Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon

We give a complete characterization of Schatten class Hankel operators H f H_f acting on weighted Segal-Bargmann spaces F squared left-parenthesis phi right-parenthesis"> F 2 ( ? stretchy="false">) encoding="application/x-tex">F^2(\varphi ) using the notion integral distance to analytic functions in alttext="double-struck upper C Superscript n"> C n encoding="application/x-tex">\mathbb {C}^n and Hörmander’s alttext="ModifyingAbove partial-differential With bar"> mathvariant="normal">? encoding="application/x-tex">\bar \partial -theory. Using our characterization, for alttext="f element-of L normal infinity"> ?<!-- ? <mml:mi>L mathvariant="normal">?<!-- ? encoding="application/x-tex">f\in L^\infty alttext="1 greater-than p 1 &gt; p encoding="application/x-tex">1&gt;p&gt;\infty , we prove that is S p"> S encoding="application/x-tex">S_p if only f overbar Baseline encoding="application/x-tex">H_{\bar {f}}\in S_p which was previously known Hilbert-Schmidt 2"> encoding="application/x-tex">S_2 standard space with alttext="phi z right-parenthesis equals alpha StartAbsoluteValue EndAbsoluteValue squared"> z = ?<!-- ? stretchy="false">| encoding="application/x-tex">\varphi (z) = \alpha |z|^2 .