Slava Rychkov
May 17, 1999  2:40pm - 3:30pm

Committee: Stein (c), Kohn, Trotter
Special topics: Harmonic Analysis, Several Complex Variables

Complex analysis (Kohn)
-----------------------
You are given an analytic function in a connected domain
in C vanishing up to infinite order at
some point. What can you say about this function? Prove it.

You are given an analytic function in a neighborhood of a
point. What is the Riemann surface of this function? What are the
points of the Riemann surface? What is the topology on it?

You are given a domain in C. Can you construct
an analytic function in this domain which cannot be extended
to any larger domain? How?
(Use general Weierstrass to construct a function with lots
of zeroes accumulating at every point of the boundary.)

Several complex variables (Kohn)
--------------------------------
You are given a domain in C^n. Can you always construct an analytic
function in it which cannot be extended to any larger domain?

State and prove the general Hartogs extension theorem.
(I was stopped with words "OK, OK, you know that" after saying
that the main idea of the proof was to take any smooth
extension inside and then to modify it to be holomorphic by solving
inhomogeneous Cauchy-Riemann equations. Actually Kohn wanted to finish
SCV section here, but Stein proposed "Ask him something tough!" Some more
question followed, but nothing actually tough, as you will see.)

What is the characterization of domains of holomorphy? (pseudoconvex)
Prove the easy direction of this theorem.
(I started proving it but there were some additional results involved,
such as the equivalence of strong and weak definition of domains
of holomorphy, and the committee got confused.
So they asked me to state all definitions and results precisely.
I then first formulated and was asked to prove the holomorphic convexity
characterization of domains of holomorphy. Then I stated
a theorem involving 4 equivalent characterizations of pseudoconvexity,
and was about to start proving it, but Kohn said he was satisfied,
and SCV were over.)

Harmonic analysis (Stein)
-------------------------
State the general form of Marcinkiewicz interpolation theorem
for linear operators.
(If T:L^{p_i,1}->L^{q_i,infty}, i=1,2,
p_i<=q_i, p_1\ne p_2, q_1\ne q_2, then T:L^p->L^q for any
0<t<1, 1/p=t/p_1+(1-t)/p_2, 1/q analogously)

What is the definition of L^{p,1}?
(I stated one via nonincreasing rearrangement.)
What is nonincreasing rearrangement?
What is special about L^{p,1} space?
(If you know that the operator is bounded in L^{p,infty} when acting
on characteristic functions of sets, then it's bounded from L^{p,1}
to L^{p,infty}.)
Do you know a characterization of L^{p,1} via some minimality property?
(No)
What is the definition of L^{p,infty}? Of L^{p,q}? Can you get a
stronger conclusion in the Marcinkiewicz interpolation using these spaces?
(Yes, you can actually conclude that T:L{p,r}-L^{q,r} for any 1<r<infty.)
Are L^{p,q} Banach spaces?
(I said I knew L^{p,q} with p,q<infty were Banach spaces, and that
L^{1,infty} was not. I was not sure about L^{p,infty} with p>1.)
Can you prove that L^{1,infty} is not a Banach space?
(There is an example in Stein&Weiss involving characteristic
functions, but I don't remember it.)
What are the inclusions between L^{p,q} spaces?
What is the relation between L^{p,1} and L^{p,infty}?
(L^{p,1} is the dual of L^{p', infty})

Do you know about Hardy spaces?
Give an example of an element of H^p for p<1.
(I said any Schwartz class function would do, but then corrected
myself by adding that certain number of moment conditions should
be satisfied.)
Give an example of a distribution in H^p.
(I decide to try the delta function, and wrote down the maximal function
definition for H^p, i.e. sup_{t>0}|\phi_t*f|\in L^p.)
What is this maximal function approximately equal to
when f is the delta function?
(It turns out to behave like 1/|x|^n.)
So where does the delta function belong to?
(It's in H^p_loc for p<1, but not in H^p.)
How can you modify this example to get a function in H^p?
(Take the difference of two delta functions at two different points.)

Do you know about spherical harmonics? Why are spherical harmonics
decompositions useful to calculate the Fourier transform of functions?
(I said that there was an identity for the Fourier transform
of a quotient of a harmonic polynomial over |x|^alpha, generalizing the fact
that (1/|x|^alpha)^=const/|x|^{n-alpha}, but I did not remember the
exact exponents.)

What is the Fourier transfrom of a Gaussian?
Do you know another identity involving spherical harmonics which
generalizes this fact?
(I remembered its name (Hecke's identity), but not the exact form of
the identity itself.
Stein said he was happy, and we went on with algebra.)

Algebra (Trotter)
-----------------

Take a ring of matrices over a field. What are the left ideals?

State the structure theorem for simple rings.

State the structure theorem for finitely generated abelian groups.
Why are the sizes of direct summands uniquely determined?

Give a nontrivial example of a subgroup of Q.
(Take all rationals with denominators involving only powers of
certain primes. At this point Trotter asked Stein if he should
ask some Galois theory, and Stein said of course.)


Adjoin 5-th root of unity to Q. What is the degree of the
extension? What is its Galois group? The same two question
for a primitive n-th root of unity.
(The degree is phi(n), since cyclotomic polynomials are irreducible;
the Galois group is (Z_n)^*)

The exam lasted about 50 minutes.