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Series: Analysis Seminar

Using integral formulas based on Green's theorem and in
particular a lemma of Uchiyama, we give simple proofs of comparisons
of different BMO norms without using the John-Nirenberg inequality
while we also give a simple proof of the strong John-Nirenberg
inequality. Along the way we prove the inclusions of BMOA in the dual
of H^1 and BMO in the dual of real H^1. Some difficulties of the
method and possible future directions to take it will be suggested at
the end.

Series: Analysis Seminar

In the first part of the talk we will give a brief survey of significant
results going from S. Brown pioneering work showing the existence of
invariant subspaces for subnormal operators (1978) to Ambrozie-Muller
breakthrough asserting the same conclusion for the adjoint of a
polynomially bounded operator (on any Banach space) whose spectrum
contains the unit circle (2003). The second part will try to give some
insight of the different techniques involved in this series of results,
culminating with a brilliant use of Carleson interpolation theory for
the last one. In the last part of the talk we will discuss additional
open questions which might be investigated by these techniques.

Series: Analysis Seminar

Series: Analysis Seminar

Motivated by mappings of finite distortion, we consider degenerate p-Laplacian equations whose ellipticity condition is satisfied by thedistortion tensor and the inner distortion function of such a mapping. Assuming a certain Muckenhoupt type condition on the weightinvolved in the ellipticity condition, we describe the set of continuity of solutions.

Series: Analysis Seminar

It is well-known that every Schur function on the bidisk can be written as
a sum involving two positive semidefinite kernels. Such decompositions,
called Agler decompositions, have been used to answer interpolation
questions on the bidisk as well as to derive the transfer function
realization of Schur functions used in systems theory. The original
arguments for the existence of such Agler decompositions were
nonconstructive and the structure of these decompositions has remained
quite mysterious.
In this talk, we will discuss an elementary proof of the existence of
Agler decompositions on the bidisk, which is constructive for inner
functions. We will use this proof as a springboard to examine the
structure of such decompositions and properties of their associated
reproducing kernel Hilbert spaces.

Series: Analysis Seminar

This is a joint work with F.~Nazarov and A.~Volberg.Let $s\in(1,2)$, and let $\mu$ be a finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$. We prove that if the lower $s$-density of $\mu$ is+equal to zero $\mu$-a.~e. in $\mathbb R^2$, then$\|R\mu\|_{L^\infty(m_2)}=\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesque measure in $\mathbb R^2$. Combined with known results of Prat and+Vihtil\"a, this shows that for any noninteger $s\in(0,2)$ and any finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have+$\|R\mu\|_{L^\infty(m_2)}=\infty$.Also I will tell about the resent result of Ben Jaye, as well as about open problems.

Series: Analysis Seminar

Truncated Toeplitz operators, introduced in full generality by Sarason a few
years ago, are compressions of multiplication operators on H^2 to subspaces
invariant to the adjoint of the shift. The talk will survey this newly
developing area, presenting several of the basic results and highlighting
some intriguing open questions.

Series: Analysis Seminar

In the talk some problems related with the famous Chernoff square root of
n - lemma in the theory of approximation of some semi-groups of operators will
be discussed. We present some optimal bounds in these approximations
(one of them is Euler approximation) and two new classes of operators,
generalizing sectorial and quasi-sectorial operators will be introduced.
The talk is based on two papers [V. Bentkus and V. Paulauskas, Letters
in Math. Physics, 68, (2004), 131-138] and [V. Paulauskas, J. Functional
Anal., 262, (2012), 2074-2099]

Series: Analysis Seminar

We will discuss several results (and open problems) related to
rearrangements of Fourier series, particularly quantitative questions about
maximal and variational operators. For instance, we show that the canonical
ordering of the trigonometric system is not optimal for certain problems in
this setting. Connections with analytic number theory will also be given.
This is based on joint work with Allison Lewko.

Series: Analysis Seminar

We discuss some recent generalizations of Euler--Maclaurin expansions for the
trapezoidal rule and of analogous asymptotic expansions for Gauss--Legendre
quadrature, in the presence of arbitrary algebraic-logarithmic endpoint
singularities. In addition of being of interest by themselves, these asymptotic
expansions enable us to design appropriate variable transformations to improve
the accuracies of these quadrature formulas arbitrarily. In general, these
transformations are singular, and their singularities can be adjusted easily to
achieve this improvement. We illustrate this issue with a numerical example
involving Gauss--Legendre quadrature.
We also discuss some recent asymptotic expansions of the coefficients of
Legendre polynomial expansions of functions over a finite interval, assuming
that the functions may have arbitrary algebraic-logarithmic interior and
endpoint suingularities. These asymptotic expansions can be used to make
definitive statements on the convergence acceleration rates of extrapolation
methods as these are applied to the Legendre polynomial expansions.