Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 28, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.28 DISCRETE STEIN-WEISS INEQUALITIES CHUNHONG LI, TIANTIAN ZHOU Abstract. This article concerns the discrete Stein-Weiss inequalities with finite terms and with infinite terms. Such inequalities can be used to study the discrete Coulomb energy, nonlinear problems appearing the crystal lattice theory and graphs in neural networks. We give the limit relations between their best constants and between their extremal sequences. In addition, we obtained analogous conclusions for the reversed discrete Stein-Weiss inequality. 1. Introduction The well-known double weighted Hardy-Sobolev-Pólya inequality states that [7, Theorem 401]∣∣∣ ∫ ∞ 0 ∫ ∞ 0 f(x)g(y), dx, dy xα|x− y|λyβ ∣∣∣ ≤ K0∥f∥Lr(0,∞)∥g∥Ls(0,∞), (1.1) for all (f, g) ∈ Lr(0,∞)× Ls(0,∞), where K0 ∈ (0,∞) is a constant, and 1 < r, s < ∞, 1/r + 1/s ≥ 1, 1/r + 1/s+ (λ+ α+ β) = 2, α < 1− 1/r, β < 1− 1/s, α+ β ≥ 0, α+ β > 0 if 1 r + 1 s = 1. Afterwards, Stein and Weiss obtained the higher dimensional results [15]∣∣∣ ∫ Rn ∫ Rn f(x)g(y), dx, dy |x|α|x− y|λ|y|β ∣∣∣ ≤ K1∥f∥Lr(Rn)∥g∥Ls(Rn), (1.2) for all (f, g) ∈ Lr(Rn)× Ls(Rn), where K1 ∈ (0,∞) is the best constant, and 1 < r, s < ∞, 0 < λ < n, 0 ≤ α+ β < n− λ, 1− 1/r − λ/n < α/n < 1− 1/r, 1 r + 1 s + λ+ α+ β 2 .. When α = β = 0, (1.2) is reduced to the Hardy-Littlewood-Sobolev inequality (se [14, Theorem 1 in Chapter 5])∣∣∣ ∫ Rn ∫ Rn f(x)g(y), dx, dy |x− y|λ ∣∣∣ ≤ K1∥f∥Lr(Rn)∥g∥Ls(Rn), ∀(f, g) ∈ Lr(Rn)× Ls(Rn), 2020 Mathematics Subject Classification. 26D15, 40B05. Key words and phrases. Discrete Stein-Weiss inequality; best constant; reversed discrete Stein-Weiss inequality; extremal sequence. ©2025. This work is licensed under a CC BY 4.0 license. Submitted September 9, 2024. Published March 14, 2025. 1 2 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 where 1 < r, s < ∞, 0 < λ < n, 1 r + 1 s + λ n = 2. Lieb applied this inequality to give the optimal estimate of upper bound of the Coulomb energy appearing in the Thomas-Fermi model (see [12]). Based on this inequality, Huang, Li and Yin obtained the discrete inequality (cf. [8])∑ i,j∈Zn,i̸=j |fi||gj | |i− j|λ ≤ K2∥f∥lr(Zn)∥g∥ls(Zn), ∀(f, g) ∈ lr(Zn)× ls(Zn), (1.3) where K2 ∈ (0,∞) is the best constant, f = (fi)i∈Zn , g = (gj)j∈Zn , and n ≥ 1, 0 < λ < n, min{r, s} > 1, 1 r + 1 s + λ n ≥ 2. When 1/r + 1/s+ λ/n > 2, they proved that K2 is attainable. When n = 1, (1.3) is the Hardy-Littlewood-Pólya inequality [7, Theorem 381]). When we cut off f = (fi)i∈Zn N and g = (gi)i∈Zn N , inequality (1.3) is reduced to∑ |i|≤N,|j|≤N,i̸=j |fi||gj | |i− j|λ ≤ KN∥f∥lr(Zn N )∥g∥ls(Zn N ), ∀(f, g) ∈ lr(Zn N )×ls(Zn N ), (1.4) where KN ∈ (0,∞) is the best constant, and Zn N := {i ∈ Zn; |i| ≤ N}. Paper [8] shows that K2 can be approximated by KN , and also shows the convergence relation between the extremal sequences when N → ∞. In 2011, Li and Villavert studied (1.4) in the critical case of λ = n = 1 (cf. [11]). They obtained the upper and the lower bounds of KN . Afterwards, Cheng and Li extended those results to the case of λ = n ≥ 2 (cf. [4]). In addition, the properties of extremal sequences were studied. Their work shows that (1.4) holds under more relaxed constraints on r, s and λ. In addition, paper [16] estimated the best constant of a discrete inequality with special double weights. In this article, we prove the discrete Stein-Weiss inequalities, and study the best constant and the extremal sequences. Theorem 1.1. Let n ≥ 1, min{r, s} > 1 and 0 < λ < n. If 0 ≤ α+ β < n− λ, α/n < 1− 1/r, β/n < 1− 1/s, 1 r + 1 s + λ+ α+ β n ≥ 2, we can find C ∈ (0,∞) such that∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ≤ C∥f∥lr(Zn 0 ) ∥g∥ls(Zn 0 ) , ∀(f, g) ∈ lr(Zn 0 )× ls(Zn 0 ). (1.5) Here Zn 0 := Zn \ {0}. We denote the best constant of (1.5) by L. Namely, L = sup (f,g)∈S { ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β } , where S := {(f, g) ∈ lr(Zn 0 )× ls(Zn 0 ); ∥f∥lr(Zn 0 ) = ∥g∥ls(Zn 0 ) = 1}. We also have the discrete inequality with finite terms. EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 3 Theorem 1.2. Let n ≥ 1, min{r, s} > 0 and λ > 0. Then there exists C ∈ (0,∞) such that ∑ i,j∈Zn 0,N ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ≤ C∥f∥lr(Zn 0,N )∥g∥ls(Zn 0,N ), (1.6) for all (f, g) ∈ lr(Zn 0,N )× ls(Zn 0,N ). Here Zn 0,N := {i ∈ Zn; 1 ≤ |i| ≤ N}. We denote the best constant of (1.6) by LN . Thus, LN = max { ∑ i,j∈Zn 0,N ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ; ∥f∥lr(Zn 0,N ) = ∥g∥ls(Zn 0,N ) = 1 } . (1.7) The following theorem shows the convergence relation between the best con- stants. Theorem 1.3. Under the assumptions of Theorem 1.1, we have lim N→∞ LN = L. Next, we consider the convergence relation between the extremal sequences. Theorem 1.4. Let n ≥ 1, min{r, s} > 1 and 0 < λ < n. If 0 ≤ α + β < n − λ, α/n < 1− 1/r, β/n < 1− 1/s, and 1/r + 1/s+ (λ+ α+ β)/n > 2, then (i) L is attainable. Namely, we can find (f∗, g∗) ∈ S such that L = ∑ i,j∈Zn 0 ,i̸=j |f∗ i ||g∗j | |i|α|i− j|λ|j|β . (ii) Denote the extremal sequences of (1.7) by (fN ∗ , gN∗ ). Then there exists a subsequence of (fN ∗ , gN∗ ) denoted by itself such that lim N→∞ (fN ∗ , gN∗ ) = (f∗, g∗), in lr(Zn 0 )× ls(Zn 0 ). Remark 1.5. Inequality (1.5) can also be found in [3, (1.3)]. There the authors did not provide a proof. Remark 1.6. Compared to Theorem 1.1, Theorem 1.2 has more relaxed constraints on r, s, α, β and λ. The reason is that the inequality only contains finite terms. Moreover, i, j in the left hand side of (1.6) can belong to Zn N (instead of Zn 0,N ) when α and β are not larger than zero. In 2015, Dou and Zhu [5] obtained the reversed Hardy-Littlewood-Sobolev in- equality (see also [1, 9, 13])∫ Rn ∫ Rn |f(x)||g(y)|, dx, dy |x− y|λ ≥ K3∥f∥Lr(Rn)∥g∥Ls(Rn), (1.8) for all (f, g) ∈ Lr(Rn)× Ls(Rn), where K3 ∈ (0,∞) is a constant, and n ≥ 1, r, s ∈ (n/(n− λ), 1), λ < 0, 1 r + 1 s + λ n = 2, Based on this inequality, [10] shows the discrete inequalities with infinite terms∑ i,j∈Zn,i̸=j |fi||gj | |i− j|λ + ∑ j∈Zn |fj ||gj | ≥ K4∥f∥lr(Zn)∥g∥ls(Zn), (1.9) 4 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 where K4 ∈ (0,∞) is a constant, and n ≥ 1, r, s ∈ (n/(n− λ), 1), λ < 0, 1 r + 1 s + λ n ≤ 2. When n = 1, [6] obtained the discrete inequalities with finite terms, and gave the limit relation between its best constant and K4. In 2018, Chen, Liu, Lu and Tao proved the reversed Stein-Weiss inequality [2, Theorem 1] ∫ Rn ∫ Rn |f(x)||g(y)|, dx, dy |x|α|x− y|λ|y|β ≥ K5∥f∥Lr(Rn)∥g∥Ls(Rn), (1.10) for all (f, g) ∈ Lr(Rn)× Ls(Rn), where K5 ∈ (0,∞) is a constant, and n ≥ 1, r, s ∈ (0, 1), λ < 0, α ∈ (−n(1− r)/r, 0], β ∈ (−n(1− s)/s, 0], 1 r + 1 s + λ+ α+ β n = 2. Now, we give the corresponding results of the discrete case. Theorem 1.7. Let n ≥ 1, r, s ∈ (0, 1), λ < 0, α ∈ (−n(1− r)/r, 0], β ∈ (−n(1− s)/s, 0], 1 r + 1 s + λ+ α+ β n ≤ 2. Then there exists C ∈ (0,∞) such that for any (f, g) ∈ lr(Zn 0 )× ls(Zn 0 ),∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0 |fj ||gj | |j|α+β ≥ C∥f∥lr(Zn 0 ) ∥g∥ls(Zn 0 ) . (1.11) Theorem 1.8. Let n ≥ 1, min{r, s} > 0 and λ < 0. Then there exists C ∈ (0,∞) such that for any (f, g) ∈ lr(Zn N )× ls(Zn N ),∑ i,j∈Zn 0,N ,i̸=j |fi||gj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0,N |fj ||gj | |j|α+β ≥ C∥f∥lr(Zn 0,N )∥g∥ls(Zn 0,N ). (1.12) We denote the best constant of (1.12) by QN . Thus, QN := min (f,g)∈S(N) { ∑ i,j∈Zn 0,N ,i̸=j |fi||gj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0,N |fj ||gj | |j|α+β } . (1.13) Theorem 1.9. Under the assumptions of Theorem 1.7, we have lim N→∞ QN = Q, where Q is the best constant in (1.11), that is, Q := inf (f,g)∈S { ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0 |fj ||gj | |j|α+β } . (1.14) With the above conclusions in mind, we need to consider whether Q can be attainable. EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 5 Theorem 1.10. Let n ≥ 1, r, s ∈ (0, 1), λ < 0, α ∈ ((r − 1)n/r, 0] and β ∈ ((s− 1)n/s, 0] satisfy 1 r + 1 s + λ+ α+ β n < 2. (1.15) Then Q is attainable. Namely, there exists (f∗, g∗) ∈ S such that Q = ∑ i,j∈Zn 0 ,i̸=j |f∗ i ||g∗j | |i|α|i− j|λ|j|β + ∑ j∈Zn 0 |f∗ j ||g∗j | |j|α+β . Meanwhile, the subsequences of extremal sequences (fN , gN ) denoted by itself in (1.13) satisfies lim N→∞ ∥fN∥lr(Zn N ) = ∥f∗∥lr(Zn), lim N→∞ ∥gN∥ls(Zn N ) = ∥g∗∥ls(Zn), (1.16) lim N→∞ ∥f∗ − (1− εN )f̃N∥lr(Zn) = lim N→∞ ∥g∗ − (1− εN )g̃N∥ls(Zn) = 0, (1.17) as long as limN→∞ εN = 0. Here f̃N and g̃N after certain translations respectively. 2. Inequalities with infinite terms In this section, we prove Theorems 1.1 and 1.7. Proof of Theorem 1.1. Assume (fj)j∈Zn 0 ∈ lr(Zn 0 ) and (gj)j∈Zn 0 ∈ ls(Zn 0 ). Let f(x) ≡ fj , g(x) ≡ gj , when |x− j| < 1/3 for j ̸= 0, f(x) = g(x) = 0, otherwise. (2.1) Therefore, ∥f∥rLr(Rn) = ∑ j∈Zn 0 ∫ |x−j|≤1/3 |fj |rdx = |Sn−1| 3nn ∑ j∈Zn 0 |fj |r. (2.2) Here Sn−1 ⊂ Rn is the unit sphere. Similarly, ∥g∥sLs(Rn) = |Sn−1| 3nn ∥g∥sls(Zn 0 ) . (2.3) When |x− i| ≤ 1/3, it follows that |i|−1/3 ≤ |x| ≤ |i|+1/3. In addition, i ∈ Zn 0 implies |i| ≥ 1. Therefore, 3 4 ≤ |i| |i|+ 1/3 ≤ |i| |x| ≤ |i| |i| − 1/3 ≤ 3 2 . (2.4) Similarly, j ∈ Zn 0 and |y − j| ≤ 1/3 imply 3 4 ≤ |j| |y| ≤ 3 2 . (2.5) When |x−i| ≤ 1/3 and |y−j| ≤ 1/3, we have |i−j|−2/3 ≤ |x−y| ≤ |i−j|+2/3. In addition, i ̸= j implies |i− j| ≥ 1. Therefore, 1 3 ≤ |x− y| |i− j| ≤ 5 3 . (2.6) 6 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Using (2.4)-(2.6), we obtain∫ Rn ∫ Rn |f(x)g(y)|, dx, dy |x|α|x− y|λ|y|β ≥ ∑ i,j∈Zn 0 ∫ B1/3(i) ∫ B1/3(j) |fi||gj |, dx, dy |x|α|x− y|λ|y|β ≥ ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ∫ B1/3(i) ∫ B1/3(j) |i|α|i− j|λ|j|β |x|α|x− y|λ|y|β , dx, dy ≥ C(α, β) ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ∫ B1/3(0) ∫ B1/3(0) |i− j|λ, dx, dy |i− j + x− y|λ ≥ C(α, β)C(λ) ( |Sn−1| 3nn )2 ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β , (2.7) where C(α, β) = min {(3 4 )α+β , (3 4 )α(3 2 )β , (3 2 )α(3 4 )β} , C(λ) = [ 1 + (5 3 )λ]−1 . Inserting (2.7) and (2.2)-(2.3) into (1.2), we obtain (1.5) with 1 r + 1 s + λ+α+β n = 2. Noting [3, Lemma 2.2], we can see that (1.5) with 1 r + 1 s + λ+α+β n > 2 still holds. □ Proof of Theorem 1.7. Assume (fj)j∈Zn 0 ∈ lr(Zn 0 ) and (gj)j∈Zn 0 ∈ ls(Zn 0 ). We take (f(x), g(x)) as in (2.1) f(x) ≡ fj , g(x) ≡ gj , when |x− j| < 1/3 for j ̸= 0, f(x) = g(x) = 0, otherwise. Therefore, by an analogous argument to the one in (2.7), we obtain by (2.4)-(2.6) that ∫ Rn ∫ Rn |f(x)g(y)| |x|α|x− y|λ|y|β , dx, dy = ∑ i,j∈Zn 0 ∫ B1/3(i) ∫ B1/3(j) |fi||gj | |x|α|x− y|λ|y|β , dx, dy ≤ C ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β ∫ B1/3(0) ∫ B1/3(0) |i− j|λ |i− j + x− y|λ , dx, dy + C ∑ j∈Zn 0 |fj ||gj | |j|α+β ∫ B1/3(0) ∫ B1/3(0) |x− y|−λ, dx, dy ≤ C ( ∑ i,j∈Zn 0 ,i̸=j |fi||gj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0 |fj ||gj | |j|α+β ) . Inserting this result and (2.2)-(2.3) into (1.10), we obtain (1.11) with 1 r + 1 s + λ+α+β n = 2. Noting [3, Lemma 2.2], we can see that (1.11) with 1 r + 1 s + λ+α+β n < 2 still holds. □ EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 7 3. Inequalities with finite terms In this