An Extended Discrete Hardy-Littlewood-Sobolev Inequality

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the"critical"case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.


Introduction
The well-known Hardy-Littlewood-Sobolev (HLS) inequality states that R n R n f (x)g(y) |x − y| µ dx dy ≤ C p,µ,n f p g q (1.1) for any f ∈ L p (R n ) and g ∈ L q (R n ) provided that 0 < µ < n, 1 < p, q < ∞ with 1 p + 1 q + µ n = 2.
C p,µ,n is the best constant for (1.1), and proved by Lieb [9] that, such C p,µ,n and corresponding maximizing pair (f, g) exists. In particular, Lieb also gave the explicit f abd C p,µ,n in the case p = q. The method Lieb used was to examine the Euler-Lagrange equation that the maximizing pair (f, g) satisfies with some techniques to exploit the symmetry of f . This idea is inherited in [8] and here to find the sharp estimate of best constant of a finite form of HLS in a critical case: p = q = 2, and hence µ = n.
Following the idea that the maximizer of HLS satisfies corresponding E-L equations, the study of the HLS inequality and weighted inequality later generalized by Stein and Weiss [11] is naturally related to the studies of various of integral equations. For recent results, see [2,12,4,3] and a brief summary can be found in [1]. These works have studied regularity and radial symmetry of solutions of such integral systems, and introduced a method of moving plane in an integral form which is proved to be a powerful tool. In [5], the result of integral system corresponding to HLS (1.1) is improved to all cases, i.e. the condition p, q ≥ 1 is removed. In this paper, we do not use the method of moving plane directly, but borrowing its idea, we use a maximum principle to deal with a discrete problem and prove the symmetry of the solution.
First, let's have a look at the discrete and 1-dimensional version of HLS inequality (1.1), the Hardy-Littlewood-Pólya (HLP) Inequality [6]: if a ∈ l p (Z) and b ∈ l q (Z) and where r, s ∈ Z and the constant C depends on p and q only. For this HLP inequality (1.2), let's consider the critical case: p = q = 2 and µ = 2 − 1 p − 1 q = 1, for which the original HLP fails, but we can compromise and get a finite form of HLP. In [8], the inequality is extended to the critical case as: If a, b ∈ l p (Z), then r =s,1≤r,s≤N where λ N is the best constant for (1.3), and λ N = 2 ln N + O(1).

Remark 1.
One of the reasons that we consider discrete version of HLS instead of the original inequality is, when µ = 1 the integrand on the left side of HLS (1.1) is not always integrable on a finite domain for L p functions. So it is not as convenient to extend 1-dimension HLS inequality (1.1) to the critical case in a similar finite form as to extend HLP (1.2) to (1.3).
As for the high dimensional discrete HLS, if a, b ∈ l p (Z n ), and 0 < µ < n, 1 < p, q < ∞ with 1 p where r, s ∈ R n and the constant C depends on p and q only. We can extend (1.4) to a finite form in the corresponding critical case: p = q = 2 and µ = n, in the following way:

So, we have an extension of HLS inequality
where the two statements below holds (ii) ∃!a N = b N and a N 2 = 1 such that the equality in (1.6) holds, and a N ∈ R L + where L = N n .
Let's call the triplet (a N = b N , λ N ) the optimizer of (1.6) since it is unique, and there are some properties of the optimizer. First, as a consequence of the uniqueness, we have symmetry property of the optimizer in the following sense, Second, the optimizer has certain monotone decaying property. For convenience of writing, let's change the range of r i from [1, N ] to [−N, N ], which makes no essential change to the results above, and we have the monotone decaying property for this special case,

and a N has a monotone decaying property from its central element: For
To prove theorem 4, we use the following maximum principle, , if a ∈ R L + then every element of a is positive. Suppose a linear equation: This Maximum Principle follows directly from standard contracting mapping iteration. It is a discrete version of maximum principle analogous to the usual versions in PDE. To see this, let's look at a typical maximum principle: let Ω ⊂ R be an open bounded and connected domain with smooth boundary ∂Ω. Let u ∈ C 2 (Ω) ∩ C(Ω) be a solution of following equation, Then by maximum principle u ≥ 0 in Ω. Actually, by strong maximum principle, Corresponding to strong maximum principle, in theorem 5 if every entry of A is strictly positive, it is easy to see that u ∈ R L + . For more general symmetric linear operators, there is also maximum principle, and one can check [7] for details.

