Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

In this paper, we establish a local regularity result for Wloc2,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2,p}_{{\mathrm {loc}}}$$\end{document} solutions to complex degenerate nonlinear elliptic equations F(DC2u)=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(D^2_{\mathbb {C}}u)=f$$\end{document} when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.


Introduction and main results
In this work, we are interested in the local regularity theory for nonlinear complex degenerate elliptic equations of the form where D 2 ℂ u denotes the complex Hessian of u. Such equations have been studied extensively in the literature, going back to the celebrated work [6] on the Dirichlet problem in the real case and its counterpart [17] for the complex setting. We recall that an important feature of degenerate equations is that, unlike uniformly elliptic equations, the C 2, -regularity may fail. Perhaps one of the most important equation of the form (1) is the complex Monge-Ampère equation: The regularity of the solution of this equation is studied in the literature by many authors and by different tools. It was first proved in pioneering work [4] that the solution u to the

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Dirichlet problem in a ball B belongs to C 1,1 (B) ∩ C 0 (B) provided that the right-hand side f ∈ C 2 (B) and the boundary data is C 2 ( B) . Another important regularity result concerning the Dirichlet problem associated with (2) in a strictly pseudoconvex domain, established in [5], is the smooth regularity up to the boundary of its solution when the right-hand side, the boundary data and the domain are all smooth (the strict positivity of the right-hand side is also essential). The local regularity of (2) (no boundary data) was also studied. A sharp result was obtained in [2] by developing some methods in [24]: solutions u ∈ W 2,p loc of (2) with f ∈ C ∞ are necessarily C ∞ whenever p > n(n − 1) , and no smaller exponent p can be expected in general.
The local regularity of other equations of the form (1) were also studied. Notably, in [9], a counterpart of [2] was proved for the so-called complex k-Hessian equation under the assumption that u belongs to W 2,p loc with p > n(k − 1). The goal of the present paper is to extend the approach of [2] and [9] to more general nonlinear complex degenerate elliptic equations. We will introduce simple conditions on the nonlinearity F to obtain general local regularity results, thus considerably broadening the field of application of this method.
In particular, we shall see that our results apply to the so-called complex k-Monge-Ampère equation or MA k -equation, k ∈ {1, … , n} , that has recently received much attention [7,8,13,23]:  [7]. In the complex setting this operator was discussed in [23] and, in the special case k = n − 1 , also in [25]. It has been shown in [8] that this operator does not satisfy an integral comparison principle, which makes the associated potential theory much harder to be developed. Finally, let us also mention that the Dirichlet problem associated with this operator was studied in [28] using a probabilistic approach.
The outline of the paper is as follows. After recalling some basic notations, we precise in Sect. 1

General notations
• All along this work, Ω ⊂ ℂ n ( n ≥ 1 ) is a nonempty open bounded connected subset. • Let ℍ n be the set of n × n Hermitian matrices. The (i, j)th entry of a matrix A ∈ ℍ n will be denoted by a ij . We recall that ℍ n is a vector space over the field ℝ of dimension n 2 . The inner product on ℍ n is the standard Frobenius inner product: The n × n identity matrix will be denoted by Id.
We will use the classical cones • The Fréchet derivative at A ∈ C of a function F ∈ C 1 (C, ℝ) , defined on a nonempty open subset C ⊂ ℍ n , will be denoted by DF(A) ∈ L(ℍ n , ℝ) . It can be identified with the matrix F a ij 1≤i,j≤n ∈ ℍ n through the formula • In this work, we use the following convention for ellipticity. We say that a function F ∶ C ⟶ ℝ is: ,j≤n , where we use the standard notations u j and ūj to denote, respectively, u The Laplacian is denoted by Δu = ∑ n j=1 u jj = Trace (D 2 ℂ u) ∈ ℝ . Note that it is the complex Laplacian, so it is one-fourth the real Laplacian. • Finally, we shall denote by C(n), C(n, Ω) , etc. a positive number that may change from line to line but that depends only on the quantities indicated between the brackets.

Main results
Throughout this work, we will always make the following assumptions: for ever > 0 and A ∈ C). (e) Concavity: F 1∕d is concave in C.

