The authors obtain conditions on the symbol \(\phi\) of the (small) Hankel operators \(h_\phi\) under which such operators are bounded when acting between Bergman spaces (on the open unit ball \(\mathbb B\) in \({\mathbb C}^n\)), \(A^p(\mathbb B)\), with possible different integrability indices \(p, q \in (0, \infty)\): \[ h_\phi : A^p(\mathbb B) \longrightarrow A^q(\mathbb B)\,. \] A characterization of the boundedness of \(h_\phi\) is obtained for the improving inequalities case (of the integrability parameters): \(q \geq p\). This result extends the previous known characterization for the case \(q=p\) (where the necessary and sufficient condition for \(h_\phi\) to be bounded is that \(\phi\) belongs to the Bloch class).
The paper also provides particular results on the boundedness of \(h_\phi\) for the case \(q < p\) (and \(p\geq 1\); note that for \(p<1\), the Hankel operator \(h_\phi\) is always improving – in the above sense).
The obtained results are mainly based on the construction of operator inequalities, the Fubini theorem, the Hölder inequality, the use of certain differential operators, and characterizations of the duals of the Bergman spaces in use.