Arbeitsgruppe Analysis und Partielle Differentialgleichungen
S5B1  Graduate Seminar on Advanced Topics in PDE (winter term 2015/2016)
Instructors: Prof. Dr. Herbert Koch,
Dr. Mariusz Mirek,
Prof. Dr. Christoph Thiele,
Dr. Roland Donninger
Lectures:
Course Description:
Talks (updated periodically):

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

Title: The HardyLittlewood majorant problem for arithmetic sets.
 Abstract: We will discuss a large class of arithmetic
subsets of integers with vanishing density for which HardyLittlewood
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 onesided 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 ParisSud 11)

Title: A firstorder approach to some elliptic boundary value problems
with rough complex coefficients and fractional regularity data.
 Abstract: We consider firstorder systems $\partial_t F + DB F = 0$
associated with secondorder 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 BesovHardySobolev spaces, we construct DBadapted BesovHardySobolev
spaces on which the functional calculus of DB is wellbehaved. When these
abstract spaces can be identified with classical spaces, we can classify
solutions to the firstorder IVPs. This provides an approach to the
aforementioned secondorder problems which works for a considerably broad
class of coefficients, including those where de GiorgiNash 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) realvalued
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 onedimensional 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)

Title: About the 2parameter maximal function on padic Heisenberg group.
 Abstract:
I present the proof of L^q boundedness, q >1, of the 2parameter maximal
function on padic Heisenberg group. I also show Cordoba’s covering argument
giving the weektype 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 BornInfeld 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 vectorvalued variant of the
CarlesonHunt 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 PoLam 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^1strong 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/timefrequency 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 MihlinHormander functional calculus:
maximal and square function estimates.
 Abstract:
We prove that the existence of a MihlinHormander 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.
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