Citation: Bahar Acay, Ramazan Ozarslan, Erdal Bas. Fractional physical models based on falling body problem[J]. AIMS Mathematics, 2020, 5(3): 2608-2628. doi: 10.3934/math.2020170
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Theory of m-subharmonic functions was recently developed by many mathematicians such as Li [20], Błocki [9], Dinew and Kołodziej [14,15], Lu [21,22], Sadullaev and Abdullaev [30], Nguyen [23,24], Åhag, Czyż and Hed [3,4] and many others. The notion of m-subharmonicity appears naturally in generalization of subharmonicity and plurisubharmonicity. For the similarities and the differences between these notions, we refer the readers to the paper [15].
A bounded domain Ω⊂Cn 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 SHm(Ω) the set of all m-subharmonic functions in Ω. Let the cones E0,m,Ep,m,Fm be defined in the similar way as in [21,25]:
E0,m={u∈SHm(Ω)∩L∞(Ω):limz→∂Ωu(z)=0 and ∫ΩHm(u)<∞},Ep,m={u∈SHm(Ω):∃ {uj}⊂E0,m,uj↓u,supj∫Ω(−uj)pHm(uj)<∞},Fm={u∈SHm(Ω):∃ {uj}⊂E0,m,uj↓u and supj∫ΩHm(uj)<∞}. |
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 E0,m,Ep,m,Fm. Define
||u||p,m=infu=u1−u2u1,u2∈Ep,m{(∫Ω(−u1−u2)pHm(u1+u2))1m+p}, | (1.1) |
with the convention that (−u1−u2)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 (δEp,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 (δFm,||⋅||0,m) is a Banach space. The authors in [12] show that (δEp,m,||⋅||p,m) can not be a Banach space. These facts are counterparts of [5,6,10,18] in m-subharmonic setting.
In Section 2, we shall show that E0,m and δE0,m are closed neither in (δEp,m,||⋅||p,m) nor in (δFm,||⋅||0,m). Moreover we prove that the inclusions E0,m⊆Fm,δE0,m⊆δFm are proper in the space (δFm,||⋅||0,m).
In Section 3, we prove that the convergence in δEp,m implies the convergence in m-capacity (Theorem 3). But the convergence in m-capacity is not a sufficient condition for the convergence in δEp,m (Example 3). Similar results in plurisubharmonic setting have been proved by Czyż in [11].
In plurisubharmonic case, the following proposition was proved in (see [11]). Let B=B(0,1)⊂Cn be the unit ball in Cn. Then the cones E0,m(B) and δE0,m(B) are not closed respectively in (δFm(B),||⋅||0,m) and (δEp,m(B),||⋅||p,m).
Proof. We define
v(z)={ln|z|if m=n,1−|z|2−2nmif 1≤m<n. |
We obtain that Hm(v):=ddc(v)∧βn−m=c(n,m)δ0, where c(n,m) is a constant depending only on n and m, δ0 is the Dirac measure at the origin 0 (see [28]). For each j∈N, define the function vj:B→R∪{−∞} by
vj(z)=max(ajv(z),−bj), |
where aj=12j,bj=1j.
We can see that vj∈E0,m(B), for each j. Therefore, the function uk:=∑kj=1vj belongs to E0,m(B). For k>l we can compute
||uk−ul||m0,m=||k∑j=l+1vj||m=∫BHm(k∑j=l+1vj)=c(n,m)(k∑j=l+1aj)m, | (2.1) |
and
||uk−ul||p+mp,m=||k∑j=l+1vj||p+mp+m=ep,m(k∑j=l+1vj)=∫B(−k∑j=l+1vj)pHm(k∑j=l+1vj)=c(n,m)k∑j1,⋯,jm=l+1[−k∑r=l+1vr(max(tj1,⋯,tjm))]paj1⋯ajm≤c(n,m)k∑j1,⋯,jm=l+1[−uk(max(tj1,⋯,tjm))]paj1⋯ajm≤c(n,m)[k∑j=l+1(−uk(tj))pmaj]m, |
where
tj={(1+bjaj)m2(m−n),if 1≤m<n,e−bjaj,if m=n. |
The last inequality is a consequence of the fact that vj is increasing function for each j. Since
vl(tj)={−1l,if 1≤l≤j,−2jj2l,if l>j, |
we have
−uk(tj)=j∑l=11l+2jjk∑l=j+112l≤j+1. |
Hence
||uk−ul||p+mp,m≤c(n,m)(k∑j=l+1(j+1)pm2j)m. | (2.2) |
Let u:B→R∪{−∞} be defined by u=limk→∞uk. Observe that u is the limit of a decreasing sequence of m-subharmonic functions and u(z)>−∞ on the boundary of the ball B(0,12). Hence u is m-subharmonic. Moreover u∉E0,m(B) since it is not bounded on B, its value is not bounded below at the origin. Equality (2.1) shows that {uk} is a Cauchy sequence in the space δFm(B). Thus the cone E0,m(B) and the space δE0,m(B) are not closed in (δFm(B),||⋅||0,m).
The series ∑∞j=1(j+1)pm2j is convergent by the ratio test. Therefore {uk} is a Cauchy sequence in δEp,m by (2.2). We have proved that the cone E0,m(B) and the space δE0,m(B) are not closed in (δEp,m(B),||⋅||p,m).
