# Estimates for the Norm of Generalized Maximal Operator on Strong Product of Graphs

Authors: Zaryab Hussain; Ghulam Murtaza; Toqeer Mahmood; Jia-Bao Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

LetG=G1×G2×⋯×Gm be the strong product of simple, finite connected graphs, and let ϕ:ℕ⟶0,∞ be an increasing function. We consider the action of generalized maximal operator MGϕ on ℓp spaces. We determine the exact value of ℓp-quasi-norm of MGϕ for the case when G is strong product of complete graphs, where 0<p≤1. However, lower and upper bounds of ℓp-norm have been determined when 1<p<∞. Finally, we computed the lower and upper bounds of MGϕp when G is strong product of arbitrary graphs, where 0<p≤1.


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## Body

## 1. Introduction

We review some of the standard facts on graphs and metric on the graphs. All the graphs considered in this paper are simple, finite, and connected. LetGVG,EG be a graph, where VG is the set of vertices and EG is the set of edges of G. The vertices which are at distance one from any vertex x∈VG are called neighbors of x. The set of neighbors of x∈VG is denoted by NGx. The degree of any vertex x∈VG is the cardinality of the set NGx and is denoted by dGx. The distance between two vertices x and y denoted by dx,y is the length of the shortest path between x and y. For more details on graph theory, we refer the readers to [1–3]. The metric (graph metric) dG:VG×VG⟶ℝ on graph G is defined as(1)dGx,y=distance betweenxandy,where x,y∈VG. This metric space G,dG is called geodesic metric space. For any function f:VG⟶ℝ, the Hardy–Littlewood maximal operator MGx:ℓp⟶ℓp [4–7] is defined as(2)MGxfq=supr≥01Bq,r∑w∈Bq,rfw,where Bq,r=s∈VG:dGq,s≤r is the ball with center q∈VG and radius r on a graph G. It contains all the vertices of G which are at distance atmost r from the vertex q. It is clear from the definition that if r=0, then Bq,r=1, and if r≥1, then Bq,r≥2. The values of metric function dG are natural numbers and radius r≥0; therefore, equation (2) can be written as(3)MGxfq=maxr∈ℕ1Bq,r∑w∈Bq,rfw.The fractional maximal operator [8] on graphs is defined as(4)MG1−t/nfq=maxr∈ℕ1Bq,r1−t/n∑w∈Bq,rfw,where 0≤t≤n. If t=0, then equation (4) reduces to equation (3). For 0<p<∞, the ℓp norm of the Hardy–Littlewood maximal operator is defined as(5)MGxℓp≔supf≠0MGxfℓpfℓp,where fℓp=∑s∈VGfsp1/p.For every functionf:VG⟶ℝ, the generalized maximal operator MGϕ:ℓp⟶ℓp [9, 10] is defined as(6)MGϕfq=maxr∈ℕ1ϕBq,r∑w∈Bq,rfw,where ϕ:ℕ⟶0,∞ is an increasing function. Note that if we take ϕx=x in equation (6), then we get the classical Hardy–Littlewood maximal operator MGx, and if we take ϕx=x1−t/n in equation (5), then we get equation (4).LetKn be complete graph on n vertices. For any vertex q∈VKn, the ball Bq,r with center q and radius r is defined as(7)Bq,r=q,forr=0,VKn,forr≥1.Therefore, the generalized maximal operator on complete graphKn takes the form(8)MKnϕfq=max1ϕ1fq,1ϕn∑v∈VKnfv.For any vertexi∈VG, the Kronecker delta function is defined as(9)δqi=1,q=i,0,q≠i.Soria and Tradacete [6] estimated the norm of maximal operator MGx in the following form.Proposition 1 (see [6]).
(i)
If0<P≤1, then(10)MKnp=1+n−1np1/p.(ii)
If1<p<∞, then(11)1+n−1np1/p≤MKnp≤1+n−1n1/p.For more details on this topic of research, see [4, 8, 10–13]. The main motivation of this paper is from [4–7, 10].The paper is structured as follows. Section2 contains the definitions which are helpful to prove the main results. Section 3 contains the main results; we find the exact value of MKϕp for the case 0<p≤1 and give lower and upper bound when 1<p<∞. An example is given to show that these bounds are not optimal. Finally, Section 4 concludes the study.


