06.10.02 · riemann-surfaces / several-variables

Plurisubharmonic functions

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Anchor (Master): Oka 1942 / Lelong 1945 (originators); Krantz Ch. 2; Hörmander §2.6; Range Ch. III

Intuition Beginner

In one complex variable the natural "energy-like" functions are the subharmonic ones: real-valued functions whose value at the centre of any small disc never exceeds their average around the rim. The modulus $lo g ∣ f ∣$ of a holomorphic function is the model example. Subharmonic functions are exactly the right test class for the geometry of regions in the plane.

In several complex variables the plane is replaced by complex space, and the right test class has to respect every complex direction at once, not just one. A plurisubharmonic function is one that stays subharmonic when you slice it along any single complex line. Slide a complex line through the region in any direction; restrict the function to that line; it must look subharmonic on that slice.

That one demand, repeated over all directions, captures the convexity that several-variable function theory actually needs. These functions are the analytic certificate that a region is well-behaved, and they are the bridge from the holomorphic-convexity picture of the previous unit to the geometry of pseudoconvex domains.

Visual Beginner

Picture a bowl-shaped surface sitting over a region of complex space. A convex bowl curves upward in every straight direction. A plurisubharmonic surface is weaker: it only has to curve upward (in the averaged, subharmonic sense) along complex lines — the special two-real-dimensional planes that respect the complex structure. Along other real directions it may dip.

The picture below shows a region with a family of complex lines threading through it. On each line the restricted function is drawn as a small subharmonic profile: the value at the centre of any disc sits at or below the rim average. When every slice passes this test, the whole function is plurisubharmonic.

The most important such surface is the height $- lo g d (z)$ , where $d (z)$ measures the distance from a point to the boundary wall of the region. As a point nears the wall, this height climbs to infinity, forming a barrier that hugs the boundary. When that barrier is plurisubharmonic, the region has no leak.

Worked example Beginner

Take the function $u (z) = ∣ z ∣^{2} = ∣ z_{1} ∣^{2} + \dots + ∣ z_{n} ∣^{2}$ on complex $n$ -space. Is it plurisubharmonic?

Slice it along a complex line. A complex line through a point $a$ in direction $b$ is the set of points $a + λb$ as $λ$ runs over the complex plane. Plug in: the restricted function becomes $∣ a + λb ∣^{2}$ , which expands to $∣ a ∣^{2} + 2 Re (⟨ a, b ⟩ \overset{ˉ}{λ}) + ∣ λ ∣^{2} ∣ b ∣^{2}$ . As a function of the single complex variable $λ$ , this is a constant plus a harmonic piece plus $∣ b ∣^{2} ∣ λ ∣^{2}$ .

The first two pieces are harmonic, and $∣ λ ∣^{2}$ is the standard subharmonic bump in one variable: its value at the centre of any disc is below the rim average. So the slice is subharmonic on every line, in every direction $b$ . Therefore $∣ z ∣^{2}$ is plurisubharmonic.

Now contrast with a function that fails. The real part $Re (z_{1} \overset{z}{ˉ}_{2})$ slices to a harmonic profile in some directions but bends the wrong way in others; averaged over all complex directions it has no definite sign, so it is not plurisubharmonic. The lesson: plurisubharmonicity is a demand made simultaneously in every complex direction, and a function can satisfy it in some slices while failing the test overall.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $Ω \subseteq C^{n}$ is open. Recall from 06.01.24 that an upper semicontinuous function $v : U \to [- \infty, \infty)$ on an open $U \subseteq C$ , not identically $- \infty$ on any component, is subharmonic if for every closed disc $\overline{D} (a, r) \subseteq U$ one has the sub-mean-value inequality $$ v(a) ;\le; \frac{1}{2\pi} \int_0^{2\pi} v(a + r e^{i\theta}), d\theta. $$

Definition (plurisubharmonic function). A function $u : Ω \to [- \infty, \infty)$ is plurisubharmonic (PSH) if

$u$ is upper semicontinuous, and
for every $a \in Ω$ and every $b \in C^{n}$ , the slice $λ ⟼ u (a + λb), λ \in {λ \in C : a + λb \in Ω},$ is subharmonic (or identically $- \infty$ ) on each component of its domain.

Write $u \in PSH (Ω)$ . We exclude $u \equiv - \infty$ on a component when convenient; the cone $PSH (Ω)$ is closed under addition, multiplication by nonnegative constants, and the pointwise maximum of finitely many members.

The Levi form. When $u \in C^{2} (Ω)$ , the slice condition has an infinitesimal form. The complex Hessian (or Levi form) of $u$ at $z$ applied to $w \in C^{n}$ is the Hermitian form $$ L_u(z; w) ;=; \sum_{j,k=1}^n \frac{\partial^2 u}{\partial z_j , \partial \bar z_k}(z), w_j \bar w_k . $$ A direct computation (carried out in the Key theorem) shows that for the slice $g (λ) = u (a + λb)$ one has $\frac{1}{4} Δ g (λ) = L_{u} (a + λb; b)$ , where $Δ$ is the real Laplacian in $λ$ . Hence a $C^{2}$ function is plurisubharmonic exactly when $L_{u} (z; w) \geq 0$ for all $z \in Ω$ and all $w \in C^{n}$ , i.e. the complex Hessian is positive semidefinite at every point.

