25.12.2 General measures in semipositive backgrounds4D81B44E0A9B

The local decomposition of measures used in lemma 5.27.10 is Cegrell's [Theorem 6.3, Ceg98], on a bounded hyperconvex domain in dimension at least two; for a non-pluripolar measure its pluripolar part vanishes. The global constructions in lemma 15.24.4, proposition 15.24.1 patch these local representations together.

The bounded-subsolution theorem theorem 15.10.1, taken from [Theorem 5.1, ALS26], solves the exponential equation whose right-hand side has a bounded density with respect to the Monge–Ampère measure of a bounded potential. Truncating the density in such a representation of a general measure brings this theorem into the construction for general measures, and domination then allows passage to the limit. The domination theorem for finite measures, theorem 15.37.6 in section 15.37, is [Theorem 3.14, ALS26]: the fixed finite measure prevents collapse, whereas the other members of the envelope class need not have finite mass. The proof of theorem 15.37.10 also allows the measure being compared to be infinite. The singular strict domination principle, theorem 15.2.2, follows [Theorem 3.11, ALS26].

The homogeneous range result theorem 15.25.1 is the sufficient direction of Kołodziej and Nguyen's [Theorem 1.3, p. 2952; proof pp. 2958–2963, KNHolder18]. Their weighted regularization estimate is [Lemma 2.8, p. 2957, KNHolder18]. The proof here first approximates a Borel density by continuous functions and then chooses smooth references diagonally. Common Hölder bounds give uniform capacity estimates and a weighted modulus for the varying solution measures; bounds above and below for the normalization constants prevent degeneration before the uniform limit is taken.

The radial bounded product in lemma 5.27.26 is a direct Bedford–Taylor calculation: the logarithmic derivative of a smooth radial maximum fixes the spherical mass. Summing these local references with geometric weights and controlling a common cutoff gives proposition 15.26.2. The homogeneous range theorem yields corollary 15.26.3, with a normalized Hölder solution and an unknown positive coefficient. The bounded-reference theorem yields theorem 15.26.4 for bounded densities against the fixed spherical probability, including singular data on backgrounds with zero lower mass. This is an individual geometric sufficient condition, distinct from solvability for every non-pluripolar measure and from the exact density-threshold question.

Passing to the limit in the measures requires that the lower envelopes do not collapse. The continuity theorem for measures dominated by a fixed measure, theorem 15.37.19, is [Lemma 4.2 and Theorem 4.1, condition (2), ALS26]; in this case positive lower mass prevents collapse, and no upper bound on the mass is used. The argument with a common envelope preceding corollary 15.37.24 gives instead the first alternative of [Lemma 4.2, ALS26]. When the approximants have the ordinary singularity type of the reference, this alternative needs no assumption of positive lower volume.

A common barrier in the ordinary singularity type also gives corollary 15.9.4, the corresponding case of [Theorem 4.1, ALS26]. The reference need not be an envelope fixed point; the condition in a smaller background suffices. Integrals of bounded potentials against varying mixed measures are handled by the torsion calculation in lemma 5.44.7, which follows Kołodziej and Nguyen [Lemma 2.3, KN22].

The resulting equations with general finite non-pluripolar measures, theorem 15.37.4, are [Theorem 5.8 and Corollary 5.9, ALS26]. The constructions with a minimal and with a prescribed reference in subsection 15.37.27, subsection 15.37.30 use the approximations of the density from that proof, and they assume both the condition (16.28.67) in a smaller background and positive lower mass. The solutions have finite mass, but for arbitrary measures they need not differ boundedly from the reference. For a Hermitian background and the minimal reference, corollary 15.37.32 identifies the scalar in the equation with the density threshold of theorem 15.28.4, that is, the largest coefficient for which the scaled density admits a subsolution. The question that remains, 24.2.2 in section 24.1, concerns zero lower mass. The weighted addition theorem theorem 15.11.1 is [Theorem 5.2, ALS26]; the proof given here localizes the passage to the equation.