A right inverse of Cauchy–Riemann operator ∂ ¯ k +a in the weighted Hilbert space L 2 (ℂ,e -|z| 2  )

Shaoyu Dai,Yifei Pan

doi:10.5802/afst.1686

Abstract

Using Hörmander L 2 method for Cauchy–Riemann equations from complex analysis, we study a simple differential operator ∂ ¯ k +a of any order (densely defined and closed) in the weighted Hilbert space L 2 (ℂ,e -|z| 2 ) and prove the existence of a right inverse that is bounded.

Highlights

In this paper, using Hörmander L2 method [2] for Cauchy–Riemann equations from complex analysis, we study the right inverse of the differential operator ∂k + a, which is densely defined and closed, in a Hilbert space by proving the following result on the existence of weak solutions of the equation ∂ku + au = f in the weighted Hilbert space L2(C, e−|z|2 )
The novelty of Theorem 1.1 is that the differential operator ∂k + a has a bounded right inverse
Remark 4.7. — It would be a natural question whether other weights would work by Hörmander L2 method, but so far we don’t know how to do

Summary

Introduction

In this paper, using Hörmander L2 method [2] for Cauchy–Riemann equations from complex analysis, we study the right inverse of the differential operator ∂k + a, which is densely defined and closed, in a Hilbert space by proving the following result on the existence of (entire) weak solutions of the equation ∂ku + au = f in the weighted Hilbert space L2(C, e−|z|2 ). The novelty of Theorem 1.1 is that the differential operator ∂k + a has a bounded (linear) right inverse. 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 ([3] and [4]) (a = 0; see [1] for a related result).

Several lemmas

Proof of theorems

Further remarks