1 Introduction
The well-known Hardy–Littlewood–Sobolev inequality (cf. [Reference Lieb16] or [Reference Stein18, Theorem 1, Chapter 5]) is
$$ \begin{align} \left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{f(x)g(y)dxdy} {|x-y|^\lambda}\right| \leq K_1\|f\|_{L^r(\mathbb{R}^n)}\|g\|_{L^s(\mathbb{R}^n)}, \quad \forall (f,g) \in L^r(\mathbb{R}^n) \times L^s(\mathbb{R}^n), \end{align} $$
where
$$ \begin{align*}1<r,s<\infty, \ 0<\lambda<n, \ 1/r+1/s+\lambda/n=2. \end{align*} $$
There are many applications of (1.1) in the theory of nonlinear PDEs and mathematical physics. Based on this inequality, Huang, Li, and Yin proved the discrete inequality (cf. [Reference Huang, Li and Yin11])
$$ \begin{align} \sum_{i,j \in \mathbb{Z}^n, i \neq j} \frac{|f_i||g_j|}{|i-j|^{\lambda}} \leq K_2\|f\|_{l^r(\mathbb{Z}^n)}\|g\|_{l^s(\mathbb{Z}^n)}, \quad \forall (f,g) \in l^r(\mathbb{Z}^n) \times l^s(\mathbb{Z}^n), \end{align} $$
where
$K_2 \in (0,\infty )$
is the best constant,
$f=(f_i)_{i \in \mathbb {Z}^n}$
,
$g=(g_j)_{j \in \mathbb {Z}^n}$
, and
$$ \begin{align*}n \geq 1, \ 0<\lambda<n, \ \min\{r,s\}>1, \ 1/r+1/s+\lambda/n \geq 2. \end{align*} $$
When
$n=1$
, (1.2) is the Hardy–Littlewood–Pólya inequality (cf. [Reference Hardy, Littlewood and Pólya9, Theorem 381]). When
$1/r+1/s+\lambda /n>2$
,
$K_2$
is attainable (see [Reference Hua and Li10] or [Reference Huang, Li and Yin11]). When
$1/r+1/s+\lambda /n=2$
, the attainability of
$K_2$
is unknown.
When we cut off
$f=(f_i)_{i \in \mathbb {Z}_N^n}$
and
$g=(g_i)_{i \in \mathbb {Z}_N^n}$
, (1.2) is reduced to
$$ \begin{align} \sum_{i,j \in \mathbb{Z}_N^n, i \neq j} \frac{|f_i||g_j|}{|i-j|^{\lambda}} \leq K_N \|f\|_{l^r(\mathbb{Z}_N^n)}\|g\|_{l^s(\mathbb{Z}_N^n)}, \quad \forall (f,g) \in l^r(\mathbb{Z}_N^n) \times l^s(\mathbb{Z}_N^n), \end{align} $$
where N is a positive integer,
$K_N \in (0,\infty )$
is the best constant, and
$\mathbb {Z}_N^n :=\{i \in \mathbb {Z}^n; |i| \leq N\}$
. Paper [Reference Huang, Li and Yin11] shows that
$K_2$
can be approximated by
$K_N$
, and also shows the convergence relation between the extremal sequences when
$N \to \infty $
. In 2011, Li and Villavert studied (1.3) in the critical case of
$\lambda =n=1$
(cf. [Reference Li and Villavert14]). They obtained the upper and the lower bounds of
$K_N$
. Afterwards, Cheng and Li extended those results to the case of
$\lambda =n \geq 2$
(cf. [Reference Cheng and Li6]). In addition, the properties of extremal sequences were studied. Their work shows that (1.3) holds under more relaxed constraints on
$r,s$
and
$\lambda $
. In addition, paper [Reference Chen and Zheng5] obtained the optimal summability intervals, and [Reference Chen and Li3] and [Reference Xu, Cheng and Fang19] estimated the best constant of discrete inequalities with double weights.
In 2015, Dou and Zhu obtained in [Reference Dou and Zhu7] the reversed Hardy–Littlewood–Sobolev inequality (see also [Reference Beckner1, Reference Chen, Liu, Lu and Tao2, Reference Lei12] and [Reference Ngo and Nguyen17])
$$ \begin{align} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)||g(y)|dxdy}{|x-y|^\lambda} \geq K_3 \|f\|_{L^r(\mathbb{R}^n)} \|g\|_{L^s(\mathbb{R}^n)},\end{align} $$
$$ \begin{align*}\forall (f,g) \in L^r(\mathbb{R}^n) \times L^s(\mathbb{R}^n), \end{align*} $$
where
$K_3 \in (0,\infty )$
is a constant, and
$$ \begin{align*}n \geq 1, \ r,s \in (n/(n-\lambda),1), \ \lambda<0, \ 1/r+1/s+\lambda/n=2. \end{align*} $$
When
$f \equiv g$
, (1.1) and (1.4) are associated with the conformal geometry (cf. [Reference Chen, Li and Ou4, Reference Li15] and [Reference Xu20]).
