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Area integral estimates for caloric functions


Author: Russell M. Brown
Journal: Trans. Amer. Math. Soc. 315 (1989), 565-589
MSC: Primary 35K05; Secondary 42B25, 45P05
DOI: https://doi.org/10.1090/S0002-9947-1989-0994163-7
MathSciNet review: 994163
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Abstract: We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a $ \left({\frac{1}{2},1}\right)$ mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in $ {L^p}(\mu)$, $ 0 < p < \infty $. The measure $ \mu $ must satisfy Muckenhoupt's $ {A_\infty }$-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0994163-7
Keywords: Heat equation, boundary behavior, nonsmooth domains
Article copyright: © Copyright 1989 American Mathematical Society

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