Sharp Inequalities for the Hardy–Littlewood Maximal Operator on Finite Directed Graphs

In this paper, we introduce and study the Hardy–Littlewood maximal operator M~ G on a finite directed graph ~ G. We obtain some optimal constants for the `p norm of M~ G by introducing two classes of directed graphs.


Introduction
The best constants for the Hardy-Littlewood maximal inequalities have always been a challenging topic of research. In 1997, Grafakos and Montgomery-Smith [1] first obtained the sharp L p (R) (1 < p < ∞) norm for the one-dimensional uncentered Hardy-Littlewood maximal operator. Since then, the best constants for Hardy-Littlewood maximal operator have been studied extensively. See [2] for the sharp L p (R) (1 < p < ∞) norm of the one-dimensional centered Hardy-Littlewood maximal operator as well as [3][4][5][6][7][8][9] for the optimal constants on the weak (1, 1) norm of the centered Hardy-Littlewood maximal operator. Recently, Soria and Tradacete [10] studied the sharp p -norm for the Hardy-Littlewoood maximal operators on finite connected graphs. It should be pointed out that geometric structure of a graph plays an important role in studying maximal operators on graphs. Given the significance of this operator, it is an interesting and natural question to ask what happens when we consider the directed graphs. It is the purpose of this paper to investigate the optimal constants for the p norm of the Hardy-Littlewood maximal operator in directed graph setting.
Let us now recall some known notations, definitions and backgrounds. Let G = (V, E) be an undirected combinatorial graph with the set of vertices V and the set of edges E. Two vertices x, y ∈ V are called neighbors if they are connected by an edge x ∼ y ∈ E. For a v ∈ V, we use the notation N G (v) to denote the set of neighbors of v. We say that G is finite if |V| < ∞. Here the notation |A| represents the cardinality of A for each subset A ⊂ V. The graph G is called connected if for any distinct x, y ∈ V, there is a finite sequence of vertices {x i } k i=0 , k ∈ N, such that x = x 0 ∼ x 1 ∼ · · · ∼ x k = y. Let d G be the metric induced by the edges in E, i.e., given u, v ∈ V, the distance d G (u, v) is the number of edges in the shortest path connecting u and v. Let B G (v, r) be the ball centered at v, with radius r on the graph G, i.e., B G (v, r) = {u ∈ V : d G (u, v) ≤ r}.
For example, B G (v, r) = {v} if 0 ≤ r < 1 and B G (v, r) = {v} ∪ N G (v) if 1 ≤ r < 2. For a function f : V → R, the Hardy-Littlewood maximal operator M G on G is defined as If G has n (n ≥ 2) vertices, the maximal operator M G can be rewritten by Over the last several years the Hardy-Littlewood maximal operator on graphs has been studied by many authors (see [10][11][12][13][14][15][16]). The Hardy-Littlewoood maximal operator on graphs was first introduced and studied by Korányi and Picardello [15] who used the above operator to explore the boundary behavior of eigenfunctions of the Laplace operator on trees. Subsequently, Cowling, Meda and Setti [12] studied the Hardy-Littlewoood maximal operator on homogeneous trees. Later, some weighted norm inequalities for the Hardy-Littlewoood maximal operators on infinite graphs were investigated by Badr and Martell [11]. Recently, Soria and Tradacete [10] studied the best constants for the p -norm of the Hardy-Littlewoood maximal operators on finite connected graphs. Later, Soria and Tradacete [16] investigated some different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal operator on infinite connected graphs. One can consult [13,14] for the variation properties of the Hardy-Littlewood maximal operator on finite connected graphs.
We now introduce the p spaces on graphs.
be a graph with the set of vertices V and the set of edges E. For 0 < p ≤ ∞, let p (V) be the set of all functions f : By Hölder's inequality, we have On the other hand, it is easy to see that Therefore, the p -boundedness for M G is trivial. Moreover, it follows from (2) that In [10], among other things, Soria and Tradacete studied the sharp constants Precisely, they established the following result.
(i) Let 0 < p ≤ 1. Then, for any graph G with n vertices, we have Moreover, (a) M G p = (1 + n−1 n p ) 1/p if and only if G = K n . Here K n denotes the complete graph with n vertices, i.e., |N K n (v)| = n − 1 for any v ∈ V. (b) M G p = (1 + n−1 2 p ) 1/p if and only if G is isomorphic to S n . Here S n denotes the star graph of n vertices, i.e., there exists a unique v ∈ V such that |N S n (v)| = n − 1 and |N S n (w)| = 1 for every w ∈ V \ {v}.
The main motivation of this paper is to extend Theorem A to the directed graph setting. Let G = (V, E) be a finite graph with the set of vertices V and the set of edges E. Given the set of right (resp., left) neighbors of v. We say that the graph G is a directed graph if every edge in E has only a unique direction and N G, In what follows, we always assume that the graph G = (V, E) with the set of vertices V and the set of edges E. Let B G (v, r) be the ball centered at v, with radius r on the graph G, equipped with the metric d G induced by the edges in E, i.e., given u, v ∈ V, the distance d G (u, v) is the number of edges in a shortest path connecting from u to v, and For a function f : V → R, we consider the Hardy-Littlewood maximal operator on G Naturally, when |V| = n, the maximal operator M G can be redefined in the way that There are some remarks as follows: This type of operator M G has its roots in the ergodic theory in infinite directed graph setting. More precisely, let G 1 = (V 1 , E 1 ), where V 1 = Z and E 1 = {i → i + 1 : i ∈ Z}.
Then M G 1 is the usual one-dimensional one-sided discrete Hardy-Littlewood maximal operator M 1 , i.e., This type of maximal operator M 1 first arose in Dunford and Schwartz's work [17] and was studied by Calderón [18]. (ii) It was pointed out in [10] that the complete graph K n whose maximal operator M K n is the smallest in the pointwise ordering among all graphs with n ≥ 2, but there is no graph G whose maximal operator M G is the largest in the pointwise ordering among all graphs with n ≥ 2. In Section 2, we point out that there is no directed graph G whose maximal operator M G is the smallest or largest in the pointwise ordering among all graphs with n ≥ 2 vertices, which is different from M G . (iii) It should be pointed out that as with M G , the maximal operator M G completely determines the graph G(see Proposition 2).
It is not difficult to see that which together with (1) leads to M G ∞ = 1 and Based on (3) and Theorem A, finding the sharp p -norm of M G is a certainly interesting issue, which is the main motivation of this work. In Section 3 we shall introduce the outward star graph − − → S O,n and the inward star graph −→ S I,n and prove that − − → S O,n and −→ S I,n are the extremal directed graphs attaining, which completely determine the lower and upper estimates of the p -norm for M G in the case 0 < p ≤ 1, respectively (see Theorem 2). We also claim that the p -norm of M G cannot determine the graph G (see Proposition 3). In Section 4, we consider the p -norm for M G in the case 1 < p < ∞. Actually, the case 1 < p < ∞ is more complicated than the case 0 < p ≤ 1, even in the finite undirected graph setting. However, some positive results are discussed. In particular, some sharp estimates of restricted type are given in Section 4.

