no. 1-2
Partial Differential Equations/Functional Analysis
An explicit counterexample for the Lp-maximal regularity problem
[Un contre-exemple explicite pour le problème de la régularité maximale Lp]
Présenté par : Gilles Pisier
Fackler, Stephan1
1 Institute of Applied Analysis, University of Ulm, Helmholtzstraße 18, 89069 Ulm, Germany
Comptes Rendus. Mathématique, Tome 351 (2013) no. 1-2, pp. 53-56

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups (Tp(t))t⩾0 such that (Tp(t))t⩾0 does not have maximal regularity for p>2. This answers negatively the open question whether maximal regularity extrapolates from L2 to the Lp-scale.

Dans cette Note, nous démontrons lʼexistence dʼune famille de semi-groupes holomorphes (Tp(t))t⩾0 telle que (Tp(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 L2 et Lp.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.01.013

Fackler, Stephan 1

1 Institute of Applied Analysis, University of Ulm, Helmholtzstraße 18, 89069 Ulm, Germany 
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Fackler, Stephan. An explicit counterexample for the $ {L}^{p}$-maximal regularity problem. Comptes Rendus. Mathématique, Tome 351 (2013) no. 1-2, pp. 53-56. doi: 10.1016/j.crma.2013.01.013

[1] Albiac, Fernando; Kalton, Nigel J. Topics in Banach Space Theory, Grad. Texts in Math., vol. 233, Springer, New York, 2006

[2] Arendt, Wolfgang Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Evolutionary Equations, vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 1-85

[3] Denk, Robert; Hieber, Matthias; Prüss, Jan R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., Volume 166 (2003) (No. 788, viii+114)

[4] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge University Press, Cambridge, 1995

[5] Dore, Giovanni Lp regularity for abstract differential equations, Kyoto, 1991 (Lecture Notes in Math.), Volume vol. 1540, Springer, Berlin (1993), pp. 25-38

[6] de Simon, Luciano Unʼapplicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Semin. Mat. Univ. Padova, Volume 34 (1964), pp. 205-223

[7] Fackler, Stephan The Kalton–Lancien theorem revisited: maximal regularity does not extrapolate, 2012 | arXiv

[8] Kalton, Nigel J.; Lancien, Gilles A solution to the problem of Lp-maximal regularity, Math. Z., Volume 235 (2000) no. 3, pp. 559-568

[9] Kunstmann, Peer C.; Weis, Lutz Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H∞-functional calculus, Functional Analytic Methods for Evolution Equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65-311

[10] Singer, Ivan Bases in Banach Spaces. I, Grundlehren Math. Wiss., vol. 154, Springer-Verlag, New York, 1970

[11] Venni, Alberto A counterexample concerning imaginary powers of linear operators, Kyoto, 1991 (Lecture Notes in Math.), Volume vol. 1540, Springer, Berlin (1993), pp. 381-387

[12] Weis, Lutz Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., Volume 319 (2001) no. 4, pp. 735-758

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