## 2. Preliminaries

LetG1, G2, … , Gm be m graphs; then, their strong product G=G1×G2×⋯×Gm is a graph having vertex set,(12)VG=u1,u2,…,um:ui∈Gi∀i=1,2,…,m,and the edge set, which is defined in the following manner; there will be an edge between u1,u2,…,um and v1,v2,…,vm in G if(a)
ui=vi and uj,vj∈EGj,j≠i(b)
ui≠vi and ui,vi∈EGi,∀iExample 1.
LetK2 be complete graph on two vertices. The strong product K=K2×K2×K2 of three K2 graphs is shown in Figure 1.
LetG be the strong product of m graphs. Then, for every function f:VG⟶ℝ, we can consider the generalized maximal operator MGϕ:ℓp⟶ℓp as(13)MGϕfu1,u2,…,um=maxr∈ℕ∑v1∈G1∑v2∈G2…∑vm∈Gmfv1,v2,…,vmϕBG1×BG2×⋯×BGm,where BGi=Bui,r,i=1,2,…,m. The norm MGϕp of the generalized maximal operator is defined as(14)MGϕp=supf≠0MGϕfpfp,where fp=∑v1∈G1∑v2∈G2…∑vm∈Gmfv1,v2,…,vmp1/p.
LetK=Kn1×Kn2×⋯×Knm be the strong product of m complete graphs with n1,n2,…,nm vertices, respectively; then,(15)Bu1,r×Bu2,r×⋯×Bum,r=u1,u2,…,um,forr=0,VK,forr≥1.
For every functionf:VK⟶ℝ, the generalized maximal operator takes the form(16)MKϕfu1,u2,…,um=max1ϕ1fu1,u2,…,um,1ϕn1×n2×⋯×nm∑v1∈Kn1∑v2∈Kn2…∑vm∈Knmfv1,v2,…,vm.
Note that the operatorMKϕ is the smallest in the pointwise ordering among all MG=G1×G2×⋯×Gmϕ, where each Gi is a graph with ni vertices for i=1,2,…,m. That is, for every nonnegative function f and every vertex u1,u2,…,um∈G, we have that(17)MKϕfu1,u2,…,um≤MGϕfu1,u2,…,um.
In particular, if0<p<∞, then(18)MKϕpp≤MGϕpp.
For any vertexu1,u2,…,um∈VK, the m Dirac delta function is defined as(19)Γu1,u2,…,umv1,v2,…,vm=δu1v1.δu2v2…δumvm.
It is easy to check that(20)Γu1,u2,…,umv1,v2,…,vm=0,foruj≠vj,for somej,1,forui=vi,∀i=1,2,…,m.Figure 1
Strong product of threeK2 graphs.


