Reflection Principle for the Complex Monge–Ampère Equation and Plurisubharmonic Functions

We study reflection principle for several central objects in pluripotential theory. First we show that the odd reflected function gives an extension for pluriharmonic functions over a flat boundary. Then we show that the even reflected function gives an extension for nonnegative plurisubharmonic functions. In particular cases odd and/or even reflected functions give extensions for classical solutions of the homogeneous complex Monge–Ampère equation. Finally, we state reflection principle for the generalized complex Monge–Ampère equation and maximal plurisubharmonic functions.


Introduction
Reflection is a method to extend functions and, in particular, solutions of homogeneous equations across a flat boundary. Classically, it is applied to some strong type equations but later on also to several weak type equations. The original reflection principle states that an analytic function given in the upper half unit disk can be extended to the whole unit disk by reflection. This result originates with Schwarz. A similar principle holds for harmonic functions in the space, see [2].
This article is part of the Topical Collection on FTHD 2018, edited by Sirkka-Liisa Eriksson, Yuri M. Grigoriev, Ville Turunen, Franciscus Sommen and Helmut Malonek.
In higher real dimensions Martio and Rickman [16] introduced the reflection principle for quasiregular mappings. Later on Martio [14] showed that the reflection principle holds for solutions of certain elliptic partial differential equations, and he also treated further the reflection principle for quasiregular mappings, see also [9]. Moreover, Martio [15] studied equivalent principle for quasiminimizers in R n . Recently, Koskenoja [12,13] considered the reflection principle for both classical and viscosity solutions of the homogeneous real Monge-Ampère equation.
In several complex variables many authors have studied the reflection principle which is well understood for holomorphic mappings and related Cauchy-Riemann equations, see expository surveys by Coupet and Sukhov [6] and by Diederich and Pinchuk [7]. We concentrate on pluriharmonic and plurisubharmonic functions and the homogeneous complex Monge-Ampère equation that are central objects in pluripotential theory, see [3,8,10,11].

Basic Properties of the Reflection
We first set central notation connected to the reflection in C n . Let G + be a domain in C n + = {z = (z 1 , . . . , z n ) ∈ C n : Im z n > 0}. Let P : C n → C n be the reflection with respect to ∂C n + , that is, P (z) = P (z 1 , . . . , z n ) = (z 1 , . . . , z n−1 ,z n ). Suppose that there is a non-empty set Then G is a domain (open and connected set) in C n . Suppose that a function u : G + → R satisfies the following boundary condition on G 0 : lim z→w u(z) = 0 for all w ∈ G 0 . (2.1) We define the odd reflected functionũ : G → R, Correspondingly, the even reflected functionû : G → R is given bŷ In [12] Koskenoja studied differentiability properties of the reflected functions in the real n-space. Most of these results and examples can be adopted straightforward to the complex n-space. It is remarkable that if we reflect a differentiable function, then it may happen that the differentiability gets broken in the reflection boundary G 0 , see examples in [12].
Recall next some standard terminology. Let Ω ⊂ C n be open. A mapping f : Ω → R is said to be differentiable (or R-differentiable) in Ω if it is differentiable in Ω with respect to the real coordinates. This means that the first order real partial derivatives of f exist at each point of Ω. Correspondingly, we say that a mapping is twice differentiable in Ω meaning that the second order real partial derivatives exist at each point of Ω. It is obvious that if u ∈ C 1 (G + ), thenũ,û ∈ C 1 (G − ), and if u ∈ C 2 (G + ), thenũ,û ∈ C 2 (G − ), see [12,Lemmas 3.1 and 3.7].
We start with giving reflection formulas for the complex partial differential operators and for the complex Hesse matrix, see [17]. These formulas are complex counterparts of the formulas given in [12, Lemmas 3.1 and 3.7 and Theorem 3.14]. We omit many calculations since the methods are rather evident and often similar to those of given earlier in the proofs.