Based on (1.4), [Reference Lei, Li and Tang13] shows a discrete inequalities with infinite terms.
Proposition 1.1 [Reference Lei, Li and Tang13, (1.17)]
Assume that
$$ \begin{align} n \geq 1, \ r,s \in (n/(n-\lambda),1), \ \lambda<0, \ 1/r+1/s+\lambda/n \leq 2. \end{align} $$
Then there is a positive constant
$K_4$
such that for all
$(f,g) \in l^r(\mathbb {Z}^n) \times l^s(\mathbb {Z}^n)$
,
$$ \begin{align} \displaystyle\sum_{i,j \in \mathbb{Z}^n, i \neq j} \frac{|f_i||g_j|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}^n} |f_j||g_j| \geq K_4\|f\|_{l^r(\mathbb{Z}^n)} \|g\|_{l^s(\mathbb{Z}^n)}. \end{align} $$
Denote the best constant in (1.6) by Q, that is,
$$ \begin{align} Q\ =\inf_{(f,g)\in \mathbb{S}} \left\{\sum_{i,j \in \mathbb{Z}^n,i \neq j} \frac{|f_i||g_j|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}^n} |f_j||g_j|\right\}, \end{align} $$
where
$$ \begin{align*}\mathbb{S}:=\{(f,g) \in l^r(\mathbb{Z}^n) \times l^s(\mathbb{Z}^n); \|f\|_{l^r(\mathbb{Z}^n)}=\|g\|_{l^s(\mathbb{Z}^n)}=1\}. \end{align*} $$
When
$n=1$
, [Reference Han and Lei8] obtained the discrete inequalities with finite terms, and gave the limit relation between its best constant and Q. In this article, we are concerned about whether the best constant in (1.6) is attainable for
$n \geq 1$
. To investigate this problem, we also give a discrete reversed Hardy–Littlewood–Sobolev inequality with finite terms, and use the convergence of the extremal sequences to study attainability of Q.
The main results of this article are listed as follows.
Theorem 1.1 Let
$n \geq 1$
,
$\min \{r,s\}>0$
and
$\lambda <0$
. Then there exists
$C \in (0,\infty )$
such that for any
$(f,g) \in l^r(\mathbb {Z}_N^n) \times l^s(\mathbb {Z}_N^n)$
,
$$ \begin{align} \sum_{i,j \in \mathbb{Z}_{N}^n,i \neq j} \frac{|f_i||g_j|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}_{N}^n} |f_j||g_j| \geq C\|f\|_{l^r(\mathbb{Z}_{N}^n)}\|g\|_{l^s(\mathbb{Z}_{N}^n)}. \end{align} $$
Compared with (1.5), Theorem 1.1 has more relaxed constraints on
$r,s$
and
$\lambda $
.
Let
$Q_N$
be the best constant of (1.8). Thus,
$$ \begin{align} Q_N=\inf \left\{\sum_{i,j \in \mathbb{Z}_{N}^n,i \neq j} \frac{|f_i||g_j|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}_{N}^n} |f_j||g_j|; \|f\|_{l^r(\mathbb{Z}_N^n)}=\|g\|_{l^s(\mathbb{Z}_N^n)}=1\right\}. \end{align} $$
We give two properties of
$Q_N$
(cf. Theorems 1.2 and 1.3).
The following theorem shows the upper and the lower bounds of
$Q_N$
.
Theorem 1.2 Let
$n \geq 1$
,
$\lambda <0$
, and
$\min \{r,s\}>0$
. Then
$$ \begin{align} Q_N \leq \min\{1,(2N)^{-\lambda}(\sum_{|i| \leq N}1)^{2-(1/r+1/s)}+(\sum_{|i| \leq N}1)^{1-(1/r+1/s)}\}. \end{align} $$
On the other hand, if (1.5) holds, we have
$$ \begin{align} Q_N \geq Q. \end{align} $$
If
$n \geq 1$
,
$\lambda <0$
,
$r,s \in (n/(n-\lambda ),1)$
satisfy
$1/r+1/s+\lambda /n>2$
, we have
$$ \begin{align} Q_N \geq Q (\sum_{|i| \leq N}1)^{2-(1/r+1/s+\lambda/n)}. \end{align} $$
The following theorem gives a convergence relation between
$Q_N$
and Q.
