Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0<\alpha<n$, and $\frac 1r+\frac 1s+\frac {n-\alpha}n\geq 2$. Indeed, we can prove that the best constant is attainable in the supercritical case $\frac 1r+\frac 1s+\frac {n-\alpha}n>2$.


Introduction
In the present paper, we investigate the attainability of the best constant of the following discrete Hardy-Littlewood-Sobolev(DHLS for abbreviation) inequality (1.1) i,j,i =j f i g j | i − j | n−α ≤ C r,s,α |f | l r |g| l s , where i, j ∈ Z n , r, s > 1, 0 < α < n, and 1 r + 1 s + n−α n ≥ 2. In fact, DHLS inequality is direct related to the classical Hardy-Littlewood-Sobolev(HLS) inequality (1.2) R n R n f (x)g(y) | x − y | n−α dxdy ≤ C r,s,α f L r g L s for any f ∈ L r (R n ) and g ∈ L s (R n ) provided that 0 < α < n, 1 < r, s < ∞ C r,s,α is the best constant for (1.2). We now provide a proof from (1.2) to get (1.1). One may consider the special case of (1.2) for Obviously, we have Then by (1.2), we get (1.1) immediately for 1 r + 1 s + n−α n = 2. For the supercritical situation, we will present a simple lemma in Section 2 to illustrate it.
It is well-known that (1.2) was studied by a remarkable paper of Lieb [14]. In [14], Lieb proved the existence of the maximizing pair (f, g), i.e. the attainability of the best constant of (1.2). In particular, Lieb also gave the explicit (f, g) and C r,s,α in the case p = q. The method Lieb used was to examine the Euler-Lagrange equation that the maximizing pair (f, g) satisfies. Also, we will analysis the Euler-Lagrange equation corresponding to (1.1). After Lieb [14], Stein and Weiss first completed Lieb's work for weighted HLS inequality. There are also many other works concerning the Eluer-Lagrange equations corresponding to HLS inequality, see [3]- [10]. Now we turn to the discrete situation. For n=1, (1.1) is just the Hardy-Littlewood-Pólya (HLP) inequality [12]. In [13], the authors considered (1.1) in a finite form under the assumptions that r = s = 2, α = 0, As this is a finite summation, (1.4) always holds by Hölder inequality for some constant λ N depending on N . From (1.1), one can see that (1.4) fails for a uniform bound as N → ∞. They proved that λ N = 2 ln N + O(1). Recently, Cheng-Li [11] generalized this result to high dimension for r = s = 2, α = 0. They pointed out that the best constant λ N satisfied here |S n−1 | represents the Lebesgue measure of the n − 1 dimensional unit sphere. The regularities of the maximizing pair (f, g) are also important in analysis. Chen-Li-Zhen [2] use the regularity lifting theorem obtained in [3] to get the optimal summation interval of the solution of the Euler-Lagrange equation of (1.1). They also get some non-existence results.
In our paper, we have the following theorem. Theorem 1.1. If r, s > 1, α ∈ (0, n), 1 r + 1 s + n−α n > 2, then the best constant C r,s,α for DHLS inequality (1.1) is attainable. Remark 1.1. In fact, the assumptions of DHLS inequality (1.1) derived from HLS inequality (1.2) should be 1 r + 1 s + n−α n = 2. Later on, we will give a simple lemma to verify that (1.1) still holds for 1 r + 1 s + n−α n ≥ 2. Remark 1.2. In the above theorem, we only proved the existence of the maximizing pair (f, g) in the supercritical case 1 r + 1 s + n−α n > 2. But, we believe it is also valid for the critical case 1 r + 1 s + n−α n = 2. The main idea to prove Theorem 1.1 is to consider a sequence of DHLS inequalities with finite elements as follows, Also we take f N , g N with |f N | r = |g N | s = 1 satisfy that We want to prove f N , g N → f, g strongly in l r , l s respectively. If it is right, we have proved Theorem 1.1. Unfortunately, this is always false as we can see DHLS inequality (1.1) is invariant under translation. We should use the Concentration Compactness ideas introduced by P.L. Lions. The following theorem is important for using Concentration Compactness ideas.
here c r,s,α is a uniform constant independent of N .