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The key assumption here is the assumption (f) that will help us to compare our general nonlinear equation F(D 2 ℂ u) = f with the complex Monge-Ampère equation and thus use the recent uniform estimates obtained in [1].
The assumption (a) is quite standard (see e.g. [6, pp. 261-262]). Note that this excludes the case of the whole space C = ℍ n .
It is important to point out that we cannot only consider homogeneous functions of degree 1 by normalizing F into F 1∕d since this normalization is not in general regular up to the boundary (e.g. Finally, by concavity and homogeneity, we can check that F satisfies (3) for every A ∈ C if and only if for every A ∈ C and P ∈ C n . In particular, F is necessarily elliptic in C (but, in general, not uniformly elliptic). We refer, for instance, to Abja et al. [1,Remark 1.3] for more details.
The first result of the present paper is the following.

Theorem 1.1 Let C and F satisfy the above standing assumptions
and  An important consequence of Theorem 1.1 and of the classical theory for uniformly elliptic real equations from [20] will be the following local regularity result: Corollary 1.2 Under the framework of Theorem 1.1, assume in addition that F satisfies the following property: (g) Sets of uniform ellipticity: there exists R 0 > 0 such that, for every R > R 0 , F is uniformly elliptic in C(R) , where sup Δu ≤ R, Then, we have the property Remark 1. 3 Note that C(R) is convex because F is concave. However, it is never a cone and it does not contain C n . Note as well that lim R→+∞ C(R) = C.
We also point out that we do not need to assume that C(R) is compact. For instance, C(R) is not bounded for the simplest example F(A) = Trace (A) on A ∈ C 1 ( n ≥ 2 ). However, this is a rather pathological case and we will see in Proposition 3.1 that C(R) is always bounded whenever

Remark 1.5
It is unclear whether similar local regularity results hold on Hermitian manifolds as well (see [26] for results related to Corollary 1.2 in this context when u ∈ C 2 ). It would be interesting to investigate this problem.
The rest of the article is organized as follows. In the next section, we prove our main results. In Sect. 3, we present some examples of Hessian equations that are covered by our framework. Finally, in Sect. 4, we discuss a bit more in detail the case of the k-Monge-Ampère equation.

Proofs of the main results
The proof of Theorem 1.1 is inspired from the proof of [24, Theorem 1] and the ideas of [2] (see also [9,Section 4]).
We introduce the normalization Clearly, G is elliptic in C . The equation F(D 2 ℂ u) = f then becomes Throughout Sect. 2, the notation L u is exclusively saved for the linearization of G about Note that L u is (degenerate) elliptic and that its coefficients are Lebesgue measurable (as composition of a continuous function with a Lebesgue measurable function).

Geometric configuration
where B R (z 0 ) ⊂ ℂ n denotes the open ball of center z 0 and radius R > 0 . Up to the transformation z ↦ (z − z 0 )∕( ∕2) , we can always assume that B ∕2 (z 0 ) is the open unit ball, that will simply be denoted by B 1 in the sequel. Thus, B (z 0 ) becomes B 2 (0) , that will simply be denoted by B 2 . Therefore, from now on, we are in the following geometric configuration:

Approximation of the Laplacian
In order not to consume too much regularity, we will need a suitable approximation of the Laplacian. This is done as in [2, p. 415] and [9, Lemma 4.2] (see also [4, Proposition 6.3]).
The first point is a direct consequence of the mean value inequality (see e.g. [

A uniform estimate for strong supersolutions
Recently, it has been obtained in [1, Theorem 1.4] some new type of Alexandrov-Bakelman-Pucci estimate for various nonlinear complex degenerate equations. Thanks to the assumptions of the present article, it applies in particular to our linearized operator L u defined in (7) for which it results in the following: where g + = max(g, 0) denotes the positive part of g.

Estimate of the operator norm
We will need the following estimate: Proof By continuity of DF on the compact The desired estimate (9) then follows from the computation ◻ We point out that it is only to show (9) that we use that F is C 1 up to the boundary.
Despite not needed, we mention that the inequality (9) also shows that the coefficients of the operator L u belong to L ∞ if d = 1 , and to L p∕(d−1) if d > 1 , which are then in L n by the assumption (4).

