2010 PDE2 Lectures

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This page will collect information related to lectures given by J. Colliander in PDE2. (These notes closely follow Terry Tao's UCLA 254A notes.)

Contents

Fourier Transform

Algebraic Properties of the Fourier Transform

Notes on Algebraic Properties of the Fourier Transform

Phase Space Localizations

Notes on Phase Space Localizations

Fourier Transform Mapping Properties

Notes on Fourier Transform Mapping Properties

Sobolev Spaces

What is a Sobolev Space?

Sobolev Spaces

Sobolev Embedding Estimate

(Proof via Level set decomposition and adapted high/low frequency truncation)

Sobolev Embedding

Gagliardo Nirenberg Estimate

Gagliardo-Nirenberg Estimate via Sobolev Embedding and Convexity

Littlewood-Paley Theory

The Littlewood-Paley Decompostion, Properties of the pieces $P_k f$, Cheap Littlewood-Paley Inequality

Littlewood-Paley

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