Department of Mathematics and Statistics
                          The University at Albany
                                      
                                 COLLOQUIUM
                                      
                           Vladimir Ya. Eiderman
                                      
            University of Kentucky and Michigan State University
                                      
                     Cartan type estimates for certain
                          vector-valued potentials
                                      
                         Friday, November 30, 2007
                                      
                             4:15 p.m., ES 143
                                      
                          Tea at 3:45 p.m., ES 152
                                      
                                  ABSTRACT
                                      
   For given complex numbers nu_{1},...,nu_{N}, points z_{1},...,z_{N} in
   the complex plane and P>0 we define the set
   
   {Z}(P,nu) = {z\inC: |SUM_{j = 1}^{N}[{nu_{j}}/{z-z_{j}}] | > P }
   
   (for nu_{j}EQUIV 1 the sum is the logarithmic derivative of the
   polynomial PROD_{j = 1}^{N}[(z-z_{j})] and has also the certain
   physical interpretation). Our goal is to estimate the size of
   \cal{Z}(P,nu) in terms of the radii of covering disks. The sharp
   estimate (up to an absolute constant) of the set
   
   {X}(P,nu) = {z \in C:SUM_{j = 1}^{N}[|{nu_{j}}/{z-z_{j}}| ] > P }
   
   can be easily obtained using a method of Cartan (1928). Thus, the
   problem is to catch the effect of the mutual cancellation of the terms
   when we pass from the sum of moduli to the modulus of the sum.
   
   In spite of the fact that the problem on estimation of {Z}(P,nu) was
   posed by Macintyre and Fuchs in 1940, it was solved only in 2005 by
   J. M. Anderson and the speaker using a tool which appeared only in the
   last 12 years in connection with the development of the theory of
   analytic capacity (Melnikov, Tolsa, Mattila, Nazarov, Treil, Volberg
   and others).
   
   In the lecture I will tell about some of the notions and facts of this
   theory as well as about a very recent generalization of the problem
   described above and its applications.