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Parabolic Potential Theory (Continued)

  • J. L. Doob
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 262)

Abstract

If \( \dot{D} \) is a nonempty open subset of \( {{\dot{\mathbb{R}}}^{N}} \) and if Г is a class of functions on \( \dot{D} \), the greatest subparabolic minorant [least superparabolic majorant] of Г, if there is one, is denoted by \( \dot{G}{{M}_{{\dot{D}}}}\Gamma \left[ {\dot{L}{{M}_{{\dot{D}}}}\Gamma } \right] \). For example, if Г is a class of superparabolic functions and if Г has a subparabolic minorant then \( \dot{G}{{M}_{{\dot{D}}}}\Gamma \) exists and is parabolic. The proof is a translation of that of Theorem III.2. The corresponding notation in the coparabolic context is \( \mathop{{\text{G}}}\limits^{*} {{{\text{M}}}_{{\dot{D}}}}\Gamma \) and \( \mathop{{\text{L}}}\limits^{*} {{{\text{M}}}_{{\dot{D}}}}\Gamma \).

Keywords

Open Subset Green Function Countable Union Parabolic Function Nonempty Open Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

  • J. L. Doob
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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