На следующей неделе в лаборатории им П.Л. Чебышева состоятся миникурсы по анализу Н. Никольского и В. Эйдермана.
Nikolai Nikolski (University of Bordeaux)
Schauder bases geometry in Hilbert spaces.
July 3-5, 14-30 - 16-30
Numerical characteristics of Hilbert space bases play an important role in Harmonic Analysis and its applications, from weighted singular integrals through numerical matrix analysis. In this short course, we focus on problems around the behaviour of the basis and unconditional basis constants, as well as Fourier-Hadamard multipliers. The McCarthy-Schwartz inequality is revisited, and its sharpness is derived from results by Spijker, Tracogna and Welfert. A Beurling-Deny theory of abstarct Besov-Dirichlet spaces is traced, culminating in a complete description of Fouruer-Muckenhoupt multipliers for weighted L² spaces with a Schoenberg weight. Applications to matrix analysis is discussed, in particular to the Kreiss matrix theorem.
Vladimir Eiderman (University of Wisconsin-Madison)
Calder\'on-Zygmund operators and related capacities
July 3-5, 17-00 - 18-35.
We will discuss some basic concepts and facts in the theory of Calder\'on-Zygmund operators which have led to the breakthrough results of Tolsa and others in the theory of the analytic capacity. Below is the brief content of the short course. 1. Painlev\'e problem on removable singularities for bounded analytic functions. 2. Analytic capacity and its many-dimensional counterparts. 3. Capacity and norms of Calder\'on-Zygmund operators. 4. Melnikov curvature of a measure and its many-dimensional analogs. 5. $T(1)$ and $T(b)$ theorems. 6. The case when curvature is ``cruelly missing''. 7. Applications and open problems. |