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: 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.
