IDA symbols and Schatten--Lorentz theory for Toeplitz operators on weighted Fock spaces

RESUMO

We study complex-symbol Toeplitz operators on weighted Fock spaces. The weight has a uniformly positive and bounded real Hessian. Local analytic distance is unchanged when an entire function is added to the symbol. We combine this distance with a complex ball average or the Berezin transform to control the local size of the symbol. Under a matching Lorentz condition on the distance, we characterize membership in every Schatten--Lorentz class. The local symbol exponent can be any $1\le q<\infty$ and is independent of the two Schatten--Lorentz indices. The proof uses positive Toeplitz estimates, Hankel estimates, and a local decomposition that preserves the initial domain. Two examples show why the local exponent and the secondary Lorentz index must be kept. We also obtain mixed mapping criteria and regular singular-value decay estimates. An error operator with a lower-order counting function does not change the leading coefficient. This gives explicit Weyl constants for quadratic weights. A radial example shows that uniform Hessian bounds alone do not imply such a constant.