General Properties for M G
It was pointed out in [10] that there exists a smallest operator M K n , in the pointwise ordering, among all M G , with G a graph of n vertices. However, there is no directed graph G whose maximal operator M G is the smallest or largest in the pointwise ordering among all graphs with n ≥ 2 vertices, which can be seen by the following result.
There exist j ∈ V, a function f : V → R and another directed graph Let us consider the function f : There exists a vertex u ∈ V such that N G,+ (u) = ∅. Let us consider the function f : V → R with f (u) = 1 and f (v) = 2 for all v ∈ N G,+ (u), and It is clear that This gives the claim (ii) by letting j = u.
Using the arguments similar to those used to derive the proof of Theorem 2.4 in [10], one can get the following properties for M G , which tells us that the operator M G completely determines the graph G.
For j ∈ V, the function δ j denotes the Kronecker delta function Then the following are equivalent:

Optimal Estimates for
In this section, we shall present some optimal estimates for M G p with 0 < p ≤ 1. To state the main results, the following lemma is needed.
Before stating our main results, let us introduce two classes of directed graphs. Let −→ S I,n be the inward star graph with n vertices, i.e., there exists a unique u ∈ V such that N−→ S I,n ,+ (u) = ∅ and N−→ S I,n , are said to be isomorphic if there is a permutation of the vertices π : However, the converse is not true (see Proposition 3).
Assume that M G p = (1 + 1 n p ) 1/p . We get by Lemma 1 that We may assume without loss of generality that It follows that Noting that n . Therefore, from (9) we see that there exists j 0 ∈ {2, . . . , n} such that M G δ 1 (j 0 ) = 1 n and M G δ 1 (j) = 0 for all j ∈ V \ {1, j 0 }. Assume that there exist which is a contradiction. Hence, we have N G,+ (j 0 ) = V \ {j 0 } and N G,+ (j) = ∅ for j = j 0 .
So G ∼ − − → S O,n . This completes the proof of part (ii).
It should be pointed out that parts (i) and (ii) in Theorem 2 show that the p -norm of M G can determine the property of graph G. However, the following proposition tells us that the p -norm of M G cannot determine the concrete graph G generally.

