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Definition and Basic Properties of Subharmonic Functions 32 Chapter I. 1. Definition. Mean Value Inequalities. If u is a Borel function on B(a, r) which is bounded above or below, we consider the mean values of u over the ball or sphere: 1 u(x) dλ(x), αm r m B(a,r) 1 u(x) dσ(x). 9) r 1 αm r m σm−1 tm−1 µS (u ; a, t) dt 0 1 tm−1 µS (u ; a, rt) dt. 5) with x = 0. We get Pr (0, y) = 1/σm−1 r m−1 and Gr (0, y) = r (|y|2−m − r 2−m )/(2 − m)σm−1 = −(1/σm−1 ) |y| t1−m dt, thus B(0,r) ∆u(y) Gr (0, y) dλ(y) = − r 1 σm−1 1 =− m 0 dt tm−1 r ∆u(y) dλ(y) |y|

B) If (Ωα )α∈I is a family of pseudoconvex open subsets of Cn , the interior of the inter◦ section Ω = is pseudoconvex. α∈I Ωα c) If (Ωj )j∈N is a non decreasing sequence of pseudoconvex open subsets of Cn , then Ω = j∈N Ωj is pseudoconvex. Proof. a) Let ϕ, ψ be smooth plurisubharmonic exhaustions of Ω, Ω′ . Then (z, w) −→ ϕ(z) + ψ(w) is an exhaustion of Ω × Ω′ and z −→ ϕ(z) + ψ(F (z)) is an exhaustion of F −1 (Ω′ ). b) We have − log d(z, ∁Ω) = supα∈I − log d(z, ∁Ωα ), so this function is plurisubharmonic.

If u ∈ C 2 (Ω, R), the subharmonicity of restrictions of u to complex lines, C ∋ w −→ u(a + wξ), a ∈ Ω, ξ ∈ Cn , is equivalent to ∂2 u(a + wξ) = ∂w∂w 1 j,k n ∂ 2u (a + wξ) ξj ξ k ∂zj ∂z k 0. Therefore, u is plurisubharmonic on Ω if and only if ∂ 2 u/∂zj ∂z k (a) ξj ξ k is a semipositive hermitian form at every point a ∈ Ω. 8) Theorem. If u ∈ Psh(Ω), u ≡ −∞ on every connected component of Ω, then for all ξ ∈ Cn ∂ 2u Hu(ξ) := ξj ξ k ∈ ′ (Ω) ∂zj ∂z k 1 j,k n is a positive measure. Conversely, if v ∈ ′ (Ω) is such that Hv(ξ) is a positive measure for every ξ ∈ Cn , there exists a unique function u ∈ Psh(Ω) locally integrable on Ω such that v is the distribution associated to u.

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Complex analytic and differential geometry by Demailly J.-P.

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