Algebra Of The Weakly Localized Operators On The Harmonic Bergman Space

The research work of this thesis can be divided into two parts.The first part is on the sharp Forelli-Rudin type estimates on the unit real ball. For the integrals and we first express them in terms of hypergeometric functions, then we discuss several properties of the hypergeometric functions in some conditions and obtain the sharp Forelli-Rudin type estimates on the unit real ball as well as a uniform version of Forelli-Rudin type estimates. Afterwards, as applications, we prove an inequality of Hilbert type on the unit real ball, simplify the proof of a sharp inequality with harmonic func-tions, discuss the constant and exponent in an inequality of Hardy-Littlewood type and estimate the norm of a class of operators which are closely related to the harmonic Bergman space.The second part is on the (p,δ)-weakly localized operators. Inspired by the work done on the holomorphic Bergman space, we introduce a class of the so-called (p,δ)-weakly localized operators on the harmonic Bergman space on the unit real ball, which forms an algebra and contains all Toeplitz operators with bounded symbols. Finally, we give a criterion of the compactness for this class of operators, i.e., T is compact on bp if and only if there exists k>0, such that...