Some weighted estimates for the ¯∂-equation and a finite rank theorem for Toeplitz operators in the Fock space
We consider the ¯∂-equation in C1 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 ¯∂-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 δ-distributions and their derivatives. The latter result generalizes the recent theorem on finite rank Toeplitz operators with symbols–functions.