Instructor
Office Hours
References
Robert Chang (rchang@reed.edu)
M 2:30–4:00, F 12:00–1:30 at Lib 390
Distributions/Fourier transform/functional analysis
[H1] Hörmander, The Analysis of Linear Partial Differential Operators, Vol I
[St] Strichartz, A Guide to Distribution Theory and Fourier Transforms
[RS] Reed and Simon, Functional Analysis
Richards and Youn, The Theory of Distributions
Microlocal analysis (the homogeneous theory)
[H3] Hörmander, The Analysis of Linear Partial Differential Operators, Vol III
[GS] Grigis and Sjöstrand, Microlocal Analysis for Differential Operators
Folland, Harmonic Analysis in Phase Space
Semiclassical analysis (with small parameter h)
[Zw] Zworski, Semiclassical Analysis
Dimassi and Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit
Week
1
Date
1/23
1/25
1/27
Topics and Reading
Intro to PsiDOs and microlocalization
Schwartz space
Crash course on function spaces
[RS] is an encyclopedic 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.
2
1/30
2/1
2/3
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].
Proof of "key lemma"
See [Zw, Lemma 3.3 on p. 31]. The last paragraph needs additional justification.
Fourier transform of a Gaussian, carefully
Stein and Shakarchi's Complex Analysis computes this on p. 43 using Cauchy's theorem.
Higher dimensional analogues are computed in [Zw, p. 29].
3
2/6
2/8
2/10
Fourier transform on Lᵖ spaces (Riemann–Lebesgue, Plancherel, Riesz–Thorin)
The chapter on Fourier transform in [RS] is an excellent reference.
Interlude: some abstract harmonic analysis on LCA groups
For starters, take a look at Terry Tao's blog post.
Standard references are Folland's A Course in Abstract Harmonic Analysis or Rudin's Fourier Analysis on Groups.
Convolutions and the soup of classical inequalities (Young's for products, Hölder's, etc.)
See [St, H1, RS], any text on PDE/analysis (e.g., Folland, Rudin, Stein–Shakarchi), or even Wikipedia.
4
2/13
2/15
2/17
Soup of classical inequalities continued (Minkowski's, generalized Hölder's, Young's for convolution)
Convolution and derivatives; approximate identity
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 worked examples.
5
2/20
2/22
2/24
Distributions: several examples
Operations on distributions
Snow day; class canceled
6
2/27
3/1
3/3
Distributions: convolution and the Fourier transform
Distributions: support
Applications to PDE: elliptic regularity, Liouville's theorem
7
3/6
3/8
3/10
Applications to PDE: heat equation (propagator and energy method)
Applications to PDE: heat equation (propagator and energy method)
Applications to PDE: wave equation in 1D (d'Alambert's formula)
Spring break
9
3/20
3/22
3/24
Stationary phase: preliminaries (FT of a complex Gaussian, Morse lemma)
Experts all quote [H1; Theorem 7.7.5] for the method of stationary phase.
I am presenting a semi-handwaving proof from [GS, § 2]; see also [Zw, § 3].
Stationary phase: quadratic phase functions
Stationary phase: general phase functions
10
3/27
3/29
3/31
Schwartz kernel theorem, oscillatory integrals as distributions
Schwartz kernel theorem can be found in any reputable PDE/distributions textbook near you.
[Zw, § 3.6] contains a general statement about oscillatory integrals as distributions.
Symbol classes
We will forgo the more general (δ,ρ)-symbols of Hörmander in favor of the Kohn–Nirenberg symbols Sᵐ.
I am roughly following [GS] and Hintz's notes for the next several lectures.
Density of residual symbols in Sᵐ, asymptotic summation
11
4/3
4/5
4/7
Quantization of elements of Sᵐ as PsiDOs, basic mapping properties
The space Ψᵐ, the symbol of a PsiDO
Left/right reduction via stationary phase
The formal computation is straightforward from the stationary phase formula; some more details (e.g., the symbol estimates which guarantee we have an honest asymptotic expansion of symbols on our hands) are found in [GS]. Alternatively, see Hintz's notes for a proof using Taylor's theorem.
12
4/10
4/12
4/14
Reduction continued
Adjoints, compositions, commutators
Singular support and pseudolocality, principal symbols
13
4/17
4/19
4/21
Principal symbol: coordinate invariance
Ellipticity, elliptic parametrix, elliptic regularity
Wavefront set: several examples
14
4/24
4/26
4/28
Microlocality, microlocal parametrix
Handwaving: propagation of singularities
Handwaving: propagation of singularities