GAMP Group 

IMF - Center for Symmetry and Deformation

Institut for Matematiske Fag
Universitetsparken 5, DK-2100 København Ø

Differential Operators and Function Spaces II (Block 4, 2012)

Lecture: Monday 1-3 pm (Aud 7)
Wednesday 1-3 pm (Aud 9)
Exercises: Wednesday 10-12 am (A103)

Office hours Heiko Gimperlein: by appointment (04.2.19)
Email: gimperlein at
Office hours Mathias Makedonski: by appointment (04.2.09)
Email: Mathias dot Makedonski at

Formal matters:
First lecture on April 23, 2012. Please refer to the description in SIS for the official content of this course.


Basic references:Terence Tao, Graduate Fourier Analysis, part 1, part 2.
(DO NOT BUY THESE!!!)E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press.
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press.
W. Rudin, Fourier analysis on groups, Interscience Publishers.

Further material:Applications of the stationary phase method
Range of the Fourier transform: the Paley-Wiener theorem
Calculus of pseudodifferential operators
The basic theorems of Fourier analysis on LCA groups
(Simple) Existence of the Haar measure on LCA groups
Nonabelian Fourier analysis

Notes: Chapter 1: Fourier transformation: Range Theorems and Interpolation
(beware of typos!) Chapter 2: Introduction to integral operators
Chapter 3: Hardy-Littlewood maximal inequality and covering theorems
Chapter 4: Calderon-Zygmund theory
Chapter 5: Pseudodifferential operators, elliptic regularity and function spaces
Chapter 6: Fourier analysis on locally compact abelian groups

Exercises: Exercises 1
(beware of typos!) Exercises 2
Exercises 3
Exercises 4
Exercises 5
Exercises 6
Exercises 7

