Arick Shao 邵崇哲

# Research / Miscellaneous

## Technical Notes

This section contains notes of a more technical nature, such as proofs or explanations of particular theorems or estimates.

### δ-Distributions in Dispersive Equations

These short notes discuss many of the manipulations involving δ-distributions and pullbacks through δ-distributions found in the study of dispersive equations. The goal is to explain in detail many points that are usually swept under the rug in the standard literature. As an application, we prove a basic bilinear estimate for solutions of the (free) Schrödinger equation.

### The Christ-Kiselev Lemma for Dispersive PDE

The Christ-Kiselev Lemma is a maximal-type estimate, which has important applications in dispersive partial differential equations. In particular, this estimate is an important tool in the application of Strichartz estimates.

Thanks to Yannis Angelopoulos for proofreading and corrections.

### Hörmander's Inequality for Wave Equations

These short notes contains a detailed account of the proof of Hörmander's inequality for wave equations in three spatial dimensions. This estimate played an essential role in establishing small data global existence for nonlinear wave equations satisfying the null condition.

## Solutions to Exercises

These are documents containing solutions to many textbook or lecture notes exercises that I have typed up at some point.

### Nonlinear Dispersive Equations (T. Tao)

This now well-known book, by Prof. Terence Tao, introduces the reader to many aspects of fairly recent research concerning dispersive partial differential equations. The first three chapters of the book, an appendix, and a detailed errata can be found here.

The book contains a multitude of exercises, many of which are quite involved, often reaching into research-level material. I am compiling and editing a list of solutions to many of these exercises. These can be found in the PDF file below. Comments, corrections, and suggestions are most certainly welcome!

Many of these exercises were solved jointly with fellow reading group members: Jordan Bell, David Reiss, Kyle Thompson.