6 research outputs found
On the -boundedness of a family of integral operators
In this paper we prove an -boundedness result for integral operators
with operator-valued kernels. The proofs are based on extrapolation techniques
with weights due to Rubio de Francia. The results will be applied by the first
and third author in a subsequent paper where a new approach to maximal
-regularity for parabolic problems with time-dependent generator is
developed.Comment: Minor revision. Accepted for publication in Rev. Mat. Iberoamerican
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Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time
In this paper we study maximal L p-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 L p-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 L p(L q)-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
Maximal regularity for non-autonomous equations with measurable dependence on time
In this paper we study maximal -regularity for evolution equations with
time-dependent operators . We merely assume a measurable dependence on time.
In the first part of the paper we present a new sufficient condition for the
-boundedness of a class of vector-valued singular integrals which does not
rely on H\"ormander 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 -th order elliptic operators
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 -theory for such equations for .
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.Comment: Application to a quasilinear equation added. Accepted for publication
in Potential Analysi