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On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders


Author: Kaj Nyström
Journal: Trans. Amer. Math. Soc. 360 (2008), 2987-3017
MSC (2000): Primary 35B65, 35C15, 35K20, 35K40, 35K50
DOI: https://doi.org/10.1090/S0002-9947-07-04328-0
Published electronically: December 20, 2007
MathSciNet review: 2379784
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Abstract: In this paper we prove results relating the (parabolic) non-tangential maximum operator and appropriate square functions in $ L^p$ for solutions to general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders $ \Omega$. In particular we prove a global as well as a local and scale invariant equivalence between the parabolic non-tangential maximal operator and appropriate square functions for solutions of our system. The novelty of our approach is that it is not based on singular integrals, the prevailing tool in the analysis of systems in non-smooth domains. Instead the methods explored have recently proved useful in the analysis of elliptic measure associated to non-symmetric operators through the work of Kenig-Koch-Pipher-Toro and in the analysis of caloric measure without the use of layer potentials.


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Additional Information

Kaj Nyström
Affiliation: Department of Mathematics, Umeå University, S-90187 Umeå, Sweden
Email: kaj.nystrom@math.umu.se

DOI: https://doi.org/10.1090/S0002-9947-07-04328-0
Keywords: Second order parabolic systems, time-varying cylinder, Carleson measure, maximal function, non-tangential maximal function, square function.
Received by editor(s): March 7, 2006
Published electronically: December 20, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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