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 mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in , . The measure must satisfy Muckenhoupt's -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