## 3. Main Results

This section details the steps to find the quasi-norm ofMKϕ, for the case 0<p≤1, and to find bounds of MKϕp, for the case of 1<p<∞. Also, we estimate the bounds of MGϕp for 0<p≤1. Moreover, some examples are presented to support the results.Lemma 1.
LetG be the strong product of m graphs, and Ω:ℓp⟶ℓp be a sublinear operator, with 0<p≤1. Then,(21)Ωp=maxu1,u2,…,um∈VGΩΓu1,u2,…,ump.Proof.
SinceΓu1,u2,…,ump=1, therefore Ωp≥maxu1,u2,…,um∈VGΩΓu1,u2,…,ump. To prove the other inequality, let h:VG⟶ℝ, with hp≤1, that is,(22)h=∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,umΓu1,u2,…,um,with ∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,ump≤1. Using Hӧlder’s inequality for 0<p≤1, it follows that(23)Ωhpp=∑v1∈G1∑v2∈G2…∑vm∈GmΩhv1,v2,…,vmp=∑v1∈G1∑v2∈G2…∑vm∈GmΩ∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,umΓu1,u2,…,umv1,v2,…,vmp≤∑v1∈G1∑v2∈G2…∑vm∈Gm∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,umΩΓu1,u2,…,umv1,v2,…,vmp≤∑v1∈G1∑v2∈G2…∑vm∈Gm∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,umΩΓu1,u2,…,umv1,v2,…,vmp=∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,ump∑v1∈G1∑v2∈G2…∑vm∈GmΩΓu1,u2,…,umv1,v2,…,vmp=∑u1∈G1∑u2∈G2…∑um∈Gmau1,u2,…,umpΩΓu1,u2,…,umpp≤maxu1,u2,…,um∈VGΩΓu1,u2,…,umpp.
It completes the proof.Theorem 1.
If0<p≤1, then(24)MKϕp=1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p,and if 1<p<∞, then(25)1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p≤MKϕp≤maxn1×n2×⋯×nmpϕpn1×n2×⋯×nm,1ϕp1+n1×n2×⋯×nm−1n1×n2×⋯×nmp−1ϕpn1×n2×⋯×nm1/p.Proof.
Letf:VK⟶ℝ be a function such that fp=1. Define m Dirac delta function Γu1,u2,…,um, where u1∈VKn1, u2∈VKn2, …, um∈VKnm. Then, for 0<p<∞, we have(26)MKϕΓu1,u2,…,ump=MKϕΓu1,u2,…,umu1,u2,…,umP+∑v1≠u1∑v2≠u2…∑vm≠umMKϕΓu1,u2,…,umv1,v2,…,vmp1/p=1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p.
AsΓp=1, so we have, for 0<p<∞,(27)1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p≤MKϕp.
For0<p≤1, using Lemma 1, we have(28)1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p=MKϕp.
Now, we will prove upper bound for1<p<∞:(29)MKϕfp=∑u1∈BG1∑u2∈BG2⋯∑um∈BGmmax1ϕ1fu1,u2,…,um,1ϕn1×n2×⋯×nm∑v1∈BG1∑v2∈BG2⋯∑vm∈BGmfv1,v2,…,vmp1/p.
After applying Hölder’s inequality, we have(30)MKϕp≤sup∑u1∈BG1∑u2∈BG2⋯∑um∈BGmmax1ϕp1fu1,u2,…,ump,1ϕpn1×n2×⋯×nmn1×n2×⋯×nmp−11/p.
Iffu1,u2,…,ump/ϕp1≤n1×n2×⋯×nmp−1/ϕpn1×n2×⋯×nm for all vertices, then we have(31)MKϕp≤n1×n2×⋯×nmpϕpn1×n2×⋯×nm1/p.
Iffu1′,u2′,…,um′p/ϕp1>n1×n2×⋯×nmp−1/ϕpn1×n2×⋯×nm for some u1′,u2′,…,um′∈VK, then we have(32)MKϕp≤1ϕp1fu1′,u2′,…,um′p+n1×n2×⋯×nm−1n1×n2×⋯×nmp−1ϕpn1×n2×⋯×nm1/p≤1ϕp1+n1×n2×⋯×nm−1n1×n2×⋯×nmp−1ϕpn1×n2×⋯×nm1/p,which completes our arguments.
The graph of the result of Theorem1 is shown in Figure 2, where ϕx=x, n1×n2×⋯×nm is from 4 to 10, and p=2 and p=3.
3D solution region for Theorem1 is shown in Figure 3, where ϕx=x, n1×n2×⋯×nm is from 4 to 12, and p is from 1 to 10.
