A note on the space of delta m -subharmonic functions

: In this note, we present some properties of a certain space of delta m -subharmonic functions. We prove that the convergence in this space implies the convergence in m -capacity.

A bounded domain Ω ⊂ C n is called m-hyperconvex if there exists an m-subharmonic function ρ : Ω → (−∞, 0) such that the closure of the set {z ∈ Ω : ρ(z) < c} is compact in Ω for every c ∈ (−∞, 0).In what follows we will always assume that Ω is an m-hyperconvex domain.Denote by S H m (Ω) the set of all m-subharmonic functions in Ω.Let the cones E 0,m , E p,m , F m be defined in the similar way as in [21,25]: For the properties and applications of these classes, see [1,21,22,25,26,27].We use the notation δK = K = K for K be one of the classes E 0,m , E p,m , F m .Define with the convention that (−u 1 − u 2 ) p = 1 if p = 0.For the reason why this quasi-norm is effective, please see [2,13,16,22,29].It was proved in [25] that (δE p,m , || • || p,m ) is a quasi-Banach space for p > 0, p 1 and it is a Banach space if p = 1.Moreover in [17] it was proved that (δF m , || • || 0,m ) is a Banach space.The authors in [12] show that (δE p,m , || • || p,m ) can not be a Banach space.These facts are counterparts of [5,6,10,18] in m-subharmonic setting.
In Section 3, we prove that the convergence in δE p,m implies the convergence in m-capacity ( Theorem 3).But the convergence in m-capacity is not a sufficient condition for the convergence in δE p,m ( Example 3).Similar results in plurisubharmonic setting have been proved by Czyż in [11].

Preliminaries
In plurisubharmonic case, the following proposition was proved in (see [11]).Let B = B(0, 1) ⊂ C n be the unit ball in C n .Then the cones E 0,m (B) and δE 0,m (B) are not closed respectively in (δF m (B), ) is a constant depending only on n and m, δ 0 is the Dirac measure at the origin 0 (see [28]).For each j ∈ N, define the function where a j = 1 2 j , b j = 1 j .We can see that v j ∈ E 0,m (B), for each j.Therefore, the function u k := k j=1 v j belongs to E 0,m (B).For k > l we can compute and where The last inequality is a consequence of the fact that v j is increasing function for each j.Since we have Hence Let u : B → R ∪ {−∞} be defined by u = lim k→∞ u k .Observe that u is the limit of a decreasing sequence of m-subharmonic functions and u(z) > −∞ on the boundary of the ball B(0, 1  2 ).Hence u is m-subharmonic.Moreover u E 0,m (B) since it is not bounded on B, its value is not bounded below at the origin.Equality (2.1) shows that {u k } is a Cauchy sequence in the space δF m (B).Thus the cone E 0,m (B) and the space δE 0,m (B) are not closed in (δF m (B), || • || 0,m ).

The series
is convergent by the ratio test.Therefore {u k } is a Cauchy sequence in δE p,m by (2.2).We have proved that the cone E 0,m (B) and the space δE 0,m (B) are not closed in (δE p,m (B), || • || p,m ).
Proof.The definition of the m-Lelong number of a function v ∈ S H m (Ω) at a ∈ Ω is the following It is easy to see that m-Lelong number is a linear functional on δF m .Moreover, as in [7, Remark 1], for a function ϕ ∈ F m then Hence, for any representation u = u 1 − u 2 of u ∈ δF m we have This implies that m-Lelong number is a bounded functional on the space δF m .We have shown that m-Lelong number is continuous on the Banach space (δF m , || • || 0,m ).We recall the definition of m-Green function with pole at a The readers can find more properties of m-Green function in [31].Assume that E 0,m = F m .Then there exists a sequence {u j } in E 0,m that converges to g m,Ω,a in the space δF m as j → ∞.The m-Lelong number of all u j at a vanishes since u j is bounded, but the m-Lelong number of g m,Ω,a at a is 1.Hence we get a contradiction.Thus, E 0,m F m .By the same argument, if δE 0,m = δF m , then there exists a sequence {u j } in E 0,m that converges to g m,Ω,a in the space δF m as j → ∞, but this is impossible since ν m,a (u j ) = 0.

The convergence in δE p,m
We are going to recall a Błocki type inequality (see [8]) for the class E p,m .Similar results for the class F m were proved by Hung and Phu in [19,Proposition 5.3] (see also [1]) and for locally bounded functions were proved by Wan and Wang [31].Assume that v ∈ E p,m and h ∈ S H m is such that Proof.See the proof of [19,Proposition 5.3].
Recall that the relative m-capacity of a Borel set E ⊂ Ω with respect to Ω is defined by We are going to recall the convergence in m-capacity.We say that a sequence {u j } ⊂ S H m (Ω) converges to u ∈ S H m (Ω) in m-capacity if for any > 0 and K Ω then we have lim j→∞ cap m,Ω (K ∩ {|u j − u| > }) = 0. Let {u j } ⊂ δE p,m be a sequence that converges to a function u ∈ δE p,m as j tends to ∞.Then {u j } converges to u in m-capacity.
Proof.Replacing u j by u j − u, we can assume that u = 0.By the definition of δE p,m , there exist functions v j , w j ∈ E p,m such that u j = v j − w j and e p (v j + w j ) → 0 as j → ∞.By [25], max(e p,m (v i ), e p,m (w j )) ≤ e p,m (v j + w j ), which implies that e p,m (v j ), e p,m (w j ) tend to 0 as j → ∞.Given > 0 and K Ω.For a function ϕ ∈ S H m (Ω), −1 ≤ ϕ ≤ 0, we have The last inequality comes from Lemma 3. Hence, by taking the supremum over all functions ϕ in inequality (3.1), we get m+p (e p,m (v j ) + e p,m (w j )) → 0 as j → ∞.Hence the sequence {u j } tends to 0 in m-capacity and the proof is finished.
A similar result for the space δF m is proved in [17].But the convergence in m-capacity is not a sufficient condition for the convergence in the space δE p,m .The following example shows that convergence in m-capacity is strictly weaker than convergences in both δE p,m and δF m .The case m = n has been showed in [11,Example 3.3].Let v(z) be the function defined in the unit ball in C n as in the proof of Proposition 2. We define Then we have u j , v j ∈ E 0,m (B) for every j, and e p,m (u j ) = c(n, m), e 0,m (v j ) = 1.These show that the sequence {u j } and {v j } do not converge to 0 in δE p,m (B) and δF m (B) respectively as j → ∞.Moreover, for fixed > 0 and K B there exists j 0 such that for all j ≥ j 0 we have This infers that both sets K ∩ {u j < − } and K ∩ {v j < − } are empty.Hence u j and v j tend to 0 in m-capacity.