Critical Schatten exponents for the Berger-Coburn endpoint problem
Abstract
Berger and Coburn showed that boundedness of a Toeplitz operator Tg on the Fock space controls the heat transform g(t) for 1/4<t<1, and conjectured that Tg is bounded if and only if g(1/4) is bounded. Looi recently disproved the forward implication by constructing a bounded Tg 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)∈ C0(ℂn), and that the forward implication fails even when Tg∈ Sp 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: |g(1/4)|∞≤ 2n|Tg|S1 for every admissible symbol g with Tg trace class. We show that the constant 2n is optimal and that the same estimate holds with Sp in place of S1 for Tg∈ Sp when 0<p≤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≤1. Finally, for 1≤ p<∞, we show that the equivalence Tg∈ Sp if and only if g(1/4)∈ Lp holds exactly when p=2, answering a question of Bauer, Coburn, and Isralowitz.
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BibTeX
@misc{virtanen2026critical,
title = {{Critical Schatten exponents for the Berger-Coburn endpoint problem}},
author = {Jani A. Virtanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.11853},
eprint = {2609.11853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11853},
}
RIS
TY - UNPB
TI - Critical Schatten exponents for the Berger-Coburn endpoint problem
AU - Jani A. Virtanen
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.11853
UR - https://arxiv.org/abs/2609.11853
AB - 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)∈ C_0(ℂ^n), and that the forward implication fails even when T_g∈ 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: |g^(1/4)|_∞≤ 2^n|T_g|_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∈ S_p when 0<p≤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≤1. Finally, for 1≤ p<∞, we show that the equivalence T_g∈ S_p if and only if g^(1/4)∈ L^p holds exactly when p=2, answering a question of Bauer, Coburn, and Isralowitz.
ER -