ASSIGNMENT 1, 18.155
DUE MIDNIGHT MONDAY 31 OCTOBER, 2011
Date: Fri Oct 28 15:30:58 2011
This assignment replaces Problem set 7 and should represent about
as much work, maybe a little more, as required for one of the Problem
sets. You can submit work at any time (electronically as usual) for me
to review and return – the mark will be based on the final submission
only. You can also ask for help etc, conditions as for a Problem set –
I do not mind what or who you consult as long as the final product is
your own.
In particular I do not mind if two, or three, people collaborate on
one project subject to the conditions above.
Here are the projects, with a list of who has asked to work on what
to date. If someone really wants to work on something else I am willing
to negtotiate.
So, what do I actually want. I will try to give you a guide in Problems6 but basically I want you to state clearly the, or a, result corresponding to the given title and provide a proof – in some cases maybe
just quoting exterior results – and some indication of the context of
the result. As I say above, I am expecting a few pages, not a thesis.
(1) Riesz representation for measures on locally compact spaces.
(2) Paley-Wiener-Schwartz Theorem. Fourier transform and some
holomorphic function theory. [Michael V.]
(3) Atiyah’s proof of Lojasiewicz-Hörmander Theorem. Fourier
transform and some Algebraic Geometry.
(4) Tempered distributional deRham cohomology of Euclidean space.
Not much topology really.[Yasha, Michael A.]
(5) Tempered Schwartz Kernel Theorem. Functional Analysis, Fourier
transform.[Jelena, Jonathan]
(6) Gagliardo-Nirenberg, that the Sobolev spaces are rings for s >
n/2 (or more generally).[Hans]
(7) Symbols and conormal distributions on Euclidean space. Fourier
transform.[Alex]
1. Riesz Representation for measures
Show that every continuous linear functional on C00 (M ) for M a locally compact metric space is given in terms of integration with respect
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ASSIGNMENT 1, 18.155 DUE MIDNIGHT MONDAY 31 OCTOBER, 2011
to a measure on the space. This is basically the first part of the old
notes, or you can get it from somewhere else. I’m assuming you would
want to give a brief account of the measure theory involved as well as
the theorem itself.
2. Paley-Wiener-Schwartz theorem
This characterizes the Fourier(-Laplace) transform of distributions
of compact support in terms of entire functions on Cn with prescribed
growth properties. There are many variants of the theorem – I would
like in n dimensions – and you might like also to describe similar results
for smooth functions of compact support and for tempered distributions, or L2 functions, with support in a half-space (which is closer to
what Wiener did for the half-line).
3. Lojasiewicz-Hörmander
First get the accents correct on Lojasiewicz.
A fundamental solution of a differential operator with constant coefficients is a distribution – in this case E ∈ S 0 (Rn ) such that
P (D)E(x) = δ(x).
Taking the Fourier transform this reduces to the ‘division problem’ of
finding Ê ∈ S 0 (Rn ) satisfying
P (ξ)Ê(ξ) = 1
i.e. at least informally Ê(ξ) = 1/P (ξ) where the zeros will be the
problem. This in fact has a solution for every non-zero polynomial,
originally proved by Lojasiewicz then improved (at least the proof) by
Hörmander and then in the late 60s Atiyah gave a conceptually simple
(but much more sophisticated) proof using resolution of singularities
following Hironaka.
Look at the paper by Atiyah and give a discuss the analytic aspects
of his proof. Can you use his method to show solvability, that for any
f ∈ S 0 (Rn ) there exists u ∈ S 0 (Rn ) such that
P (ξ)u(ξ) = f (ξ) in S 0 (Rn )?
4. deRham theorem with tempered coefficients
Since we can differentiate tempered distributions it is straight forward to define the deRham differential on distributional-forms
X
XX
d
uI dxI =
∂j uI dxj ∧ dI , uI ∈ S 0 (Rn ),
I
I
j
ASSIGNMENT 1, 18.155
DUE MIDNIGHT MONDAY 31 OCTOBER, 2011
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assuming you can do this in the smooth case. Proving a Poincaré
Lemma allows one to compute the tempered cohomology of Rn . If you
then want to push on further you can think about the same sort of
thing on manifolds (if you have the necessary background).
5. Schwartz kernel theorem for tempered distibutions
Show that there is a natural bijection between the space of linear
operators
T : S(Rm ) −→ S 0 (Rn )
which are continuous in the sense that of φj → φ in S(Rm ) then
T φj → T φ weakly in S 0 (Rn ), and the space S 0 (Rn × Rm ). This can
be done using the Fourier transform and Schwartz Representation theorem. You might like to discuss what this means in terms of tensor
products (or not). Note that a ‘kernel’ A ∈ S 0 (Rn × Rm ) gives rise
to an operator TA (there is something to prove here of course) via the
formula
(1)
(TA φ)(ψ) = A(ψ φ)
where the ‘exterior tensor product’ here just means ψ(x)φ(y) ∈ S(Rn ×
Rm ).
6. Gaglardo-Nirenberg
The Sobolev spaces H s (Rn ) for s > n/2 are subspaces of C00 (Rn ),
the bounded continuous functions on Rn with supremum norm. In fact
they are also subalgebras, i.e. are closed under pointwise products:
H s (Rn ) · H s (Rn ) ⊂ H s (Rn ), s > n/2
and indeed form a C ∞ -algebra, meaning that if f is a smooth function
on Rk then f (u1 , . . . , uk ) ∈ H s (Rn ) if all the ui ∈ H s (Rn ). This result
follows from the Gagliardo-Nirenberg estimates.
You maybe should start with the case s ∈ N when the estimates are
more straightforward.
7. Symbols and conormal distibutions
The standard space of symbols (with bounds) usually written S m (Rn )
consists of those smooth functions satisfying the estimates
|∂xα a(x)| ≤ Cα (1 + |x|)m−|α|
for all multiindices α. Describe in detail the Fourier transform of these
symbols (conormal distributions at the origin) and how these are related to Sobolev spaces. There is plenty more to do about these if you
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ASSIGNMENT 1, 18.155 DUE MIDNIGHT MONDAY 31 OCTOBER, 2011
want (the beginning of microlocal analysis). It is difficult to get a precise characterization of the Fourier transform of S m but you should at
least characterize the union over m for instance in terms of regularity:
If a ∈ S m (Rn ) there exists s = s(m) such that
ξ β Dξα â ∈ H s+|β|−|α| (Rn )
and conclude that singsupp(â) ⊂ {0}.