Jensen's inequality in convex subsets of ℍ n
The following version of Jensen's inequality will be needed (see e.g. [

Proof of Theorem 1.1
We are now ready to prove our first main result.
(1) Strategy of the proof. Let 0 < < 1 . For , ∈ [2, +∞) (to be determined later) let us introduce an auxiliary function w given by where Note that: • w is unambiguously defined since T u ≥ 0 by item (i) of Lemma 2.1 and subharmonicity of u, which follows from the assumptions that D 2 thanks to the regularity of u and Sobolev embeddings (using that p ≥ n).
The goal will be to bound sup B 1 w from above by a positive number depending only on the quantities indicated in the statement of Theorem 1.1. This will also provide an upper bound for sup̃T u since (z) ≥ (dist ( , Ω)∕4) 2 > 0 for z ∈̃ (recall (8)). This will in turn imply that Δu ∈ L ∞ (̃) with the same bound as for T u . Indeed, denoting by C a bound for T u we will have shown that T u ∈ S , where We can check that this set is closed in L p (̃) (using, for instance, the partial converse of the Lebesgue dominated convergence theorem [21,Theorem 3.12]). Since it is also clearly convex, it is then weakly closed in L p (̃) (see e.g. [22,Theorem 3.12]). As T u ⇀ Δu weakly in L p (̃) (item (iii) of Lemma 2.1) and T u ∈ S , it follows that Δu ∈ S as well, which is exactly what we want. Now, in order to bound sup B 1 w we are going to use the condition (4) on p to show that (L u (−w)) + ∈ L q (B 1 ) for some q > n , with estimate for some C > 0 depending only on the quantities indicated in the statement of Theorem 1.1. The conclusion will then follow from the uniform estimate of Theorem 2.2 with g = (L u (−w)) + . Note that w = 0 on B 1 (in fact, this is the (only) obstruction to simply take = 1).
We have and Consequently, where (using also the identity v i = v̄i for any real-valued function v) and (3) Estimate of the term I.
To estimate this term, we will make use of the ellipticity and of the estimate (9). Let us introduce the following sesquilinear form on ℂ n : This form is nonpositive by ellipticity of G. Therefore, we can use the basic inequality  2ℜ (a, b) ≤ − (a, a) − (b, b) with a i = 1 √ t i and b j = √ t(T u) j ( t > 0 to be chosen below), to obtain Taking now t = ( − 1) ∕T u > 0 , this removes the third term in I to give Let Direct computations show that where ij denotes the Kronecker delta. Since ≤ 1 , we obtain Consequently, using (9), we have Note that we cannot use the better estimate (3) since B ∉ C n . (4) Estimate of the term II.
For this term, we will use the concavity of F to show that To prove such an estimate, we would like to use the concavity inequality with But first observe that B = (D 2 ℂ u) (z) (see Lemma 2.1) and apply Jensen's inequality (Lemma 2.4) to see that indeed B ∈ C for a.e. z ∈ B 1 with, in addition, We can now use (11) to obtain the desired estimate: This yields the desired estimate (10). ◻