Theorem 1.3 Assume that (1.5) holds, we have
$$ \begin{align*}\lim_{N \to \infty} Q_N =Q. \end{align*} $$
Next, we consider whether Q is attainable.
Theorem 1.4 Let
$n \geq 1$
,
$\lambda <0$
, and
$r,s \in (n/(n-\lambda ),1)$
satisfy
$$ \begin{align} 1/r+1/s+\lambda/n < 2. \end{align} $$
Then Q is attainable. Namely, there exists
$(f^*, g^*) \in \mathbb {S}$
such that
$$ \begin{align*}Q= \sum_{i,j \in \mathbb{Z}^n,i \neq j} \frac{|f_i^*||g_j^*|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}^n} |f_j^*||g_j^*|. \end{align*} $$
When
$1/r+1/s+\lambda /n = 2$
, the attainability of Q is still open.
2 Proof of Theorems 1.1–1.3
Proof of Theorem 1.1
Proof Write
$a_i=|f_i|$
,
$b_i=|g_i|$
, and
$a=(a_i)$
,
$b=(b_i)$
for
$|i| \leq N$
. Set
$$ \begin{align*}\mathbb{R}_+^N :=\{x=(x_i)_{i \in \mathbb{Z}_N^n}; x_i \geq 0\}, \end{align*} $$
and the multivariate function
$$ \begin{align*}E(a,b)=\sum_{i,j \in \mathbb{Z}_{N}^n, i \neq j} \frac{a_ib_j}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}_{N}^n} {a_jb_j}. \end{align*} $$
Since
$$ \begin{align*}\mathbb{S}(N):=\left\{(a,b);\|a\|_{l^r(\mathbb{Z}_{N}^n)} = \|b\|_{l^s(\mathbb{Z}_{N}^n)}=1\right\} \end{align*} $$
is bounded and closed in
$\mathbb {R}_+^N \times \mathbb {R}_+^N$
, we can find a minimizer
$(a(N),b(N))$
of
$E(a,b)$
in
$\mathbb {S}(N)$
. To investigate the minimizer, we consider
$$ \begin{align*}J(a,b)=E(a,b) -\lambda_1(\|a\|_{l^r(\mathbb{Z}_N^n)}-1)-\lambda_2 (\|b\|_{l^s(\mathbb{Z}_N^n)}-1), \end{align*} $$
where
$\lambda _1$
and
$\lambda _2$
are Lagrange multipliers. According to the theory of the constrained extremum, both the partial derivatives of J are equal to zero at
$(a(N),b(N))$
. Therefore, for all
$(a,b) \in \mathbb {R}_+^N \times \mathbb {R}_+^N$
and
$t \in \mathbb {R}$
, there holds
$$ \begin{align*}\left[\frac{d}{dt} J(a(N)+ta,b(N))\right]_{t=0}= \left[\frac{d}{dt} J(a(N),b(N)+tb)\right]_{t=0}=0, \end{align*} $$
which implies
$$ \begin{align*}\begin{cases} \displaystyle \lambda_{1}=a(N)_i^{1-r}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{b(N)_j}{|i-j|^\lambda}+{b(N)_i}\right),\\[3mm] \displaystyle \lambda_{2}=b(N)_i^{1-s}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{a(N)_j}{|i-j|^\lambda}+{a(N)_i}\right). \end{cases} \end{align*} $$
Multiplying two equations above by
$a(N)_i^{r}$
and
$b(N)_i^{s}$
, respectively and summing for i, and noting that
$(a(N),b(N)) \in \mathbb {S}(N)$
is the minimizer, we can see that
$\lambda _{1}=\lambda _{2}=Q_N$
. Namely,
$$ \begin{align} \begin{cases} \displaystyle Q_{N}=a(N)_i^{1-r}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{b(N)_j}{|i-j|^\lambda}+{b(N)_i}\right),\\[3mm] \displaystyle Q_{N}=b(N)_i^{1-s}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{a(N)_j}{|i-j|^\lambda}+{a(N)_i}\right). \end{cases} \end{align} $$
Noting
$(a(N),b(N))\neq (0,0)$
(because of
$(a(N),b(N))\in \mathbb {S}(N)$
), from (2.1) we can see
$$ \begin{align} Q_N>0. \end{align} $$
Write
$e=(1,0,\cdots ,0) \in \mathbb {Z}_{N}^n$
, and take
$$ \begin{align*}\bar{a}_i=\bar{b}_i=\left\{ \begin{array}{@{}l@{}ll} & 1, \quad &when \ \ i=e;\\ & 0, \quad &when \ \ i \in \mathbb{Z}_{N}^n \setminus \{e\}. \end{array} \right. \end{align*} $$
Therefore,
$(\bar {a},\bar {b}) \in \mathbb {S}(N)$
. Thus,
$Q_{N}$
has an upper bound estimate
$$ \begin{align} Q_{N} \leq \bar{a}_e\bar{b}_e=1. \end{align} $$
Thus, (1.8) holds for
$C=Q_N$
.
