L. Vega
April 20, 15:00
Un resultado de unicidad para la ecuación de Schrödinger

Abstract: Presentaremos un trabajo realizado en colaboración con L. Escauriaza, C. Kenig, y G. Ponce en el que se demuestra que una solución de la ecuación de Schrödinger con un potencial acotado que decaiga mas rápido que una gaussiana en dos tiempos distintos ha de ser cero. Este resultado generaliza el bien conocido de G.H. Hardy que afirma que una funcion tal que ella y su transformada de Fourier decaigan mas rápido que una gaussiana ha de ser trivial.

J. Bennett
April 21, 11:30
Multilinear Kakeya and k-plane transform estimates

Abstract: We will discuss certain self-similarity properties of the classical multilinear Brascamp-Lieb inequalities and their k-plane transform generalisations.

J. Garnett
April 25, 11:30
Problems on analytic capacity, Lipschitz harmonic capacity and bilipschitz invariance

April 27, 11:30
Interpolating Blaschke products and approximation problems

C. Pérez
April 26, 12:00
On the two weight problem for the Hilbert transform

Abstract: In this talk we plan to survey on some recent work with D. Cruz Uribe and J.M. Martell where we prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.

K. Astala

Abstract: In these two talks we discuss recent topics in quasiconformal mapping in two dimension. The first, joint work with Iwaniev and Martin, considers mapping which minimize the mean of the distortion function and related questions in minimizing the Dirichlet energies among suitable classes of homeomorphisms.
If time permits we also cover results for mapping of finite distortion, needed for the second talk.
The second talk, joint work with Rohde, Sackman and Tao, discusses solutions to the Beltrami equations when the coefficients are chosen randomly.

Abstract: We will show some new asymptotics of orthogonal polynomials when the measure is a small perturbation of the measure in Szego-Kolmogorov-Krein theorem. The reasoning is based on Koosis type estimates. Then we derive the asymptotics of orthogonal polynomials in case of LARGE perturbation of Szego-Kolmogorov-Krein theorem. But this large perturbation should be rather specific. We show the relation with the problem of finding the sum rules for Jacobi matrices and with works of Killip, Simon, Deift.

May 9, 15:00
Global regularity for critically dissipative quasi-geostrophic equation

Abstract: In this joint work with A. Kiselev and F. Nazarov it is shown that the solution of quasi-geostrophic equation with critical order of dissipation is smooth for any time if the initial data is smooth. There is no assumption on the smallness of initial data.

S. Treil
May 10, 11:30 & May 11, 11:30
Two weight estimates and perturbations of self-adjoint operators

Abstract: In the first talk I am going to present a special regularization of the Cauchy integral, appearing in the theory of rank one perturbations of self-adjoint operators. I will show how the interplay between operator theory and harmonic analysis allows to obtain new result in both areas. In particular, as a corollary of an elementary theory of two weight estimates I will get a new result in the perturbation theory, namely, a sufficient condition of the disappearance of the singular spectrum.
The second talk will be devoted to the two-weight estimates of the so-called "well localized" operators, which include all martingale transforms (Haar multipliers), as well as the so-called Haar shift (the latter operator is especially important, because taking the average of Haar shifts over all dyadic lattices one gets a non-zero multiple of the Hilbert Transform). I will present a necessary and sufficient condition of Sawyer type for the two-weight estimates for such well-localized operators. This part is based on a joint work with F. Nazarov and A. Volberg.

L. Grafakos
May 16, 15:00
L^2 boundedness versus L^p boundedness for Calderon-Zygmund singular integral operators

Abstract: We study the L^p boundedness of Calderon-Zygmund singular integral operators on $R^n$ given by convolution with a kernel of the form $\Omega(x/|x|)|x|^{-n}$, where $\Omega$ is even and integrable function on the sphere $S^{n-1}$ with mean-value zero. In particular, we are interested in the question as to whether an $L^2$ bounded such operator must necessarily be $L^p$ bounded for $1<p\neq2<\infty$. This work is joint with Petr Honzik and Dmitry Ryabogin.

R. Brown

Abstract: We consider a magnetic Schrödinger operator $ L_W = \sum (D_j + W_j)^2$ in a $C^1$-domain, $\Omega$, in $\reals^n$ with $n\geq 3$. We assume that the vector potential is continuous. We let $\Lambda _W$ denote the Dirichlet-to-Neumann map associated to $L_W$. We show how to recover the tangential component of $W$ at the boundary from the Dirichlet-to-Neumann map. Our method is constructive and provides a stability estimate. This work is joint with Mikko Salo.

S. Hofmann
May 18, 15:00
Stability of layer potentials and $L^{2}$ Solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty}$ coefficients

Abstract: We consider divergence form elliptic operators of the form $L=-\dv A(x)\nabla$, defined in $\mathbb{R}^{n+1} = \{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertiblity of the corresponding layer potential operators on $L^2(\mathbb{R}^{n}) = L^2(\partial\mathbb{R}_{+}^{n+1})$, is stable under complex, $L^{\infty}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2(\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\Vert A-A_{1}\Vert_{\infty}$ is small enough and $A_{1}$ is real, symmetric, $L^{\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$ data, for small complex perturbations of a real symmetric matrix. Previously, these sorts of $L^2$ results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices, or in the special case that the coefficients $A_{j,n+1}=0=A_{n=1,j}$, $1\leq j\leq n$. In the latter case, the result is equivalent to the analytic perturbation theory for Kato's square root operators. Joint work with M. Alfonseca, P. Auscher, A. Axelsson and S. Kim

S. Kurylev
May 24, 12:00 & May 25, 15:00
Dirac operators with boundary data: index theorems and inverse problems

Abstract: We consider index theorems and inverse problems for the Dirac-type operators associated with Clifford connections The main assumption is the existence of chirality. This makes it possible to introduce a superstructure on the Dirac bundle. The superstructure makes it possible to sometimes introduce the decomposition of the Dirac operator, with proper boundary conditions, into two Fredholm operators. It also makes it possible to control the propagation of waves in each channel. We use as boundary data for the inverse problem in the form of either the Cauchy data of the solution of the non stationary Dirac equation, or the boundary spectral data for the corresponding self-adjoint elliptic operator. we then obtain formulae connecting these data with the index of the Dirac operator and develop a method to reconstruct the Dirac bundle up to a natural group of transformations. Joint work with M.Lassas