A Class of Integral Operators Induced by Harmonic Bergman-Besov kernels on Lebesgue Classes

Abstract: We provide a full characterization in terms of the six parameters involved the boundedness of all standard weighted integral operators induced by harmonic Bergman-Besov kernels acting between different Lebesgue classes with standard weights on the unit ball of R^{n}. These operators in some sense generalize the harmonic Bergman-Besov projections. To obtain the necessity conditions, we use a technique that heavily depends on the precise inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball. This fruitful technique is new. It has been used first with holomorphic Bergman-Besov kernels by Kaptano\u{g}lu and \"Ureyen. Methods of the sufficiency proofs we employ are Schur tests or H\"older or Minkowski type inequalities which also make use of estimates of Forelli-Rudin type integrals.