Remark 2.1
Although (2.1) and (2.2) show that
$a(N)_i>0$
and
$b(N)_i>0$
for each
$i =1,2,\cdots ,N$
, there may hold that
$$ \begin{align} \lim_{N \to \infty} a(N)_i =\lim_{N \to \infty} b(N)_i=0, \quad for \ \ each \ \ i. \end{align} $$
Once (2.4) is true, the attainability of Q seems difficult to be verified. In next subsection, we prove that (2.4) does not happen (cf. Lemma 3.1).
Proof of Theorem 1.2
Take
$a_i^r=b_i^s=(\sum _{|i| \leq N}1)^{-1}$
. Then
$(a,b) \in \mathbb {S}(N)$
. Therefore, by
$|i-j| \leq 2N$
as
$i, j \in \mathbb {Z}_N^n$
, there holds
$$ \begin{align*}\begin{aligned} Q_N &\leq \sum_{i,j \in \mathbb{Z}_N^n,i \neq j} \frac{a_ib_j}{|i-j|^{\lambda}}+\sum_{i \in \mathbb{Z}_N^n}a_ib_i\\ &\leq (\sum_{|i| \leq N}1)^{-(1/r+1/s)} (\sum_{i,j \in \mathbb{Z}_N^n} |i-j|^{-\lambda}+\sum_{|i| \leq N}1)\\ &\leq (2N)^{-\lambda}(\sum_{|i| \leq N}1)^{2-(1/r+1/s)}+(\sum_{|i| \leq N}1)^{1-(1/r+1/s)}. \end{aligned} \end{align*} $$
Combining with (2.3), we have
$$ \begin{align*}Q_N \leq \min\{1,(2N)^{-\lambda}(\sum_{|i| \leq N}1)^{2-(1/r+1/s)}+(\sum_{|i| \leq N}1)^{1-(1/r+1/s)}\}. \end{align*} $$
This is (1.10).
Moreover, when
$1/r+1/s+\lambda /n \leq 2$
, multiplying (2.1)
$_1$
by
$a(N)_i^r$
and summing for i, we get
$$ \begin{align} Q_N = \sum_{i,j \in \mathbb{Z}_N^n,i \neq j} \frac{a(N)_ib(N)_j}{|i-j|^{\lambda}}+\sum_{i \in \mathbb{Z}_N^n}a(N)_ib(N)_i. \end{align} $$
Set
$$ \begin{align*}\bar{a}_i=\left\{ \begin{array}{@{}l@{}ll} &a(N)_i, &\ when\ |i| \leq N;\\ &0, & \ when \ |i|>N. \end{array} \right. \end{align*} $$
Similarly,
$\bar {b}_i$
is defined by the same way. In view of
$(a(N),b(N)) \in \mathbb {S}(N)$
, there holds
$(\bar {a},\bar {b}) \in \mathbb {S}$
. Therefore, from (2.5) and (1.7), it follows that
$$ \begin{align*} \begin{aligned} Q_N& =\sum_{i,j \in \mathbb{Z}_N^n,i \neq j} \frac{\bar{a}_i\bar{b}_j}{|i-j|^{\lambda}}+\sum_{i \in \mathbb{Z}_N^n}\bar{a}_i\bar{b}_i\\ &\geq Q\|\bar{a}\|_{l^r(\mathbb{Z}^n)}\|\bar{b}\|_{l^s(\mathbb{Z}^n)} = Q. \end{aligned} \end{align*} $$
This is (1.11).