Strict plurisubharmonicity. $u \in C^{2} (Ω)$ is strictly plurisubharmonic if $L_{u} (z; w) > 0$ for all $z$ and all $w \neq = 0$ — the complex Hessian is positive definite. The function $∣ z ∣^{2}$ has $L_{∣ z ∣^{2}} (z; w) = ∣ w ∣^{2}$ , so it is strictly plurisubharmonic everywhere.

Exhaustion functions. A function $φ : Ω \to R$ is an exhaustion of $Ω$ if the sublevel sets $Ω_{c} = {z \in Ω : φ (z) < c}$ are relatively compact in $Ω$ for every $c \in R$ . A PSH exhaustion is one that is also plurisubharmonic. The existence of a continuous PSH exhaustion is the analytic definition of pseudoconvexity, developed in 06.10.03.

Counterexamples to common slips

Plurisubharmonicity is strictly stronger than ordinary subharmonicity as a function on the underlying real $2 n$ -space. Subharmonicity tests the full real Laplacian $\sum_{j} \partial^{2} / \partial z_{j} \partial \overset{z}{ˉ}_{j}$ (the trace of $L_{u}$ ); plurisubharmonicity tests every diagonal entry $L_{u} (z; w)$ , a stronger demand. A function can be subharmonic in $R^{2 n}$ yet fail to be PSH.
The slice direction $b$ ranges over all of $C^{n}$ , so it includes complex multiples; testing only the $n$ coordinate axes is insufficient. The form $L_{u}$ must be positive semidefinite, not merely have nonnegative diagonal in a fixed frame.
PSH functions are not closed under multiplication by negative constants: $- u$ is PSH only when $u$ is pluriharmonic (both $u$ and $- u$ PSH), which forces $L_{u} \equiv 0$ and means $u$ is locally the real part of a holomorphic function.

Key theorem with proof Intermediate+

Theorem (characterisations of plurisubharmonicity). Let $Ω \subseteq C^{n}$ be open and $u : Ω \to [- \infty, \infty)$ upper semicontinuous, not identically $- \infty$ on any component. The following hold.

(a) If $u \in C^{2} (Ω)$ , then $u$ is plurisubharmonic if and only if the Levi form $L_{u} (z; w) \geq 0$ for all $z \in Ω$ , $w \in C^{n}$ .

(b) (Regularisation) If $u$ is plurisubharmonic, then on $Ω_{ε} = {z : d (z, \partial Ω) > ε}$ the convolution $u_{ε} = u * χ_{ε}$ with a radially symmetric mollifier $χ_{ε}$ is smooth, plurisubharmonic, and decreases to $u$ pointwise as $ε ↓ 0$ .

(c) (Maximum principle) If $u$ is plurisubharmonic on a connected $Ω$ and attains a global maximum at an interior point, then $u$ is constant.

The pivot is the computation relating the real Laplacian of a slice to the complex Hessian ^{[Krantz Ch. 2]}.

Lemma (Laplacian of a slice). For $u \in C^{2} (Ω)$ , $a \in Ω$ , $b \in C^{n}$ , set $g (λ) = u (a + λb)$ for $λ$ near $0$ . Then $\frac{1}{4} Δ_{λ} g (0) = L_{u} (a; b)$ , where $Δ_{λ} = 4 \partial^{2} / \partial λ \partial \overset{ˉ}{λ}$ is the Laplacian in the single variable $λ$ .

Proof of Lemma. Write $z = a + λb$ , so $z_{j} = a_{j} + λ b_{j}$ and $\partial z_{j} / \partial λ = b_{j}$ , $\partial \overset{z}{ˉ}_{k} / \partial \overset{ˉ}{λ} = \overset{ˉ}{b}_{k}$ , while $\partial z_{j} / \partial \overset{ˉ}{λ} = 0$ since each $z_{j}$ is holomorphic in $λ$ . By the chain rule, $$ \frac{\partial g}{\partial\lambda} = \sum_j \frac{\partial u}{\partial z_j}, b_j, \qquad \frac{\partial^2 g}{\partial\lambda,\partial\bar\lambda} = \sum_{j,k} \frac{\partial^2 u}{\partial z_j,\partial\bar z_k}, b_j \bar b_k = L_u(z; b), $$ using that the holomorphic derivatives $\partial / \partial z_{j}$ commute past the $b_{j}$ and the mixed second derivative picks out exactly the $\partial z_{j} \partial \overset{z}{ˉ}_{k}$ block. Multiplying by $4$ gives $Δ_{λ} g = 4 L_{u} (z; b)$ , and at $λ = 0$ this reads $\frac{1}{4} Δ_{λ} g (0) = L_{u} (a; b)$ . $□$

Proof of Theorem.

