Abstract: Weighted Carleson embedding (weighted
paraproduct estimates in another language) lies in the core of
various harmonic analysis and PDE results. Not much is known about
it in multi-parameter situation, while one parameter is completely
understood. I will formulate several new results on weighted
multi-parameter Carleson embedding on multi-trees and their
corollaries as embeddings of Hilbert spaces of analytic functions on
poly-discs. I will also formulate corresponding Poincar\'e
inequalities on multi-trees and poly-discs. Some of those results
are final, but even embedding of Hardy space on bi-disc is not
completely described. My presentation is based on joint works with
N. Arcozzi, I. Holmes, P. Mozolyako, P. Zorin-Kranich.
Tuesday
27th September 2022, 15:00–16:00 TST (GMT+8), online on
Zoom
Title:
Classical multiplier theorems and their sharp variants
Abstract: The question of finding good sufficient
conditions on a bounded function $m$ guaranteeing the
$L^p$-boundedness of the associated Fourier multiplier operator is a
long-standing open problem in harmonic analysis. In this talk, I
will recall the classical multiplier theorems of H\"ormander and
Marcinkiewicz and present their sharp variants in which the
multiplier belongs to a certain fractional Sobolev space. The talk
is based in part on a joint work with L. Grafakos and M. Masty\l o.
Tuesday 30th September 2022, 9:00–10:00 TST (GMT+8), online on
Zoom
Abstract: We prove the global existence of the
non-negative unique mild solution for the Cauchy problem of the
cutoff Boltzmann equation for soft potential model −1<=γ<0 with the
small initial data in three dimensional space. Thus our result fixes
the gap for the case γ=−1 in three dimensional space in the authors'
previous work where the estimate for the loss term was improperly
used. The other gap there for the case γ=0 in two dimensional space
is recently fixed by Chen, Denlinger and Pavlović. The initial data
f0 is non-negative, small in weighted L3_{x,v} and finite in
weighted L15/8_{x,v}. We also show that the solution scatters with
respect to the kinetic transport operator. The novel contribution of
this work lies in the exploration of the symmetric property of the
gain term in terms of weighted estimate. It is the key ingredient
for solving the model −1<γ<0 when applying the Strichartz estimates.
Tuesday 25th October 2022, 9:00–10:00 TST (GMT+8), online on
Zoom
Abstract: We discuss two ways that analysis and
number theory have recently teamed up, using a back and
forth interplay to make progress on two different types of
counting problems. First we will count equilateral triangles
in Euclidean space. Second we will determine how often a
random polynomial fails to have "full" Galois group. Though
easy to state, these questions have generated a lot of
interesting techniques through the years, which we will
glimpse during this talk.
Tuesday 1st November 2022, 15:00–16:00 TST (GMT+8), online on
Zoom
Abstract: Nonlocal variational problems arise in
various applications, such as continuum mechanics, the
theory of phase transitions, or image processing. Naturally,
the presence of nonlocalities leads to new effects, and the
standard methods in the calculus of variations, which tend
to rely intrinsically on localization arguments, do not
apply. In this talk, we address questions arising from the
existence theory for three different classes of variational
functionals: integrals depending on Riesz fractional
gradients, double integrals, and double supremals - and find
qualitatively very different results. Regarding the
characterization of weak lower semicontinuity, it may be
surprising that quasiconvexity, which is well-known from the
classical local setting, also provides the correct convexity
notion for the fractional integrals. Our proof relies on a
translation mechanism that allows switching between
classical and fractional gradients. In the case of double
supremals, we show that the natural guess of separate level
convexity fails in the vectorial case, and introduce the new
Cartesian level convexity. As for relaxation, we discuss the
central issue of why one cannot expect these nonlocal
functionals, in contrast to their local counterparts, to be
structure-preserving. This is based on joint work with
Antonella Ritorto, Hidde Schönberger (both KU
Eichstätt-Ingolstadt), and Elvira Zappale (Sapienza
University of Rome).
Tuesday 8th November 2022, 15:00–16:00 TST (GMT+8), online on
Zoom
Abstract: We prove a general principle, called the
principal alternative, which yields an easily verifiable
necessary and sufficient condition for the existence or the
non-existence of an optimal Orlicz space in a wide variety
of specific tasks including boundedness of operators. We
show that the key relation is the positioning of certain
rearrangement-invariant space, characteristic for the task
in question, to its fundamental Orlicz space. The main
motivation stems from the imbalance between the
expressivity, meaning the richness and versatility, of
certain class of function spaces, and its accessibility,
i.e., its complexity and technical difficulty. More
precisely, while an optimal rearrangement-invariant space in
a given task often exists, it might be too complicated or
too implicit to be of any practical value. Optimal Orlicz
spaces, on the other hand, are simpler and more manageable
for applications, but they tend not to exist at all. We
apply the general abstract result to several specific tasks
including continuity of Sobolev embeddings or boundedness of
integral operators such as the Hardy-Littlewood maximal
operator and the Laplace transform. The proof of the
principal alternative is based on relations of endpoint
Lorentz spaces to unions or intersections of Orlicz spaces.
This is a joint work with Vít Musil (Brno) and Jakub Takáč
(Warwick).
Tuesday 22nd November 2022, 15:00–16:00 TST (GMT+8), online on
Zoom
Abstract: In this talk we will describe several
qualitative and quantitative unique continuation properties
for the fractional discrete Laplacian. We will show that, in
contrast to the fractional continuous Laplacian, global
unique continuation fails to hold in general for fractional
discrete elliptic equations. We will also discuss
quantitative versions of unique continuation which
illustrate how the properties in the continuous setting may
be recovered if exponentially small (in terms of the lattice
size) correction factors are added. Joint work with Aingeru
Fernández-Bertolin and Angkana Rüland.
Tuesday 29th November 2022, 9:00–10:00 TST (GMT+8), online on
Zoom
Abstract: The aim of these lectures is to give a
gentle introduction to Strichartz estimates, with an emphasis on
particular cases such as the linear Schr\"odinger and wave
equations. The associated dispersive estimates play a highly
important role in the theory of Strichartz estimates so I will begin
in Lecture 1 by proving the required dispersive estimates.
Next, in Lecture 2, I will prove the homogeneous Strichartz
estimates in all admissible cases, including the so-called Keel--Tao
endpoint case. Building on the content of the first two lectures, in
Lecture 3, I will discuss the situation regarding inhomogeneous
Strichartz estimates.
Abstract: The aim of these lectures is to give a
gentle introduction to Strichartz estimates, with an emphasis on
particular cases such as the linear Schr\"odinger and wave
equations. The associated dispersive estimates play a highly
important role in the theory of Strichartz estimates so I will begin
in Lecture 1 by proving the required dispersive estimates.
Next, in Lecture 2, I will prove the homogeneous Strichartz
estimates in all admissible cases, including the so-called Keel--Tao
endpoint case. Building on the content of the first two lectures, in
Lecture 3, I will discuss the situation regarding inhomogeneous
Strichartz estimates.
Abstract: The aim of these lectures is to give a
gentle introduction to Strichartz estimates, with an emphasis on
particular cases such as the linear Schr\"odinger and wave
equations. The associated dispersive estimates play a highly
important role in the theory of Strichartz estimates so I will begin
in Lecture 1 by proving the required dispersive estimates.
Next, in Lecture 2, I will prove the homogeneous Strichartz
estimates in all admissible cases, including the so-called Keel--Tao
endpoint case. Building on the content of the first two lectures, in
Lecture 3, I will discuss the situation regarding inhomogeneous
Strichartz estimates.
