Some weighted estimates for the dbar- equation and a finite rank theorem for Toeplitz operators in the Fock space

TL;DR

This paper establishes weighted estimates for the dbar-equation with Gaussian decay in complex space and applies these results to prove a finite rank theorem for Toeplitz operators with distributional symbols in the Fock space.

Contribution

It introduces new weighted estimates for the dbar-equation with Gaussian decay and extends finite rank theorems for Toeplitz operators to distributional symbols.

Findings

Existence of solutions with Gaussian decay under certain orthogonality conditions.

Finite rank Toeplitz operators have symbols as finite combinations of delta distributions and derivatives.

Generalization of finite rank theorems to distributional symbols in the Fock space.

Abstract

We consider the $\dbar-$ equation in $\C^1$ in classes of functions with Gaussian decay at infinity. We prove that if the right-hand side of the equation is majorated by $\exp(-q|z|^2)$, with some positive $q$, together with derivatives up to some order, and is orthogonal, as a distribution, to all analytical polynomials, then there exists a solution with decays, together with derivatives, as $\exp(-q'|z|^2)$, for any $q'<q/e$. This result carries over to the $\dbar$-equation in classes of distributions, again, with Gaussian decay at infinity, in some precisely defined sense. The properties of the solution are used further on to prove the finite rank theorem for Toeplitz operators with distributional symbols in the Fock space: the symbol of such operator must be a combination of finitely many $\delta$-distributions and their derivatives. The latter result generalizes the recent theorem on finite rank Toeplitz operators with symbols-functions.