(a). Suppose $u \in C^{2}$ and $L_{u} \geq 0$ . Fix $a, b$ ; the slice $g (λ) = u (a + λb)$ has $Δ_{λ} g = 4 L_{u} (a + λb; b) \geq 0$ , so $g$ is subharmonic in one variable (a $C^{2}$ function is subharmonic exactly when its Laplacian is nonnegative, by 06.01.24). Hence $u$ is plurisubharmonic. Conversely, if $u$ is plurisubharmonic then each slice is subharmonic, so $Δ_{λ} g (0) \geq 0$ , which by the Lemma gives $L_{u} (a; b) \geq 0$ . As $a, b$ were arbitrary, $L_{u} \geq 0$ .

(b). The mollifier $χ_{ε} (z) = ε^{- 2 n} χ (z / ε)$ is radial, nonnegative, with $\int χ_{ε} = 1$ . Smoothness of $u_{ε}$ on $Ω_{ε}$ is standard. For plurisubharmonicity, restrict to a complex line: $u_{ε} (a + λb) = \int u (a + λb - ζ) χ_{ε} (ζ) d V (ζ)$ , an average of translates of the subharmonic slice $λ \mapsto u (a + λb - ζ)$ ; averages of subharmonic functions against a nonnegative weight are subharmonic, so each slice of $u_{ε}$ is subharmonic and $u_{ε} \in PSH$ . The sub-mean-value inequality applied on spheres, together with radial symmetry of $χ_{ε}$ , makes $ε \mapsto u_{ε} (z)$ monotone decreasing with limit $u (z)$ as $ε ↓ 0$ (upper semicontinuity supplies $lim sup u_{ε} \leq u$ ; the sub-mean-value inequality supplies $u_{ε} \geq u$ ).

(c). Let $M = sup_{Ω} u$ be attained at an interior $z_{0}$ , and let $E = {z : u (z) = M}$ . Upper semicontinuity makes $E$ closed in $Ω$ . To see $E$ is open, take $z_{1} \in E$ and a small ball $B (z_{1}, r) \subseteq Ω$ . For any direction $b$ the slice $g (λ) = u (z_{1} + λb)$ is subharmonic with $g (0) = M = max g$ ; the one-variable maximum principle from 06.01.24 forces $g \equiv M$ on a disc, so $u = M$ on a neighbourhood of $z_{1}$ in the $b$ -line. Sweeping $b$ over all directions covers $B (z_{1}, r)$ , so $E$ is open. By connectedness $E = Ω$ and $u \equiv M$ . $□$

Bridge. This characterisation builds toward the pseudoconvexity theory of 06.10.03 and appears again in the $L^{2}$ existence theorem for $\overset{ˉ}{\partial}$ , where the Levi form of the weight is exactly what makes the Hörmander estimate close. The foundational reason the Levi form governs everything is the slice Lemma: it identifies the one-variable Laplacian of a complex slice with the complex Hessian, so the global condition $L_{u} \geq 0$ is precisely the infinitesimal shadow of subharmonicity-on-every-line. This is exactly the convexity that holomorphic functions respect, and it generalises the one-variable picture where the single quantity $\partial^{2} / \partial z \partial \overset{z}{ˉ}$ already carries all of subharmonicity. Putting these together, plurisubharmonicity is dual to holomorphy in the sense that $lo g ∣ f ∣$ is the universal PSH function while $∣ z ∣^{2}$ is the universal strictly PSH one; the central insight is that the bridge is the boundary-distance height $- lo g d (z, \partial Ω)$ , whose plurisubharmonicity will be shown to identify pseudoconvexity with the holomorphic convexity of 06.10.01.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Let $f \in O (Ω)$ be holomorphic and not identically $0$ on any component. Show that $lo g ∣ f ∣$ and, for every $p > 0$ , $∣ f ∣^{p}$ are plurisubharmonic.

Hint

Restrict to a complex line and use the one-variable facts: $lo g$ of the modulus of a one-variable holomorphic function is subharmonic, and a convex increasing function of a subharmonic function is subharmonic.

Answer

On any complex line the restriction $f (a + λb)$ is a one-variable holomorphic function $h (λ)$ . By the one-variable theory, $lo g ∣ h ∣$ is subharmonic (harmonic off the zeros, with $- \infty$ logarithmic poles at zeros, and the sub-mean-value inequality at zeros holds since the value $- \infty$ is below any average). Thus $lo g ∣ f ∣$ slices to subharmonic and is PSH. For $∣ f ∣^{p} = exp (p lo g ∣ f ∣)$ , the map $t \mapsto e^{pt}$ is convex and increasing, and a convex increasing function composed with a subharmonic function is subharmonic; so each slice of $∣ f ∣^{p}$ is subharmonic and $∣ f ∣^{p} \in PSH (Ω)$ . Rubric: full credit for the slice reduction plus the convex-increasing composition.

Exercise 4 (medium, short-answer).

Show that a $C^{2}$ function $u (z) = u (∣ z_{1} ∣, \dots, ∣ z_{n} ∣)$ depending only on the moduli $r_{j} = ∣ z_{j} ∣$ , with $u = ϕ (lo g r_{1}, \dots, lo g r_{n})$ for some $ϕ \in C^{2} (R^{n})$ , is plurisubharmonic on $(C^{*})^{n}$ if and only if $ϕ$ is convex.

Hint

In the coordinates $t_{j} = lo g ∣ z_{j} ∣$ , the Levi form of such a "Reinhardt" function is a constant multiple of the real Hessian of $ϕ$ .

