On Grand and Small Bergman Spaces

Grand and small Bergman spaces of functions holomorphic in the unit disc are introduced. The boundedness of the Bergman projection operator on grand Bergman spaces is proved. The main result consists of estimates for functions in grand and small Bergman spaces near the boundary, which differ from those in the case of the classical Bergman space by a logarithmic multiplier with positive (for grand spaces) or negative (for small spaces) exponent.