25.1.9 Bounded products and nonlinear capacityE66D40EC5904

The recursive definition of products of currents $\mathrm{dd}^{\mathrm{c}}u$ with $u$ locally bounded and plurisubharmonic, the Chern–Levine–Nirenberg estimates and continuity along decreasing sequences are developed in [Chapter III, Section 3, (3.1)–(3.11), pp. 144–150, DemBook]; see theorem 7.2.5, theorem 7.3.7. In our proof of decreasing continuity, the integration by parts formula lemma 7.3.3 for functions that coincide near the boundary of a ball is combined with an induction on the number of factors, which also gives the symmetry of the product at each stage. The weighted estimate theorem 7.4.1 concerns only the measures $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \beta ^{n-q}$; a version relative to a closed positive current, for one factor at a time, is lemma 13.4.1.

For continuity along increasing sequences, the behavior of products under maxima and the comparison principle (theorem 7.6.1, proposition 7.3.10, theorem 7.3.11) compare [Theorem 3.23, pp. 89–91, and Theorems 3.27 and 3.29, pp. 93–94, GZ17]. The domination principle corollary 7.3.12 is [Proposition 1.2.10, XiaSampleBook]; we prove it by a strictly plurisubharmonic quadratic perturbation and the comparison principle, applied on the given bounded open set itself.

The Monge–Ampère capacity, quasi-continuity and convergence in capacity (definition 7.4.2, theorem 7.5.2, theorem 7.5.8) appear in [Proposition 4.19, Theorem 4.20 and Lemma 4.21, pp. 110–112, and Theorem 4.26, pp. 114–115, GZ17]. Our capacity is defined with competitors taking values between zero and one, whereas the competitors in [Definition 4.16, p. 108, GZ17] take values between $-1$ and $0$; since adding a constant does not change Monge–Ampère measures, the two normalizations give the same capacity. Both Demailly's book and [GZ17] use the operator $\mathrm{dd^ c}=(\mathrm{i}/\pi )\partial \bar\partial $, twice ours, so their Monge–Ampère measures and capacities are $2^ n$ times ours.

The estimate of proposition 7.4.5 is attributed to Błocki in [Proposition 3.21, pp. 88–89, GZ17]. The power $M^{n+1}$ of the oscillation printed there should be $M^ n$: the proof uses $n$ integrations by parts, and the left-hand side is homogeneous of degree $n$ in $v$. We prove the estimate with $M^ n$, under the assumption that $u=h$ near the boundary of the ball, first for smooth functions and then, by regularization and corollary 7.3.8, for bounded ones. Corollary 7.6.2 shows that the set where an upper envelope differs from its regularization has capacity zero; this is used in section 5.47 to show that such sets are pluripolar.

The solution of the homogeneous Dirichlet problem on the ball, theorem 7.7.4, goes back to [BT76]. Averaging over automorphisms that move the center exactly to a prescribed point is carried out for polydiscs in [Section 2.3, Theorem 2.3.2 and Proposition 2.3.3, pp. 50–52, Blo02Pluripotential]; lemma 7.7.1 provides the corresponding automorphisms of the ball. Compare also [Theorems 5.13–5.14, pp. 143–149, GZ17]. In Step 3 of the proof of theorem 7.7.4, the determinant of the complex Hessian of the solution is shown to vanish almost everywhere by comparing the solution with its second-order Taylor polynomial at points where this expansion holds. For functions with bounded real Hessian, the expansion holds almost everywhere by proposition 2.8.5, which is proved with the differentiation of integrals and the Poincaré inequality on cubes. The homogeneous balayage of corollary 7.7.5, together with the comparison principle, shows that a locally bounded plurisubharmonic function is maximal if and only if its Monge–Ampère measure vanishes (theorem 7.7.7); in particular, maximality is a local property for locally bounded functions.

