Maximal q-Subharmonicity in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{C}^{n}$\end{document}

In this paper, we study maximal q-subharmonic functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{n}$\end{document}. We prove that maximality of q-subharmonic functions is a local notion and give a condition to check the maximality of C2q-subharmonic functions.

Moreover, a recent result in [3] claimed that maximality is a local notion for locally bounded plurisubharmmonic functions (see Corollary 1 in [3]). Also in [3], Blocki extended the above result for the class E(Ω) introduced and investigated by Cegrell in [4] recently.
The aim of this paper is to extend the class of maximal plurisubharmonic functions to the class of maximal q-subharmonic functions, where q is an integer with 1 ≤ q ≤ n and to give some results about maximal q-subharmonic functions. Notice that for studying maximality of plurisubharmonic functions one often approaches by using the complex Monge-Ampère operator. But defining the complex Monge-Ampère operator for q-subharmonic functions is impossible. Hence one needs to find another approach for studying maximal qsubharmonic functions. In this paper, by using a new method, we prove that the maximality of q-subharmonic functions is equivalent to a local notion. We also provide a condition to check when a C 2 q-subharmonic function u is maximal q-subharmonic.
The note is organized as follows. Besides the introduction, the note has two sections. Section 2 is devoted to study q-subharmonic functions and to establish some results concerning this class. Section 3 deals with maximal q-subharmonic functions and proves the local property of this class.

q-Subharmonic Functions in C n
First, we recall the following definition of q-subharmonic functions which has been introduced by H. Ahn and N.Q. Dieu in [1] (also see [5]).

Definition 2 Let
Ω be an open set in C n . An upper semicontinuous function u : Ω −→ [−∞, ∞), u ≡ −∞ is called q-subharmonic if for every q-dimensional complex plane L in C n , u| L is a subharmonic function on L ∩ Ω. This means that for every compact subset K L ∩ Ω and every continuous harmonic function h on K such that u ≤ h on ∂K it follows that u ≤ h on K.
The set of q-subharmonic functions on Ω is denoted by SH q (Ω). Compared with subharmonic and plurisubharmonic functions in potential theory and pluripotential theory, it is easy to see that 1-subharmonic functions are plurisubharmonic and n-subharmonic functions are subharmonic.
The following basic properties of q-subharmonic functions can be proved in the same way as for subharmonic functions.

Proposition 1
Let Ω be an open set in C n and 1 ≤ q ≤ n. Then the following hold: where Ω ε = {z ∈ Ω : d(z, ∂Ω) > ε} and ε = (z/ε)/|ε| 2n , is a nonnegative smooth function in C n vanishing outside the unit ball and satisfying C n dV n = 1. Moreover, u * ε decreasingly tends to u when ε ↓ 0. 5. If χ is a convex increasing function in R and u is q-subharmonic in Ω then so is χ • u. 6. If u is a q-subharmonic function then for any unitary change of coordinates ϕ : C n → C n , the function u • ϕ ∈ SH q (Ω).

Let u be a q-subharmonic function in Ω and G relatively compact open subset of Ω and
Then the function Now we give the following.

Proposition 2
Let u be an upper-semicontinuous function on Ω ⊂ C n and u ∈ L 1 (Ω, loc), where L 1 (Ω, loc) denotes the set of locally integrable functions on Ω. Then the following statements are equivalent: In particular, if u ∈ C 2 (Ω) then u ∈ SH q (Ω) if and only if its complex Hessian has the sum of q smallest eigenvalues nonnegative at each point.
Proof First, we assume that u ∈ C 2 (Ω). Let u ∈ SH q (Ω) and z 0 ∈ Ω. We can choose a system of coordinates (z 1 , . . . , z n ) of C n such that the Hessian Then by the hypothesis u| Ω∩L is subharmonic on Ω ∩ L and so k∈K Conversely, assume that i∂∂u(z) ∧ ω q−1 ≥ 0 for all z ∈ Ω. Let L be a q-dimensional subspace of C n . Since i∂∂u(z) ∧ ω q−1 ≥ 0, z ∈ Ω it follows that u ∈ SH(Ω ∩ L). Hence, u ∈ SH q (Ω). Thus the conclusion is true in the case u ∈ C 2 (Ω).
Assume that u is as in the statement of the proposition. By putting u ε = u * ε and applying the above results to u ε , we obtain the desired conclusion.
Example 1 We give an example of a q-subharmonic function which is not plurisubharmonic. Let d > 1 and 1 < q ≤ d. Consider the function It is easy to see that q j =1 ∂ 2 ϕ ∂z j ∂z j (z) = 0 and by (b) of Proposition 2 it follows that ϕ is q-subharmonic. However, ϕ is not plurisubharmonic. Indeed, let = {(z 1 , 0, . . . , 0)} ⊂ C d be a complex line. Then ϕ| = (1 − q)|z 1 | 2 is not subharmonic, and the desired conclusion follows.

Maximal q-Subharmonic Functions
The following definition is similar as in the situation of maximal plurisubharmonic functions presented in [2] and [7].
The set of all maximal q-subharmonic functions in Ω is denoted by MSH q (Ω).
We give the following.

