Parabolic Potential Theory (Continued)
If Ḋ is a nonempty open subset of ℝ̇ N and if Γ is a class of functions on Ḋ, the greatest subparabolic minorant [least superparabolic majorant] of Γ, if there is one, is denoted by ĠMḊΓ [ĿMḊΓ]. For example, if Γ is a class of superparabolic functions and
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Parabolic Potential Theory (Continued)
1. Greatest Minorants and Least Majorants If D is a nonempty open subset of ~N and if r is a class of functions on D, the greatest subparabolic minorant [least superparabolic majorant] of r, if there is one, is denoted by GMDr [LMDr]. For example, if r is a class of superparabolic functions and if r has a subparabolic minorant then GMDr exists and is parabolic. The proof is a translation of that of Theorem 11I.2. The corresponding notation in the coparabolic context is GMDr and
LMvr.
EXAMPLE.
Let D be either ~N (N;?: I) or an interval in ~N. Then GMVGD(·,~)
= GMDGD(~'·) = 0 for every point ~ in D. In fact, say for GMVGD(·'~) when D is an interval, the parabolic minorant in question is positive, is majorized by GD(·, ~), and so has limit 0 at every lateral and lower boundary point of D. This minorant therefore vanishes identically, according to the
parabolic function maximum-minimum theorem. More generally it will follow from the Riesz decomposition of a positive superparabolic function on a nonempty open subset D of ~N that the parabolic potential of a measure on D if finite on a dense subset of Dis superparabolic on D and has greatest subparabolic minorant O.
2. The Parabolic Fundamental Convergence Theorem (Preliminary Version) and the Reduction Operation The proofof the following counterpart ofthe first version of the Fundamental Convergence Theorem (Theorem 111.3) in the classical context follows the proof of Theorem 111.3 and is therefore omitted. Theorem. Let r: {ua , ct E I} be a family of superparabolic functions on an open subset of ~N, locally uniformly bounded below, and define u(~) = infa~l ua(~)' Then ~ :s; u, . (2.1) J.L. Doob, Classical Potential Theory and Its Probably Counter © Springer-Verlag Berlin Heidelberg 2001
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I. XVII. Parabolic Potential Theory (Continued)
and
(a) (b) (c) (d)
Uis superparabolic. + ~ = Uon each open set on which is superparabolic. ~ = u IN+I almost everywhere. There is a countable subfamily ofr whose infimum has smoothing U.
u
+
Application: The Natural Order Decomposition Theorem As application of this simple version of the Fundamental Convergence Theorem in the parabolic context we remark that the classical context Natural Order Decomposition Theorem (Theorem III.7) translates directly into the parabolic context: If U, uI , u2 are positive superparabolic functions on D with U :s; ul + u2 , then there are positive superparabolic functions u~, Uz on D for which u~ :s; UI ' Uz :s; U2' U= U~ + uz. The classical context proof requires only trivial changes. Observe that this decomposition and its proof are also valid for relative superharmonic and superparabolic functions. Alternatively the decomposition theorem for superharmonic and superparabolic functions implies trivially the decomposition theorem in the relative contexts.
3. The Parabolic Context Reduction Operations If D is a nonempty open subset of ~N coupled with a boundary oD provided by a metric compactification, if A c D U oD, and if v is a positive superparab
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