Berger-Coburn端点问题的临界Schatten指数

中文摘要 AI 辅助

Berger和Coburn证明了Fock空间上Toeplitz算子$T_g$的有界性控制着热变换$g^{(t)}$（对于$1/4<t<1$），并猜想端点$g^{(1/4)}$刻画有界性。Looi最近通过构造一个有界$T_g$但$g^{(1/4)}$无界的例子否定了这一猜想。我们证明$p=1$是精确的Schatten指数，使得$T_g\in S_p$蕴含$g^{(1/4)}$有界。出现在Berger和Coburn迹公式中的迹类算子，其迹范数在$t\downarrow1/4$时发散，但强收敛于$2^n J$，由此可得$$ g^{(1/4)}(a)=2^n\operatorname{tr} \bigl(T_gW_aJW_a^*\bigr),\qquad \\|g^{(1/4)}\\|_\infty\leq2^n\norm{T_g}_{S_1}, $$ 对每个满足$T_g$为迹类的容许符号$g$成立，其中$J$为宇称算子，$W_a$为Weyl平移。我们证明$2^n$是最优的，且同样的界对$0<p\leq1$时的$T_g\in S_p$成立。对于$p>1$，即使对紧支撑光滑符号，也不存在相应的$S_p$估计。此外，Baire范畴论证表明存在一个容许符号$g$，使得$T_g\in S_p$对所有$p>1$成立，但$g^{(1/4)}$无界，尽管该论证未给出显式符号。我们还确定了$1/4<t\leq1$时Berger-Coburn估计中的最优常数。

英文摘要

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 the endpoint $g^{(1/4)}$ characterizes boundedness. Looi recently disproved this by constructing a bounded $T_g$ with unbounded $g^{(1/4)}$. We disprove the converse implication by showing that there exists a real-valued admissible symbol $g$ such that $g^{(1/4)}\in C_0(\mathbb C^n)$ while $T_g$ has no bounded extension. Despite this two-sided failure, we show that the forward implication has a sharp Schatten-class form, that is, $p=1$ is the exact Schatten exponent for which $T_g\in S_p$ forces $g^{(1/4)}$ to be bounded. The trace-class operators appearing in Berger and Coburn's trace formula have divergent trace norms as $t\downarrow1/4$, yet converge strongly to $2^n J$, and it follows that $g^{(1/4)}(a)=2^n\operatorname{tr}(T_gW_aJW_a^*)$ and $\|g^{(1/4)}\|_\infty\leq2^n\|T_g\|_{S_1}$ for every admissible symbol $g$ with $T_g$ trace class, where $J$ is the parity operator and $W_a$ is Weyl translation. We prove that $2^n$ is optimal, and the same bound holds for $T_g\in S_p$ with $0<p\leq1$. For $p>1$, no corresponding $S_p$ estimate is possible, even for compactly supported smooth symbols. Moreover, a Baire category argument shows there is an admissible symbol $g$ with $T_g\in S_p$ for every $p>1$ while $g^{(1/4)}$ is unbounded, though it yields no explicit symbol. We also determine the optimal constants in the Berger--Coburn estimates for $1/4<t\leq1$.

发表机构

• University of Eastern Finland（东芬兰大学）
• University of Reading（雷丁大学）
• University of Helsinki（赫尔辛基大学）

机构由 AI 辅助整理，请以论文原文为准。