Georgia Institute of Technology College of Sciences

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.

The classification theorem for a C_0 operator describes its quasisimilarity class by means of its Jordan model. The purpose of this talk will be to investigate when the relation between the operator and its model can be improved to similarity. More precisely, when the minimal function of the operator T can be written as a product of inner functions satisfying the so-called (generalized) Carleson condition, we give some natural operator theoretic assumptions on T that guarantee similarity.

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

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.

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

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.

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 $\mathbb R^2$.

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]

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

We continue with the proof of a real variable characterization of the two weight inequality for the Hilbert transform, focusing on a function theory in relevant for weights which are not doubling.