Unbounded symbols, heat flow, and Toeplitz operators

TL;DR

This paper shows a fundamental separation between two Toeplitz realizations on the Bargmann–Fock space for unbounded symbols: the form-defined operator $T_g$ can remain bounded under heat-flow regularity, while the natural-domain operator $U_g$ can be unbounded. It identifies a precise Carleson-measure criterion for the boundedness of $U_g$ via $|g|^2 d\mu$, and proves that this criterion is strictly stronger than the linear averaging that governs $T_g$. The authors construct a smooth radial symbol $g$ satisfying the coherent-state domain hypothesis with bounded heat transforms for all $t>0$, yet with $U_g$ unbounded, illustrating a global, geometry-at-infinity obstruction. They further show that heat-flow regularity is irreversible in this setting and that bootstrapping heat-regularity cannot resolve the gap between sufficiency and the critical time, thereby refining the Berger–Coburn heat-flow picture. Overall, the work delineates the limits of transferring heat-flow regularity to the natural-domain Toeplitz operator and provides sharp, kernel-based criteria distinguishing the two realizations.

Abstract

We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish that the form-defined operator $T_g$ and the natural-domain operator $U_g$ decouple in the unbounded symbols regime: while $T_g$ is governed by linear averaging, $U_g$ is controlled by the quadratic intensity of $|g|^2$. We construct a smooth, nonnegative radial symbol $g$ satisfying the coherent-state admissibility hypothesis with bounded heat transforms for all time $t>0$; for this symbol, $T_g$ is bounded, yet $U_g$ is unbounded. This is a strictly global phenomenon: under the coherent-state hypothesis, local singularities are insufficient to cause unboundedness, leaving the ``geometry at infinity'' as the sole obstruction. Boundedness of $U_g$ is equivalent to the condition that $|g|^2 dμ$ is a Fock--Carleson measure, a condition strictly stronger than the linear average $g dμ$ governing $T_g$. Finally, regarding the gap between the known sub-critical sufficiency condition and the critical heat time, we prove that heat-flow regularity is irreversible in this context and show that bootstrapping strategies cannot resolve the gap between sufficiency and critical time.

Theorems & Definitions (27)

• Theorem 1.1: Main separation phenomenon
• Corollary 1.2: Purely global failure of the $|g|^2$ kernel test
• Theorem 1.3: Characterization of bounded $U_g$
• Theorem 1.4: Two kernel tests
• Remark 1.5
• Proposition 1.6: Power symbols
• Lemma 2.1
• proof
• Lemma 2.2
• proof