Answer

Write $t_{j} = lo g ∣ z_{j} ∣ = \frac{1}{2} lo g (z_{j} \overset{z}{ˉ}_{j})$ , so $\partial t_{j} / \partial z_{k} = δ_{j k} / (2 z_{j})$ and $\partial t_{j} / \partial \overset{z}{ˉ}_{k} = δ_{j k} / (2 \overset{z}{ˉ}_{j})$ . Then $\partial^{2} u / \partial z_{j} \partial \overset{z}{ˉ}_{k} = \frac{1}{4} (z_{j} \overset{z}{ˉ}_{k})^{- 1} \partial^{2} ϕ / \partial t_{j} \partial t_{k}$ (the first-order terms cancel because $\partial^{2} t_{j} / \partial z_{j} \partial \overset{z}{ˉ}_{j}$ contributes only to the diagonal of a vanishing gradient piece when $ϕ$ is evaluated, and the surviving block is the Hessian of $ϕ$ ). So $L_{u} (z; w) = \frac{1}{4} \sum_{j, k} (Hess ϕ)_{j k} (w_{j} / z_{j}) \overline{(w_{k} / z_{k})}$ , which is the Hessian of $ϕ$ evaluated on the rescaled vector $(w_{j} / z_{j})_{j}$ . As the rescaling is an isomorphism of $C^{n}$ , $L_{u} \geq 0$ for all $w$ iff $Hess ϕ \geq 0$ , i.e. $ϕ$ convex. Rubric: full credit for the change of variables and the equivalence $L_{u} \geq 0 ⟺ Hess ϕ \geq 0$ .

Exercise 5 (medium, short-answer).

Let ${u_{j}}$ be a sequence of plurisubharmonic functions decreasing pointwise to a function $u$ with $u > - \infty$ on a dense set. Show $u$ is plurisubharmonic.

Hint

A decreasing limit of upper semicontinuous functions is upper semicontinuous, and a decreasing limit of subharmonic functions on a line is subharmonic.

Answer

Each $u_{j}$ is upper semicontinuous; a decreasing limit of upper semicontinuous functions is upper semicontinuous, so $u$ is. On a complex line, $u_{j} (a + λb)$ is subharmonic and decreases to $u (a + λb)$ ; the sub-mean-value inequality passes to the decreasing limit by the monotone convergence theorem applied to the rim integrals (each side decreases, and the inequality $u_{j} (a) \leq avg u_{j}$ survives in the limit). Since $u$ is not identically $- \infty$ (it is finite on a dense set), each slice is subharmonic and $u \in PSH$ . Rubric: full credit for upper semicontinuity of the limit plus the monotone passage of the sub-mean-value inequality.

Exercise 6 (hard, short-answer).

Prove the gluing lemma: if $u$ is PSH on $Ω$ , $v$ is PSH on an open $ω \subseteq Ω$ , and $lim sup_{z \to ζ} v (z) \leq u (ζ)$ for every $ζ \in Ω \cap \partial ω$ , then $$ w = \begin{cases}\max(u, v) & \text{on } \omega,\ u & \text{on } \Omega\setminus\omega\end{cases} $$ is plurisubharmonic on $Ω$ .

Hint

The boundary inequality is exactly what makes $w$ upper semicontinuous across $\partial ω$ ; away from $\partial ω$ , $w$ is locally a max of PSH functions or equals $u$ .

Answer

On $ω$ , $w = max (u, v)$ is PSH (Exercise 2). On the interior of $Ω ∖ ω$ , $w = u$ is PSH. The work is at points $ζ \in Ω \cap \partial ω$ . Upper semicontinuity: for $z \to ζ$ with $z \in ω$ , $lim sup w (z) = lim sup max (u, v) (z) \leq max (u (ζ), lim sup v) \leq u (ζ) = w (ζ)$ using the hypothesis; for $z \to ζ$ outside $ω$ , $lim sup w = lim sup u \leq u (ζ)$ . So $w$ is upper semicontinuous at $ζ$ . For the sub-mean-value inequality on a complex line through $ζ$ , note $w = max (u, \tilde{v})$ where $\tilde{v} = v$ on $ω$ and $\tilde{v} = - \infty$ on $Ω ∖ \overline{ω}$ ; the boundary hypothesis makes $\tilde{v}$ upper semicontinuous and subharmonic on each slice (an extension of a subharmonic function by $- \infty$ that respects the boundary $lim sup$ is subharmonic), so $w$ slices to a max of two subharmonic functions, hence subharmonic. Therefore $w \in PSH (Ω)$ . This gluing construction is the engine for building local PSH peak functions in the Levi problem 06.10.05. Rubric: full credit for the boundary upper-semicontinuity check and the slice subharmonicity of the $- \infty$ -extension.

Exercise 7 (hard, short-answer).

Let $Ω \subseteq C^{n}$ be a bounded domain and $φ (z) = - lo g d (z, \partial Ω) + ∣ z ∣^{2}$ . Show that $φ$ is a continuous exhaustion of $Ω$ , and identify precisely the condition on $Ω$ under which $φ$ is plurisubharmonic.

