A counterexample to the Berger--Coburn conjecture
Sam Looi
TL;DR
The paper proves that the endpoint heat-transform criterion proposed by Berger and Coburn does not characterize bounded Toeplitz operators on the Bargmann–Fock space for general measurable symbols. It constructs a sequence of translated, bounded blocks with summable Toeplitz norms whose heat profiles at time t=1/4 stay uniformly large, and then sums them to obtain a symbol g with T_g bounded but g^{(1/4)} unbounded. A Hilbert-Schmidt/Weyl-quantization framework and a summation lemma are used to transfer block-level controls to the full operator, while forcing unbounded growth of g^{(1/4)} at a sparse set of points. This refutes the Berger–Coburn conjecture in all complex dimensions n ≥ 1 and highlights limitations of endpoint heat-flow criteria in general symbol classes.
Abstract
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 $t=\tfrac14$, 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 $g\in L^2(\mathbb C^n,dμ)$ such that $gk_a\in L^2(dμ)$ for every normalized reproducing kernel $k_a$, and the associated Toeplitz form extends to a bounded operator on $H^2(\mathbb C^n,dμ)$, but the heat transform $g^{(1/4)}$ is unbounded on $\mathbb C^n$. The example is obtained by summing translated bounded "blocks" whose Toeplitz norms are summable while their $t=\tfrac14$ 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.