Area integral estimates for caloric functions

Russell M. Brown

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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 languageEnglish
Pages (from-to)565-589
Number of pages25
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - Oct 1989


  • Boundary behavior
  • Heat equation
  • Nonsmooth domains

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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