The graph presented in Figure3 shows the results of Theorem 1 that are not optimal. It is quite difficult task to calculate the exact value of MKϕp for the case 1<p<∞. The following example explains the situation.Figure 2
Estimation forp=2 and p=3.
(a)(b)Figure 3
3D view of estimation forn=4…12 and p=1…10.Example 2.
The estimates we obtained in Theorem1 for 1<p<∞ is not optimal in general. For example, if we take graph K2×K2 and ϕx=x. Consider the function f:1,3,1,4,2,3,2,4⟶ℝ. We suppose that f1,3=f1,2=f2,3, MK2×K2xf1,3=MK2×K2xf1,4=MK2×K2xf2,3=3f1,3+f1,4/4, and MK2×K2xf2,4=f2,4. Then, f2,4≥f1,3. If we denote f1,3/f2,4 by λ, then, for every 0<p<∞, we have(33)MK2×K2xfpfp=33f1,3+f1,4/4p+f1,4p3f1,3p+f1,4p1/p=1433λ+1p+4p3λp+11/p,which leads to(34)MK2×K2xp=14sup0≤λ≤133λ+1p+4p3λp+11/p.
It is easy to see that, for1<p<∞, the supremum is attained at the unique root λp∈0,1 of the equation(35)1+3λp−1=3λp−13λ+1p+4pλp−19λp+3.
In particular, if we takep=2, then we get λ=0.246, and from equation (34), we get MK2×K2x2=1.151. If we calculate it from Theorem 1, we get 1.090≤MK2×K2x2≤1.323. This shows that the estimation in Theorem 1 is not optimal in general for 1<p<∞. Now, in the next theorem, we find the estimates of G.Theorem 2.
LetG be the strong product of m graphs and 0<p≤1; then, we have(36)1ϕp1+n1×n2×⋯×nm−1ϕpn1×n2×⋯×nm1/p≤MGϕp≤1ϕp1+n1×n2×⋯×nm−1ϕp2m1/p.Proof.
Lower bound is trivial. For the upper bound, letu1,u2,…,um∈VG and consider the m Dirac delta function Γu1,u2,…,um. Then, we have(37)MGϕΓu1,u2,…,ump=MGϕΓu1,u2,…,umu1,u2,…,ump+∑v1≠u1∑v2≠u2…∑vm≠umMGϕΓu1,u2,…,umv1,v2,…,vmp1/p=1ϕp1+∑v1≠u1∑v2≠u2…∑vm≠um1ϕBG1×BG2×⋯×BGm∑w1∑w2…∑wmΓu1,u2,…,umw1,w2,…,wmp1/p.
As eachGi is connected, BGi≥2 for each i and radius r≥1. Hence,(38)MGϕΓu1,u2,…,ump≤1ϕp1+n1×n2×⋯×nm−1ϕp2m1/p.
By using Lemma1, we obtain(39)MGϕp≤1ϕp1+n1×n2×⋯nm−1ϕp2m1/p.
If we takeϕx=x and m=1, then Theorems 1 and 2, respectively, yields the same results obtained in [9]. This shows that the results presented in this paper are the generalized form of the results in [6].
We have graph for the result of Theorem2 in Figure 4, where p=0.5, ϕx=x, and n1×n2×⋯×nm is from 4 to 10.
Some particular examples to support the result of Theorem2 are given below.Figure 4
Estimation forp=0.5.Example 3.
LetW5 be a wheel graph on five vertices and consider the strong product K2×W5 of K2 with W5. Take ϕx=x, f=Γ, and p=1. Then, MK2×W5x=2.100.
LetVK2=1,2 and VW5=3,4,5,6,7, where 7 is the central vertex of W5. Now, 6∉NW53 and 5∉NW54. Then, the strong product K2×W5 has a vertex set(40)VK2×W5=1,3,1,4,1,5,1,6,1,7,2,3,2,4,2,5,2,6,2,7.
Note thatK2×W5 has 37 edges and d1,7K2×W5=d2,7K2×W5=9, while all other vertices of this graph have degree 7. Hence,(41)MK2×W5xΓ1,3=1,for1,3,110,for1,6,1,7,2,6,2,7,18,otherwise,with MK2×W5xΓ1,3=2.025. It is easy to see that MK2×W5xΓ1,4=MK2×W5xΓ1,5=MK2×W5xΓ1,6=MK2×W5xΓ2,3=MK2×W5xΓ2,4=MK2×W5xΓ2,5=MK2×W5xΓ2,6=2.025.
