This semester, starting the week 14-21 of February, I’m giving a course in Harmonic Analysis on the PhD level as part of the Doctoral Program in Mathematics at the Department of Mathematics at IST. I will use this blog in order to coordinate the course, post comments and notes, as well as a means of communicating with whoever is interested in following the course. For this I expect the comment function of the blog to play a central role and give everyone an easy way to comment on the content of a lecture or ask questions related to the course in general.
Although there might be small changes, the main plan for the course is the following:
I. Introduction: We will start by setting up the main environment for our studies, that is, the appropriate function spaces where our functions will live and our operators will act. There will always be an underlying measure space . As a typical example you should think of as the Euclidean space , as the -algebra of Borel, or Lebesgue measurable sets, and as the Lebesgue measure on . We will however put things in a more general context whenever it is useful or necessary. We will usually consider appropriate spaces of functions . The most typical example here would be the space of functions whose -th powers are integrable with respect to the measure , that is the spaces and will usually lie in the interval . Another relevant space of importance is the space of functions that marginally fail to be in , that is the weak- spaces. These, as we will see, are defined in terms of the measure of the distribution function of the function . We will also extensively use the spaces of infinitely differentiable functions with compact support, the space of Schwartz functions, that is the space of infinitely differentiable functions whose partial derivatives of every order (including the -order derivative, that is the function itself) decay faster than any polynomial power at infinity, the space of continuous functions that tend to zero at infinity and so on. I will assume that most of the audience is familiar with these notions on some level or another. However, this will be our starting point; we will recall these notions from measure theory (or real analysis if you want) and take them one step further. A recurring theme in this course will be the study of operators acting on these function spaces and, in particular, their boundedness and mapping properties. For this we will oftentimes use classical inequalities in measure spaces as for example Hölder’s inequality, Minkowski’s inequality and Young’s inequality, as well as slightly more sophisticated tools, that is, different forms of interpolation of operators (e.g. Marcinkiewicz interpolation theorem, Riesz-Thorin interpolation theorem), Schur’s test, convolution inequalities and duality arguments. We will review the classical inequalities and introduce the more sophisticated tools just mentioned. However, I do not plan to exhaust all the possible tools from real analysis here; that would be impossible. We will go on with our main agenda and digress a bit whenever necessary.
II. The Fourier transform: We will introduce the Fourier transform of appropriate functions and study its main properties on the corresponding spaces. Special mention will be made on the Fourier transform on the space of finite measures on , on , on as well as on the Schwartz space . Although the latter function space seems pretty limited, its dual, the space of tempered distributions, is rich enough to allow us to extend the definition of the Fourier transform (in a weak sense) to a wide variety of objects, including spaces for . The space of tempered distributions will not be central in this course but we will rely on it in order to define operators (as for example the Fourier transform, or the derivative) on functions that do not possess the necessary regularity. We will give examples of classical Fourier transforms, like the Fourier transform of the Gaussian, and discuss how one can reconstruct the original function from its Fourier transform, that is we will see when, how, and in what sense we can `invert’ the Fourier transform. Some time will be given to the discussion of bounded linear operators that commute with translations. We will see that these operators are convolution operators with an appropriate distribution.
III. The Hardy-Littlewood Maximal function: We will introduce (or recall) the Maximal function of Hardy and Littlewood and prove its main boundedness properties. This will be done in different ways; we will use the classical approach that is prove the to weak inequality by means of a covering lemma and then interpolate between this bound and the trivial bound. We will also study the relevance of the maximal function to the standard Calderón-Zygmund decomposition. We will also discuss the Marcinkiewicz integral which we will use in the study of singular integrals.
IV. Singular Integrals: We will discuss the boundedness properties of Singular integral operators, starting our discussion from the Hilbert transform which is the primordial example. The main theme here will be how to conclude that a singular integral operator is bounded on -spaces, starting from the hypothesis that it is in fact bounded on . A further step will be to substitute the -boundedness hypothesis with a suitable (seemingly weaker) condition on the kernel of our operator that will allow us to conclude that the operator is bounded on .
V. Littlewood-Paley theory and multiplier operators: This concluding section of the course aims mainly at introducing the dyadic decomposition of a function in terms of its Fourier transform, and prove the Littlewood-Paley inequalities. Roughly speaking, these inequalities allow us to decompose a function to different pieces which have localized frequencies in dyadic multidemensional `intervals’, and behave almost orthogonally to each other. In the Hilbert space this is precise. The Littlewood-Paley inequalities provide us with a substitute in , . Given time we will discuss multiplier operators and give two fundamental theorems: the Mikhlin-Hörmander multiplier theorem and the Marcinkiewicz multiplier theorem.
The preceding description gives the main topics I would like to cover in the course. On the other hand I plan to touch upon some special subjects as for example, oscillatory integral estimates, Sobolev inequalities and relation to PDE’s, weighted norm inequalities, Fourier transform on different groups, Fourier series and so on. There will be relevant exercises in your homework giving you a flavor of these subjects (with appropriate guidance of course!) as well as examples in the classroom. There is also a possibility to substitute part of the grade of the written exam by studying and presenting a special subject we will choose together. This will play a double role. Firstly give you the chance to study a slightly more involved subject and understand its intricacies as well as giving a flavor to the rest of the audience on other aspects of Harmonic Analysis that time will not permit us to cover in the main course. Look a bit below at the exam and grading description.
The exact place of the lectures is not yet known. I will post it here when I have more information. The time of the lectures will be arranged between us. I encourage to already e-mail me with preferences and/or restrictions on your weekly schedule, baring in mind the following: there will be two one hour and a half lectures every week. My intention is to have a small break during each lecture, but this will depend on the overall time-logistics and schedule of the participants to the course, available classrooms and so on. I also intend to have an extra lecture of one hour or one hour an a half, depending again on the same logistics, where we will discuss your homework.
3. Grading, Homework and Exams
Let’s move to the subject of exams. There will be a set of exercises given to you as homework (approximately every two weeks). You will have to hand in your solutions in two weeks’ time. This will amount to of the total grade. There will be two written exams, let’s say one mid-term and one final that will amount to the rest of the grade. Alternatively you can get of the final grade from studying and presenting before the rest of classroom a special subject in harmonic analysis that we will choose together. That last combination will split your final grade to (homework)(presentation)(two written exams).
I will try to keep a course calendar right here on this blog, where you can also use comments to ask questions, clarify things or discuss any related issue; you can certainly do that in the classroom but I expect you to be aware of what’s going on here in this blog, as well as check your e-mails on a regular basis for course related issues. Check also my web site where all my contact information is available.
I will suggest some books that I think will be of great help throughout the course. This list however is neither restrictive nor exhaustive. I would encourage you to use any book or online resource that you feel can help you. Check also the links on the sidebar of this blog. I plan to follow roughly [SW] for the first parts of the course (I,II) and [S], [D] for the remaining material (III,IV,V).
Tuesday – 14:10 to 15:55 – classroom: V1.25 (1st floor of the Civil Engineering building).
Thursday – 14:10 to 15:55 – classroom: P9 (2nd floor of the Math building).
Friday – 13:10 to 14:55 – classroom: P9 (2nd floor of the Math building).
7. To Do List
[update 15 Feb 2011: schedule of the course and code DMat0101 added.]
[update 17 Mar 2011: schedule updated, Friday class moved one hour earlier.]