An explicit counterexample for the $ {L}^{p}$-maximal regularity problem

Stephan Fackler

doi:10.1016/j.crma.2013.01.013

Partial Differential Equations/Functional Analysis

An explicit counterexample for the

L^{p}

-maximal regularity problem
[Un contre-exemple explicite pour le problème de la régularité maximale

L^{p}

]

Présenté par : Gilles Pisier

Stephan Fackler ¹

¹ Institute of Applied Analysis, University of Ulm, Helmholtzstraße 18, 89069 Ulm, Germany

Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 53-56.

Résumé
Abstract

Dans cette Note, nous démontrons lʼexistence dʼune famille de semi-groupes holomorphes $(T_{p} {(t))}_{t ⩾ 0}$ telle que $(T_{p} {(t))}_{t ⩾ 0}$ nʼa pas la régularité maximale pour $p > 2$ . De cette façon, nous répondons négativement à la question ouverte qui consiste à savoir si la régularité maximale extrapole entre $L^{2}$ et $L^{p}$ .

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups $(T_{p} {(t))}_{t ⩾ 0}$ such that $(T_{p} {(t))}_{t ⩾ 0}$ does not have maximal regularity for $p > 2$ . This answers negatively the open question whether maximal regularity extrapolates from $L^{2}$ to the $L^{p}$ -scale.

Reçu le : 2012-12-14
Accepté le : 2013-01-29
Publié le : 2013-01-01

DOI : 10.1016/j.crma.2013.01.013

Affiliations des auteurs :

Stephan Fackler ¹

¹ Institute of Applied Analysis, University of Ulm, Helmholtzstraße 18, 89069 Ulm, Germany

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Stephan Fackler. An explicit counterexample for the $ {L}^{p}$-maximal regularity problem. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 53-56. doi : 10.1016/j.crma.2013.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.013/

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