA groups
For starters, take a look at
Terry Tao's blog. Standard references are Folland's
A Course in Abstract Harmonic Analysis or Rudin's
Fourier Analysis on Groups.
2/10
Convolutions and the soup of classical inequalities (Young's for products, Hölder's)
See [St, H1, RS], any text on PDE/analysis (e.g., Folland, Rudin, Stein–Shakarchi), or even Wikipedia.
2/13
Soup of classical inequalities continued (Minkowski's, generalized Hölder's, Young's for convolution)
2/15
Convolution and derivatives; approximate identity
2/17
Distributions: definitions and continuity in terms of seminorm bounds
[St] is a good place to start and [H1] remains the encyclopedic reference. A happy medium is [RS, § V], which contains several exmaples.
2/20
Distributions: several examples
2/22
Operations on distributions
2/27
Distributions: convolution and the Fourier transform
3/1
Distributions: support
3/3
Applications to PDE: gain of $L^2$ derivatives (elliptic regularity), Liouville's theorem
I am stealing from the excellent Partial Differential Equations by Rauch
3/6
Heat equation: propagator and energy method
3/8
Heat equation: propagator and energy method
3/10
Wave equation in 1D: d'Alambert's formula
3/20
Stationary phase: preliminaries (FT of a complex Gaussian, Morse lemma)
Experts all quote [H1; Theorem 7.7.5] for this result. I am doing a semi-hand-waving version from [GS, § 2]. Alternative proofs and more details (e.g., a careful proof of the Fourier transform of a complex Gaussian) is in [Zw, § 3].
3/22
Stationary phase: quadratic phase functions
3/24
Stationary phase: general phase functions
3/27
Schwartz kernel theorem, oscillatory integrals as distributions
Schwartz kernel theorem can be found in any repuatble PDE/distributions textbook near you. [Zw, § 3.6] contains a general statement about oscillatory integrals as distributions.
3/29
Symbol classes
I am roughly following [GS] and Hintz's notes. Symbol classes of type $(\rho, \delta)$ were introduced by Hörmander. We will simplify things and work with classes $S^m := S^m_{1,0}$.
3/31
Density of $S^{-\infty}$, asymptotic summation
4/3
Quantization as PsiDOs, basic mapping properties
4/5
PsiDOs: quantizing $S^m$
4/7
Left/right reduction for $S^{-\infty}$
4/10
Left/right reduction for $S^m$ via stationary phase
The formal computation is straightforward from the stationary phase formula; some more details (e.g., the symbol estimates that show what we get is indeed an honest asymptotic expansion of symbols) are found in [GS]. Alternatively, see Hintz's notes for a proof using Taylor's theorem.
4/12
Reduction conitued, adjoints, compositions
Again, we take the highway that is formal stationary phase, leaving additional justification to [GS]. Hintz's notes, which are rigorous, uses Taylor's theorem.
4/14
Principal symbol: commutators, coordinate invariance
Once again, see [GS] for a more careful version of stationary phase, or Hintz's notes (equation (5.9) is the so-called Kuranishi trick).
4/17
Principal symbol: coordinate invariance continued
4/19
Handwaving: some differential geometry, singular support
4/21
Handwaving: ellipticity, elliptic parametrix, mapping properties of PsiDOs
4/24
Handwaving: wavefront set, microlocalization
4/26
Handwaving: propagation of singularities
4/28
Handwaving: quantum mechanics