Parabolic Potential Theory (Continued)

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


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 \( \mathop{{\text{L}}}\limits^{*} {{{\text{M}}}_{{\dot{D}}}}\Gamma \).


Open Subset Green Function Countable Union Parabolic Function Nonempty Open Subset 
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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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