When
$1/r+1/s+\lambda /n>2$
, we choose
$p' \in (r,1)$
and
$q' \in (s,1)$
satisfying
$$ \begin{align} 1/p'+1/q'+\lambda/n=2. \end{align} $$
From (1.9) and (1.11), it follows that
$$ \begin{align} \sum_{i,j \in \mathbb{Z}_N^n,i \neq j} \frac{{a}_i {b}_j}{|i-j|^{\lambda}}+\sum_{i \in \mathbb{Z}_N^n} {a}_i {b}_i \geq Q\| {a}\|_{l^{p'}(\mathbb{Z}_N^n)}\|{b}\|_{l^{q'}(\mathbb{Z}_N^n)} \end{align} $$
for all
$(a,b) \in l^r(\mathbb {Z}_N^n) \times l^s(\mathbb {Z}_N^n)$
. Using the reversed Hölder inequality (cf. [Reference Hardy, Littlewood and Pólya9, Theorem 13, (2.8.4)]), we see that
$$ \begin{align*}\|a\|_{l^{p'}(\mathbb{Z}_N^n)} \geq (\sum_{i \in \mathbb{Z}_N^n}1)^{1/p'-1/r} \|a\|_{l^r(\mathbb{Z}_N^n)}; \ \|b\|_{l^{q'}(\mathbb{Z}_N^n)} \geq (\sum_{i \in \mathbb{Z}_N^n}1)^{1/q'-1/s} \|b\|_{l^s(\mathbb{Z}_N^n)}. \end{align*} $$
Inserting these into (2.7) yields
$$ \begin{align*}\sum_{i,j \in \mathbb{Z}_N^n,i \neq j} \frac{{a}_i {b}_j}{|i-j|^{\lambda}}+\sum_{i \in \mathbb{Z}_N^n} {a}_i {b}_i \geq Q(\sum_{i \in \mathbb{Z}_N^n}1)^{1/p'+1/q'-1/r-1/s} \|{a}\|_{l^r(\mathbb{Z}_N^n)}\|{b}\|_{l^s(\mathbb{Z}_N^n)}. \end{align*} $$
Proof of Theorem 1.3
Proof Let (1.5) hold. First, (1.11) shows that
$Q_N \geq Q$
. Since
$Q_N$
is decreasing, we know that
$\lim _{N \to \infty } Q_N$
exists.
Noting the definition of Q, we can find a minimizing sequence
$(f^m,g^m) \in \mathbb {S}$
such that
$$ \begin{align*}\displaystyle\sum_{i,j \in \mathbb{Z}^n, i \neq j} \frac{|f_i^m||g_j^m|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}^n} {|f_j^m||g_j^m|} \leq Q+\frac{1}{m}. \end{align*} $$
This implies that the series of the left hand side in the result above is convergent. Therefore, we can find
$N_m$
satisfying
$N_m \to \infty $
when
$m \to \infty $
such that
$$ \begin{align} \displaystyle\sum_{i,j \in \mathbb{Z}_{N_m}^n, i \neq j} \frac{|f_i^m||g_j^m|}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}_{N_m}^n} {|f_j^m||g_j^m|} \leq Q+\frac{2}{m}. \end{align} $$
On the other hand,
$(f^m,g^m) \in \mathbb {S}$
implies
$$ \begin{align} \|f^m\|_{l^r(\mathbb{Z}_{N_m}^n)}^r \geq 1-1/m, \quad \|g^m\|_{l^s(\mathbb{Z}_{N_m}^n)}^s \geq 1-1/m, \end{align} $$
as long as
$N_m>m$
is sufficiently large. Therefore, noting that
$$ \begin{align*}\left(\frac{f^m}{\|f^m\|_{l^r(\mathbb{Z}_{N_m}^n)}}, \frac{g^m}{\|g^m\|_{l^s(\mathbb{Z}_{N_m}^n)}}\right) \in \mathbb{S}(N_m), \end{align*} $$
from (2.8) and (2.9) we deduce that
$$ \begin{align*}\begin{aligned} Q_{N_m} &\leq \sum_{i,j \in \mathbb{Z}_{N_m}^n,i \neq j} \frac{|f_i^m||g_j^m|}{\|f^m\|_{l^r(\mathbb{Z}_{N_m}^n)} \|g^m\|_{l^s(\mathbb{Z}_{N_m}^n)}|i-j|^\lambda} \\ &\quad +\sum_{j \in \mathbb{Z}_{N_m}^n} \frac{|f_j^m||g_j^m|}{\|f^m\|_{l^r(\mathbb{Z}_{N_m}^n)} \|g^m\|_{l^s(\mathbb{Z}_{N_m}^n)}}\\ &\leq \left(Q+\frac{2}{m}\right)\left(1-\frac{1}{m}\right)^{-1/r-1/s}. \end{aligned} \end{align*} $$
Combining with (1.11) and letting
$m \to \infty $
, we can complete the proof.
