Integral-type operators acting on a Hilbert-Bergman space of logarithmic weights on the unit ball

Abstract

For a given holomorphic function g on the open unit ball in C^N, we consider the following integral operators T<inf>g</inf> f(z) = ^Z<inf>0</inf>¹ f(tz)Rg(tz)^dt<inf>t</inf> and I<inf>g</inf> f(z) = ^Z<inf>0</inf>¹ R f(tz)g(tz)^dt t on a Hilbert-Bergman space of logarithmic weights. By using an estimate for the norm of the Hilbert-Bergman space in terms of the radial derivative, we describe necessary and sufficient conditions for the boundedness of the operators. We also estimate the essential norm of these operators via the boundary behavior of some quantities that involve a symbol function g.