The characterization of the inequality $(\mathrm{dd}^{\mathrm{c}}u)^ n\geq f\eta ^ n$, where $\eta =\mathrm{dd}^{\mathrm{c}}|z|^2$, by upper test functions and the linearization of the determinant (proposition 7.8.3) are proved in [Proposition 1.5 and Lemma 1.7, pp. 5–7, EGZ10ViscosityV1], the first arXiv version of the later article by Eyssidieux, Guedj and Zeriahi. The passage from upper test functions to distributional inequalities, which the proof in that preprint quotes from Hörmander, is lemma 7.8.2. The trace matrices used there have determinant $n^{-n}$; multiplying them by $n$ gives our matrices of determinant one.

The inequality for mixed products of smooth functions and its extension to bounded functions whose Monge–Ampère measures have densities with respect to Lebesgue measure are stated in [Theorems 1.1–1.2, pp. 1–2, Din07MixedV1], the first arXiv version of Dinew's article; the extension is attributed there to Kołodziej and Błocki. We prove the case of continuous densities, corollary 7.8.4, by linearization and decreasing convolution, and the case of a common measure in corollary 7.12.6. At fixed potentials, the smooth measure displayed by Dinew is $(4\pi )^ n$ times ours.

The Dirichlet problem on a ball with continuous density, theorem 7.9.2, follows the argument of [Theorems 5.13–5.14 and Corollary 5.15, pp. 143–149, GZ17]. Since the automorphisms of lemma 7.7.1 move the center exactly to a prescribed point, the preliminary Lipschitz comparison with translations used there is not needed. Under such an automorphism $S$ the $n$-th root of the density transforms into $(f^{1/n}\circ S)|\det _{\mathbb C}DS|^{2/n}$; on p. 145 this factor is written first without the absolute value and later without the derivative. The symmetric error is quadratic, and the half-sum is taken only once; this corrects the formulas printed on pp. 145–146. The stability estimate (7.9.3) contains a term for the boundary data, which is omitted on p. 149, and its proof does not require monotone approximation of the boundary data and of the density. Densities are taken with respect to $\eta ^ n=(\mathrm{dd}^{\mathrm{c}}|z|^2)^ n$; at fixed potentials, the operator of [GZ17] gives $2^ n$ times our measures.

The extension to bounded densities (lemma 7.10.1, theorem 7.10.2) corresponds to [Theorem 5.17 and Lemma 5.18, pp. 150–151, GZ17]. In lemma 7.10.1 the comparison is made on strict sublevel sets, and only the shifted sublevel set is required to be relatively compact. In the proof of theorem 7.10.2 the approximating densities have an explicit common bound. Convergence in capacity of the approximating solutions is proved without assuming that the solution for a bounded measurable density is continuous inside the ball, by applying Hartogs' lemma on compact sets outside the exceptional sets of quasi-continuity. Approximating the $n$-th roots of the densities by convolution extends the stability estimate (7.9.3) to bounded densities.

The class $\mathcal E_0(B)$ of definition 7.11.1 is due to Cegrell: its definition on hyperconvex domains is recalled in [Section 3, p. 161, Ceg04], where the class is said to have been introduced in [Ceg98], and it is also recalled in [Section 2, p. 3, Din07MixedV1]. The moment condition used in Dinew's article to approximate a measure by its averages over cubes is stated in Theorem 2.2, pp. 4–5, whose conclusions on approximation and stability are quoted from other sources. Theorem 7.11.2 establishes the moment condition for measures dominated by the Monge–Ampère measure of a bounded plurisubharmonic function; it follows from Błocki's estimate proposition 7.4.5 by integration over level sets, after truncating the function with zero boundary values and passing to a smaller ball. The constants depend explicitly on the mass of the measure and on the oscillation of the subsolution, and the estimate also holds in dimension one.

By proposition 7.11.5, which combines this estimate with the local $L^ p$ bound proposition 5.10.8, the cube averages satisfy the moment condition uniformly. Applied to the solutions with zero boundary values, this gives uniform bounds for their local energy moments, and applied to the measure $\eta ^ n$, a constant multiple of Lebesgue measure, it gives compactness in $L^1$ along subsequences. At fixed potentials, Dinew's operator multiplies measures and local energy moments by $(4\pi )^ n$; because of the power of the mass on its right-hand side, both sides of (7.11.4) are multiplied by the same factor.

