The reduced Bergman kernel and its properties (Sahil Gehlawat - Université de Lille)

For a planar domain D, the reduced Bergman kernel of D can be thought of as
the reproducing kernel of a closed subspace R(D) of the classical Bergman
space A^2(D) of D. This subspace R(D) is the collection of L^2 integrable
holomorphic functions on D which admits a primitive on D. The reduced Bergman
kernel was first introduced by S. Bergman himself along with M. Schiffer.
Later on, M. Sakai and J. Burbea worked on the higher order weighted versions
of this kernel. In this talk, we will revisit these higher order reduced
Bergman kernels and prove some important properties like transformation
formula, Ramadanov-type theorems, and localization results for these kernels,
which are already known for the classical Bergman kernel. We will also make
some observations about their boundary behaviour. This is a joint work with
Dr. Amar Deep Sarkar and Aakanksha Jain.