Positive Toeplitz operators on pluriharmonic Fock spaces: Schatten class criteria, sharp norm comparisons, and generalized weights

Abstract

Let $\mu$ be a positive Borel measure on $\mathbb C^n$. We prove that, for every $0<p<\infty$, the Toeplitz operator $T_\mu^{\mathrm{ph}}$ induced by $\mu$ on the pluriharmonic Fock space belongs to the Schatten class $S_p$ if and only if the local mass function $z\mapsto\mu(B(z,r))$ belongs to $L^p(\mathbb C^n)$ for one, or equivalently every, $r>0$. For $n\geq2$, this resolves a conjecture of Jaguzovi'c and Vujadinovi'c, and the range $0<p<1$ is new in every dimension. Writing $T_\mu$ for the corresponding holomorphic Toeplitz operator, we obtain the sharp estimates $$\|T_\mu\|_{S_p}^p \leq \|T_\mu^{\mathrm{ph}}\|_{S_p}^p \leq 2^{\max\{1,p\}} \|T_\mu\|_{S_p}^p. $$ We also prove a sharp comparison with constant two in every symmetrically normed ideal and an exact trace formula, using positivity and a $2\times2$ block decomposition whose diagonal blocks are $T_\mu$ and an antiunitary copy of its compression to the functions orthogonal to constants. We then consider generalized Fock weights satisfying $m \, dd^c|z|^2\le dd^c\phi\le M \, dd^c|z|^2$. For the canonical holomorphic and antiholomorphic direct sum norm, the same Schatten and symmetrically normed ideal estimates hold. For the norm inherited from $L^2(\mathbb C^n,e^{-2\phi}dV)$, the local mass criterion also holds whenever $e^{-2\phi}$ is comparable to a generalized Fock weight invariant under the scalar circle action. Without further assumptions, the local mass criterion can fail for the inherited norm: in one complex dimension, we construct a weight of the form $\phi(z)=|z|^2/2+\operatorname{Re}q(z)$, with $q$ entire, and a finite positive measure whose local masses belong to every $L^p$, although the Toeplitz form on the inherited pluriharmonic space is unbounded.