A right inverse of Cauchy–Riemann operator $\protect \bar{\partial }^k+a$ in the weighted Hilbert space $L^2(\protect \mathbb{C},e^{-|z|^2})$

Shaoyu Dai; Yifei Pan

doi:10.5802/afst.1686

A right inverse of Cauchy–Riemann operator

{\bar{\partial}}^{k} + a

in the weighted Hilbert space

L^{2} (ℂ, e^{- {| z |}^{2}})

Shaoyu Dai ¹; Yifei Pan ²

¹ Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China
² Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne 46805-1499, USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 619-632.

Abstract
Abstract

Using Hörmander $L^{2}$ method for Cauchy–Riemann equations from complex analysis, we study a simple differential operator ${\bar{\partial}}^{k} + a$ 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.

Nous utilisons la méthode des estimées $L^{2}$ de Hörmander pour les équations de Cauchy–Riemann pour étudier un opérateur différentiel simple ${\bar{\partial}}^{k} + a$ de tout ordre (fermé et densément défini) dans l’espace de Hilbert à poids $L^{2} (ℂ, e^{- {| z |}^{2}})$ . Nous montrons l’existence d’un inverse à droite qui est borné.

Received: 2019-07-11
Accepted: 2019-10-02
Published online: 2021-10-26

DOI: 10.5802/afst.1686

Author's affiliations:

Shaoyu Dai ¹; Yifei Pan ²

¹ Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China
² Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne 46805-1499, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Shaoyu Dai and Yifei Pan},
     title = {A right inverse of {Cauchy{\textendash}Riemann} operator $\protect \bar{\partial }^k+a$ in the weighted {Hilbert} space $L^2(\protect \mathbb{C},e^{-|z|^2})$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Shaoyu Dai; Yifei Pan. A right inverse of Cauchy–Riemann operator $\protect \bar{\partial }^k+a$ in the weighted Hilbert space $L^2(\protect \mathbb{C},e^{-|z|^2})$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 619-632. doi : 10.5802/afst.1686. https://afst.centre-mersenne.org/articles/10.5802/afst.1686/

References
Cited by

[1] Haakan Hedenmalm On Hörmander’s solution of the $\bar{\partial}$ -equation. I, Math. Z., Volume 281 (2015) no. 1-2, pp. 349-355 | DOI | MR | Zbl

[2] Lars Hörmander $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[3] Lars Hörmander An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7, North-Holland, 1990, 92 pages | MR | Zbl

[4] Lars Hörmander Notions of convexity, Progress in Mathematics, 127, Birkhäuser, 1994, 257 pages | MR | Zbl

[5] Warren P. Johnson The curious history of Faà di Bruno’s formula, Am. Math. Mon., Volume 109 (2002) no. 3, pp. 217-234 | MR | Zbl

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