Small Hankel operators on weighted Besov spaces

Abstract

We characterize bounded small Hankel operators from $B_p(\om)$ to $\overline{B_q(\om)}$ on the polydisc for $0<p,q<\infty$, under explicit conditions relating the kernel parameter to the weight. When $q<p$, boundedness is equivalent to a weighted Besov condition on the symbol; when $p\le q$, it is equivalent to a weighted supremum condition on a shifted radial derivative. In particular, the diagonal case is characterized by the product Bloch space. The proof combines weighted Bergman duality and atomic factorization with local kernel estimates and randomized atomic tests.