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 (1/2, 1) mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in LP(μ), 0 < p < ∞. The measure μ. must satisfy Muckenhoupt’s A∞-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.
Original language | English |
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Pages (from-to) | 565-589 |
Number of pages | 25 |
Journal | Transactions of the American Mathematical Society |
Volume | 315 |
Issue number | 2 |
DOIs | |
State | Published - Oct 1989 |
Keywords
- Boundary behavior
- Heat equation
- Nonsmooth domains
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics