Abstract
In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an Lp(Lq)-theory for such equations for p,q∈(1,∞). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.
Original language | English |
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Pages (from-to) | 1-41 |
Number of pages | 41 |
Journal | Potential Analysis |
DOIs | |
Publication status | Published - 2016 |
Keywords
- Singular integrals
- Maximal Lp-regularity
- Evolution equations
- Functional calculus
- Elliptic operators
- Ap-weights
- R-boundedness
- Extrapolation
- Quasi-linear PDE