section, we prove Theorems 1.2 and 1.8. Proof of Theorem 1.2. Let f0 = g0 = 0. We write ai = |fi|, bi = |gi|, and a = (ai)i∈Zn 0,N , b = (bi)i∈Zn 0,N . We set RN + := {x = (x1, x2, · · · , xN );xi ≥ 0, i = 1, 2, · · · , N}, and the multivariate function E1(a, b) = ∑ i,j∈Zn 0,N ,i̸=j aibj |i|α|i− j|λ|j|β . Since S(N) := { (a, b) : ∥a∥lr(Zn 0,N ) = ∥b∥ls(Zn 0,N ) = 1 } is bounded and closed in RN +×RN + , there exists a maximizer (a(N), b(N)) of E1(a, b) in S(N). We write J1(a, b) = E1(a, b)− λ1(∥a∥lr(Zn 0,N ) − 1)− λ2(∥b∥ls(Zn 0,N ) − 1), where λ1 and λ2 are Lagrange multipliers. Applying the theory of the constrained extremum, both the partial derivatives of J1(a, b) are equal to zero at (a(N), b(N)), that is, [ d dt J1(a(N) + ta, b(N)) ] t=0 = [ d dt J1(a(N), b(N) + tb) ] t=0 = 0, for all (a, b) ∈ RN + × RN + and t ∈ R. This means λ1a(N)r−1 i = 1 |i|α ∑ j∈Zn 0,N ,j ̸=i b(N)j |i− j|λ|j|β , λ2b(N)s−1 i = 1 |i|β ∑ j∈Zn 0,N ,j ̸=i a(N)j |i− j|λ|j|α . Multiply the two equations above by a(N)i and b(N)i respectively and sum for i. Then noting that (a(N), b(N)) ∈ S(N) is the maximizer, we can see that λ1 = λ2 = LN . Namely, LNa(N)r−1 i = 1 |i|α ∑ j∈Zn 0,N ,j ̸=i b(N)j |i− j|λ|j|β , LNb(N)s−1 i = 1 |i|β ∑ j∈Zn 0,N ,j ̸=i a(N)j |i− j|λ|j|α . (3.1) In view of (a(N), b(N)) ∈ S(N), we have (a(N), b(N)) ̸= (0, 0). Therefore, from (3.1) we can deduce by a contradiction argument that LN > 0. (3.2) Without loss of generality, assume that iN ∈ Zn 0,N satisfies a(N)iN = max 1≤|i|≤N {a(N)i, b(N)i}. 8 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Noting (3.2), from (3.1)1 we can find a positive constant CN which only depends on iN and N such that LN a(N)r−2 iN ≤ 1 |iN |α ∑ 1≤|j|≤N,j ̸=iN 1 |j − iN |λ|j|β ≤ CN , (3.3) which implies LN ≤ a(N)2−r iN CN . (3.4) In view of ∑ 1≤|i|≤N a(N)ri = 1, we obtain( ∑ 1≤|i|≤N 1 )−1 ≤ a(N)riN ≤ 1. (3.5) Inserting this result into (3.4), we see LN < ∞. The proof is complete. □ Proof of Theorem 1.8. Let f0 = g0 = 0. We write ai = |fi|, bi = |gi|, and a = (ai)i∈Zn 0,N , b = (bi)i∈Zn 0,N . We set E2(a, b) = ∑ i,j∈Zn 0,N ,i̸=j aibj |i|α|i− j|λ|j|β + ∑ j∈Zn 0,N ajbj |j|α+β . Since S(N) is bounded and closed in RN +×RN + , there exists a minimizer (a(N), b(N)) of E2(a, b) in S(N). We write J2(a, b) = E2(a, b)− λ3(∥a∥lr(Zn 0,N ) − 1)− λ4(∥b∥ls(Zn 0,N ) − 1), where λ3 and λ4 are Lagrange multipliers. Applying the theory of the constrained extremum, both the partial derivatives of J2(a, b) are equal to zero at (a(N), b(N)), that is, [ d dt J2(a(N) + ta, b(N)) ] t=0 = [ d dt J2(a(N), b(N) + tb) ] t=0 = 0, for all (a, b) ∈ RN + × RN + and t ∈ R. This means λ3 = a(N)1−r i ( 1 |i|α ∑ j∈Zn 0,N ,j ̸=i b(N)j |i− j|λ|j|β + b(N)i |i|α+β ) , λ4 = b(N)1−s i ( 1 |i|β ∑ j∈Zn 0,N ,j ̸=i a(N)j |i− j|λ|j|α + a(N)i |i|α+β ) . Multiply two equations above by a(N)ri and b(N)si respectively and sum for i. Then noting that (a(N), b(N)) ∈ S(N) is the maximizer, we can see that λ3 = λ4 = QN . Namely, QN = a(N)1−r i ( 1 |i|α ∑ j∈Zn 0,N ,j ̸=i b(N)j |i− j|λ|j|β + b(N)i |i|α+β ) , QN = b(N)1−s i ( 1 |i|β ∑ j∈Zn 0,N ,j ̸=i a(N)j |i− j|λ|j|α + a(N)i |i|α+β ) . (3.6) Noting (a(N), b(N)) ̸= (0, 0) (because of (a(N), b(N)) ∈ S(N)), from (3.6) we can see QN > 0. (3.7) EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 9 In addition, QN is decreasing with respect to N , and hence QN ≤ Q1. Write e = (1, 0, . . . , 0) ∈ Zn 0,N , and take āi = b̄i = { 1, when i = e, 0, when i ∈ Zn 0,N \ {e}. Therefore, (ā, b̄) ∈ S(1), and hence Q1 ≤ āeb̄e = 1. Thus, QN has an upper bound QN ≤ 1. (3.8) Thus, (1.12) holds for C = QN . □ Remark 3.1. Under assumptions (3.1)-(3.2) and (3.6)-(3.7), we obtain a(N)i > 0 and b(N)i > 0 for i = 1, 2, . . . , N, which implies lim N→∞ a(N)i = lim N→∞ b(N)i = 0, for i = 1, 2, . . . , N. (3.9) However, we show that (3.9) is not valid by Lemmas 5.1 and 5.2. 