## S5B1 - Graduate Seminar on Advanced Topics in PDE (winter term 2015/2016)

Lectures:

Friday, 14 (c.t.) - 16, Seminar room 0.011

Course Description:

We will discuss in this seminar various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to a Master or PhD thesis. Students interested in participating should contact one of the organizers. Link to the basis page

Talks (updated periodically):

• October 30, 2015: Mariusz Mirek (University of Bonn)

Title: The Hardy-Littlewood majorant problem for arithmetic sets.

Abstract: We will discuss a large class of arithmetic subsets of integers with vanishing density for which Hardy-Littlewood majorant property holds.

• November 6, 2015: Olli Saari (Aalto University)

Title: Weights from parabolic equations.

Abstract: In this talk, I will discuss a generalization of one-sided Muckenhoupt weights from the real line to higher dimensions. The resulting class of weights has a tight connection to the regularity theory of parabolic partial differential equations. These parabolic weights share many properties with the classical Muckenhoupt weights (with a relation to elliptic PDEs), but there are also many remarkable differences. The talk is based on a joint work with J. Kinnunen.

• November 13, 2015: Polona Durcik (University of Bonn).

Title: Norm oscillation of bilinear averages.

Abstract: We quantify norm convergence of certain bilinear averages by controlling the number of jumps of certain size. Joint with Vjekoslav Kovac, Kristina Ana Skreb and Christoph Thiele.

• November 20, 2015: Alex Amenta (University of Paris-Sud 11)

Title: A first-order approach to some elliptic boundary value problems with rough complex coefficients and fractional regularity data.

Abstract: We consider first-order systems $\partial_t F + DB F = 0$ associated with second-order elliptic operators with rough complex coefficients on the upper half space $\mathbb{R}^{n+1}_+$. 'Classical' solvability methods for associated boundary value problems via layer potentials are not applicable in such generality, and we must resort to more abstract methods involving holomorphic functional calculus. With the aim of classifying solutions to these problems with boundary data in Besov-Hardy-Sobolev spaces, we construct DB-adapted Besov-Hardy-Sobolev spaces on which the functional calculus of DB is well-behaved. When these abstract spaces can be identified with classical spaces, we can classify solutions to the first-order IVPs. This provides an approach to the aforementioned second-order problems which works for a considerably broad class of coefficients, including those where de Giorgi-Nash regularity fails. The technical heart of this material consists of boundedness results for operators constructed in terms of the holomorphic functional calculus of DB in weighted tent spaces and their real interpolants.

• November 27, 2015: Diogo Oliveira e Silva (University of Bonn)

Title: On the root uncertainty principle.

Abstract: This talk will focus on a recent result of Bourgain, Clozel and Kahane. One of its versions states that a (properly normalized) real-valued function which equals its Fourier transform on the whole real line and vanishes at the origin necessarily has a root which is larger than c>0, where the best constant c satisfies 0.41< c <0.64. A similar result holds in higher dimensions. I will show how to improve the one-dimensional result to 0.45< c <0.60, and the lower bound in higher dimensions. Time permitting, I will also argue that extremizers possessing only a finite number of double roots cannot exist. This is joint work with Stefan Steinerberger.

• December 4, 2015: Bartosz Trojan (University of Wroclaw)

Abstract: I present the proof of L^q boundedness, q >1, of the 2-parameter maximal function on p-adic Heisenberg group. I also show Cordoba’s covering argument giving the week-type estimates for functions from Orlicz space Llog L.

• December 11, 2015: Ziping Rao (University of Bonn)

Title: An overview of Minimal surfaces in Minkowski Space.

Abstract: This is the first talk for my Master thesis seminar. I will start from introducing the concept of minimal surfaces and focus on the case in Minkowski space, which also relates to the Born-Infeld equations in string theory. I will then explain some different formulation of the problem and the PDEs arising from them, then discuss some geometric properties of the minimal surfaces.

• December 18, 2015: Joris Roos (University of Bonn).

Title: A Carleson operator along monomial curves in the plane.

Abstract: We prove partial $L^p$ bounds for a polynomial Carleson operator along monomial curves $(t,t^m)$ in the plane with a phase polynomial consisting of a single monomial. These bounds are partial in the sense that we only consider linearizing functions depending on one variable. Moreover, we can only deal with certain combinations of curves and phases. For some of these cases we use a vector-valued variant of the Carleson-Hunt theorem as a black box. As an ingredient of the proofs we use refined variants of Stein and Wainger's method for phases consisting of one and two fractional monomials. Joint with Shaoming Guo, Lillian Pierce and Po-Lam Yung.

• January 15, 2016: Victor Lie (Purdue University)

Title: The pointwise convergence of Fourier Series near L^1. Historical evolution; main questions; recent developments; implications.

Abstract: In this talk we focus on the old and celebrated question regarding the pointwise behavior of Fourier Series near L^1. We start with several brief historical remarks on the subject, describing the context and the formulation of the main problem(s). We then present the evolution of the main negative and positive results from early 20th century to present day. The main part of our talk addresses the most recent progress on the topic: 1) the resolution of Konyagin's conjecture (ICM, Madrid 2006) on the pointwise convergence of Fourier Series along lacunary subsequences; 2) the L^1-strong convergence of Fourier Series along lacunary subsequences. We end with several considerations on the relevance/impact of the above two items on the subject of the pointwise convergence of Fourier Series.

• January 22, 2016: Gennady Uraltsev (University of Bonn).

Title: Time frequency analysis below L^2 using iterated outer measure spaces.

Abstract: In the seminal 2013 paper "Lp theory for outer measures and two themes of Lennart Carleson united" Do and Thiele introduced function spaces that gave intuitive sense to the complicated combinatorial arguments arising in time frequency analysis, in particular when proving boundedness of the Bilinear Hilbert Transform and on the (variational) Carleson Operator. However outer L^p techniques broke down below L^2 even when the abovementioned operators were shown to be bounded by other means. Thanks to an insight from a recent work of Di Plinio and Ou we formulate iterated time/time-frequency outer L^p space theorems that provide a direct proof in the complete Banach triangle for BHT and for all parameters of (variational) Carleson avoiding indirect methods like interpolation between restricted weak type estimates typical of previous works.

• February 5, 2016: Blazej Wrobel (University of Bonn).

Title: On the consequences of a Mihlin-Hormander functional calculus: maximal and square function estimates.

Abstract: We prove that the existence of a Mihlin-Hormander functional calculus for an operator L implies the boundedness on Lp of both the maximal operators and the continuous square functions build on spectral multipliers of L. The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.