A counterexample to the Berger--Coburn conjecture

Berger and Coburn proposed an endpoint boundedness criterion for Toeplitz operators on the Bargmann–Fock space in which the decisive quantity is the heat transform of the symbol at the borderline time $t=\frac{1}{4}$, the time naturally singled out by the Weyl calculus under the Bargmann transform. We show that this criterion fails for general measurable symbols in every complex dimension $n\ge 1$. Concretely, we construct a measurable symbol $g\in L^2({\mathbb{C}}^n,d\mu )$ such that $g{k}_a\in L^2(d\mu )$ for every normalized reproducing kernel $k_a$, and the associated Toeplitz form extends to a bounded operator on $H^2({\mathbb{C}}^n,d\mu )$, but the heat transform $g^{(1\mathrm{/}4)}$ is unbounded on ${\mathbb{C}}^n$. The example is obtained by summing translated bounded “blocks” whose Toeplitz norms are summable while their $t=\frac{1}{4}$ heat profiles have fixed size. The blocks are produced by combining a Hilbert–Schmidt estimate for Weyl quantization with the Bargmann correspondence between Weyl and Toeplitz operators.