3 Proof of Theorem 1.4
The proof of Theorem 1.4 is based on the ‘Concentration Compactness’ argument which is also applied to show existence of an extremizer of the discrete Hardy–Littlewood–Sobolev inequalities in [Reference Huang, Li and Yin11].
Lemma 3.1 Assume that the assumptions of Theorem 1.4 are true. If
$(a,b) \in (\mathbb {R}_+^N \times \mathbb {R}_+^N) \cap \mathbb {S}(N)$
solves (2.1), we can find a positive constant
$\sigma $
which is independent of N, such that
$$ \begin{align*}\sigma_N:=\min\{ \max_{|i| \leq N} a_i, \max_{|i| \leq N} b_i \} \geq \sigma. \end{align*} $$
Proof Define
$\eta :=\max\{b_i; |i| \leq N\}$
. From (2.1), we see that for any
$t>0$
,
$$ \begin{align} \eta^t Q_N \geq a_i^{1-r}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{b_j^{1+t}}{|i-j|^\lambda}+{b_i^{1+t}} \right). \end{align} $$
In addition, (1.13) implies that there exists
$m, w>0$
satisfying
$$ \begin{align} \frac{1}{m}+\frac{1}{s}=\frac{1}{w}+\frac{1}{r} =2-\frac{\lambda}{n} \end{align} $$
such that
$\tilde {b}\in l^w(\mathbb {Z}^n_N)$
, where
$\tilde {b}_i=b_i^{1+t}$
with
$t=s/w-1$
. By (1.13) and (3.2) it follows
$$ \begin{align*}r>m \quad and \quad s>w. \end{align*} $$
Therefore,
$t>0$
and it follows from
$(a,b) \in \mathbb {S}(N)$
that
$$ \begin{align} \|\tilde{b}\|_{l^{w}(\mathbb{Z}_{N}^n)}^w =\|b\|_{l^{s}(\mathbb{Z}_{N}^n)}^s=1. \end{align} $$
Multiply (3.1) by
$a_i^{r}$
and sum for
$i \in \mathbb {Z}_{N}^n$
. In view of
$(a,b) \in \mathbb {S}(N)$
,
$$ \begin{align} \eta^t Q_N \geq \sum_{i, j \in \mathbb{Z}_{N}^n, j \neq i} \frac{a_ib_j^{1+t}}{|i-j|^\lambda}+\sum_{i \in \mathbb{Z}_{N}^n}{a_ib_i^{1+t}}. \end{align} $$
We use (1.6) with
$K_4=Q$
and (3.2) for a and
$\tilde {b}$
. Then the right hand side of (3.4) is equal to
$$ \begin{align*}\sum_{i, j \in \mathbb{Z}_{N}^n, j \neq i} \frac{a_i\tilde{b}_j}{|i-j|^\lambda}+\sum_{i \in \mathbb{Z}_{N}^n}{a_i\tilde{b}_i} \geq Q. \end{align*} $$
Here we use
$(a,b) \in \mathbb {S}(N)$
and (3.3). Combining this result and (3.4) with (2.3), we get
$$ \begin{align*}\eta \geq Q^{1/t}. \end{align*} $$
Similarly, we can also obtain
$$ \begin{align*}\max\{a_i; |i| \leq N\} \geq Q^{1/t'}, \end{align*} $$
where
$t'=r/m-1$
. Take
$$ \begin{align*}\sigma=\min\{Q^{1/t},Q^{1/t'}\}. \end{align*} $$
Thus,
$\sigma>0$
is independent of N. This completes the proof of Lemma 3.1.
Proof of Theorem 1.4
Proof Step 1. Existence of limit pair.