Some possible projects: Complex interpolation (Anna)
based on M.E. Taylor, Partial Differential Equations, Vol. 1
abstract approach to interpolation, interpolation of Lp and Sobolev spaces, Sobolev spaces of arbitrary order on domains
Applications of interpolation
based on Tao's notes and C. Bennett and R. C. Sharpley, Interpolation of Operators
weak-type Schur test, Hardy-Littlewood-Sobolev fractional integration inequality, the Hilbert transform on Lp
Estimating oscillatory integrals - method of stationary phase (Jakob)
based on one of the books by Dumassi and Sjöstrand, Grigis and Sjöstrand or L. Hörmander, The Analysis of Linear Partial Differential Operators I
some applications can be found here
The uncertainty principle (Jonas)
based e.g. on this article, Tao and Hörmander.
Uncertainty principles of Heisenberg, Hardy and Beurling
The Lieb-Thirring inequality (Christina)
based on notes by Jan Philip Solovej plus elementary facts about eigenvalues
This project uses basic Fourier analysis to count the negative eigenvalues of Schrödinger operators (-Delta + V).
Applications of the Hardy-Littlewood maximal inequality
based e.g. on Tao's notes (Chapter 3), which cover applications to (Hardy) spaces of holomorphic functions and ergodic theory (convergence of ergodic averages).
The Schwartz kernel theorem
based e.g. on L. Hörmander, The Analysis of Linear Partial Differential Operators I
This theorem shows that every bounded linear operator mapping test functions to distributions is a (generalized) integral operator, possibly with a distribution rather than a function as its kernel. For example, the kernel of the identity operator is the Dirac distribution. The project is more closely related to the DifFun1 course than the other topics.
The Tomas-Stein restriction theorem
based on Stein's Harmonic Analysis book, notes by Tao and/or these notes.
Restricting an Lp function in the plane to a curve does not make sense, as one may modify the function on sets of measure 0. However, in certain cases the restriction of its Fourier transform is well defined. This fact has numerous relations to basic geometric questions and to hyperbolic PDE (Strichartz estimates). Aims: Tomas-Stein restriction theorem, outline some applications or related problems.
Restriction theorems and elementary geometry
This project would extend the previous one, focusing on interactions between harmonic analysis (restriction theorems, Fourier inversion) and constructions of highly pathological subsets of Rd.
Unbounded ball multpliers
The Plancherel theorem assures that Fourier multipliers with bounded, measurable symbol induce bounded operators on L2. Surprisingly, Fefferman has shown that the Fourier multiplier whose symbol is the characteristic function of a ball fails to be bounded on Lp(Rd) if p is different from 2 and d > 1. The proof has close relations to the previous two projects..
Applications of the Calderon-Zygmund theorem
The Calderon-Zygmund theorem can be used to prove the boundedness of various interesting operators. This project would introduce a few specific operators, explain their relevance and discuss their basic properties.
Lidskii's theorem and traces of integral operators
based e.g. on P. Lax, Functional Analysis.
Given a trace-class operator A on a Hilbert space, one hopes to compute its trace as in finite dimensions by summing the diagonal entries < A e_i, e_i> over an orthonormal basis e_i. Lidskii's theorem makes this idea rigorous and, applied to integral operators, frequently allows to compute the trace as the integral of the kernel k(x,y) over the diagonal x=y.
Dyadic characterizations of function spaces
The Littlewood-Paley inequality (Section 5 of Chapter 4) characterizes the Lp (and Sobolev) norm of a function f in terms of the components of f with frequency 2^j. Similar characterizations are available for most of the classical function spaces such as Hölder-continuous functions. They allow for a unified approach to these spaces and are immensely useful to study their fine properties, such as embedding and restriction theorems, and the solutions to PDE.
Existence of fundamental solutions
This project investigates if a PDE admits a solution in the neighborhood of a point. For partial differential operators with constant coefficients, the Malgrange-Ehrenpreis theorem asserts the existence of a fundamental solution. It is explicitly constructed using the Fourier transform. If there is time, you can also discuss Lewy's example of a PDE without any local solutions.
based e.g. on Folland, Introduction to PDE.
Topics on pseudodifferential operators
Pseudodifferential operators (e.g. functions of a differential operator) will be briefly discussed later in the course. This project would discuss further aspects of the theory, either analytic or more geometric.
The index of a Fredholm operator (Camilla)
Fredholm operators are operators which are invertible up to a compact operator, e.g. compact perturbations of the identity. They have finite dimensional kernels and cokernels, and the difference of these dimensions is called the index. The project looks at the basic properties of this invariant.
The index of elliptic operators on the circle
This project continues the previous one by studying pseudodifferential operators on a particularly simple manifold, the circle (it can also be combined with the previous one). The main aim is to show that elliptic operators on the circle are Fredholm and that the index can be computed from the symbol.
Topics on Fourier analysis on abelian groups
Fourier analysis on locally compact abelian groups will be discussed at the end of the term. This project would expand on further aspects of your choice.
Existence of the Haar measure on LCA groups
On locally compact abelian groups there is a simple, functional analytic proof for the existence of an invariant Radon measure. Based on these notes.
Poisson summation formula
The classical Poisson summation formula is an identity connecting sums over a lattice of a function to similar sums over the dual lattice of the Fourier transform. It proves useful whereever such sums arise, e.g. for sampling theorems, identities of theta and similar number theoretic functions, solid state physics etc. This project intends to prove the formula for LCA groups and outline some applications. Based e.g. on A. Deitmar, S. Echterhoff, Principles of Harmonic Analysis.
Fourier methods in additive combinatorics
The Fourier transform on groups such as Z, Z/NZ or R/Z is a crucial tool in additive combinatorics and related aspects of number theory: The Fourier transform of the indicator function of a set A contains information about the structure and size of A. For example, do the prime numbers contain arbitrarily long arithmetic progressions? How many? Given A, what can we say about A+A+A or A*A*A? Based on T. Tao, V. Vu, Additive Combinatorics, Ch. 4, and related notes.
Further topics
It is unlikely that we have time to talk about much of Chapters 6, 7 and 8 in Tao's notes. Feel free to choose one of the topics as a project.


Files of exercise sheets
Further material