4. Convergence of the best constants In this section, we prove Theorems 1.3 and 1.9. Proof of Theorem 1.3. Since S(N) ⊂ S(N +1) ⊂ S, LN is monotonically increasing with respect to N and LN ≤ L, (4.1) as long as the assumptions in Theorem 1.1 hold. Thus, lim N→∞ LN exists. (4.2) From the definition of L, we can find a maximizing sequence (fm, gm) ∈ S such that ∑ i,j∈Zn 0 ,i̸=j |fm i ||gmj | |i|α|i− j|λ|j|β ≥ L− 1 m . In view of (1.5), the series of the left hand side in the result above is convergent. Therefore, we can find Nm → ∞ (when m → ∞) such that∑ i,j∈Zn 0,Nm ,i̸=j |fm i ||gmj | |i|α|i− j|λ|j|β ≥ L− 2 m . (4.3) On the other hand, ∥fm∥lr(Zn N ) ≤ 1 and ∥gm∥ls(Zn N ) ≤ 1. Therefore, by (1.6) with C = LN and (4.1) we obtain ∑ i,j∈Zn 0,Nm ,i̸=j |fm i ||gmj | |i|α|i− j|λ|j|β ≤ L. Combining with (4.3) and letting m → ∞, we can see LNm → L when Nm → ∞. In view of (4.2), we can complete the proof. □ 10 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Proof of Theorem 1.9. Under the assumptions of Theorem 1.7, we see easily that QN ≥ Q. (4.4) Since QN is decreasing, we know that limN→∞ QN exists. Since Q is the best constant in (1.11), we can find a minimizing sequence (fm, gm) ∈ S such that∑ i,j∈Zn 0 ,i̸=j |fm i ||gmj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0 |fm j ||gmj | |j|α+β ≤ Q+ 1 m . This implies that the series of the left hand side in the result above is convergent. Therefore, we can find Nm → ∞ (when m → ∞) such that∑ i,j∈Zn 0,Nm ,i̸=j |fm i ||gmj | |i|α|i− j|λ|j|β + ∑ j∈Zn 0,Nm |fm j ||gmj | |j|α+β ≤ Q+ 2 m . (4.5) On the other hand, (fm, gm) ∈ S implies ∥fm∥rlr(Zn 0,Nm ) ≥ 1− 1/m, ∥gm∥sls(Zn 0,Nm ) ≥ 1− 1/m, (4.6) as long as Nm > m is sufficiently large. Therefore, noting that( fm ∥fm∥lr(Zn 0,Nm ) , gm ∥gm∥ls(Zn 0,Nm ) ) ∈ S(Nm), from (4.5) and (4.6) we deduce that QNm ≤ ∑ i,j∈Zn 0,Nm ,i̸=j |fm i ||gmj | ∥fm∥lr(Zn 0,Nm )∥gm∥ls(Zn 0,Nm )|i|α|i− j|λ|j|β + ∑ j∈Zn 0,Nm |fm j ||gmj | ∥fm∥lr(Zn 0,Nm )∥gm∥ls(Zn 0,Nm )|j|α+β ≤ ( Q+ 2 m )( 1− 1 m )−1/r−1/s . Combining this with (4.4) and letting m → ∞, we complete the proof. □ 5. Convergence of the extremal sequences In this section, we prove Theorems 1.4 and 1.10. Now we provide two lemmas needed later. Lemma 5.1. Assume that the conditions of Theorem 1.4 hold. If (a, b) ∈ (RN + × RN + )∩ S(N) solves (3.1), we can find a positive constant σ which is independent of N , such that σN := min { max 1≤|i|≤N ai, max 1≤|i|≤N bi } ≥ σ. Proof. Since LN is increasing with respect to N , LN ≥ L1. Take e1 = (1, 0, . . . , 0) and e2 = (−1, 0, . . . , 0). Set āi = { 1, when i = e1; 0, when i ∈ Zn 0,N \ {e1}, EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 11 and b̄i = { 1, when i = e2; 0, when i ∈ Zn 0,N \ {e2}. Therefore, (ā, b̄) ∈ S(1), and hence L1 ≥ āe1 b̄e2/|e1 − e2|λ. Thus, LN has a lower bound LN ≥ 2−λ. (5.1) We denote the max{bi; 1 ≤ |i| ≤ N} by η. From (3.1) we see that for any t ∈ (0, 1), LNar−1 i ≤ ηt ∑ j∈Zn 0,N ,j ̸=i b1−t j |i|α|i− j|λ|j|β . Taking the result above to the power of r/(r − 1) at both sides and summing for i ∈ Zn 0,N , we obtain L r r−1 N ≤ η tr r−1 ∑ i∈Zn 0,N ( ∑ j∈Zn 0,N ,j ̸=i b1−t j |i|α|i− j|λ|j|β ) r r−1 . (5.2) Let p, q > 0 satisfy 1 r + 1 q = 1 s + 1 p = 2− λ+ α+ β n . By the conditions of Theorem 1.4 we know that p > r and q > s. For each b̃ ∈ lq(Zn 0,N ), define operator T as follows (T b̃)i = ∑ j∈Zn 0,N ,j ̸=i b̃j |i|α|i− j|λ|j|β . Therefore, (5.2) becomes LN ≤ ηt∥T b̃∥lr/(r−1)(Zn 0,N ) with b̃i = b1−t i . (5.3) From the definition of the norm and (1.5), we obtain ∥T b̃∥lr/(r−1)(Zn 0,N ) = sup ∥a∥lr(Zn 0,N )=1 ∑ i,j∈Zn 0,N ,j ̸=i aib̃j |i|α|i− j|λ|j|β ≤ L ∥b̃∥lq(Zn 0,N ). (5.4) Choosing b̃i = b1−t i and t = 1− s/q, we have ∥b̃∥qlq(Zn 0,N ) = ∥b∥sls(Zn 0,N ) = 1. (5.5) Combining (5.1) with (5.3)-(5.5), we obtain 2−λ ≤ LN ≤ ηtL which implies η ≥ (2λL)−1/t. Similarly, we can obtain max{ai; 1 ≤ |i| ≤ N} ≥ (2λL)−1/t′ , where t′ = 1− r/p. Thus, the proof is completed if we take σ = min{(2λL)−1/t, (2λL)−1/t′}. □ 12 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Lemma 5.2. Assume that the conditions of Theorem 1.10 hold. If (a, b) ∈ (RN + × RN + )∩ S(N) solves (3.6), we can find a positive constant σ which is independent of N , such that σN := min { max 1≤|i|≤N ai, max 1≤|i|≤N bi } ≥ σ. Proof. We define η := max{bi; 1 ≤ |i| ≤ N}. From (3.6), we see that for each t ∈ (0, 1), QNηt ≥ a1−r i ( ∑ j∈Zn 0,N ,j ̸=i b1+t j |i|α|i− j|λ|j|β + b1+t i |i|α+β ) . In addition, there exists m,w > 0 satisfying 1 m + 1 s = 1 w + 1 r = 2− λ+ α+ β n . such that b̃ ∈ lw(Zn), where b̃i = b1+t i and t = s/w − 1. By the conditions of Theorem 1.10 we know that r > m and s > w. Therefore, ∥b̃∥wlw(Zn 0,N ) = ∥b∥sls(Zn 0,N ) = 1. (5.6) Multiply the above inequality by ari in both sides and sum for i ∈ Zn 0,N . Applying (1.12) with C = QN for a and b̃, and combining (4.4) with (5.6), we obtain QNηt ≥ ∑ i,j∈Zn 0,N ,j ̸=i aib 1+t j |i|α|i− j|λ|j|β + aib 1+t i |i|α+β ≥ Q. Noticing (3.8), it holds η−tQ ≤ QN ≤ 1 which implies η ≥ Q1/t. Similarly, we can obtain max{ai; 1 ≤ |i| ≤ N} ≥ Q1/t′ , where t′ = r/m− 1. Thus, the proof is completed if we take σ = min{Q1/t, Q1/t′}. □ Proof of Theorem 1.4. We denote the extremal sequences of (1.7) by (fN ∗ , gN∗ ). Write a(N) = (a(N)i)i∈Zn 0 , b(N) = (b(N)i)i∈Zn 0 , where a(N)i = |(fN ∗ )i|, b(N)i = |(gN∗ )i|, when 1 ≤ |i| ≤ N, a(N)i = b(N)i = 0, when |i| > N. Thus, (a(N), b(N)) ∈ S (5.7) because of (fN ∗ , gN∗ ) ∈ S(N). By the Bolzano-Weierstrass theorem, we can find a subsequence of (a(N), b(N)) denoted by itself such that lim N→∞ a(N)i = a∗i , lim N→∞ b(N)i = b∗i , (5.8) for any given i ∈ Zn 0 . In addition, we also see that (a(N)i, b(N)i) → (a∗i , b ∗ i ) weakly in lr(Zn 0 )× ls(Zn 0 ) (5.9) when N → ∞. We write a∗ = (a∗i )i∈Zn 0 and b∗ = (b∗i )i∈Zn 0 . According to Lemma 5.1, (a∗, b∗) ̸= (0, 0). We will verify that (a∗, b∗) is an extremal sequence. EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 13 Since (fN ∗ , gN∗ ) is the extremal sequence of (1.7), (a(N), b(N)) solves (3.1). Therefore, LNa(N)r−1 i = 1 |i|α ( ∑ 1≤|j|≤M,j ̸=i b(N)j |i− j|λ|j|β + ∑ |j|>M,j ̸=i b(N)j |i− j|λ|j|β ) , where M > 2|i| is a large integer which is independent of N . Letting N → ∞ in the result above and using Theorem 1.3 and (5.8), we obtain L(a∗i ) r−1 = 1 |i|α ( ∑ 1≤|j|≤M,j ̸=i b∗j |i− j|λ|j|β + lim N→∞ ∑ |j|>M b(N)j |i− j|λ|j|β ) . (5.10) Applying the Hölder inequality and noting ∥b(N)∥ls(Zn 0 ) = 1 (implied by (5.7)), we have ∑ |j|>M b(N)j |i− j|λ|j|β ≤ ( ∑ |j|>M [b(N)j ] s )1/s( ∑ |j|>M ( 1 |i− j|λ|j|β ) s s−1 )1−1/s ≤ ( ∑ |j|>M ( 2λ |j|λ+β ) s s−1 )1−1/s ≤ C(Mn− s(λ+β) s−1 )1−1/s. Here we used |i− j| ≥ |j| − |i| ≥ |j|/2 because of |j| > M > 2|i|. From the conditions of Theorem 1.4: α/n < 1−1/r and 1/r+1/s+(λ+α+β)/n > 2, we can see that n < s(λ+ β)/(s− 1). Therefore, the result above shows that∑ |j|>M b(N)j |i− j|λ|j|β → 0 when M → ∞. Inserting this result into (5.10), we obtain L(a∗i ) r−1 = 1 |i|α ∑ j∈Zn 0 ,j ̸=i b∗j |i− j|λ|j|β . (5.11) Similarly, we have L (b∗i ) s−1 = 1 |i|β ∑ j∈Zn 0 ,j ̸=i a∗j |i− j|λ|j|α . (5.12) Multiplying (5.11) and (5.12) by a∗i and b∗i respectively, we see that ∥a∗∥rlr(Zn 0 ) = ∥b∗∥sls(Zn 0 ) . (5.13) We denote ∥a∗∥lr(Zn 0 ) by γ. Although Lemma 5.1 shows σN > σ (and hence γ > 0), it cannot be excluded that a∗i = 0 for some i ∈ Zn 0 . Therefore, we claim γ ≤ 1. In fact, B := {(a, b) ∈ lr(Zn 0 )× ls(Zn 0 ); ∥a∥lr(Zn 0 ) ≤ 1, ∥b∥ls(Zn 0 ) ≤ 1} is the weakly closed subset of lr(Zn 0 ) × ls(Zn 0 ) since it is the convex closed subset. Thus, from (5.9) it follows (a∗, b∗) ∈ B, and hence γ ≤ 1. Furthermore, we claim γ = 1. (5.14) Otherwise, 0 < γ < 1. Multiplying (5.11) by a∗i and summing for i ∈ Zn 0 we obtain L ∑ i∈Zn 0 (a∗i ) r = ∑ i,j∈Zn 0 ,i̸=j a∗i b ∗ j |i|α|i− j|λ|j|β . 