Remark 2.2.
We could prove formulas (2.4), (2.6), (2.5) and (2.7) also without complex chain rules just by using the corresponding formulas for the first order partial differential operators given in [12]. For example, since by [12, The complex Hesse matrix (or the complex Hessian) of a twice differentiable function u at a point z is the n × n matrix where the entries are given by the second order complex partial differential operator Vol. 29 (2019) Reflection Principle for the Complex Monge-Ampère Equation

Reflection Principle for Pluriharmonic Functions
Let Ω ⊂ C n be open. A C 2 -function u : Ω → R is pluriharmonic in Ω if for all j, k = 1, . . . , n, at every z ∈ Ω. Equivalently, a function u is pluriharmonic in Ω if and only if for every z ∈ Ω and w ∈ C n \ {0} the function λ → u(z + λw) is harmonic on {λ ∈ C : z + λw ∈ Ω}. This means that a pluriharmonic function is harmonic when restricted to any complex line inside Ω. Pluriharmonic functions form an invariant class under biholomorphic mappings. Therefore, in several complex variables pluriharmonic functions is a more important class of functions than harmonic functions which are not invariant under holomorphic mappings or even under complex linear mappings.
Pluriharmonic functions are harmonic in the sense of real coordinates. Hence the classical reflection principle for harmonic functions imply that a pluriharmonic function given in an open G + ⊂ C n + has a harmonic extensionũ to the reflected domain G but it is not straightforward thatũ is pluriharmonic in G. However, using the classical reflection principle for harmonic functions, it is simple to prove that the reflection principle holds for pluriharmonic functions. The crucial observation in the proof is that the reflected functioñ u is C ∞ in G, and hence the second order derivatives exist in G 0 and they are limits of the second order derivatives in G + ∪ G − . In general, if a function u is C 2 in G − , it may happen thatũ is not differentiable in a point z 0 ∈ G 0 , see Example 5.1. Proof. Suppose that u : G + → R is pluriharmonic and the boundary condition (2.1) holds. Since u is harmonic in G, the classical reflection principle for harmonic functions implies thatũ is harmonic in G. It follows thatũ is for every j, k = 1, . . . , n − 1. Sinceũ is C ∞ in G, we may change the order of the partial differentiation, and we have again by formula (2.13) that ∂ 2ũ ∂z n ∂z n (z) = − ∂ 2 u ∂z n ∂z n (P (z)) = − ∂ 2 u ∂z n ∂z n (P (z)) = 0.

Remark 3.2.
Armitage [1] showed that the classical reflection principle for h harmonic in G + holds when one assumes (instead of the boundary condition (2.1), that is, h tending to 0 at each point of G 0 ⊂ ∂R n + ) that h converges locally in mean to 0 on G 0 , that is, for all (x, 0) ∈ G 0 there exists r > 0 such that lim t→0+ |y−x|<r h(y, t) dy = 0.

Adv. Appl. Clifford Algebras
It is clear that the boundary condition (2.1) is stronger than the boundary condition (3.2). Therefore, in Theorem 4.1, it is sufficient to assume that the boundary condition (3.2) holds.
The classical reflection principle for harmonic functions can be proved by using the mean value principle of harmonic functions, see [2, Proof of Theorem 1.3.6]. It is remarkable that for the proof of the reflection principle for pluriharmonic functions (Theorem 4.1) the mean value principle can not be applied in G 0 . This is because if we take points z 0 ∈ G 0 and z ∈ G + , then the reflected point P (z) ∈ G − is not usually in an arbitrary complex line passing through the points z 0 and z.

Reflection Principle for Plurisubharmonic Functions
Let Ω be an open set in C n . An upper semicontinuous function u : Ω → R ∪ {−∞} which is not identically −∞ on any component of Ω is said to be plurisubharmonic in Ω if for each z ∈ Ω and w ∈ C n , the function λ −→ u(z + λw) is subharmonic or identically −∞ on every component of the set We need to restrict our considerations here to plurisubharmonic functions having nonnegative values only. The set of nonnegative reals is denoted by R + = {x ∈ R : x 0}. Proof. Upper semicontinuity of u in G + implies thatû is upper semicontinuous in G − . Thereforeû is upper semicontinuous in G = G + ∪ G 0 ∪ G − sincê u is continuous in a neighbourhood of G 0 . If z ∈ G − and w ∈ C n are such that {z + λw : λ ∈ C, |λ| 1} ⊂ G − , then 1 2π Remark 4.2. Since the even reflection preserves upper semicontinuity, it follows from (4.1) that the following more general observation is valid: If u is plurisubharmonic in an open set U ⊂ C n + , thenû is plurisubharmonic in P U ⊂ C n − .