The following proposition shows that the closure of the cone E0,m (resp. δE0,m) is strictly smaller than Fm (resp. δFm) in the space (δFm,||⋅||0,m). We have ¯E0,m⊊Fm and ¯δE0,m⊊δFm in the space (δFm,||⋅||0,m).
Proof. The definition of the m-Lelong number of a function v∈SHm(Ω) at a∈Ω is the following
νm,a(v)=limr→0+∫|z−a|≤rddcv∧[ddc(−|z−a|2−2nm)]m−1∧βn−m |
It is easy to see that m-Lelong number is a linear functional on δFm. Moreover, as in [7, Remark 1], for a function φ∈Fm then
νm,a(φ)≤(Hm(φ)({a}))1m≤(Hm(φ)(Ω))1m. |
Hence, for any representation u=u1−u2 of u∈δFm we have
|νm,a(u)|≤(Hm(u1+u2)(Ω))1m. |
This implies that m-Lelong number is a bounded functional on the space δFm. We have shown that m-Lelong number is continuous on the Banach space (δFm,||⋅||0,m). We recall the definition of m-Green function with pole at a
gm,Ω,a(z)=sup{v∈SH−m(Ω):u(z)+|z−a|2−2nm≤O(1) as z→a}. |
The readers can find more properties of m-Green function in [31]. Assume that ¯E0,m=Fm. Then there exists a sequence {uj} in E0,m that converges to gm,Ω,a in the space δFm as j→∞. The m-Lelong number of all uj at a vanishes since uj is bounded, but the m-Lelong number of gm,Ω,a at a is 1. Hence we get a contradiction. Thus, ¯E0,m⊊Fm. By the same argument, if ¯δE0,m=δFm, then there exists a sequence {uj} in E0,m that converges to gm,Ω,a in the space δFm as j→∞, but this is impossible since νm,a(uj)=0.
We are going to recall a Błocki type inequality (see [8]) for the class Ep,m. Similar results for the class Fm 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∈Ep,m and h∈SHm is such that −1≤h≤0. Then
∫Ω(−v)m+pHm(h)≤m!∫Ω(−v)pHm(v). |
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
capm,Ω(E)=sup{∫EHm(u):u∈SHm(Ω),−1≤u≤0}. |
We are going to recall the convergence in m-capacity. We say that a sequence {uj}⊂SHm(Ω) converges to u∈SHm(Ω) in m-capacity if for any ϵ>0 and K⋐Ω then we have
limj→∞capm,Ω(K∩{|uj−u|>ϵ})=0. |
Let {uj}⊂δEp,m be a sequence that converges to a function u∈δEp,m as j tends to ∞. Then {uj} converges to u in m-capacity.
Proof. Replacing uj by uj−u, we can assume that u=0. By the definition of δEp,m, there exist functions vj,wj∈Ep,m such that uj=vj−wj and ep(vj+wj)→0 as j→∞. By [25],
max(ep,m(vi),ep,m(wj))≤ep,m(vj+wj), |
which implies that ep,m(vj),ep,m(wj) tend to 0 as j→∞. Given ϵ>0 and K⋐Ω. For a function φ∈SHm(Ω), −1≤φ≤0, we have
∫{|vj|>ϵ}∩KHm(φ)≤1ϵp+m∫Ω(−vj)p+mHm(φ)≤m!ϵp+mep,m(vj). | (3.1) |
The last inequality comes from Lemma 3. Hence, by taking the supremum over all functions φ in inequality (3.1), we get
capm,Ω({|vj|>ϵ}∩K)≤m!ϵm+pep,m(vj). | (3.2) |
Similarly,
capm,Ω({|wj|>ϵ}∩K)≤m!ϵm+pep,m(wj). | (3.3) |
From (3.2), (3.3) we obtain
capm,Ω({|uj|>ϵ}∩K)≤capm,Ω({|vj|>ϵ2}∩K)+capm,Ω({|wj|>ϵ2}∩K)≤m!2m+pϵm+p(ep,m(vj)+ep,m(wj))→0 as j→∞. |
Hence the sequence {uj} tends to 0 in m-capacity and the proof is finished.
A similar result for the space δFm is proved in [17]. But the convergence in m-capacity is not a sufficient condition for the convergence in the space δEp,m. The following example shows that convergence in m-capacity is strictly weaker than convergences in both δEp,m and δFm. The case m=n has been showed in [11, Example 3.3]. Let v(z) be the function defined in the unit ball in Cn as in the proof of Proposition 2. We define
uj(z)=max(jpmv(z),−1j), vj(z)=max(v(z),−1j) |
Then we have uj,vj∈E0,m(B) for every j, and ep,m(uj)=c(n,m),e0,m(vj)=1. These show that the sequence {uj} and {vj} do not converge to 0 in δEp,m(B) and δFm(B) respectively as j→∞. Moreover, for fixed ϵ>0 and K⋐B there exists j0 such that for all j≥j0 we have
uj=vj=−1j on K. |
This infers that both sets K∩{uj<−ϵ} and K∩{vj<−ϵ} are empty. Hence uj and vj tend to 0 in m-capacity.
The authors would like to thank Rafał Czyż for many valuable comments and suggestions for this manuscript. We are grateful to the referee whose remarks and comments helped to improve the paper.
The authors declare no conflict of interest.
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