Estimates for the Polyharmonic Bergman Kernel and Their Applications

Abstract

We consider the space of all square integrable polyharmonic functions of degree m on the unit ball which is called the weighted m-polyharmonic Bergman space. We give an explicit orthonormal basis of the weighted true m-polyharmonic Bergman space and estimates for the reproducing kernel of the true m-polyharmonic Bergman space. Moreover, we introduce an operator $$S_{m,\alpha } = \Delta _{\alpha }^{m-1} ( 1-|x|^2)^{2(m-1)}$$ and give the correspondence between the weighted harmonic Bergman space and the weighted true m-polyharmonic Bergman space. As applications, we also mention the Gleason problem and the Lipschitz type characterization for the m-polyharmonic Bergman space.