Hint

The $∣ z ∣^{2}$ term and boundedness force the sublevel sets compact; plurisubharmonicity of $- lo g d (\cdot, \partial Ω)$ is the analytic form of pseudoconvexity.

Answer

Continuity: $d (\cdot, \partial Ω)$ is continuous and positive on $Ω$ , so $- lo g d$ is continuous; adding $∣ z ∣^{2}$ keeps continuity. Exhaustion: a sublevel set ${φ < c}$ forces both $- lo g d (z, \partial Ω) < c$ (so $d (z, \partial Ω) > e^{- c}$ , keeping $z$ away from the boundary) and $∣ z ∣^{2} < c$ (bounding $z$ , automatic here since $Ω$ is bounded); the set is thus closed in $C^{n}$ , bounded, and at positive distance from $\partial Ω$ , hence relatively compact in $Ω$ . Plurisubharmonicity: $∣ z ∣^{2}$ is strictly PSH, so $φ$ is PSH if and only if $- lo g d (z, \partial Ω)$ is PSH. This last condition is precisely Hartogs pseudoconvexity of $Ω$ , and is equivalent (for domains with $C^{2}$ boundary) to Levi pseudoconvexity, as developed in 06.10.03. Rubric: full credit for the exhaustion argument and the identification "PSH $⟺$ $- lo g d$ PSH $⟺$ pseudoconvex".

Advanced results Master

Plurisubharmonic functions are the analytic backbone of several-variable complex analysis. The results below sharpen the cone-theoretic and potential-theoretic structure and connect it to the holomorphic convexity of 06.10.01 and the pseudoconvexity of 06.10.03.

Composition and convexity. If $u_{1}, \dots, u_{m} \in PSH (Ω)$ and $χ : R^{m} \to R$ is convex and increasing in each variable, then $χ (u_{1}, \dots, u_{m}) \in PSH (Ω)$ . The proof reduces to a complex line, where it is the one-variable fact that a convex increasing function of subharmonic functions is subharmonic, combined with the regularisation of the Key theorem to handle non-smooth $u_{j}$ . Two consequences organise the examples: $∣ f ∣^{p} = (e^{l o g ∣ f ∣})^{p}$ is PSH for any holomorphic $f$ and any $p > 0$ (Exercise 3), and $lo g (∣ f_{1} ∣^{2} + \dots + ∣ f_{k} ∣^{2})$ is PSH for holomorphic $f_{1}, \dots, f_{k}$ , since it is $lo g$ of a sum of squared moduli and $lo g \sum e^{2 t_{j}}$ is convex increasing.

Potential-theoretic regularity. A plurisubharmonic function is locally integrable (it cannot equal $- \infty$ on a set of positive measure unless identically $- \infty$ on a component) and defines a positive distribution $i \partial \overset{ˉ}{\partial} u \geq 0$ — a closed positive $(1, 1)$ -current. This is the entry point to pluripotential theory: the complex Monge–Ampère operator $(i \partial \overset{ˉ}{\partial} u)^{n}$ , defined for bounded PSH $u$ by Bedford–Taylor, measures the "total Levi mass" and is the several-variable analogue of the Laplacian-as-measure $Δ u$ of one-variable potential theory. The Riesz representation of a subharmonic function as a logarithmic potential generalises to the Lelong–Poincaré formula $i \partial \overset{ˉ}{\partial} lo g ∣ f ∣ = 2 π [Z_{f}]$ , identifying the Levi mass of $lo g ∣ f ∣$ with the integration current over the zero divisor of $f$ .

The boundary-distance function. For $Ω \subseteq C^{n}$ open, set $δ (z) = d (z, \partial Ω)$ and $φ = - lo g δ$ . The single most important theorem about PSH functions in this chapter is that $φ$ is plurisubharmonic if and only if $Ω$ is pseudoconvex, and this in turn is equivalent to $Ω$ admitting some continuous PSH exhaustion. The "if and only if" is the Hartogs–Oka–Lelong characterisation; one direction (PSH exhaustion $\Rightarrow$ pseudoconvex) is direct, the other (Levi pseudoconvex $\Rightarrow$ $- lo g δ$ PSH) uses the Levi form of a defining function. This is the precise sense in which PSH functions certify the geometry that 06.10.01 characterised function-theoretically.

Approximation from above. Every PSH function is a decreasing limit of smooth PSH functions on relatively compact subdomains (the regularisation of the Key theorem), and on a pseudoconvex $Ω$ this can be globalised: every continuous PSH function is a decreasing limit of smooth strictly PSH functions on all of $Ω$ . This approximation is what lets the Hörmander $L^{2}$ method of 06.10.04 assume its weight $φ$ is smooth and strictly PSH without loss, since the estimates pass to the decreasing limit.

