25.1.8 Plurisubharmonic functions and invariant minimaE49C1A0D4682

The theorem that a function which is subharmonic or identically $-\infty $ on complex lines is automatically upper semicontinuous is proved in the unsigned manuscript [Theorem 1.1 and its proof, pp. 1–6, PSHD], and our proof of theorem 5.8.4 completes the argument given there. In that manuscript, Lemma 1.2, pp. 1–2, proves measurability by induction on the dimension, using means over bounded fibers; Proposition 2.1, pp. 4–5, identifies a finite function that is subharmonic on lines with the canonical representative; and the final proof, p. 6, uses truncation. The incidence results needed for this argument are proved in section 2.9, and proposition 5.4.4 describes the restrictions of the canonical representative to complex lines without using the theorem. We keep the usual definition of plurisubharmonic functions (definition 5.8.1), which requires upper semicontinuity; theorem 5.8.4, proved shortly afterwards, shows that this condition is redundant.

The distributional characterization, upper envelopes and compactness of plurisubharmonic functions (theorem 5.9.1, theorem 5.10.2, theorem 5.10.4) are deduced from the corresponding results for subharmonic functions in section 3.4, section 4.1. At this stage the set where an upper envelope differs from its regularization is only known to be real polar, that is, polar as a subset of $\mathbb R^{2n}$ in local charts; its pluripolarity is proved in section 5.47. The extension theorem (theorem 5.11.3) gives both local boundedness and extension across a relatively closed set that is locally $\mathcal H^{2n-3}$-null; to apply it to analytic sets of codimension at least two (corollary 9.3.18) one needs the local estimate for the Hausdorff measure of analytic sets in terms of their dimension.

For $L^ p$ integrability and convergence in $L^ p_{\mathrm{loc}}$ compare [Theorem 1.48, parts 1–2 and proof Steps 1–2, pp. 30–32, GZ17]. As there, we pass from discs to polydiscs (lemma 5.10.5, proposition 5.10.8, theorem 5.10.10), but we use the Riesz decomposition with a compactly supported measure instead of the Poisson–Jensen formula on the boundary. Since our Riesz measure is $\Delta v$, its logarithmic potential carries the factor $1/(2\pi )$. The bound (5.10.9) gives uniform estimates on smaller polydiscs, and convergence in $L^ p_{\mathrm{loc}}$ then follows from Hölder's inequality. The identity with the factor two in the truncation display of [p. 32, GZ17] is incorrect, as is also noted in an unofficial list of errata for that book.

The local exponential integrability estimates (lemma 5.10.11, proposition 5.10.12) are proved with the same compactly supported Riesz decomposition and Jensen's inequality (lemma 2.2.4), with an exponent that depends only on a local $L^1$ bound; a finite covering by charts then gives the uniform estimate for normalized quasi-plurisubharmonic functions on a compact manifold, corollary 10.3.3.

The partial Legendre transform is due to Kiselman [Kis78]. For his minimum principle and the regularization argument in its proof compare [Chapter I, Section 7.B, Theorem (7.5) and its proof, p. 57, DemBook]. In theorem 5.12.1 the existence of an invariant exhaustion is a hypothesis, and the smooth case is proved with a Schur complement. Such an exhaustion is constructed for product tubes over open convex sets in corollary 5.12.2; the case of general pseudoconvex tubes, corollary 6.3.6, uses in addition the plurisubharmonicity of $-\log $ of the boundary distance.