C ∞ regularity
In this section, we show how to deduce Corollary 1.2 from Theorem 1.1 and the classical theory for uniformly elliptic real equations from [20].
We follow the presentations of [26] and [27]. Let 2n be the space of 2n × 2n real symmetric matrices. The inner product on 2n is the standard Frobenius inner product.
In this section, a function u of n complex variables (z 1 , … , z n ) will also be a considered as a function of 2n real variables (x 1 , … , x n , y 1 , … , x n ) , where z j = x j + iy j . The real Hessian of u is then denoted by The open ball in ℝ 2n of center 0 radius r > 0 will be denoted by B r . Corollary 1.2 is essentially a consequence of the following fundamental general result and Schauder's estimates: Theorem 2.5 Let F ∶ 2n ⟶ ℝ be concave and uniformly elliptic, and let f ∈ C 0, (B 1 ) ( 0 < ≤ 1 ). If u ∈ W 2,p loc (B 1 ) is a strong solution to and p ≥ 2n , then there exist ∈ (0, 1) and ∈ (0, 1) such that We recall that "strong solution" simply means that the equation holds almost everywhere. This theorem is a consequence of [20, Theorem 8.1] (see e.g. [27, Section 2] for details). In this reference, the result deals in fact with C-viscosity solutions, but we recall that a strong solution u ∈ W 2,p loc with p ≥ 2n is a C-viscosity solution, see [18,Section III]. In order to apply this result, we need two preliminary observations: firstly, we have to associate with our equation F(D 2 ℂ u) = f an equation for the real Hessian F (D 2 ℝ u) = f and, secondly, this F has to be defined over the whole space 2n . This is done in a standard way (see, e.g., [3,26,27], etc.): • We identify n × n Hermitian matrices with the subspace of 2n given by matrices invariant by the canonical complex structure: where the map ∶ ℍ n ⟶ 2n is given by Let us also introduce the projection ∶ 2n ⟶ (ℍ n ) given by The complex and real Hessians are then related by the identity As a result, if u solves F(D 2 ℂ u) = f , then it solves as well where F ∶ E(R) ⟶ ℝ is given by and E(R) ⊂ 2n is defined by We recall that C(R) is defined in (5) and that it is not a cone, so that the factor 2 cannot be removed. It is clear that E(R) is an open convex set and that F is concave in E(R) .
Since F is uniformly elliptic in C(R) by assumption of Corollary 1.2, it is also clear that F is uniformly elliptic in E(R). • Let us now extend F to 2n . This is done by considering F ∶ 2n ⟶ ℝ defined by u ∈ C 2, (B ).
where 0 < m ≤ M denote the ellipticity constants of F . We can check that F is concave and uniformly elliptic in 2n , and that we have F =F in E(R) (see e.g. [26, Lemma 4.1]). Corollary 1.2 is now an easy consequence of the previous results.
Proof of Corollary 1.2 After translation and dilation of the coordinates if necessary, it is sufficient to show that u ∈ C ∞ (B ) for some ∈ (0, 1) when B 1 ⊂⊂ B 2 ⊂⊂ Ω.
From Theorem 1.1, we know that, for some R 1 > 0, On the other hand, by our assumptions on f, there exists R 2 > 0 such that Consequently, for any R > max R 0 , R 1 , R 2 we have It now follows from the previous discussion that u is a strong solution to the concave and uniformly elliptic real equation Besides, u ∈ W 2,p (B 1 ) for every 1 ≤ p < ∞ by Calderón-Zygmund estimates since u, Δu ∈ L ∞ (B 2 ) . We can then apply Theorem 2.5 and obtain that u ∈ C 2, (B ) for some ∈ (0, 1) and ∈ (0, 1). As a result, u ∈ C 2, (B ) solves F (D 2 ℝ u) = f in B . Since it follows that u ∈ C ∞ (B ) from the classical Schauder's estimates, see e.g. [20, Proposition 9.1] (the proof in this reference is carried out for E(R) = 2n but all the arguments go through after noticing that is a compact set of 2n which is included in E(R) ). ◻

Application to Hessian equations
Let us now present some examples covered by our framework. We emphasize that our results Theorem 1.1 and Corollary 1.2 do not require that our equations are Hessian, but all the examples of application that we present here will be Hessian equations. Let us first recall some notations.
• For A ∈ ℍ n , its eigenvalues (which are real) will always be sorted as follows: We then introduce ∶ ℍ n ⟶ ℝ n defined by • A function F ∶ C ⟶ ℝ is said to be a Hessian operator if there exist a set Γ ⊂ ℝ n and a function F ∶ Γ ⟶ ℝ such that The notation Γ will always be saved for sets of ℝ n whereas the notation C will always be saved for sets of ℍ n . • For k ∈ {1, … , n} , we introduce the classical cones where k is the kth elementary symmetric polynomial: where, here and in the rest of this article, we use the notation for every ( 1 , … , n ) ∈ S and every permutation .
Let us now provide practical conditions on Γ and F to guarantee that C = −1 (Γ) and F =F• satisfy the assumptions of our main results. We emphasize that the map has few nice properties, so this result is far from being trivial. The proof of Proposition 3.1 is postponed to the end of this section for the sake of the presentation.