Synthesis. Plurisubharmonicity is the foundational reason several-variable complex analysis has a working potential theory at all: the slice-and-Levi-form definition identifies a global convexity-type condition with the positivity of a single Hermitian form, and this is exactly the data that the $\overset{ˉ}{\partial}$ operator sees through its weights. Putting these together, three pictures coincide — the function-theoretic (holomorphic convexity, 06.10.01), the geometric (Levi pseudoconvexity of the boundary, 06.10.03), and the potential-theoretic (existence of a PSH exhaustion) — and the central insight is that the boundary-distance height $- lo g d (z, \partial Ω)$ is the universal certificate linking all three. The cone $PSH (Ω)$ is dual to the holomorphic functions in the sense that $lo g ∣ f ∣$ generates it while $∣ z ∣^{2}$ supplies strict positivity, and the bridge from these examples to the global theory is the Bedford–Taylor Monge–Ampère current $(i \partial \overset{ˉ}{\partial} u)^{n}$ , which generalises the one-variable Riesz mass and feeds the Lelong–Poincaré identification of Levi mass with zero divisors. This pattern recurs throughout the chapter: every later theorem — the $L^{2}$ estimate, the Levi-problem solution, the kernel constructions — runs on a plurisubharmonic weight whose Levi form is the hypothesis that makes the analysis close.

Full proof set Master

Proposition 1 (sub-mean-value over spheres). Let $u \in PSH (Ω)$ and $\overline{B} (a, r) \subseteq Ω$ . Then $$ u(a) ;\le; \fint_{\partial B(a, r)} u, d\sigma ;\le; \fint_{B(a,r)} u, dV, $$ where $\fint$ denotes the average and $d σ$ the surface measure on the sphere.

Proof. It suffices to treat smooth $u$ ; the general case follows by the decreasing regularisation $u_{ε} ↓ u$ of the Key theorem and monotone convergence. For smooth $u$ , the Levi form $L_{u} \geq 0$ implies the real Laplacian $Δ u = 4 \sum_{j} \partial^{2} u / \partial z_{j} \partial \overset{z}{ˉ}_{j} = 4 tr L_{u} \geq 0$ , so $u$ is subharmonic as a function on $R^{2 n}$ . Green's identity on the ball gives, for the spherical average $A (r) = \fint_{\partial B (a, r)} u d σ$ , $$ A'(r) = \frac{1}{\sigma_{2n-1} r^{2n-1}} \int_{B(a,r)} \Delta u, dV \ge 0, $$ so $A (r)$ is nondecreasing in $r$ with $A (0^{+}) = u (a)$ , giving $u (a) \leq A (r)$ . Integrating the spherical averages in $r$ against the radial measure yields the solid average bound $A (r) \leq \fint_{B (a, r)} u d V$ from the same monotonicity. $□$

Proposition 2 (PSH is biholomorphically invariant). Let $F : Ω^{'} \to Ω$ be holomorphic and $u \in PSH (Ω)$ . Then $u \circ F \in PSH (Ω^{'})$ . In particular plurisubharmonicity is preserved by biholomorphisms.

Proof. Upper semicontinuity of $u \circ F$ follows from continuity of $F$ . For the slice condition, fix $a^{'} \in Ω^{'}$ , $b^{'} \in C^{m}$ (with $Ω^{'} \subseteq C^{m}$ ) and consider $λ \mapsto (u \circ F) (a^{'} + λ b^{'}) = u (F (a^{'} + λ b^{'}))$ . The map $λ \mapsto F (a^{'} + λ b^{'})$ is holomorphic from a disc in $C$ into $Ω \subseteq C^{n}$ , so the composite $u \circ (holomorphic disc)$ is the restriction of $u$ to a holomorphic image of a disc. By the case of smooth $u$ and the chain-rule computation $\partial^{2} (u \circ F) / \partial λ \partial \overset{ˉ}{λ} = \sum_{j, k} (\partial^{2} u / \partial z_{j} \partial \overset{z}{ˉ}_{k}) (\partial F_{j} / \partial λ) \overline{(\partial F_{k} / \partial λ)} = L_{u} (F; F^{'} (λ) b^{'}) \geq 0$ , the slice is subharmonic; the general $u$ follows by regularisation. Hence $u \circ F$ is PSH. Since holomorphy of $F$ — not merely smoothness — is used (the $\partial F_{j} / \partial \overset{ˉ}{λ}$ terms must vanish), this is a genuinely complex-analytic invariance with no real-variable analogue. $□$

Proposition 3 (Hartogs lemma on upper limits). Let ${u_{j}} \subseteq PSH (Ω)$ be locally uniformly bounded above, and suppose $lim sup_{j} u_{j} (z) \leq C$ for every $z$ in $Ω$ . Then for every compact $K \subseteq Ω$ and every $ε > 0$ there is $J$ with $u_{j} (z) \leq C + ε$ for all $z \in K$ and $j \geq J$ .

