Lec #  TOPICS  KEY DATES 

1  Space of Continuous Functions, Dual Space, Positivity  
2  Outer Measures and Measures  
3  Caratheodory’s Theorem  
4  Measurable Functions and the Integral  Including Lebesgue’s Theorem of Dominated Convergence  
5  Riesz Representation, L^{p} Spaces and Completeness, L^{2}(X,μ) and Hilbert Space  Problem set 1 due 
6 
Riesz Representation for Hilbert Space Differentiability and Schwartz Space of Test Functions 
Problem set 2 due 
7 
Properties of S(R^{n}) Tempered Distributions Differentiation and Differential Operators Fourier Transform 

8 
Bump Functions Characterization of δ Fourier Inversion Plancherel Formula 
Problem set 3 due 
9 
Convolution and Density Fourier Transform on L^{2}(R^{n}) 

10 
Sobolev Spaces and Sobolev Embedding Duality between Sobolev Spaces 

11 
Schwartz Representation Theorem Fundamental Solution of ∂x + i∂y Support of a Distribution; Distributions of Compact Support (Start) 
Problem set 4 due 
12 
Compact Supports Convolution of Distributions supp(u_{*}v) ⊂ supp(u)+ supp(v) if one, at least, has Compact Support Fundamental Solutions 
Problem set 5 due 
13  Singular Support, Hypoellipticity, Ellipticity  Parametrices for Elliptic Operators  
14 
Fundamental Solution of the Heat Operator, Hypoellipticity, Initial Value Problem Homogeneity The Distributions x^{z}_{±}, z ∈ C\(N) 

15 
Distributions Supported at 0 Homogeneous Distributions of Order on the Line
Hadamard Regularization Cone Supports 

16 
Singular Support and Products Conic Support and Convolution 

17 
Wavefront Set Refines Singular Support Scattering Wavefront Set Product and Wavefront Set The Wave Equation 
Problem set 6 due 
18 
Fundamental Solution of the Wave Equation Solution to the Cauchy Problem 

19  Operators and Schwartz’ Kernel Theorem  Problem set 7 due 
2024 
Lidskii’s Theorem Δ on the Torus SelfAdjointness of Δ+V Spectral Decomposition Wave Equation on the Torus Wave Equation on Torus with Potential Δ+V on R^{n} with V ∈ C_{c}(R^{n}) 

25 
Questions Trace as Integral of the Kernel over the Diagonal Microlocal Analysis 
Calendar
Course Info
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Lecture Notes
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Problem Sets with Solutions