Remark 3.2
The concavity assumption is satisfied by a large class of polynomials, namely, the hyperbolic polynomials. We recall that a polynomial F homogeneous of degree d ∈ ℕ * (A) = ( 1 (A), … , n (A)).
is called hyperbolic with respect to e ∈ ℝ n if F (e) ≠ 0 and if, for every ∈ ℝ n , the onevariable polynomial t ∈ ℂ ⟼F( − te) has only real roots. In this case, F 1∕d is concave in the open convex cone (called hyperbolicity cone) We refer to [6, Section 1] and [12] for more details.
Let us now finally present some concrete examples.
• The complex Monge-Ampère equation: The degree of homogeneity is obviously d = n and the domination property (12) is trivial. This polynomial is hyperbolic with respect to e = (1, 1, … , 1) . We can check that the associated hyperbolicity cone is indeed Γ n . It clearly satisfies all the desired assumptions. • The complex k-Hessian equation: for k ∈ {1, … , n} , The degree of homogeneity is obviously d = k and the domination property (12) follows from Maclaurin's inequality: for any k ≥ 2 , This polynomial is hyperbolic with respect to e = (1, 1, … , 1) . We can check that the associated hyperbolicity cone is indeed Γ k . It clearly satisfies all the desired assumptions ( k > 1).
• For s ∈ [0, 1) , let The cones Γ 2−s interpolate between Γ 2 and Γ 1 . The degree of homogeneity is obviously d = 2 and the domination property (12) is trivial since F ( ) ≥ (1 − s) 2 1 2 . The cone clearly satisfies all the desired assumptions ( F is also a hyperbolic polynomial with respect to e = (1, 1) , but the concavity of F 1∕2 is easily checked by a direct computation here).
• The complex k-Monge-Ampère equation: for k ∈ {1, … , n} , where The degree of homogeneity is d = C k n . The domination property (12) is a consequence of the following inequality: for any k ≥ 2 , This property will be detailed in Sect. 4.2. This polynomial is again hyperbolic with respect to e = (1, 1, … , 1) . We can check that the associated hyperbolicity cone is indeed Γ � k . It clearly satisfies all the desired assumptions ( k < n). We conclude this section with the proof Proposition 3.1.

Proof of Proposition 3.1
We only establish the nonobvious properties.
(1) Regularity of F. Since F is a symmetric polynomial, by Newton's fundamental theorem of symmetric polynomials we can write for some polynomial p. Since all the functions A ∈ ℍ n ⟼ k ( (A)) are C ∞ (ℍ n , ℝ) (recall that they are equal to (−1) k ( n−k h∕ t n−k )(0, A) where h ∈ C ∞ (ℝ × ℍ n , ℝ) is the function h(t, A) = det(tId − A) ), we deduce that F ∈ C ∞ (ℍ n , ℝ) by composition. (2) Convexity of C and concavity of F 1∕d . We point out that the concavity of F 1∕d is also shown in [6, Section 3] but we would like to present a different proof here. Let A, B ∈ C and t ∈ [0, 1] be fixed. We have to show that To this end, it is convenient to make use of the theory of majorization, for which we refer to [19]. The starting point is that, for every k ∈ {1, … , n} , the function From this property, we have that t (A) + (1 − t) (B) "majorizes" (tA + (1 − t)B) , that is It follows that (see e.g. [ (14) implies that Using again the concavity of F 1∕d , we obtain the concavity of F 1∕d . (3) Uniform ellipticity of F in C(R).
Let R > 0 be fixed. Let We first show that the assumption Γ ∩ Γ 1 = {0} implies that Γ(R) is bounded. To this end, it is sufficient to prove that there exists > 0 such that To show this property, we argue by contradiction and assume that there exists a sequence ( ) >0 ⊂ Γ such that Since Γ ⊂ Γ 1 , we have in particular that ≠ 0 and we can normalize the sequence by considering ̃= ∕| | . We have ̃∈ Γ since Γ is a cone. We can extract a subsequence, still denoted by (̃) >0 , that converges to some ̃∈ Γ . Besides, ̃≠ 0 since | |̃| | = 1 . Passing to the limit → 0 in (15), we obtain that Since Γ ⊂ Γ 1 , it follows that ̃∈ Γ ∩ Γ 1 and thus ̃= 0 by assumption, a contradiction. As a result, Γ(R) is compact and there exist real numbers m ≤ M such that, for every i ∈ {1, … , n} , Let us now prove that m > 0 . To this end, it is sufficient to show that The inclusion Γ(R) ⊂ Γ immediately follows from the assumption that F = 0 on Γ . Let us now prove the inequality. We use some ideas of the proof of [6,Corollary,. Let i ∈ {1, … , n} be fixed. Let e i denote the ith canonical vector of ℝ n . We have e i ∈ Γ n ⊂ Γ . From the concavity and homogeneity of Ĝ =F 1∕d , which belongs to C 1 (Γ) ∩ C 0 (Γ) since F > 0 in Γ , we have Let us now prove that this inequality is strict. Assume not, and let then * ∈ Γ be such that (Ĝ∕ i )( * ) = 0 . Since Γ is open, there exists > 0 such that * + te i ∈ Γ for every t ∈ [0, ] . By concavity, we then have As a result, Since F > 0 in Γ , this is equivalent to By analyticity, we deduce that this identity holds for any t ∈ ℝ , in particular for negative ones. Let then Since Γ ⊂ Γ 1 , we necessarilyhave T( * ) > −∞ . Integrating the identity (F∕ i )( * + te i ) = 0 over [T( * ), 0] then yields This is a contradiction since F > 0 in Γ ∋ * , whereas F = 0 on Γ ∋ * + T( * )e i . Therefore, (16) holds with m > 0 . To conclude, it remains to show that this implies the uniform ellipticity of F =F• in C(R) . Let then A, B ∈ C(R) with A ≤ B . We denote the eigenvalues of A (resp. B) by ( 1 , … , n ) (resp. ( 1 , … , n ) ). By definition, we have to show that for some 0 < m ′ ≤ M ′ (that do not depend on A, B). We only prove the first inequality, the other one being proved similarly. Since B − A ≥ 0 , the first inequality is equivalent to for some 0 < m ′′ . Since Trace (B − A) = Trace (B) − Trace (A) , this is equivalent to the following property for F : Let us then prove this inequality. By definition, we have , ∈ Γ(R) . Assume first that (16) and integrating over t ∈ [0, 1] , we have (16), iterating this process and summing all the obtained inequalities, we eventually obtain (17) with m �� = m . The general case can be deduced from the case (18) by an approximation argument (using the compactness of Γ(R)). ◻