Proposition 3
Let Ω be an open subset in C n and let u ∈ SH q (Ω) ∩ L ∞ loc (Ω). Then the following conditions are equivalent: and the desired conclusion follows. Now we are in a position to prove the local property of maximal q-subharmonic functions. Namely, we have the following.

Theorem 1
Let Ω ⊂ C n be an open set, q be an integer with 1 ≤ q ≤ n and u ∈ SH q (Ω) ∩ L ∞ loc (Ω). Then u is maximal q-subharmonic if and only if u is local maximal q-subharmonic in Ω (i.e., for every z ∈ Ω there is an open neighborhood V z ⊂ Ω of z such that u| Vz is maximal q-subharmonic on V z ).
Proof The proof of the necessity is obvious. Now we give the proof of the sufficiency. Assume that G Ω and v is a q-subharmonic function on Ω such that v ≤ u on Ω\G.
We have to prove that v ≤ u on G. Choose z j ∈ Ω, j = 1, . . . , m and open subsets K j V z j Ω, j = 1, . . . , m such that z j ∈ V z j , G ⊂ m j =1 K z j and u is maximal q-subharmonic on V z j for all j = 1, . . . , m.
We split the proof into two steps.
Step 1. We prove that if K j , j = 1, . . . , m are open subsets in Ω and K j , j = 1, . . . , m are relatively compact open subsets such that K j K j V j Ω, j = 1, . . . , m then the following inequality holds It suffices to prove that (2) holds for m = 2. Let V = V 1 ∪ V 2 . To get a contradiction, without loss of generality we may assume that For ε > 0, put u (ε) (z) = u(z) − ε|z| 2 , z ∈ V and choose ε sufficiently small such that Then there exists a sequence Now we consider the following two cases.
and we get a contradiction. Case 2. p ∈ K 1 ∪K 2 . We may assume that p ∈ K 1 K 1 . We choose balls B = B(p, r 2 ) ⊂ B 0 = B(p, r) ⊂ B 1 = B(p, r 1 ) B 2 = B(p, r 2 ) K 1 . Moreover, we may assume that p j ∈ B for all j ≥ 1. For all j we can write Note that since Re z, p j is pluriharmonic and u is a maximal q-subharmonic function in V 1 , hence, by (b) of Proposition 3,u ε On the other hand, because z ∈ B 2 \ B 1 , it follows that |z − p j | 2 ≥ r 2 4 for all j ≥ 1. 2 4 for all z ∈ B 2 \ B 1 and for all j ≥ 1. It follows that Letting j → ∞, we infer that and we get a contradiction. Hence, (2) is proved.
Step 2. Let x 0 ∈ G. We need to prove By Step 1, we get This shows that (3) is true and the proof is complete.
From the above theorem we get the following useful corollary.

Corollary 1 Assume that u ∈ C(Ω). Then u ∈ MSH q (Ω) if and only if for every open subset
G Ω and every v ∈ SH q (G) ∩ C 2 (G) the following holds where K G is an arbitrary relatively compact open subset of G.
Proof From Theorem 1 and (d) of Proposition 3, it follows that the necessity is clear. Thus it suffices to prove that if (4) holds then u ∈ MSH q (Ω). Let v ∈ SH q (Ω), G Ω. By (d) of Proposition 3, we have to prove To get a contradiction, we assume that for all ε sufficiently small. Hence by (4) we have and we get a contradiction. Thus, and Proposition 3 implies that u ∈ MSH q (Ω). The proof is complete.
Compared to Proposition 1.4.9 in [2] we have the following.
is a decreasing sequence of maximal q-subharmonic functions in Ω. Then u = lim j →∞ u j either is a maximal q-subharmonic function or ≡ −∞ on Ω.
Proof Assume that u ≡ −∞. By (d) of Proposition 3, it is enough to prove that for every open subset G Ω and every v ∈ SH q (Ω) we have Let Thus (5) is proved and the desired conclusion follows.
The following fact is well known. Lemma 1 Assume that A = (a jk ) n j,k=1 is a complex n × n-matrix such that A = A t . Put B = (a jk ) n−1 j,k=1 . Let λ 1 ≤ λ 2 ≤ · · · ≤ λ n be the eigenvalues of A and μ 1 ≤ μ 2 ≤ · · · ≤ μ n−1 the eigenvalues of B. Then In particular, if the matrix B has one nonnegative eigenvalue then so does the matrix A.
The proof is complete.
We need the following fact.

Lemma 2
Let u ∈ C 2 (Ω) and assume that its complex Hessian has least one nonnegative eigenvalue at each point. Then for every open subset G G 1 Ω we have Proof From the hypothesis and Lemma 2.6 in [6], it follows that u is an (n − 1)plurisubharmonic function in Ω. The desired conclusion of the lemma follows from Lemma 2.7 in [6].
Next we give a condition under which a C 2 q-subharmonic function is maximal qsubharmonic.

Theorem 2 Let
Ω be an open set in C n and let q be an integer with 1 ≤ q ≤ n − 1. Assume that u ∈ C 2 (Ω). Then u ∈ MSH q (Ω) if and only if its complex Hessian has the sum of q smallest eigenvalues equal to 0 at each point.