Remarks on the $L^p$ convergence of Bessel–Fourier series on the disc

Ryan Luis Acosta Babb

doi:10.5802/crmath.464

Harmonic analysis, Partial differential equations

Remarks on the

L^{p}

convergence of Bessel–Fourier series on the disc

Ryan Luis Acosta Babb ¹

¹ University of Warwick, United Kingdom

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1075-1080.

Abstract
Résumé

The $L^{p}$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p \neq 2$ . After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba regarding the $L^{p}$ convergence of the Bessel–Fourier series in the mixed norm space $L_{rad}^{p} (L_{ang}^{2})$ on the disk for the range $\frac{4}{3} < p < 4$ . We then describe how to modify their result to obtain $L^{p} (𝔻, r d r d t)$ norm convergence in the subspace $L_{rad}^{p} (L_{ang}^{q})$ ( $\frac{1}{p} + \frac{1}{q} = 1$ ) for the restricted range $2 \leq p < 4$ .

La convergence $L^{p}$ des développements en fonctions propres du Laplacien dans des domaines du plan est largement inconnue lorsque $p \neq 2$ . Après avoir discuté des séries de Fourier classiques sur le tore, nous passons au disque, dont les fonctions propres sont explicitement calculables comme étant le produit des fonctions trigonométriques et de Bessel. Nous résumons un résultat de Balodis et Córdoba concernant la convergence $L^{p}$ de la série de Bessel–Fourier dans l’espace de norme mixte $L_{rad}^{p} (L_{ang}^{2})$ dans le disque pour l’intervalle $\frac{4}{3} < p < 4$ . Nous décrivons ensuite comment on peut modifier leur résultat pour obtenir la convergence dans la norme $L^{p} (𝔻, r d r d t)$ dans le sous-espace $L_{rad}^{p} (L_{ang}^{q})$ ( $\frac{1}{p} + \frac{1}{q} = 1$ ) pour l’intervalle $2 \leq p < 4$ .

Received: 2022-02-25
Revised: 2022-05-10
Accepted: 2023-01-10
Published online: 2023-10-24

DOI: 10.5802/crmath.464

Classification: 42-02, 42C05, 33C10, 34L10

Author's affiliations:

Ryan Luis Acosta Babb ¹

¹ University of Warwick, United Kingdom

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Ryan Luis Acosta Babb},
     title = {Remarks on the $L^p$ convergence of {Bessel{\textendash}Fourier} series on the disc},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1075--1080},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.464},
     language = {en},
}

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Ryan Luis Acosta Babb. Remarks on the $L^p$ convergence of Bessel–Fourier series on the disc. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1075-1080. doi : 10.5802/crmath.464. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.464/

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