MIT OpenCourseWare

Browse materialsMIT 18.155 · Fall 2004

Differential Analysis

Subject: Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor: Prof. Richard Melrose
Term: Fall 2004
Level: Graduate
What's inside: 1 solution · 2 lecture notes · 25 other files · 3 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.155 Differential Analysis, Fall 2004, Prof. Richard Melrose — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

18 files

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8 files

Other materials from this course

MIT OpenCourseWare, Prof. Richard Melrose, 18.155 Differential Analysis, Fall 2004.

License: CC BY-NC-SA 4.0 · Edition: Fall 2004 · Open the original course on OCW ↗

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Differential Analysis

Before you start

What's here, and what isn't

Materials by section

Lecture Notes

Assignments

Other materials from this course

Lecture 1: Space of Continuous Functions, Dual Space, Positivity

Lecture 2: Outer Measures and Measures

Lecture 3: Caratheodory’s Theorem

Lecture 4: Measurable Functions and the Integral - Including Lebesgue’s Theorem of Dominated Convergence

Lecture 5: Riesz Representation, L p Spaces and Completeness, L ²( X ,μ) and Hilbert Space

Lecture 6: Riesz Representation for Hilbert Space Differentiability and Schwartz Space of Test Functions

Lecture 7: Properties of S (Rⁿ) Tempered Distributions Differentiation and Differential Operators Fourier Transform

Lecture 8: Bump Functions Characterization of δ Fourier Inversion Plancherel Formula

Lecture 9: Convolution and Density Fourier Transform on L ²(Rⁿ)

Lecture 10: Sobolev Spaces and Sobolev Embedding Duality between Sobolev Spaces

Lecture 11: Schwartz Representation Theorem Fundamental Solution of ∂x + i∂y Support of a Distribution; Distributions of Compact Support (Start)

Lecture 12: Compact Supports Convolution of Distributions supp(u*v) ⊂ supp(u)+ supp(v) if one, at least, has Compact Support Fundamental Solutions

Lecture 13: Singular Support, Hypoellipticity, Ellipticity - Parametrices for Elliptic Operators

Lecture 14: Fundamental Solution of the Heat Operator, Hypoellipticity, Initial Value Problem Homogeneity The Distributions xz±, z ∈ C\(-N)

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

Lecture 16: Singular Support and Products Conic Support and Convolution

Lecture 17: Wavefront Set Refines Singular Support Scattering Wavefront Set Product and Wavefront Set The Wave Equation

Lecture 18: Fundamental Solution of the Wave Equation Solution to the Cauchy Problem

Lecture 19: Operators and Schwartz’ Kernel Theorem

Lecture 25: Questions Trace as Integral of the Kernel over the Diagonal Microlocal Analysis