\nopagenumbers
\font\tenmsy=msbm10
\font\sevenmsy=msbm7
\font\fivemsy=msbm5
\newfam\msbfam \textfont\msbfam=\tenmsy \scriptfont\msbfam=\sevenmsy
\scriptscriptfont\msbfam=\fivemsy
\def\hexnumber#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or A\or
B\or C\or D\or E\or F\fi}\edef
\msyy{\hexnumber\msbfam}
\mathchardef\C="0\msyy43
\def\dbar{\overline\partial}

\centerline{\bf Syllabus for J. Tedor's Oral Qualifying Exam}
\par \noindent {\it Committee:} 
Sagun Chanillo, Stephen Greenfield, Xiaojun Huang, Fran\c cois Treves

\medskip

\noindent I.\ {\bf Complex Analysis}

\smallskip

\item{1.} The $\dbar$ equation in one variable
\itemitem{a.} Cauchy's integral formula in discs and its applications
\itemitem{b.} The Runge approximation theorem
\itemitem{c.} The Mittag-Leffler theorem
\itemitem{d.} The Weierstra\ss\ theorem

\smallskip

\item{2.} The $\dbar$ equation in several variables

\itemitem{a.} Cauchy's integral formula in polydiscs and its applications
\itemitem{b.} Hartogs' extension theorem

\smallskip

\item{3.} Domains of holomorphy

\itemitem{a.} Holomorphic convexity
\itemitem{b.} Reinhardt domains and convergence of power series

\smallskip

\item{4.} Pseudoconvexity and plurisubharmonic functions

\itemitem{a.} The plurisubharmonicity of $-\log \delta(z, ^c \Omega)$,
$\Omega$ a domain of holomorphy
\itemitem{b.} Plurisubharmonic exhaustion functions
\itemitem{c.} Plurisubharmonic convexity
\itemitem{d.} The Levi form
\itemitem{e.} Analytic discs and the continuity principle

\smallskip

\item{5.} $L^2$ estimates and existence theorems for the
$\overline\partial$
operator

\itemitem{a.} H\"ormander's solution and the basic estimate
\itemitem{b.} The Levi problem

\smallskip

\noindent (Reference: Chapters 1, 2, 4 %one, two and four
of Lars H\"ormander,
{\it An Introduction to Complex Analysis in Several Variables})

\medskip

\noindent II.\ {\bf Harmonic Analysis}

\smallskip

\item{1.} The Fourier transform

\itemitem{a.} The $L^1$ theory and the inversion theorem
\itemitem{b.} The $L^2$ theory and the Plancherel theorem
\itemitem{c.} The class of tempered distributions
\itemitem{d.} Operators that commute with translations


\smallskip

\item{2.} Real-variable theory

\itemitem{a.} The Hardy-Littlewood maximal function
\itemitem{b.} The Lebesgue differentiation theorem
\itemitem{c.} The integral of Marcinkiewicz
\itemitem{d.} The Calder\'on-Zygmund decomposition

\smallskip

\item{3.} Basic interpolation theorems

\itemitem{a.} The Marcinkiewicz interpolation theorem
\itemitem{b.} The Riesz-Thorin interpolation theorem

\smallskip

\item{4.} Singular integrals of convolution type

\itemitem{a.} The gradient condition
\itemitem{b.} The H\"ormander condition
\itemitem{c.} Homogeneous singular integrals

\smallskip

\item{5.} Examples of the theory

\itemitem{a.} The Hilbert transform
\itemitem{b.} The Riesz transforms
\itemitem{c.} Poisson integrals

\smallskip

\noindent (References: Chapter one of Elias Stein and Guido Weiss, {\it Introduction to Fourier
Analysis on Euclidean Spaces\/} and chapters 1--3 %one, two and three 
of Elias Stein, {\it Singular Integrals and Differentiability Properties of
Functions\/})

\bye
