• Harmonic Analysis and Partial Differential Equations
• Introductory Workshop

• Schedule
• Participants

We shall present some recent works developed in successive collaborations with A. McIntosh and A. Rosen (formerly Axelsson), with S. Stahlhut and with M. Mourgoglou. The goal is to obtain representations for solutions of boundary value problems for some second order elliptic systems of Dirichlet or Neumann types in various topologies for the data. Without knowing whether or not these problems have (unique) solutions, it is still possible to obtain a priori representation on the gradient of those solutions. We want to explain the mechanism for obtaining such representations and some ideas of proofs. Those representations completely elucidate some issues concerning solvability vs uniqueness in such problems.

Lecture 1: We present the second order elliptic systems under considerations and give some examples. Then we discuss the Auscher-Axelsson-McIntosh first order approach for studying boundary value problems for L² data and construct solutions via a certain semigroup. We give some elements of holomorphic functional calculus needed for such an approach. The central estimate is an elaborate consequence of the solution of the Kato conjecture.

Lecture 2: In order to extend the first order approach to BVP with L^p data in the sense of Kenig-Pipher, we need to extend our semigroups to L^p setting. Unfortunately, our semigroups are seldom bounded on all of L^p. They turn out to be bounded on some abstract Hardy spaces associated to a first order operator and defined via area functional techniques. We give the elements of this theory originating from works of Auscher-McIntosh-Russ and Hofmann-Mayboroda. We explain that there is a range of exponents p for which those abstract Hardy spaces are, in fact, subspaces of L^p (when p ≤ 1, L^p is replaced by the Hardy space H^p) and give some area functional and non-tangential maximal characterisation of such spaces.

Lecture 3: We turn to the converse problem. Having a solution of our second order system in some class, is it possible to obtain a representation in terms of the semigroup we have constructed? More precisely, does the gradient of such solutions have a trace and is it representable applying the semigroup to its trace? For p = 2, this was positively answered in an earlier work Auscher-Axelsson. For p ≠ 2, the answer is also yes when we work in the range of exponents p and use all the a priori estimates obtained in lecture 2. This kind of results is by now well understood in the theory of harmonic functions for example. However, we note here the absence of positivity assumptions allowing to use maximum principles or Fatou type results, so that our techniques are completely different and seem interesting on their own.

Pascal Auscher video 2 http://www.youtube.com/watch?v=9x3hNwZcyfY

Pascal Auscher video 3 http://www.youtube.com/watch?v=UtJXrIpd8XM

Uniform distribution theory, which originated from a famous paper of H. Weyl, from the very start has been closely connected to Fourier analysis. One of the most interesting examples of such relations is an intricate similarity between the behavior of discrepancy (a quantitative measure of equidistribution) and lacunary Fourier series. We will discuss some old and recent results in this directions, as well as the interplay with some other areas of mathematics.

Dmitryi Bilyk video http://www.youtube.com/watch?v=UAvgPXqAZL4

Sebastian Herr Video 1 <https://www.youtube.com/watch?v=nvs-6UxNq0g>

Sebastian Herr Video 2  http://www.youtube.com/watch?v=FDKXMnJ7LnU

Sebastian Herr Video 3 http://www.youtube.com/watch?v=j6fYTfThIU4

I will discuss some recent work on the water waves equation in 2 dimensions. We will be interested in several aspects: formulation of the evolution problem, the local regularity theory, existence of global solutions, and the dynamical formation of singularities.

Peer Kunstmann Video http://www.youtube.com/watch?v=GISyxKTt614

One of the simplest and the most important results in elliptic theory is the maximum principle. It provides sharp estimates for the solutions to elliptic PDEs in in terms of the corresponding norm of the boundary data. It holds on arbitrary domains for all (real) second order divergence form elliptic operators . The well-posedness of boundary problems in , , is a far more intricate and challenging question, even in a half-space, . In particular, it is known that some smoothness of in , the transversal direction to the boundary, is needed.

In the present talk we shall discuss the well-posedness in for elliptic PDEs associated to matrices independent on the transversal direction to the boundary and their perturbations. The sharp results for the Dirichlet / regularity set of problems on the complete range of , Sobolev, and Besov spaces will be presented for operators with real coefficients, including the case of the non-homogeneous equations. The proofs invoke an intricate interplay of harmonic analysis and PDEs: singular integral operators, harmonic measure techniques, square function / non-tangential maximal function estimates, Hodge decomposition, and the Kato problem estimates.

This is joint work with A. Barton, S. Hofmann, C. Kenig, and J. Pipher.

Estimation of oscillatory integrals is a powerful tool in harmonic analysis; correspondingly the estimation of exponential sums (and more generally character sums) is a powerful tool in analytic number theory. In fact, many well-known problems and conjectures in number theory can be reduced to bounding such sums - but in general, it is quite difficult to prove appropriate bounds. In this talk we will introduce the notion of character sums, describe a number of applications for them, and survey recent work that has made new progress on bounding such sums in arbitrary dimensions, analogous to bounds for oscillatory integrals. No background in number theory will be assumed.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton-Jacobi-Bellman PDE. We show how this technique (the reduction to a Hamilton-Jacobi-Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of A₂ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation.

As an illustration we also compute the numerical value of the norm in L^p of the real and imaginary parts of the Ahlfors-Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors-Beurling operator itself and explain the connection with Morrey's problem.

If time permits we also show how HJB approach allows us to build counterexamples to weak Muckenhoupt conjecture and some of its relatives.

Alexander Volberg video http://www.youtube.com/watch?v=Ocw2AAiS-I0