In this paper we first characterize the Schatten p-class and Schatten h-class Hankel and Toeplitz operators on Bergman spaces \(A_{\omega _{1,2}}^2({\mathbb {M}})\) induced by regular weights \(\omega _{1,2}\) of the annulus \({\mathbb {M}}\) with full range \(p\in (0,\infty )\) and h being a continuous increasing convex function on \((0,\infty )\) . As an application, we then establish essential norm estimates for bounded Hankel operators from Bergman spaces \(A_{\omega _{1,2}}^p({\mathbb {M}})\) to Lebesgue spaces \(L_{\omega _{1,2}}^q({\mathbb {M}})\) for all possible \(p,q\in (1, \infty )\) . Moreover, Schatten p-class properties and essential norm estimates for Hankel operators on Bergman spaces over the unit disk \({\mathbb {D}}\) induced by regular weights are also obtained, which can be viewed as a further application of boundedness and compactness of Hankel operators proved by Hu and Jin (J Geom Anal 29:3494–3519, 2019). To establish these desired characterizations, the diagonal and off-diagonal decompositions, various careful estimates for reproducing kernels, Berezin transforms, Carleson measures and the solution of \({\bar{\partial }}\) -equations are crucial tools in our proofs.
