Critical Schatten exponents for the Berger-Coburn endpoint problem

Abstract

Berger and Coburn showed that boundedness of a Toeplitz operator $T_g$ on the Fock space controls the heat transform $g^{(t)}$ for $1/4<t<1$, and conjectured that $T_g$ is bounded if and only if $g^{(1/4)}$ is bounded. Looi recently disproved the forward implication by constructing a bounded $T_g$ with unbounded $g^{(1/4)}$. We prove that the reverse implication also fails, even for a real-valued admissible symbol $g$ with $g^{(1/4)}\in C_0({\mathbb{C}}^n)$, and that the forward implication fails even when $T_g\in S_p$ for every finite $p>1$. Our counterexamples come from Baire category and closed graph arguments, which yield no explicit symbols. Despite this two-sided failure, the forward implication has a sharp Schatten-class form: $\Vert g^{(1/4)}{\Vert }_{\infty }\le 2^n\Vert T_g{\Vert }_{S_1}$ for every admissible symbol $g$ with $T_g$ trace class. We show that the constant $2^n$ is optimal and that the same estimate holds with $S_p$ in place of $S_1$ for $T_g\in S_p$ when $0<p\le 1$. For $p>1$, no corresponding uniform estimate holds, even for smooth compactly supported symbols. We also determine the optimal constants in the Berger--Coburn estimates for $1/4<t\le 1$. Finally, for $1\le p<\infty$, we show that the equivalence $T_g\in S_p$ if and only if $g^{(1/4)}\in L^p$ holds exactly when $p=2$, answering a question of Bauer, Coburn, and Isralowitz.