Best Constant Estimate in High Dimension Space
Proof of part (i) of theorem 2. Step 1.
By the definition of λ N , we have J(a, b), i.e. we will maximize J(a, b) under the constraints a 2 = b 2 = 1 (in fact, we use 1 2 a 2 2 = 1 2 b 2 2 = 1 2 ). Therefore, we conduct Euler-Lagrange equations and by compactness: where r, s ∈ R n and 1 ≤ r i , s i ≤ N . For convenience, write (2.2) in matrix form, Left multiply the first equation of (2.3) by a T , the second equation by b T , and by the fact that A is symmetric and a N 2 = b N 2 = 1, one sees that |r−s 0 | n , which leads to Part (ii) will be shown later in section 3.

Lemma 1.
If (a, b, λ N ) satisfies a 2 = b 2 = 1 and makes the equality of (1.6) hold, then a, b ∈ R L + ∪ R L − . Notice that if there is a sign change among the elements of a and b, (a, b) must not be an optimizer since | a i b i | < |a i ||b i |. So the lemma holds, and it means that we can assume the triplet (a N , b N , λ N ) above to satisfy a N , b N ∈ R L + . Now, let's introduce a notation, • The equality of (1.6) holds is called an optimizer or solution of optimization of (1.6).
Obviously, (a N , b N , λ N ) is an optimizer. Next, we are going to prove part(ii) of theorem 2, i.e., the optimizer is unique in positive cone and a N = b N .

Uniqueness of The Optimizer
From previous discussion we see that, an optimizer of (1. where a 2 = b 2 = 1, r = (r i ) ∈ R n , and 1 ≤ r i ≤ N , 1 ≤ i ≤ n. a, b ∈ R L , where L = N n . By lemma 1, we only need to study solution of (3.1) in the positive cone R L + . In the proof, we will use the following simple map, Theorem 6. If (a, b, λ 1 , λ 2 ) is a solution of (3.1), where a, b ∈ R L + , then λ 1 = λ 2 = λ N , and a = b ∈ R L + is unique. Proof.
Step 2. a, b ∈ R L + . Since we have λ 2 a = Ca, where C = A T A and A is a symmetric matrix. So C is non-negative definite.
Proof of part (ii) of theorem 2. The same as the 3rd argument of step 4 above, since an optimizer (a N , b N , λ N ) is a solution of (3.1), part (ii) follows from theorem 6.

Remark 7.
At the time of this writing, thanks to Professor Dongsheng Li of Jiaotong University in Xi'an, we find that uniqueness follows directly from Perron's theorem [10]. So the proof above can be much simplified.

Corollary 1. λ is increasing as N increases.
Proof. Let λ N and A N be a solution and coefficient matrix of (3.1). So, where (ξ N , 0) means (ξ N , 0) ∈ R L and L = (N + 1) n , and arranging ξ N to take the first N n entries and stuffing the rest with zeros. Then calculate in blocks of matrices.

Symmetry of The Optimizer
From section 2 we see the uniqueness of the optimizer of (1.6) in positive cone. So, from this point, if it is clear in context, we use (a = b, λ) instead of (a N , b N , λ N ) when referring the optimizer of (1.6) for simplicity. Proof of theorem 3. From (3.1) we have is also a solution to (3.1). Then, by uniqueness of the solution,ā = a. So, a Φ(r) = a r . Example 4.1. If a is an optimizer, then a (r i ,r ′ ) = a (N −r i +1,r ′ ) for 1 ≤ i ≤ N .

Monotone Property of The Optimizer
For convenience of writing, we change the range of r i 's from 1 ≤ r i ≤ N to −N ≤ r i ≤ N which makes no change to the results above essentially.
Proof of Theorem 4. We are only going to show (1.7) is true for i = 1 for simplicity. Consider d Then by applying theorem 3, we have It is easy to see that entries of A and F are non-negative. So, A : R L + → R L + , where L = N (2N + 1) (n−1) , and F ∈ R L + . Therefore, provided A < 1, then by Theorem 5 (Maximum Principle) we get d (1) ∈ R L + , hence (1.7) is proved. So, the only thing left to prove is A < 1. Notice that if C, D are symmetric matrices such that C, D : R L + → R L + , for some positive integer L, then C ≤ C + D , because where A N is the matrix of (3.1) of the case that −N ≤ r i ≤ N and 1 ≤ i ≤ n, so A N = λ. So, A < 1.