14 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Using (1.5) with C = L to estimate the right hand side of the result above, and noting (5.13) we obtain Lγr ≤ L∥a∗∥lr(Zn 0 ) ∥b∗∥ls(Zn 0 ) = Lγ1+r/s, which implies γr−1−r/s ≤ 1. (5.15) From the conditions of Theorem 1.4, it follows that 1/r+1/s > 2−(λ+α+β)/n > 1. Thus, r − 1 − r/s = r(1 − 1/r − 1/s) < 0. Therefore, (5.15) contradicts with 0 < γ < 1. Namely, (5.14) is true. Multiplying (5.11) by a∗i , summing for i ∈ Zn 0 , and using (5.14) we see that L = ∑ i,j∈Zn 0 ,i̸=j a∗i b ∗ j |i|α|i− j|λ|j|β . This shows that L is attainable and (a∗, b∗) is an extremal sequence. Namely, (i) is proved. Finally, in view of (fN ∗ , gN∗ ) ∈ S(N) and (5.14), we have lim N→∞ ∥fN ∗ ∥lr(Zn 0 ) = ∥a∗∥lr(Zn 0 ) , lim N→∞ ∥gN∗ ∥ls(Zn 0 ) = ∥b∗∥ls(Zn 0 ) . Therefore, by the Brezis-Lieb lemma [8, Lemma 4.1], from (5.8) we can complete the proof of (ii). □ Proof of Theorem 1.10. Step 1. Existence of limit pair. According to [8], we intro- duce a new translation pair by using Lemma 5.2. Denote the extremal sequences of (1.13) by (fN , gN ) and |fN i1 | = max{|fN i |; |i| ≤ N}, |gNi2 | = max{|gNi |; |i| ≤ N} We write a(N) = (a(N)i)i∈Zn 0 , b(N) = (b(N)i)i∈Zn 0 , where a(N)i = |fN i+i1 |, in Ω1 N , a(N)i = 0, in Zn \ Ω1 N , and b(N)i = |gNi+i2 |, in Ω2 N , b(N)i = 0, in Zn \ Ω2 N . Here Ωk N = {i + ik; |i| ≤ N} (k = 1, 2). According to Lemma 5.2, for each N it holds a(N)0 ≥ σ, b(N)0 ≥ σ. (5.16) Since (fN , gN ) ∈ S(N), it follows that (a(N), b(N)) ∈ S. (5.17) By the Bolzano-Weierstrass theorem, we can find a subsequence of (a(N), b(N)) denoted by itself such that for each i, lim N→∞ a(N)i = a∗i , lim N→∞ b(N)i = b∗i . (5.18) We write a∗ = (a∗i )i∈Zn and b∗ = (b∗i )i∈Zn . When a∗i > 0, (5.18) implies a(N)i − a∗i /2 ≥ 0 for large N . When a∗i = 0, this result still holds. Thus, using the reversed EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 15 Minkowski inequality [7, Theorem 166] to a(N) = [a(N)− a∗/2] + a∗/2, by (5.17) we obtain 1 = ∥a(N)∥lr(Zn) ≥ ∥a(N)− a∗/2∥lr(Zn) + ∥a∗∥lr(Zn)/2 ≥ ∥a∗∥lr(Zn)/2, which implies a∗ ∈ lr(Zn 0 ). (5.19) Similarly, we have b∗ ∈ ls(Zn 0 ). (5.20) According to Lemma 5.2, (a∗, b∗) ̸= (0, 0). We will verify that limit pair (a∗, b∗) is an extremal sequence of (1.14). Step 2. Equations satisfied by limit pair. Since (fN , gN ) is the extremal sequence of (1.13), from (3.6) it follows that (a(N), b(N)) satisfies QN = a(N)1−r i+i1 ( 1 |i|α ∑ j∈Zn 0,N ,j ̸=i b(N)j+j2 |i− j|λ|j|β + b(N)i+i2 |i|α+β ) , QN = b(N)1−s i+i2 ( 1 |i|β ∑ j∈Zn 0,N ,j ̸=i a(N)j+j1 |i− j|λ|j|α + a(N)i+i1 |i|α+β ) . (5.21) The first equation in (5.21) shows that QN = a(N)1−r i+i1 1 |i|α ( ∑ 1≤|j|≤U,j ̸=i b(N)j+j2 |i− j|λ|j|β + ∑ |j|>U,j ̸=i b(N)j+j2 |i− j|λ|j|β + b(N)i+i2 |i|β ) , where U > 0 is a large integer which is independent of N . Letting N → ∞ in the result above and using (1.9) and (5.18), we obtain Q = (a∗i+i1) 1−r 1 |i|α ( ∑ 1≤|j|≤U,j ̸=i b∗j+j2 |i− j|λ|j|β + lim N→∞ ∑ |j|>U,j ̸=i b(N)j+j2 |i− j|λ|j|β + b∗i+i2 |i|β ) . (5.22) In addition, from the second equation in (5.21) it follows that QN ≥ b(N)1−s j+j2 a(N)0|j|−β |j + i1|−λ|i1|−α for |j| > 0. Combining this with (3.8) and (5.16), for large |j|, we have b(N)j ≤ C(σ, i1)|j|− λ+β s−1 . (5.23) Thus, we obtain that for each i,∑ |j|>U b(N)j |i− j|λ|j|β ≤ C ∑ |j|>U |j|− λ+β s−1 −(λ+β) ≤ CUn−λ+β s−1 −(λ+β). From the conditions in Theorem 1.10: λ < 0, β ∈ ((s− 1)n/s, 0] and s ∈ (0, 1), we have n− λ+β s−1 < λ+ β, which indicates that∑ |j|>U b(N)j |i− j|λ|j|β → 0, as U → ∞. 