Advances in Differential Equations

Weighted norm estimates and maximal regularity

S. Blunck and P. C. Kunstmann

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We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate for the semigroup $(e^{tA})$ and it is strictly weaker than the assumption that the $e^{tA}$ are integral operators whose kernels satisfy Gaussian estimates. As an application we present new results for the maximal regularity of Schr\"odinger operators with singular potentials, elliptic higher order operators with bounded measurable coefficients, and elliptic second order operators with singular lower order terms. Moreover, we prove a similar result for maximal regularity of the discrete time evolution equation $ u_{n+1} - Tu_n = f_n ,$ $n\in\mathbb N_0 ,$ $u_0=0 $.

Article information

Adv. Differential Equations Volume 7, Number 12 (2002), 1513-1532.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34G10: Linear equations [See also 47D06, 47D09]
Secondary: 35K90: Abstract parabolic equations 42B25: Maximal functions, Littlewood-Paley theory 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Blunck, S.; Kunstmann, P. C. Weighted norm estimates and maximal regularity. Adv. Differential Equations 7 (2002), no. 12, 1513--1532.

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