Optimal Estimates for M G p with 1 < p < ∞
This section is devoted to presenting some positive results for the M G p with 1 < p < ∞. Before formulating the main results, let us give the following observation, which is useful in our proof.

Lemma 2.
Let (X, · X ), (Y, · Y ) be two normed spaces and let T : X → Y be a sublinear operator, with 0 < p < ∞. Then the following is valid: At first, we present the p -norm for M−→ S I,n with 1 < p < ∞.
Proof. At first, we shall prove part (i). Without loss of generality, we may assume that −→ We now prove M−→ S I,n p < 1 + n − 1 2 1/p , for 1 < p < ∞.
Invoking Lemma 2, one has For a given sequence {a i } n i=1 with ∑ n i=1 a p i = 1 and all a i ≥ 0 (i = 1, . . . , n). We set By the Jensen's inequality we have a i +a 1 This proves (10). Next, we prove part (ii). Let f = ∑ n j=1 a j δ j with each a j ≥ 0. If a j ≥ a 1 for all j = 2, . . . , n, then M−→ S I,n f (i) = f (i) for all i = 1, . . . , n. It follows that M−→ S I,n f 2 (V) ≤ f 2 (V) . Otherwise, there exists j 0 ∈ {2, . . . , n} such that a j 0 < a 1 . Without loss of generality we may assume that a 2 ≤ . . . ≤ a k ≤ a 1 ≤ a k+1 ≤ . . . ≤ a n .
In particular, for fixed k ∈ [2, n], let f = ∑ n j=1 a j δ j , where a j = x 0 for all j ∈ {2, . . . , k} and a j = 1 for all j ∈ {k + 1, . . . , n}. One can easily check that This together with (17) yields the conclusion of part (ii).
As applications of Theorem 3, we get . It is clear that . Therefore, applying Theorem 3 we get We note that the equality in (18)  2 . Therefore, we get by Theorem 3 that It should be pointed out that the equality in (19) is attained if and only if This proves part (ii) and completes the proof.
The following result presents the estimates for M−−→ S O,n p with 1 < p < ∞.
Proof. At first, we shall prove part (i). We may assume without loss of generality that We now prove M−−→ S O,n p < (1 + n −1 ) 1/p , for 1 < p < ∞. (20) Then we have Therefore, to prove (20), it suffices to show that for any sequence {a i } n i=1 with ∑ n i=1 |a i | p = 1. Given a sequence {a i } n i=1 with ∑ n i=1 |a i | p = 1, we consider two cases: (a) If |a 1 | ≥ 1 n ∑ n j=1 |a j |. Then This proves (21) in this case.
(b) If |a 1 | < 1 n ∑ n j=1 |a j |. By the Jensen's inequality, we have This proves (21) in this case. Next, we prove part (ii). Let f = ∑ n i=1 a i δ i with each a i ≥ 0 (i = 1, . . . , n) and ∑ n j=1 a 2 j = 1. Without loss of generality we may assume that Assume that a 1 < 1 n ∑ n j=1 a j . Then we have Here the equality in (22) is attained if and only if a i = α for all 1 ≤ i ≤ k and a j = β for all k + 1 ≤ j ≤ n. Moreover, α, β satisfy kα 2 + (n − k)β 2 = 1 and α < β. Please note that the AM-GM inequality holds: for all x ∈ (0, ∞), where the above equality holds if and only if β = xα. This combines with (22) leads to for all x ∈ (0, ∞). Here the equality in (23) is attained if and only if β = xα. There exists a unique x 0 ∈ (0, ∞) such that where It follows from (22)-(24) that Here the equality in (25) is attained if and only if a i = α > 0 for all 1 ≤ i ≤ k and a j = β > 0 for all k + 1 ≤ j ≤ n. Moreover, β = x 0 α and α < β. This proves part (ii).
To obtain some sharp constants for M G p in the range 1 < p < ∞, we consider the following restricted-type estimate: where G = (V, E). From the definition we see that We have the following estimate for M−→ S I,n p,rest .