As in [Reference Huang, Li and Yin11, Section 3], we can use Lemma 3.1 to introduce a new translation pair. Denote the extremal sequences of (1.9) by
$(f^N,g^N)$
, and
$$ \begin{align*}f_{i_1}^N=\max\{|f_i^N|;|i| \leq N\},\quad g_{i_2}^N=\max\{|g_i^N|;|i| \leq N\}. \end{align*} $$
Write
$a(N)=(a(N)_i)_{i \in \mathbb {Z}^n}$
,
$b(N)=(b(N)_i)_{i \in \mathbb {Z}^n}$
, where
$$ \begin{align*}\left \{\!\!\!\!\!\!\!\begin{array}{lll} &a(N)_i = f_{i+i_1}^N, \quad &in \quad \Omega_N^1, \\[3mm] &a(N)_i=0,\quad &in \quad \mathbb{Z}^n \setminus \Omega_N^1, \end{array} \right. \end{align*} $$
$$ \begin{align*}\left \{\!\!\!\!\!\!\!\begin{array}{lll} &b(N)_i = g_{i+i_2}^N, \quad &in \quad \Omega_N^2, \\[3mm] &b(N)_i=0,\quad &in \quad \mathbb{Z}^n \setminus \Omega_N^2, \end{array} \right. \end{align*} $$
and
$\Omega _N^k=\{i+i_k; |i| \leq N\} (k=1,2)$
. By Lemma 3.1, it follows that
$$ \begin{align} a(N)_0 \geq \sigma \quad and \quad b(N)_0 \geq \sigma \end{align} $$
for each N. In view of
$(f^N,g^N) \in \mathbb {S}(N)$
, it follows
$$ \begin{align} (a(N),b(N)) \in \mathbb{S}. \end{align} $$
By the Bolzano–Weierstrass theorem, there exists a subsequence of
$(a(N),b(N))$
denoted by itself such that
$$ \begin{align} \lim_{N \to \infty}a(N)_i=a_i^*, \quad \lim_{N \to \infty}b(N)_i=b_i^*. \end{align} $$
Write
$a^*=(a_i^*)_{i \in \mathbb {Z}^n}$
and
$b^*=(b_i^*)_{i \in \mathbb {Z}^n}$
. When
$a_i^*>0$
, (3.7) implies
$a(N)_i-a_i^*/2 \geq 0$
for large N. When
$a_i^*=0$
, this result still holds. Thus, using the reversed Minkowski inequality (cf. [Reference Hardy, Littlewood and Pólya9, Theorem 166]) to
$a(N)=[a(N)-a^*/2]+a^*/2$
, by (3.6) we obtain
$$ \begin{align*}1=\|a(N)\|_{l^r(\mathbb{Z}^n)} \geq \|a(N)-a^*/2\|_{l^r(\mathbb{Z}^n)} +\|a^*\|_{l^r(\mathbb{Z}^n)}/2 \geq \|a^*\|_{l^r(\mathbb{Z}^n)}/2, \end{align*} $$
which implies
$$ \begin{align} a^* \in l^{r}(\mathbb{Z}^n). \end{align} $$
Similarly, we also have
$$ \begin{align} b^* \in l^{s}(\mathbb{Z}^n). \end{align} $$
According to Lemma 3.1,
$(a^*,b^*) \neq (0,0)$
. We will verify that limit pair
$(a^*,b^*)$
is an extremal sequence of (1.7).
Step 2. Equations satisfied by limit pair.
Since
$(f^N,g^N)$
is the extremal sequence of (1.9), it satisfies (2.1). Therefore,
$(a(N),b(N))$
also solves (2.1).
$$ \begin{align} \begin{cases} \displaystyle Q_{N}=a(N)_{i+i_1}^{1-r}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{b(N)_{j+j_2}}{|i-j|^\lambda}+b(N)_{i+i_2}\right),\\[3mm] \displaystyle Q_{N}=b(N)_{i+i_2}^{1-s}\left(\sum_{j \in \mathbb{Z}_{N}^n, j \neq i} \frac{a(N)_{j+j_1}}{|i-j|^\lambda}+a(N)_{i+i_1}\right). \end{cases} \end{align} $$
The first equation in (3.10) shows that
$$ \begin{align*}Q_N =a(N)_{i+i_1}^{1-r} \left(\sum_{|j| \leq U, j \neq i} \frac{b(N)_{j+j_2}}{|i-j|^\lambda} +\sum_{|j|>U, j \neq i} \frac{b(N)_{j+j_2}}{|i-j|^\lambda}+b(N)_{i+i_2}\right), \end{align*} $$
where
$U>0$
is a large positive integer which is independent of N. Letting
$N \to \infty $
in the result above and using Theorem 1.3 and (3.7), we get
