An integral operator induced by a Zygmund function

We introduce a kernel function induced by a continuous function f 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 f belongs to the big Zygmund class Λ ∗ , the little Zygmund class λ ∗ or the Sobolev class H 3 2 , 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.