Abstract
Let be the strong product of simple, finite connected graphs, and let be an increasing function. We consider the action of generalized maximal operator on spaces. We determine the exact value of -quasi-norm of for the case when is strong product of complete graphs, where . However, lower and upper bounds of -norm have been determined when . Finally, we computed the lower and upper bounds of when is strong product of arbitrary graphs, where .
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. Let be a graph, where is the set of vertices and is the set of edges of . The vertices which are at distance one from any vertex are called neighbors of . The set of neighbors of is denoted by . The degree of any vertex is the cardinality of the set and is denoted by . The distance between two vertices and denoted by is the length of the shortest path between and . For more details on graph theory, we refer the readers to [1–3]. The metric (graph metric) on graph is defined aswhere . This metric space is called geodesic metric space. For any function , the Hardy–Littlewood maximal operator [4–7] is defined aswhere is the ball with center and radius on a graph . It contains all the vertices of which are at distance atmost from the vertex . It is clear from the definition that if , then , and if , then . The values of metric function are natural numbers and radius ; therefore, equation (2) can be written as
The fractional maximal operator [8] on graphs is defined aswhere . If , then equation (4) reduces to equation (3). For , the norm of the Hardy–Littlewood maximal operator is defined aswhere .
For every function , the generalized maximal operator [9, 10] is defined aswhere is an increasing function. Note that if we take in equation (6), then we get the classical Hardy–Littlewood maximal operator , and if we take in equation (5), then we get equation (4).
Let be complete graph on vertices. For any vertex , the ball with center and radius is defined as
Therefore, the generalized maximal operator on complete graph takes the form
For any vertex , the Kronecker delta function is defined as
Soria and Tradacete [6] estimated the norm of maximal operator in the following form.
Proposition 1 (see [6]). (i)If , then(ii)If , then
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. Section 2 contains the definitions which are helpful to prove the main results. Section 3 contains the main results; we find the exact value of for the case and give lower and upper bound when . An example is given to show that these bounds are not optimal. Finally, Section 4 concludes the study.
2. Preliminaries
Let , , , be graphs; then, their strong product is a graph having vertex set,and the edge set, which is defined in the following manner; there will be an edge between and in if(a) and (b) and
Example 1. Let be complete graph on two vertices. The strong product of three graphs is shown in Figure 1.
Let be the strong product of graphs. Then, for every function , we can consider the generalized maximal operator aswhere . The norm of the generalized maximal operator is defined aswhere .
Let be the strong product of complete graphs with vertices, respectively; then,For every function , the generalized maximal operator takes the formNote that the operator is the smallest in the pointwise ordering among all , where each is a graph with vertices for . That is, for every nonnegative function and every vertex , we have thatIn particular, if , thenFor any vertex , the Dirac delta function is defined asIt is easy to check that
3. Main Results
This section details the steps to find the quasi-norm of , for the case , and to find bounds of , for the case of . Also, we estimate the bounds of for . Moreover, some examples are presented to support the results.
Lemma 1. Let be the strong product of graphs, and be a sublinear operator, with . Then,
Proof. Since , therefore . To prove the other inequality, let , with , that is,with . Using Hӧlder’s inequality for , it follows thatIt completes the proof.
Theorem 1. If , thenand if , then
Proof. Let be a function such that . Define Dirac delta function , where , , , . Then, for , we haveAs , so we have, for ,For , using Lemma 1, we haveNow, we will prove upper bound for :After applying Hölder’s inequality, we haveIf for all vertices, then we haveIf for some , then we havewhich completes our arguments.
The graph of the result of Theorem 1 is shown in Figure 2, where , is from 4 to 10, and and .
3D solution region for Theorem 1 is shown in Figure 3, where , is from 4 to 12, and is from 1 to 10.
The graph presented in Figure 3 shows the results of Theorem 1 that are not optimal. It is quite difficult task to calculate the exact value of for the case . The following example explains the situation.
(a)
(b)
Example 2. The estimates we obtained in Theorem 1 for is not optimal in general. For example, if we take graph and . Consider the function . We suppose that , , and . Then, . If we denote by , then, for every , we havewhich leads toIt is easy to see that, for , the supremum is attained at the unique root of the equationIn particular, if we take , then we get , and from equation (34), we get . If we calculate it from Theorem 1, we get . This shows that the estimation in Theorem 1 is not optimal in general for . Now, in the next theorem, we find the estimates of .
Theorem 2. Let be the strong product of graphs and ; then, we have
Proof. Lower bound is trivial. For the upper bound, let and consider the Dirac delta function . Then, we haveAs each is connected, for each and radius . Hence,By using Lemma 1, we obtainIf we take and , 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 Theorem 2 in Figure 4, where , , and is from 4 to 10.
Some particular examples to support the result of Theorem 2 are given below.
Example 3. Let be a wheel graph on five vertices and consider the strong product of with . Take , , and . Then, .
Let and , where 7 is the central vertex of . Now, and . Then, the strong product has a vertex setNote that has 37 edges and , while all other vertices of this graph have degree 7. Hence,with . It is easy to see that .
Also,with , , and .
Example 4. Consider the graph used in Example 3. Take , , and . Then, . We havewith and
In a similar way, we havewith and . This implies that .
Example 5. Let be star graph on three vertices and consider the strong product . Take , , and . Then, .
Let , , and with 7 as a central vertex of . Then, the strong product is a graph with vertex setNote that there are 50 edges in this graph and , while all the other vertices of the graph have degree 7. We havewith . It is easy to see that .
Also,with . Similarly, . This implies that .
Example 6. Consider the graph used in example 4, , , and ; then, .
Here, we havewith . Similarly, .
Now,So, . Similarly, . .
If we take the same conditions which we used in examples 3–6 in the result of Theorem 2, then we get , , , and . 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 on spaces and calculated the quasi-norm for . We gave the lower bound and upper bound for the quasi-norm , where . Finally, we have proved that and are the lower bound and upper bound, respectively.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The second author and third author thank University of Management and Technology, Lahore, and National Textile University, Faisalabad, for their support.