Also,(42)MK2×W5xΓ1,7=1,for1,7,110,for2,7,18,otherwise,with MK2×W5xΓ1,7=2.100, MK2×W5xΓ2,7=2.100, and MK2×W5x=2.100.Example 4.
Consider the graph used in Example3. Take ϕx=x2, f=Γ, and p=1. Then, MK2×W5x2=1.135. We have(43)MK2×W5x2Γ1,3=1,for1,3,1100,for1,6,1,7,2,6,2,7,164,otherwise,with MK2×W5x2Γ1,3=1.118 and MK2×W5x2Γ1,4=MK2×W5x2Γ1,5=MK2×W5x2Γ1,6=MK2×W5x2Γ2,3=MK2×W5x2Γ2,4=MK2×W5x2Γ2,5=MK2×W5x2Γ2,6=1.118.
In a similar way, we have(44)MK2×W5x2Γ1,7=1,for1,7,1100,for2,7,164,otherwise,with MK2×W5x2Γ1,7=1.135 and MK2×W5x2Γ2,7=1.135. This implies that MK2×W5x2=1.135.Example 5.
LetS3 be star graph on three vertices and consider the strong product K2×K2×S3. Take ϕx=x, f=Γ, and p=1. Then, MK2×K2×S3x=2.250.
LetVK2=1,2, VK2=3,4, and VS3=5,6,7 with 7 as a central vertex of S3. Then, the strong product K2×K2×S3 is a graph with vertex set(45)VK2×K2×S3=1,3,5,1,3,6,1,3,7,1,4,5,1,4,6,1,4,7,2,3,5,2,3,6,2,3,7,2,4,5,2,4,6,2,4,7.
Note that there are 50 edges in this graph andd1,3,7K2×K2×S3=d1,4,7K2×K2×S3=d2,3,7K2×K2×S3=d2,4,7K2×K2×S3=11, while all the other vertices of the graph have degree 7. We have(46)MK2×K2×S3xΓ1,3,5=1,for1,3,5,18,for1,4,5,2,3,5,2,4,5,112,otherwise,with MK2×K2×S3xΓ1,3,5=2.042. It is easy to see that MK2×K2×S3xΓ1,3,6=MK2×K2×S3xΓ1,4,5=MK2×K2×S3xΓ1,4,6=MK2×K2×S3xΓ2,3,5=MK2×K2×S3xΓ2,3,6=MK2×K2×S3xΓ2,4,5=MK2×K2×S3xΓ2,4,6=2.042.
Also,(47)MK2×K2×S3xΓ1,3,7=1,for1,3,7,112,for1,4,7,2,3,7,2,4,7,18,otherwise,with MK2×K2×S3xΓ1,3,7=2.250. Similarly, MK2×K2×S3xΓ1,4,7=MK2×K2×S3xΓ2,3,7=MK2×K2×S3xΓ2,4,7=2.250. This implies that MK2×K2×S3x=2.250.Example 6.
Consider the graph used in example4, ϕx=1+lnx, f=Γ, and p=1; then, MK2×K2×S31+lnx=4.459.
Here, we have(48)MK2×K2×S31+lnxΓ1,3,5=1,for1,3,5,11+ln8,for1,4,5,2,3,5,2,4,5,11+ln12,otherwise,with MK2×K2×S31+lnxΓ1,3,5=4.271. Similarly, MK2×K2×S31+lnxΓ1,3,6=MK2×K2×S31+lnxΓ1,4,5=MK2×K2×S31+lnxΓ1,4,6=MK2×K2×S31+lnxΓ2,3,5=MK2×K2×S31+lnxΓ2,3,6=MK2×K2×S31+lnxΓ2,4,5=MK2×K2×S31+lnxΓ2,4,6=4.271.
Now,(49)MK2×K2×S31+lnxΓ1,3,7=1,for1,3,7,11+ln12,for1,4,7,2,3,7,2,4,7,11+ln8,otherwise.
So,MK2×K2×S31+lnxΓ1,3,7=4.459. Similarly, MK2×K2×S31+lnxΓ1,4,7=MK2×K2×S31+lnxΓ2,3,7=MK2×K2×S31+lnxΓ2,4,7=4.459. ⇒‖MK2×K2×S31+lnx‖=4.459.
If we take the same conditions which we used in examples3–6 in the result of Theorem 2, then we get 1.900≤MG1×G2x≤3.250, 1.090≤MG1×G2x2≤1.563, 1.917≤MG1×G2×G3x≤2.375, and 4.156≤MG1×G2×G31+lnx≤4.572. This implies that the examples 3–6 verify the result of Theorem 2.


## 4. Conclusion

In this paper, we have considered the action of generalized maximal operator onℓp spaces and calculated the quasi-norm MKϕp for 0<p≤1. We gave the lower bound and upper bound for the quasi-norm MKϕp, where 1<p<∞. Finally, we have proved that 1/ϕp1+n1×n2×⋯×nm−1/ϕpn1×n2×⋯×nm1/p and 1/ϕp1+n1×n2×⋯×nm−1/ϕp2m1/p are the lower bound and upper bound, respectively.



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