Reflection Principle for the Classical Homogeneous Complex Monge-Ampère Equation
In this section we study reflection principle for the classical homogeneous complex Monge-Ampère equation For the even reflected functionv we have for each and hence the first order derivative ∂v ∂z2 (z) = − 1 2i . But since for each z ∈ C 2 + we have ∂v ∂z2 (z) = 1 2i , the first order derivative ∂v ∂z2 (z 0 ) does not exist in any z 0 ∈ ∂C 2 + . Consequently,v is not C 1 in any neighbourhood of ∂C 2 + .

Reflection Principle for the Generalized Complex Monge-Ampère Equation
Finally, we consider reflection principle for the homogeneous generalized complex Monge-Ampère equation, analogously to solutions of certain elliptic partial differential equations, see [9] and [14]. A plurisubharmonic and locally bounded function u : Ω → [−∞, ∞) can be operated by the generalized complex Monge-Ampère operator (dd c ) n , see [11,Section 3.4]. If u ∈ C 2 (Ω), then (dd c u) n = 4 n n! det ∂ 2 u ∂z j ∂z k dV, (6.1) where dV is the volume form in C n . The primary definition of the generalized complex Monge-Ampère operator was given by Bedford and Taylor [4]. Later on, Cegrell [5] introduced a slightly more general and in some sense optional definition of (dd c ) n . Then the plurisubharmonic function u is not required to be locally bounded but the definition is still not valid for all plurisubharmonic functions given in a general open set Ω ⊂ C n . If u is plurisubharmonic and locally bounded in Ω, then (dd c u) n gives a nonnegative Borel measure on Ω. Contrary to this, if u is plurisuperharmonic and locally bounded in Ω, then (dd c u) n gives a non-positive Borel measure on Ω. A

Adv. Appl. Clifford Algebras
The notion of maximal plurisubharmonic functions is due to Sadullaev [18]. In one complex variable, the maximal plurisubharmonic functions are precisely the harmonic functions, hence solutions to the Laplace equation Δu = 0 and consequently they belong to C ∞ (Ω). In more than one variable, the class contains, for example, all plurisubharmonic functions which depend on n − 1 variables only.
It is known that a locally bounded plurisubharmonic function u in Ω satisfies the homogeneous generalized complex Monge-Ampère equation (6.2) if and only if u is maximal [11,Theorem 4.4.2]. Moreover, we observe that if a locally bounded plurisubharmonic function u in G + is nonnegative and satisfies the boundary property (2.1), then it is plurisuperharmonic in Ω − . Therefore our main result regarding the reflection principle for the generalized complex Monge-Ampère equation can be stated for the even reflected functions only. Theorem 6.2. Let u be a nonnegative, locally bounded and plurisubharmonic function in G + such that the boundary condition (2.1) holds. If u is maximal in G + , thenû is maximal in G.
Proof. Let u be maximal in G + . By Theorem 4.1 the even reflected function u is plurisubharmonic in G. It is clear that the restrictionû| G− is plurisubharmonic and maximal in G − .
Suppose by contradiction thatû is not maximal in G. Then there exist a plurisubharmonic function v in G, an open set D G and a point z 0 ∈ D such that v û on ∂D but v(z 0 ) >û(z 0 ). If z 0 ∈ G + , then u =û| G+ is not maximal in G + , and if z 0 ∈ G − , thenû| G+ is not maximal in G − , which are both contradictions. Finally, suppose that z 0 ∈ G 0 . Since v is upper semicontinuous, there is a point z 1 ∈ G + ∩D such that v(z 1 ) >û(z 1 ) = u(z 1 ). By considering the function w(z) = max{v(z), u(z)}, z ∈ G + , which is plurisubharmonic in G + , we see that u is not maximal in G + . This is again a contradiction.