Additional properties for the MA k -equation
In this section, we detail the case of the k-Monge-Ampère equation, which is important for applications. Let us first state explicitly the local regularity result that we have obtained for this equation.
We recall once again that, unless k = 1 (the Monge-Ampère equation), it is not known whether this condition on p is sharp. However, we will provide in Sect. 4.3 an example that shows that a threshold does exist for such a result to be valid.

k-Plurisubharmonic functions
The importance in the study of the operator MA k lies in its connection with the notion of k-plurisubharmonic function.
semi-continuous and if it is subharmonic whenever it is restricted to any affine complex plane of dimension k. We denote by k-plurisubharmonic the vector space of such functions.
For instance, 1-plurisubharmonic functions are the plurisubharmonic functions, and n-plurisubharmonic functions are the subharmonic functions.
To make clearer the link between k-plurisubharmonic functions and the operator MA k , we first introduce some notations.

Definition 4.3 Let
A ∈ ℂ n×n and k ∈ {1, … , n} . The operator D A ∶ Λ k ℂ n ⟶ Λ k ℂ n is the linear action of A as derivation on the space Λ k ℂ n of k-vectors. On simple k-vectors, this means that By choosing the canonical basis of the complex space Λ k ℂ n (recall that dim Λ k ℂ n = C k n ), D A can be identified with a matrix of size C k n × C k n , that will still be denoted by D A .  This cone is linked to the notion of k-plurisubharmonic function as follows: when u ∈ C 2 (Ω) , we have the characterization For more information about k-plurisubharmonic functions, we refer, for instance, the reader to [14,15].

Comparison between MA k and MA k−1
In this section, we show that the crucial domination property (12) holds for MA k . In fact, we prove the more precise property (13) that allows to compare MA k and MA k−1 . Let k ∈ {2, … , n} be fixed. Let us first show the inclusion Let then ∈ Γ � k−1 and let us show that i 1 + ⋯ + i k > 0 . For any (i 1 , … , i k ) ∈ E k n , we denote by I k = i 1 , … , i k and Then, the claim follows from the identity To show this identity, we have to count how many i 1 , i 2 , … are in the sum of the left-hand side. For i 1 , the only possibility is to have j 1 = i 1 . Therefore, there are C k−1 k−2 = k − 1 such terms. For i 2 , there are two and only two possibilities, namely j 1 = i 2 or j 2 = i 2 (in this second case, necessarily j 1 = i 1 ). This gives C k−1 k−3 + C k−2 k−3 terms, which is equal to C k−1 k−2 by Pascal's rule. Repeating this reasoning leads to (19).
MA k ( ) ≤ MA k−1 ( ), 1 3 • The sum i 1 + ⋯ + i k is of the form |z ′′ | 2 if so is each element of the sum. There are C k m possibilities for this situation to happen ( C k m = 0 if m < k). • In all the other cases, the sum i 1 + ⋯ + i k is of the form |z �� | 2 −2 . Since there are C k n products overall in MA k , this means that such sums appear C k n − C k m times.
In summary, we have for some smooth positive function . Consequently, MA k ( (D 2 ℂ u)) ∈ C ∞ (ℂ n ) when the power of |z ′′ | is exactly equal to zero, which gives the desired condition on , namely For this value of , we also have MA k ( (D 2