An integral operator induced by a Zygmund function

We introduce a kernel function induced by a continuous function ff on the unit circle, and show that the corresponding integral operator on the standard Bergman space is a bounded, compact or Hilbert–Schmidt operator precisely when ff belongs to the big Zygmund class Λ∗Λ∗, the little Zygmund class λ∗λ∗ or the Sobolev class H32, respectively. This may be considered as the infinitesimal version of the main result obtained in Hu and Shen (2012) [9] by two of the authors.