On Hörmander’s solution of the $$\bar{\partial }$$ ∂ ¯ -equation. I

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

“…We are going to prove the following statement. Hedenmalm, the latter proposed a short proof of a result, closely related to Theorem 4.1, see [8]. In fact, in [8], the solvability and weighted integral estimates for the∂-equation for functions, dual to (1.1), have been proved for a wide class of weights.…”

“…Hedenmalm, the latter proposed a short proof of a result, closely related to Theorem 4.1, see [8]. In fact, in [8], the solvability and weighted integral estimates for the∂-equation for functions, dual to (1.1), have been proved for a wide class of weights. This result does not replace our Theorem 4.1, since we need here the solvability and estimates for the∂-equation in some special classes of distributions, not only functions.…”

“…This result does not replace our Theorem 4.1, since we need here the solvability and estimates for the∂-equation in some special classes of distributions, not only functions. Probably, using the result in [8] might somewhat shorten the proof of our theorem. We, nevertheless, keep here the original proof, since, by our opinion, certain ideas and technical tricks used here might turn useful in further studies of the∂-equation.…”

A right inverse of differential operator dkdxk+a in weighted Hilbert sp

“…For the first order ∂ := ∂ 1 , the Cauchy-Riemann operator, we have the following slight extension of the simplest case of Hörmander's theorem in the complex plane ( [2] and [3]) (a = 0; see [4] for a related result). Note that ∆ = 4∂∂.…”

Serre duality and Hörmander's solution of the ∂¯-equation

“…In the case n = 1, H. Hedenmalm [3] proved the following theorem Theorem 1.1 Let ω ∈ L 2 0,1 (C, e ϕ ) then there is u ∈ L 2 (C, c ϕ e ϕ ) such that∂u = ω, and u L 2 (C,cϕe ϕ ) ≤ C ω L 2 (C,e ϕ ) , provided that ω ⊥ H 0 (C, e −ϕ ).…”