Georgia Institute of Technology College of Sciences

Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in-particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's. In this talk, I will discuss several results concerning the two weight inequalities for various Calder\'on-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss.

The two-weight conjecture for singular integral operators T was first raised by Nazarov, Treil and Volberg on finding the real variable characterization of the two weights u and v so that T is bounded on the weighted $L^2$ spaces. This conjecture was only solved completely for the Hilbert transform on R until recently. In this talk, I will describe our result that resolves a part of this conjecture for any Calder\'on-Zygmund operator on the spaces of homogeneous type by providing a complete set of sufficient conditions on the pair of weights. We will also discuss the existence of similar analogues for multilinear Calder\'on-Zygmund operators.

We estimate the  Riesz basis (RB) bounds obtained in Hruschev, Nikolskii and Pavlov' s classical characterization of exponential RB. As an application, we  improve previously known estimates of the RB bounds in some classical cases, such as RB obtained by an Avdonin type perturbation, or RB which are the zero-set of sine-type functions. This talk is based on joint work with S. Nitzan

 In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any  non-zero square integrable function $g$ and  subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved. 

In this talk,  I will then introduce an inductive approach to investigate the conjecture, by attempting to answer the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$?  I will report on a recent joint work with V.~Oussa in which we use this approach to prove the conjecture when the initial configuration  $\Lambda=\{(a_k, b_k)\}_{k=1}^N$  is either a subset of the unit lattice $\mathbb{Z}^2$ or a subset of a line $L$.   

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix  $A$ with real, bounded, and possibly non-symmetric coefficients. If a proper {$L^1$-mean oscillation} of the coefficients of $A$ satisfies suitable Dini-type assumptions, we prove the following: if $\mu$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$,   and

$$T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y)\, d\mu(y)$$

denotes the gradient of the single layer potential associated with $L_A$, then

$$1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},$$

where $\mathcal R_\mu$ indicates the $n$-dimensional Riesz transform. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for $\mathcal R_\mu$, which were recently extended to  $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.