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Series: Analysis Seminar

We will discuss the fine notion of the pointwise convergence of ergodic averages
in setting where one the ergodic transformation is a Z^d action, and the averages
are over more exotic sets than just cubes. In this setting, pointwise convergence
does not follow from the usual ergodicity arguments. Bourgain, in his study of the
polynomial ergodic averages invented the variational technique, which we extend to
our more exotic averages.

Series: Analysis Seminar

In this talk I will describe an Lp theory for outer measures,
which could be used to connect two themes of Lennart Carleson's work:
Carleson measures and time frequency analysis.
This is joint work with Christoph Thiele.

Series: Analysis Seminar

Complex analysis in several variables is very different from the
one variable theory. Hence it is natural to expect that operator theory on
Bergman spaces of pseudoconvex domains in $\mathbb{C}^n$ will be different
from the one on the Bergman space on the unit disk. In this talk I will
present several results that highlight this difference about compactness of
Hankel operators. This is joint work with Mehmet Celik and Zeljko Cuckovic.

Series: Analysis Seminar

In 1980, T. M. Wolff has given the following version of the ideal membership for finitely generated ideals in $H^{\infty}(\mathbb{D})$: \[\ensuremath{\mbox{If \,\,}\left\{ f_{j}\right\} _{j=1}^{n}}\subset H^{\infty}(\mathbb{D}),\, h\in H^{\infty}(\mathbb{D})\,\,\mbox{and }\]\[\vert h(z)\vert\leq\left(\underset{j=1}{\overset{n}{\sum}}\vert f_{j}(z)\vert^{2}\right)^{\frac{1}{2}}\,\mbox{for all \ensuremath{z\in\mathbb{D},}}\]then \[h^{3}\in\mathcal{I}\left(\left\{ f_{j}\right\} _{j=1}^{n}\right),\,\,\mbox{the ideal generated by \ensuremath{\left\{ f_{j}\right\} _{j=1}^{n}}in \ensuremath{H^{\infty}}\ensuremath{(\mathbb{D})}. }\]In this talk, we will give an analogue of the Wolff's ideal problem in the multiplier algebra on weighted Dirichlet space. Also, we will give a characterization for radical ideal membership.

Series: Analysis Seminar

For dimensions n greater than or equal to 3, and integers N greater than 1, there is a
distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm. In particular the same distribution will have optimal L^p norms, for 1 < p < \infty. The collection P is a random digit shift of the examples of W.L. Chen and M. Skriganov.

Series: Analysis Seminar

On the Hardy space, by means of an elegant and ingenious argument, Widom showed that the spectrum of a bounded Toeplitz operator is always connected and Douglas showed that the essential spectrum of a bounded Toeplitz operator is also connected. On the Bergman space, in 1979, G. McDonald and the C. Sundberg showed that the essential spectrum of a Toeplitz operator
with bounded harmonic symbol is connected if the symbol is either real
or piecewise continuous on the boundary. They asked whether the essential spectrum of a Toeplitz operator on the Bergman space with bounded harmonic symbol is connected. In this talk, we will show an example that the spectrum and the essential spectrum of a Toeplitz operator with bounded harmonic symbol is disconnected. This is a joint work with Carl Sundberg.

Series: Analysis Seminar

A recent conjecture in harmonic analysis that was exploredin the past 20 years was the A_2 conjecture, that is the sharp bound onthe A_p weight characteristic of a Calderon-Zygmund singular integraloperator on weighted L_p space. The non-sharp bound had been knownsince the 1970's, but interest in the sharpness was spurred recentlyby connections to quasiconformal mappings and PDE. Finally solved infull by Hytonen, the proof is complex, intricate and lengthy. A new "simple" approach using local mean oscillation and positive operatorbounds was published by Lerner. We discuss this and some recent progress in the area, including our new proof for spaces of homogeneoustype, in the style of Lerner (Joint work with Armen Vagharshakyan).

Series: Analysis Seminar

In this talk, we will characterize the compact operators on Bergman spaces of the ball and polydisc. The main result we will discuss shows that an
operator on the Bergman space is compact if and only if its
Berezin transform vanishes on the boundary and additionally
this operator belongs to the Toeplitz algebra.
We additionally will comment about how to extend these results to
bounded symmetric domains, and for "Bergman-type" function spaces.

Series: Analysis Seminar

We consider a so-called "One sided bump conjecture", which gives asufficient condition for two weight boundedness of a Calderon-Zygmundoperator. The proof will essentially use the Corona decomposition, which isa main tool for a first proof of $A_2$ (also, $A_p$ and $A_p-A_\infty$)conjecture. We will focus on main difficulty, that does not allow to get afull proof of our one sided bump conjecture.

Series: Analysis Seminar

We will study one and two weight inequalities for several different
operators from harmonic analysis, with an emphasis on vector-valued
operators. A large portion of current research in the area of one weight
inequalities is devoted to estimating a given operators' norm in terms
of a weight's A_p characteristic; we consider some related problems
and the extension of several results to the vector-valued setting. In
the two weight setting we consider some of the difficulties of
characterizing a two weight inequality through Sawyer-type testing
conditions.