25.7 Envelopes and comparisonCA53357795CC

CA53357795CC

Narzillaev studies weighted extremal functions of a compact set $K\subset \mathbb C^ n$ with a bounded weight $\psi $ on $K$ and logarithmic growth $\delta \log ^+\lVert z\rVert $, $\delta >0$. The concavity in the growth parameter and the alternatives for regular contact in theorem 16.28.14 are stated in [v1, Lemma 3.1 and Theorem 1.1, p. 6, NarzillaevWeighted26]; the proof given here also shows directly, by scaling, that the set of parameters at which the unregularized envelope attains the obstacle on $K$ is empty or a ray. Lemma 16.28.9, Proposition 16.28.10 derive the correspondence with compact envelopes on projective space, the finiteness of the envelope and its almost-everywhere order from the compactness and extension results proved earlier in the book. In the book's convention, the growth $\delta \log ^+\lVert z\rVert $ corresponds to the projective background $(\delta /2)\omega _{\mathrm{FS}}$ and to the obstacle $\psi -\delta \log (1+\lVert z\rVert ^2)/2$; Narzillaev's operator $\mathrm{dd}^{\mathrm{c}}=\mathrm i\partial \bar\partial /2$ is $\pi $ times the book's. None of the theorems of that preprint on products, plurithinness or weighted counterexamples is used in these proofs.

For the zero obstacle, corollary 16.28.13 gives the case of compact sets of the classical fact, recalled in [v1, Section 2, equation (4), p. 4, NarzillaevWeighted26], that removing a pluripolar set does not change the extremal function. For bounded weights, the proof uses the logarithmic pole barrier of lemma 16.28.12. Proposition 5.40.6 carries out the corresponding argument for compact sets in an arbitrary smooth big background, using the correction of pointwise competitors in lemma 5.40.5.

The estimates for increments in the growth parameter in lemma 16.28.18 are [v1, Lemma 3.1, p. 6, NarzillaevWeighted26], and at a point of regular contact proposition 16.28.20, corollary 16.28.24 recover [v1, Proposition 3.2, p. 7, NarzillaevWeighted26]. The proof given here keeps the estimates for the initial and the later slopes at every point of $\mathbb C^ n$ and every positive growth parameter, including the uniform error for large increments. The representation by probability measures and the attained derivative in theorem 16.28.27, theorem 16.28.29 come from [v1, Theorems 3.3–3.4, pp. 8–9, NarzillaevWeighted26]; the same argument with the finite envelope extends them to every point and every positive parameter. The passage to weak limits of these probability measures uses the continuity of the weight. The joint response to changes of the weight and of the growth, and the criteria for differentiability, in lemma 16.28.33, proposition 16.28.35, corollary 16.28.37, corollary 16.28.39, follow directly from the attained representation and the compactness of its set of active pairs. Example 16.28.40 gives an explicit corner in the dependence on the weight, on the closed unit disk.

The relative extremal function of definition 5.27.4 follows the convention for compact sets in [v1, Section 2, equation (8), p. 5, NarzillaevWeighted26]. Lemma 16.28.43, which shows that the relative extremal function equals $-1$ at a point exactly when the global extremal function vanishes there, is proved by an explicit logarithmic gluing and the pole barrier above. Lemma 16.28.45, Theorem 16.28.46 give [v1, Lemma 4.1 and Theorem 4.2, p. 10, NarzillaevWeighted26], including the local strictness hypothesis $\mathrm{dd}^{\mathrm{c}}\Psi \geq c\omega _0$ near the point, with $\omega _0=\mathrm{dd}^{\mathrm{c}}\lVert z\rVert ^2$, a global weight $\Psi $ in the given growth class extending the weight on $K$, and the closed compact pieces inside the ball. Narzillaev's operator and Euclidean form are both $\pi $ times those of the book, so the coefficient $c$ and the bound for the defect are unchanged. The pointwise consequence for the Hessian in corollary 16.28.48 is the observation following that theorem on p. 10.