Proof. Fix $K$ and $ε$ . By Proposition 1, for any ball $\overline{B} (a, r) \subseteq Ω$ , $$ u_j(a) \le \fint_{B(a,r)} u_j, dV. $$ The hypothesis $lim sup_{j} u_{j} \leq C$ together with the uniform upper bound and Fatou's lemma (applied to $M - u_{j}$ for an upper bound $M$ ) gives $lim sup_{j} \fint_{B (a, r)} u_{j} d V \leq \fint_{B (a, r)} (lim sup_{j} u_{j}) d V \leq C$ . Choose $r$ so that $\overline{B} (a, r) \subseteq Ω$ uniformly for $a \in K$ (possible by compactness, with $r < d (K, \partial Ω)$ ). Then for $a \in K$ , $lim sup_{j} u_{j} (a) \leq lim sup_{j} \fint_{B (a, r)} u_{j} d V \leq C$ . A compactness/equicontinuity argument on the solid averages — they are continuous in $a$ and converge uniformly on $K$ by the dominated bound — upgrades the pointwise $lim sup$ to the uniform statement: there is $J$ with $\fint_{B (a, r)} u_{j} d V \leq C + ε$ for all $a \in K$ , $j \geq J$ , hence $u_{j} (a) \leq C + ε$ on $K$ . $□$

Proposition 4 (maximum on the hull). Let $Ω \subseteq C^{n}$ be a domain of holomorphy, $K ⋐ Ω$ compact, and $\hat{K}_{Ω}$ its holomorphically convex hull from 06.10.01. Then for every $u \in PSH (Ω)$ that is continuous, $$ \sup_{\hat K_\Omega} u ;=; \sup_K u. $$

Proof. The inequality $sup_{K} u \leq sup_{\hat{K}_{Ω}} u$ holds since $K \subseteq \hat{K}_{Ω}$ . For the reverse, first suppose $u$ has the special form $u = lo g ∣ f ∣$ with $f \in O (Ω)$ . By definition of the hull, every $z \in \hat{K}_{Ω}$ satisfies $∣ f (z) ∣ \leq sup_{K} ∣ f ∣$ , hence $lo g ∣ f (z) ∣ \leq sup_{K} lo g ∣ f ∣$ , giving $sup_{\hat{K}_{Ω}} lo g ∣ f ∣ \leq sup_{K} lo g ∣ f ∣$ . For a general continuous PSH $u$ , approximate: on a domain of holomorphy a continuous PSH function is a decreasing limit of functions of the form $max_{i} (lo g ∣ f_{i} ∣ + c_{i})$ with $f_{i} \in O (Ω)$ (a standard consequence of the holomorphic-convexity machinery, since the maximum of such log-moduli is dense from above in the PSH cone on $Ω$ ). Each approximant obeys the hull identity by the special case and max-stability, and the identity passes to the decreasing limit. Therefore $sup_{\hat{K}_{Ω}} u = sup_{K} u$ . This identifies the holomorphically convex hull with the plurisubharmonic hull — the points where every PSH function is controlled by its values on $K$ — closing the loop with 06.10.01. $□$

Connections Master

Domains of holomorphy and holomorphic convexity 06.10.01. Proposition 4 identifies the holomorphically convex hull $\hat{K}_{Ω}$ with the plurisubharmonic hull: on a domain of holomorphy a continuous PSH function attains its supremum over the hull already on $K$ . Plurisubharmonic functions are the analytic certificate of the no-leak condition that 06.10.01 characterised through holomorphic convexity, and $- lo g d (z, \partial Ω)$ is the universal exhaustion linking the two.
Dirichlet problem and the Perron method 06.01.24. Plurisubharmonicity is the several-variable refinement of the subharmonicity that drives the Perron method in one variable. The sub-mean-value inequality, the maximum principle, and upper semicontinuity are inherited slice-by-slice from the one-variable theory developed there; the Perron upper-envelope construction has its pluripotential analogue in the Bedford–Taylor solution of the complex Monge–Ampère Dirichlet problem.
Holomorphic functions of several variables 06.07.01. The universal examples $lo g ∣ f ∣$ and $∣ f ∣^{p}$ are built from the holomorphic functions of that unit, and the slice computation that proves their plurisubharmonicity uses the polydisc Cauchy theory there. The Levi form is the second-order shadow of holomorphy: it vanishes on real parts of holomorphic functions and is positive on $∣ z ∣^{2}$ .
Pseudoconvexity and the Levi form 06.10.03. This unit supplies the function-class — PSH functions and their exhaustions — in terms of which pseudoconvexity is defined. The equivalence " $- lo g d (z, \partial Ω)$ PSH $⟺$ Levi pseudoconvex $⟺$ admits a PSH exhaustion" is the bridge built there on the foundation laid here.
The $\overset{ˉ}{\partial}$ problem with $L^{2}$ estimates 06.10.04 (when it ships). The Hörmander weighted $L^{2}$ existence theorem runs on a strictly plurisubharmonic weight $φ$ , and the positivity of its Levi form is exactly the term that makes the Bochner–Kodaira–Morrey–Kohn identity yield an a-priori estimate. The smooth-strictly-PSH approximation proved here is what licenses assuming the weight regular.

Historical & philosophical context Master

The class of plurisubharmonic functions was introduced and named in 1942–1945 independently by Kiyoshi Oka and Pierre Lelong, who recognised that the one-variable notion of subharmonicity had to be replaced by a condition tested along every complex line. Lelong's systematic development appeared in Les fonctions plurisousharmoniques ^{[Lelong 1945]} (Annales Scientifiques de l'École Normale Supérieure (3) 62, 301–338), which gave the upper-semicontinuous, slice-subharmonic definition, proved the regularisation and maximum-principle theorems, and isolated $- lo g d (z, \partial Ω)$ as the canonical example. Oka had arrived at the same class — under the name fonctions pseudoconvexes — in his sixth memoir ^{[Oka 1942]} (Tôhoku Mathematical Journal 49, 15–52), where the functions were the technical instrument for attacking the Levi problem; the convergence of the two viewpoints is why the subject's central equivalence (pseudoconvex $⟺$ PSH-exhaustible) carries both names.

