Advances in Differential Equations

Weighted norm estimates and maximal regularity

S. Blunck and P. C. Kunstmann

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Abstract

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

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

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651585

Mathematical Reviews number (MathSciNet)
MR1920543

Zentralblatt MATH identifier
1045.34031

Subjects
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]

Citation

Blunck, S.; Kunstmann, P. C. Weighted norm estimates and maximal regularity. Adv. Differential Equations 7 (2002), no. 12, 1513--1532. https://projecteuclid.org/euclid.ade/1356651585.


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