Distribution function inequalities for the area integral

  • Burgess Davis
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


Let A be the area integral of a function u harmonic in the Euclidean half-space R n x (0, ∞). Information about the distribution function of a localized version of A is obtained that leads to a general integral inequality between A and the nontangential maximal function of u and provides a convenient approach to the study of the pointwise behavior of u near the boundary. In addition, the general integral inequality of [2] between the nontangential maximal function of u and that of a properly chosen conjugate is shown to hold also in the case n > 1.


Harmonic Function Boundary Point Maximal Function Lipschitz Domain Measurable Subset 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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