The convergence of the energy integrals of the approximating solutions is asserted in [Theorem 2.2(3), p. 5, Din07MixedV1], where it is quoted from Cegrell. Theorem 7.11.11 proves it for compactly supported measures dominated by the Monge–Ampère measure of a bounded plurisubharmonic function, along every subsequence of solutions that converges in $L^1$. One ingredient is the continuity of integrals against a fixed measure that does not charge sets of capacity zero, lemma 7.11.9.

Convergence in capacity, as a consequence of the convergence of the energy integrals, is quoted in [Theorem 2.3, p. 5, Din07MixedV1]. In theorem 7.11.24 we derive it from the comparison of capacities of strict sublevel sets (lemma 7.10.1) and the bound on second moments. By lemma 7.11.19, truncation preserves the total Monge–Ampère mass of a function in $\mathcal E_0(B)$; the lemma also bounds the change of the measure caused by truncation in terms of the energy, and shows that Monge–Ampère measures with bounded energy are uniformly absolutely continuous with respect to capacity.

The definition of the solution as an envelope of subsolutions is recalled in [Definition 2.4 and the following paragraph, p. 5, Din07MixedV1]. We solve the Dirichlet problem with zero boundary values, and identify the solution with this envelope, for compactly supported measures on a ball that are dominated by the Monge–Ampère measure of a bounded plurisubharmonic function (theorem 7.11.30).

The mixed energy inequalities used in the approximation argument are recorded in [Section 2, p. 4, Din07MixedV1] and attributed there to Cegrell. For their proof by integration by parts and the Cauchy–Schwarz inequality compare [Lemma 5.4 and Theorem 5.5, pp. 168–170, Ceg04]; Theorem 5.5 there has a fixed weight and exponents $1/n$. Theorem 7.11.33 is the version with $n+1$ functions, with an explicit but weaker constant. If $\mathrm{dd}^{\mathrm{c}}$ is multiplied by a constant $c>0$, both sides of (7.11.34) are multiplied by $c^ n$. Corollary 7.11.38 uses this bound to discard the sets where the functions are very negative before continuity in capacity is applied; the limit functions there are locally bounded.

Dinew's inequality for bounded functions and a common measure is [Theorem 3.3, p. 9, Din07MixedV1]. Its proof by approximation (Proposition 3.1 and Lemma 3.2 there, pp. 7–9) relies on results on the local Dirichlet problem, energies and continuity in capacity that are quoted from other sources. Theorem 7.12.1 allows $n$ different functions; its proof uses the same approximation by cube averages, with these inputs proved in the preceding sections. The inequality for mixed discriminants, lemma 7.8.6, follows from the reversed Cauchy–Schwarz inequality given by lemma 5.3.4, and linearization of the determinant then gives the case of continuous densities, corollary 7.8.8. In this bounded setting the common measure may charge pluripolar sets. The extension to unbounded functions in Cegrell's classes, [Theorem 3.4, Din07MixedV1], is not treated here.

The computation on contact sets in [Lemma 2.4.7, Step 1, XiaSampleBook] uses Stampacchia's theorem that weak derivatives vanish almost everywhere on level sets, but it also asserts that a function in $W^{2,1}$ has a classical real Hessian. A complete argument is given in [Lemma 3.1 and Theorem 3.2, pp. 4–8, DNT2019v1] (first arXiv version). There $C^{1,\overline1}$ means that the Laplacian with respect to a metric is bounded, not that the full real Hessian is bounded, and elliptic $W^{2,p}$ regularity is used.

In this book the vanishing of weak derivatives on level sets is lemma 7.13.1, proved with a smooth truncated chain rule, and functions with bounded Euclidean Laplacian have real $W^{2,2}$ derivatives by lemma 7.13.2. With these lemmas we prove proposition 7.13.3, proposition 7.13.4 for regular potentials and obstacles in $W^{2,1}_{\mathrm{loc}}$; the density on the contact set does not require the determinant of the obstacle to be integrable on the whole domain. The operator used in [DNT2019v1] is $\mathrm{dd^ c}=\mathrm i\partial \bar\partial $, which is $2\pi $ times ours.