A note on maximal regularity in relation with measure theory
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Keywords:
Approximation techniques, Non-autonomous equations, Maximal regularity, SemigroupsAbstract
We present a synthetic method based on double approximation (multilevel approach) to state the maximal regularity of nonautonomous evolution equations, of non-autonomous evolution problems, mainly those driven by closed operators arising from sesquilinear forms, which enjoy some analytic properties. The infinite product of semigroups and elementary results, mainly some classical remarkable sets, in measure theory will play a central role in technical calculus.
References
P. Acquistapace and Brunello Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rendiconti del seminario matematico della Università di Padova 78 (1987), 47–107.
M. Achache and El Maati Ouhabaz. Lions’ maximal regularity problem with h12-regularity in time. Journal of Differential Equations, 266(6):3654–3678, 2019.
M. Achache and El Maati Ouhabaz. Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces Arabian Journal of Mathematics, 11, pages 539–547 (2022).
H. Amann, Maximal regularity for nonautonomous evolution equations, Advanced Nonlinear Studies 4 (2004), no. 4, 417–430.
Amansag et al. Staffans-Weiss perturbations for maximal Lp-regularity in Banach spaces.Jou.Evol. Equ.(JEE), vol. 22. n. 15, (2022).
Akira Ichikawa, Stability of semilinear stochastic evolution equations, Journal of Mathematical Analysis and Applications 90 (1982), no. 1, 12–44.
W. Arendt, Ralph Chill, Simona Fornaro, and César Poupaud, Lp-maximal regularity for non-autonomous evolution equations, Journal of Differential Equations 237 (2007), no. 1, 1–26.
W. Arendt, D. Dier and E. M. Ouhabaz. Invariance of convex sets for non-autonomous evolution equations governed by forms. J. Lond. Math. Soc., II. Ser. 89, No. 3 (2014) 903–916. Zbl 1295.35285, MR3217655.
A. Balmaseda et.al. On a sharper bound on the stability of non-autonomous SchrÃűdinger equations and applications to quantum control.J. Funct. Anal 287 (110163).
B-Eberhardt and G-Greiner, Baillon’s theorem on maximal regularity, Positive Operators and Semigroups on Banach Lattices, Springer, 1992, pp. 47–54.
O. El-Mennaoui, V. Keyantuo, H. Laasri. Infinitesimal product of semigroups. Ulmer Seminare. Heft 16 (2011), 219–230.
O. Elmennaoui , Keyantuo, V. and Sani, A. Fractional integration of imaginary order in vector-valued Holder spaces. Arch. Math. 120, 493–506 (2023). https://doi.org/10.1007/s00013-023-01841-6
S. Fackler, The kalton-lancien theorem revisited: maximal regularity does not extrapolate, Journal of Functional Analysis 266 no. 1, 121–138. (2014).
Falconer, Kenneth, The Geometry of Fractal Sets, Cambridge University Press (1985).
B. Haak, Duc-Trung Hoang, and El-Maati Ouhabaz, Controllability and observability for non-autonomous evolution equations: the averaged hautustest, Systems and Control Letters 133 (2019), 104524.
Hagen Neidhardt, Artur Stephan, and Valentin A Zagrebnov, Trotter product formula and linear evolution equations on hilbert spaces, Analysis and Operator Theory, Springer, 2019, pp. 271–299.
C. Hilmi and A. Sani Integral product: definition and stability PMJ. Vol 12(1), (2023).
Iqra Altaf et.al. Hausdorff dimension of Besicovitch sets of Cantor graphs. Mathematica (2024). Available at https://doi.org/10.1112/mtk.12241
N. J. Kalton and P. Portal. Remarks on l1 and l∞ maximal regularity for power-bounded operators. J. Aust. Math. Soc.84 (2008), 345–365. doi:10.1017/S1446788708000712
N J Kalton and Gilles Lancien, A solution to the problem of lp− maximal regularity, Mathematische Zeitschrift 235, no. 3, (2000) 559–568.
H. Laasri. (2012). Problémes d’évolution et intégrales produits dans les espaces de Banach. These de Doctorat, Faculté des sciences, Agadir.
J.L. Lions. Equations Différentielles Opérationnelles et Problèmes aux Limites.
Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961. Zbl 0098.31101, MR0153974
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin, 1983. Zbl 0516.47023, MR0710486.
SG Pyatkov, On the maximal regularity property for evolution equations., Azerbaijan Journal of Mathematics 9 (2019), no. 1.
J. Prüss and Roland Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, Journal of mathematical analysis and applications 256 (2001), no. 2, 405–430.
A. Slavick. Product Integration Its History And Applications. Matfyzpress, Praha, (2007). Zbl 1216.28001, MR2917851.
Salvatore Federico, Giorgio Ferrari, Frank Riedel, and Michael Röckner, On a class of infinite-dimensional singular stochastic control problems, arXiv preprint arXiv:1904.11450 (2019).
Sani, A. (2014). Intégrale produit et régularité maximale du problème de Cauchy non autonome (Doctoral dissertation, Thèse de Doctorat, Faculté des science Agadir).
Schmidt, E, Georg. On multiplicative Lebesgue integration and families of evolution operators. Mathematica Scandinavica, 1971, vol. 29, no 1, p. 113–133.
A, Sani and H, Laasri, Evolution equations governed by lipschitz continuous nonautonomous forms, Czechoslovak Mathematical Journal 65 (2015), no. 2, 475–491.
H. Tanabe. Equations of Evolution. Pitman 1979. Zbl 0417.35003, MR0533824.
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