Schatten limits for semi-commutators in Toeplitz quantization on Fock space

Abstract

We solve a problem posed by Hu and Virtanen by characterizing, for every $0 2n$, the limit is characterized by relative weak compactness in $L^1(\mathbb R^{2n})$ of normalized local mean oscillation densities. For $p\ge2$, uniform convergence over $\|g\|_\infty\le1$ is equivalent to vanishing density mass. A continuous compactly supported counterexample for $p>2n$ shows that bounded density mass and convergence for $g=\bar f$ do not imply convergence for every fixed bounded $g$. The symbol classes are weight-independent unital algebras, yielding limits for finite Toeplitz products and commutators; quantitative Hölder estimates also give finite-rank approximation bounds. The Gaussian case transfers to anti-Wick quantization by unitary equivalence. On compact Kähler manifolds with positive line bundles, we prove a Hölder sufficient condition, including the endpoint, for each fixed bounded second symbol. A projective-line example shows that trace norm convergence can fail for a smooth first symbol and a fixed bounded second symbol. The proofs combine local holomorphic approximation, Schatten and Sobolev estimates, singular integral methods, and the Dunford--Pettis criterion.