Theorem 1.2 tells us that after a translationf
This excludes the casef N ,ḡ N → 0. We will have after translation, The present paper is organized as follows. In Section 2, we will prove Theorem 1.2 and the first part of Theorem 1.3. This is the main part of this paper and the Concentration Compactness ideas is used in this section. We will prove Theorem 1.1 in Section 3 and the last part of Theorem 1.3 in Section 4.

Concentration Compactness property
This section is devoted to prove Theorem 1.2. First we shall illustrate (1.1) with the following lemma.
Lemma 2.1. Suppose a ∈ l p (Z n ), then |a| l q ≤ |a| l p for ∀q ≥ p.
Proof. For simplicity, we may assume |a| l p = 1 which means |a i | ≤ 1, i ∈ Z n . This implies that i∈Z n This ends the proof of the present lemma.
By Lemma 2.1, one can directly get (1.1) from the critical case 1 r + 1 s + n−α n = 2. A directly computation easily yields the Euler-Lagrange equation for (1.5): If we multiply the first equation of (2.1) by f N i , the second equation by g N i and sum up both sides, we can find out that r λ = s µ = C r,s,α,N . From the definition of C r,s,α,N , it is easy to see that C r,s,α,N > 0 is non-decreasing with respect to N . Moreover, we have the following lemma which corresponds to the first part of Theorem 1.3. Proof. It is obvious that lim N →∞ C r,s,α,N ≤ C r,s,α .
Now we choose a maximizing sequence f (m) , g (m) > 0 with |f (m) | l r = |g (m) | l s = 1 such that Then we can choose N m large enough depending on m such that Here f From the cut-off above, we have Passing m → ∞, we get the desired result.
By Lemma 2.2, it is true that for some uniform constants c 0 , C 0 . Therefore without loss of generality, we may assume C r,s,α,N = 1 in the proof of Theorem 1.2, since we only use the uniform up bound and lower bound of C r,s,α,N . The proof for Theorem 1.2: Taking the equation of f N for instance, by (2.1) we have Here 0 < < 1 is a parameter to be determined later. This means that Now we define an operator T satisfying: Then by DHLS inequality, we have |T f | l p ≤ C|f | l q for 1 q + n−α n = 1 + 1 p . We take p = r r−1 , q = s 1− , then we can get the righthand side of (2.3), To guarantee (2.4), we need r − 1 r By the assumption of Theorem 1.2, we see > 0. Also, as 1 r + n−α n < 2, we must have < 1. From (2.4), one can get Or we have max The proof for the lower bound of max |k|≤N (f N k ) is just the same, we omit the details here.2 1 r + 1 s + n−α n > 2 plays an important role. This is also the reason why we can't prove the existence of the maximizing pair (f, g) in the critical case in the present paper.
3. The existence of maximizing pair (f, g) Since Theorem 1.2, we can have g, weakly in l r , l s respectively and f (N ) → f, g (N ) → g, pointwise in Z n . It is easy to see that f 0 , g 0 ≥ c > 0 and |f | l r , |g| l s ≤ 1. Now we can have the following lemma.
Proof. We only need to show the first part of (3.1) is right. As f i > 0, we can see f (N ) i > 0 for N large, then for any fixed M , We can pass the limit in the left-hand side and the first part of right-hand side of (3.2) since it is finite summation.
In getting the last inequality, we have used 1 r + 1 s + n−α n > 2 which means 1 s > α n . Lemma 3.2.
Proof. If it's not true, by Lemma 3.1, we can easily see that 0 < |f | r l r = |g| s l s < 1. Set Then which is a contradiction to the definition of best constant. The last inequality follows In fact, Lemma 3.2 implies Theorem 1.1 with (f, g) as the maximizing pair. Although in passing the limit to get the maximizing pair (f, g), we may only have f i , g i > 0 for i ∈ Ω ⊂ Z n . But in fact, as we know (f, g) are maximizing pair, they should satisfy (3.1) for all i ∈ Z n which means f, g > 0.
4. The strong convergence off N ,ḡ N This section is denoted to prove the second part of Theorem 1.3. The following lemma is a special case of Theorem 2 in [1]. We provide a simple proof here. Hence for N ≥ max(N M , N ) = N 0 , |i|>M |f N i | r ≤ 3 |f | r r .

Now we have
for N ≥ N 0 . Passing → 0, we have finished the proof of the present lemma.
The second part of Theorem 1.3 is the direct conclusion of Lemma 3.2 and Lemma 4.1.