16 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Inserting the above result into (5.22) yields Q = (a∗i+i1) 1−r 1 |i|α ( ∑ j∈Zn 0 ,j ̸=i b∗j+j2 |i− j|λ|j|β + b∗i+i2 |i|β ) . (5.24) Similarly, we have Q = (b∗i+i2) 1−s 1 |i|β ( ∑ j∈Zn 0 ,j ̸=i a∗j+j1 |i− j|λ|j|α + a∗i+i1 |i|α ) . (5.25) Multiplying (5.24) by (a∗i+i1 )r and (5.25) by (b∗i+i2 )s respectively and then summing for i ∈ Zn 0 , we see that ∥a∗∥rlr(Zn 0 ) = ∥b∗∥sls(Zn 0 ) . (5.26) Step 3. Define γ := ∥a∗∥lr(Zn 0 ) . We claim that γ = 1. (5.27) In fact, multiplying (5.24) by (a∗i+i1 )r and then summing for i ∈ Zn 0 , we see that Q ∑ i∈Zn 0 (a∗i+i1) r = ∑ i,j∈Zn 0 ,j ̸=i a∗i+i1 b∗j+j2 |i|α|i− j|λ|j|β + ∑ i∈Zn 0 a∗i+i1 b∗i+i2 |i|α+β . Applying (1.11) with C = Q, from the result above and (5.26), we have Qγr ≥ Q∥a∗∥lr(ZN 0 )∥b∗∥ZN 0 = Qγ1+r/s, which implies γr(1−1/r−1/s) ≥ 1. Noting that r, s ∈ (0, 1), we have 1−1/r−1/s < 0. This means γ ≤ 1. (5.28) According to (5.18), when |i| = 1, there exists N1 > 0 for any ε > 0 such that a∗i ≥ (1− ε)a(t)i, for each t ≥ N1. (5.29) When |i| = 2, for this ε, there exists N2 ≥ N1 such that a∗i ≥ (1− ε)a(t)i, for each t ≥ N2. Combining this with (5.29), for |i| ≤ 2, we obtain that a∗i ≥ (1− ε)a(t)i, for each t ≥ N2. By induction, for |i| ≤ m, there exists Nm ≥ Nm−1 such that a∗i ≥ (1− ε)a(t)i, for each t ≥ Nm. Namely, a∗i ≥ (1− ε)a(Nm)i, (5.30) which implies (1 − ε)a(Nm)i − a∗i → 0− for each |i| ≤ m as m → ∞. Therefore, we can find t̃ > 0 satisfying ∥a(t̃)∥lr(Zn 0,t̃ ) = 1 such that a(Nt̃)i ≥ a(t̃)i for |i| ≤ t̃. In view of (5.30), applying the reversed Minkowski inequality to a∗ = [a∗ − (1 − ε)a(Nt̃)] + (1− ε)a(Nt̃), by (5.17) we obtain γ = ∥a∗∥lr(Zn 0 ) ≥ ∥a∗ − (1− ε)a(Nt̃)∥lr(Zn 0 ) + (1− ε)∥a(Nt̃)∥lr(Zn 0 ) ≥ (1− ε)∥a(t̃)∥lr(Zn 0,t̃ ) = 1− ε. (5.31) EJDE-2025/28 DISCRETE STEIN-WEISS INEQUALITIES 17 Letting ε → 0, we can see γ ≥ 1. Combining this result with (5.28), we immediately obtain (5.27). Step 4. Complete the proof. Multiply both sides of (5.24) by (a∗i+i1 )r and sum for i ∈ Zn 0 . Then applying (5.27) and (5.26), we derive that Q = ∑ i,j∈Zn 0 ,i̸=j a∗i+i1 b∗j+j2 |i|α|i− j|λ|j|β + ∑ j∈Zn 0 a∗j+j1 b∗j+j2 |j|α+β . This shows that Q is attainable and (ā∗, b̄∗) is an extremal sequence. Here ā∗i = a∗i+i1 and b̄∗i = b∗i+i2 . At the same time, noticing (5.17) and (5.27), we have lim N→∞ ∥a(N)∥lr(Zn 0 ) = ∥a∗∥lr(Zn 0 ) . Similarly, we have lim N→∞ ∥b(N)∥ls(Zn 0 ) = ∥b∗∥ls(Zn 0 ) . Namely, (1.16) is proved. According to (5.31) and (1.16), we obtain (1.17). 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Stein; Singular integrals and differentiability properties of functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. [15] E. M. Stein, G. Weiss; Fractional integrals in n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503–514. [16] J. Xu, Z. Cheng, Y. Fang; An extension of discrete weighted Hardy-Littlewood-Sobolev in- equality in space dimension one, Sci. Sin. Math., 45 (2015), 129–140. 18 CHUNHONG LI, TIANTIAN ZHOU EJDE-2025/28 Chunhong Li College of Mathematics and Computer Science, Guangxi Science and Technology Nor- mal University, Laibin, 546100, Guangxi, China Email address: lichunhong@gxstnu.edu.cn Tiantian Zhou Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal Univer- sity, Nanjing 210023, China Email address: zhoutiantiannj@163.com 1. Introduction 2. Inequalities with infinite terms 3. Inequalities with finite terms 4. Convergence of the best constants 5. Convergence of the extremal sequences Acknowledgements References