The deeper motivation was the Levi problem, posed implicitly by Eugenio Elia Levi's 1910 analysis of the boundary condition a domain of holomorphy must satisfy. Levi had found a second-order positivity condition — what is now the Levi form — on the boundary of a domain of holomorphy; the question was whether this local condition was also sufficient. Oka and Lelong understood that the right global object was not the boundary form alone but a function on the whole domain whose complex Hessian carried the same positivity, and plurisubharmonic functions were exactly that object. The reframing turned a boundary-geometry question into a function-theoretic one and supplied the exhaustion functions on which Oka's 1953 solution of the Levi problem, and later Hörmander's 1965 $L^{2}$ method, would run.

Philosophically, plurisubharmonicity marks the point where several-variable theory stops imitating one-variable theory and acquires its own potential theory. Subharmonicity in one variable is governed by a single elliptic operator, the Laplacian; plurisubharmonicity is governed by a Hermitian form, the complex Hessian, whose positivity is a genuinely higher-rank condition with no one-variable shadow. The complex Monge–Ampère operator of Bedford and Taylor, the Lelong currents of integration over analytic sets, and the modern pluripotential theory of Demailly all descend from this one shift in viewpoint: that the correct test functions for $C^{n}$ are the ones subharmonic on every complex line.

Bibliography Master

@article{Lelong1945,
  author  = {Lelong, Pierre},
  title   = {Les fonctions plurisousharmoniques},
  journal = {Ann. Sci. \'Ecole Norm. Sup. (3)},
  volume  = {62},
  year    = {1945},
  pages   = {301--338}
}

@article{Oka1942,
  author  = {Oka, Kiyoshi},
  title   = {Sur les fonctions analytiques de plusieurs variables. VI.
             {Domaines} pseudoconvexes},
  journal = {T\^ohoku Math. J.},
  volume  = {49},
  year    = {1942},
  pages   = {15--52}
}

@book{KrantzSCV,
  author    = {Krantz, Steven G.},
  title     = {Function Theory of Several Complex Variables},
  edition   = {2},
  series    = {AMS Chelsea Publishing},
  volume    = {340},
  publisher = {American Mathematical Society},
  year      = {2001}
}

@book{HormanderSCV,
  author    = {H\"ormander, Lars},
  title     = {An Introduction to Complex Analysis in Several Variables},
  edition   = {3},
  series    = {North-Holland Mathematical Library},
  volume    = {7},
  publisher = {North-Holland},
  year      = {1990}
}

@book{RangeSCV,
  author    = {Range, R. Michael},
  title     = {Holomorphic Functions and Integral Representations in
               Several Complex Variables},
  series    = {Graduate Texts in Mathematics},
  volume    = {108},
  publisher = {Springer},
  year      = {1986}
}

@article{BedfordTaylor1976,
  author  = {Bedford, Eric and Taylor, B. A.},
  title   = {The {Dirichlet} problem for a complex {Monge-Amp\`ere}
             equation},
  journal = {Invent. Math.},
  volume  = {37},
  year    = {1976},
  pages   = {1--44}
}

@article{Levi1910,
  author  = {Levi, Eugenio Elia},
  title   = {Studii sui punti singolari essenziali delle funzioni
             analitiche di due o pi\`u variabili complesse},
  journal = {Ann. Mat. Pura Appl. (3)},
  volume  = {17},
  year    = {1910},
  pages   = {61--87}
}

Prerequisites

06.10.01
06.07.01
06.01.24
06.01.01

Tier anchors

beginner: Krantz *Function Theory of Several Complex Variables* Ch. 2, informal sections; Range *Holomorphic Functions and Integral Representations* Ch. III intro
intermediate: Krantz *Function Theory of Several Complex Variables* §2.6; Hörmander *An Introduction to Complex Analysis in Several Variables* §2.6
master: Oka 1942 / Lelong 1945 (originators); Krantz Ch. 2; Hörmander §2.6; Range Ch. III

References

Lelong 1945 — Les fonctions plurisousharmoniques · Ann. Sci. École Norm. Sup. (3) 62, 301–338 (originator paper naming and developing plurisubharmonic functions)
Oka 1942 — Sur les fonctions analytiques de plusieurs variables, VI: Domaines pseudoconvexes · Tôhoku Math. J. 49, 15–52 (independent originator; 'fonctions pseudoconvexes')
Krantz — Function Theory of Several Complex Variables · AMS Chelsea 340, 2nd ed., Ch. 2 §2.6 (plurisubharmonic functions and the Levi form)
Hörmander — An Introduction to Complex Analysis in Several Variables · North-Holland, 3rd ed. 1990, §2.6 (plurisubharmonic functions, pseudoconvexity)
Range — Holomorphic Functions and Integral Representations in Several Complex Variables · Springer GTM 108, Ch. III (plurisubharmonicity and pseudoconvexity)

Estimated time

beginner: 15m
intermediate: 40m
master: 66m