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Series: Analysis Seminar

Series: Analysis Seminar

Series: Analysis Seminar

Cotlar’s identity provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms. E. Ricard and I discovered a more flexible version of this identity, in the recent study of the boundedness of Hilbert transforms on the free groups. In this talk, I will try to introduce this version of Cotlar’s identity and the Lp Fourier multipliers on free groups.

Series: Analysis Seminar

Series: Analysis Seminar

TBA

Series: Analysis Seminar

In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

Series: Analysis Seminar

Dynamical sampling is a new area in
sampling theory that deals with signals that evolve over time under the
action of a linear operator. There are lots of studies on various
aspects of the dynamical sampling problem. However, they all focus on
uniform
discrete time-sets $\mathcal T\subset\{0,1,2,\ldots, \}$. In our study,
we concentrate on the case $\mathcal T=[0,L]$. The goal of the
present work is to study the frame property of the systems
$\{A^tg:g\in\mathcal G, t\in[0,L] \}$. To this end, we also
characterize the completeness and Besselness properties of these
systems.

Series: Analysis Seminar

In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.

Series: Analysis Seminar

We prove a criterion for nondoubling parabolic measure to satisfy a weak reverse H¨older inequality
on a domain with time-backwards ADR boundary, following a result of Bennewitz-Lewis for nondoubling
harmonic measure.

Series: Analysis Seminar

Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider
the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions
of this type arise in physics as modes of periodic vibration
of drums and membranes. They also represent stationary states of a free
quantum particle on a Riemannian manifold. In the first part of the
lecture, I will give a survey of results which demonstrate how the
geometry of $M$ affects the behaviour of these special
functions, particularly their “size” which can be quantified by
estimating $L^p$ norms.
In joint work with Malabika Pramanik (U. British Columbia),
I will present in the second part of my lecture a result on the $L^p$ restriction of
these eigenfunctions to random Cantor-type
subsets of $M$. This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results
of Burq-Gérard-Tzetkov ’06 (and later work of other authors). Our
method includes concentration inequalities from probability theory in
addition to the analysis of singular Fourier integral operators on
fractals.