The planar disks and the quantitative barrier of lemma 16.28.50 come from [v1, Lemma 6.1, pp. 12–13, NarzillaevWeighted26]. Proposition 16.28.51 proves part of the product construction in [v1, Lemma 6.2, pp. 13–14, NarzillaevWeighted26], namely the contact defect along the exceptional fiber and the failure of plurithinness at the interior point specified there. The logarithmic series is constructed by convergence of canonical representatives on compact sets, and an explicit pole barrier produces competitors that satisfy the obstacle everywhere. The proof needs neither a general product formula nor a classification at all boundary points.

The formula on normal fibers for the pluricomplex Green function with a divisorial pole, proposition 5.40.10, is the case of the unit ball of Dao, Nguyen and Tran's [v1, Proposition 4.2, p. 25, DNTGreen26], with logarithms normalized as $\log |\zeta |^2$ in the normal coordinate $\zeta $; the proof uses the maximum principle in one variable. The bounded minorants of proposition 5.40.12 converge at every interior point, including the points of the divisor, with an explicit error bound. This is a special model for the approximation by bounded envelopes in [Theorem 1.7, pp. 5–6, DNTGreen26], whose general result assumes an inward deformation in a sector and uses B-regularity for continuity. The calculation of boundary cluster values in example 5.40.15 concerns the trace of the divisor on the boundary, whereas that theorem asserts continuity inside the domain.

Contact sets connect the order of potentials with the order of their Monge–Ampère measures. For semipositive big backgrounds, theorem 16.28.5 is due to Alehyane, Lu and Salouf [Theorem 2.2, ALS26]; the passage from bounded to unbounded potentials in the book removes the bigness assumption and a uniform mass assumption. For a general big background, Boucksom, Guedj and Lu [Theorem 2.9 and Proposition 2.11, BGL25] prove that the measure of the envelope is concentrated on the contact set under bounded mass, and local replacement together with a strict singular barrier gives theorem 15.8.6 without that assumption.

The measure estimate for envelope fixed points, proposition 16.28.7, is [Proposition 2.12, ALS26]. Let $\theta $ be smooth and semipositive, and let $\varphi \in \mathrm{PSH}(X,\theta )$ satisfy $P_\theta [\varphi ]=\varphi $ for the ordinary envelope of definition 5.40.4. The proof given here uses the identity $P_\theta (\varphi +1,0)=\varphi $, where $P_\theta (\varphi +1,0)$ is the largest $\theta $-plurisubharmonic function lying below $\min (\varphi +1,0)$ (section 5.40); contact with the first obstacle occurs only on the pluripolar set $\{ \varphi =-\infty \} $, so only contact with zero contributes to the measure. For bounded potentials, the proof of the domination principle by envelopes, theorem 16.28.56, is due to Guedj and Lu [Proposition 2.8, GL22].

Difference envelopes give a relative class in which singular potentials can still be compared. The construction appears for semipositive big backgrounds in [Proposition 2.4 and Lemma 2.5, ALS26] and for arbitrary big backgrounds in Xuan Li's [Section 2.1, XL26]; the estimate for sublevel sets shows that the envelope is not identically $-\infty $ as soon as its family of competitors is nonempty. The relative envelope class of definition 16.28.61 follows [Definition 3.2, ALS26], and the proof of its closure under convex stretching, lemma 16.28.63, is the argument of [Proposition 3.5, ALS26].

The condition (16.28.67) on the reference and its consequence, positivity of the mass of each individual potential in the class, appear in [Proposition 2.10 and the discussion after Definition 2.13, ALS26]; that paper does not assume bounded mass for its equations with $L^ p$ data in a fixed ordinary type. This positivity for individual potentials must be distinguished from a positive infimum of the mass over the whole ordinary singularity type. The mechanism giving the lower bound for the mass in theorem 16.28.70 is that of [Proposition 3.8, ALS26].

For lower semicontinuous obstacles the contact argument requires an approximation. The proof of proposition 16.28.3 follows Guedj and Lu [Theorem 2.3, GL22] and uses the Bedford–Taylor convergence theorem for increasing bounded sequences [Theorem 7.4, BT82]; smooth local corrections and an expansion against the remaining background extend it to nonclosed backgrounds.

The modified comparison principle, theorem 14.19.6, is due to Kołodziej and Nguyen [Theorem 0.2, KN15]. The proof given here follows the finite induction of Guedj and Lu [Theorem 1.5, GL22], with larger dimensional constants.