$$ \begin{align} Q =(a_{i+i_1}^*)^{1-r} \left(\sum_{|j| \leq U, j \neq i} \frac{b_{j+j_2}^*}{|i-j|^\lambda} +\lim_{N \to \infty}\sum_{|j|>U,j \neq i} \frac{b(N)_{j+j_2}}{|i-j|^\lambda}+b^*_{i+i_2}\right). \end{align} $$
Clearly, we can deduce from the second equation in (3.10) that
$$ \begin{align*}Q_N \geq b(N)_{j+j_2}^{1-s} [a(N)_0] |j+i_1|^{-\lambda} \end{align*} $$
for
$|j|>0$
. Therefore, applying (3.5) and (2.3) we get
$$ \begin{align*}b(N)_j \leq \sigma^{\frac{1}{s-1}}(2|j|)^{\frac{\lambda}{1-s}} \quad for \ \ large \ \ |j|. \end{align*} $$
Thus, we can find
$C>0$
(independent of N and U) such that for each fixed i,
$$ \begin{align*}\sum_{|j|>U} \frac{b(N)_j}{|i-j|^\lambda} \leq C\sum_{|j|>U} |j|^{\frac{\lambda}{1-s}-\lambda} \leq CU^{n+\frac{\lambda}{1-s}-\lambda}. \end{align*} $$
In view of
$s \in (n/(n-\lambda ),1)$
, we see
$\lambda /(1-s)<\lambda -n$
. Therefore,
$$ \begin{align*}\lim_{U \to \infty}\lim_{N \to \infty}\sum_{|j|>U} \frac{b(N)_j}{|i-j|^\lambda}=0. \end{align*} $$
Inserting the above result into (3.11) yields
$$ \begin{align} Q =(a_{i+i_1}^*)^{1-r} \left(\sum_{j \in \mathbb{Z}^n, j \neq i} \frac{b_{j+j_2}^*}{|i-j|^\lambda}+{b_{i+i_2}^*}\right). \end{align} $$
Similarly, we also have
$$ \begin{align} Q =(b_{i+i_2}^*)^{1-s} \left(\sum_{j \in \mathbb{Z}^n, j \neq i} \frac{a_{j+j_1}^*}{|i-j|^\lambda}+{a_{i+i_1}^*} \right). \end{align} $$
Step 3. Define
$\gamma :=\|a^*\|_{l^{r}(\mathbb {Z}^n)}$
. We claim that
$$ \begin{align} \gamma=1. \end{align} $$
Multiplying (3.12) by
$(a_{i+i_1}^*)^{r}$
and (3.13) by
$(b_{i+i_2}^*)^s$
respectively, and then summing for
$i\in \mathbb {Z}^n$
, we see
$$ \begin{align} \|a^*\|_{l^{r}(\mathbb{Z}^n)}^{r}=\|b^*\|_{l^s(\mathbb{Z}^n)}^s. \end{align} $$
First we prove that
$$ \begin{align} \gamma \leq 1. \end{align} $$
In fact, multiplying (3.12) by
$(a_{i+i_1}^*)^r$
and summing for i, we have
$$ \begin{align*}Q \ \sum_{i \in \mathbb{Z}^n} (a_{i+i_1}^*)^r= \left(\sum_{i,j \in \mathbb{Z}^n, j \neq i} \frac{a_{i+i_1}^*b_{j+j_2}^*}{|i-j|^\lambda} +\sum_{i \in \mathbb{Z}^n}{a_{i+i_1}^*b_{i+i_2}^*}\right). \end{align*} $$
Applying (1.6) with
$K_4=Q$
, from the result above and (3.15) we get
$$ \begin{align*}Q\gamma^r \geq Q \|a^*\|_{l^{r}(\mathbb{Z}^n)} \|b^*\|_{l^{s}(\mathbb{Z}^n)} =Q\gamma^{1+r/s}, \end{align*} $$
which implies
$\gamma ^{r(1-1/r-1/s)} \geq 1$
. In view of
$1-1/r-1/s<0$
(now
$r,s \in (0,1)$
), we obtain (3.16).
On the other hand, in view of (3.7), when
$|i|=0$
, for any
$\varepsilon>0$
, we can find
$N_0>0$
such that
$$ \begin{align} a_0^\ast \geq (1-\varepsilon)a(k)_0, \quad for \ \ each\ \ k \geq N_0. \end{align} $$
When
$|i|=1$
, for this
$\varepsilon $
, we can find
$N_1 \geq N_0$
such that
$$ \begin{align*}a_i^\ast \geq (1-\varepsilon)a(k)_i, \quad for \ \ each\ \ k \geq N_1. \end{align*} $$
Combining this with (3.17) we have that when
$|i| \leq 1$
,
$$ \begin{align*}a_i^\ast \geq (1-\varepsilon)a(k)_i, \quad for \ \ each\ \ k \geq N_1. \end{align*} $$
By induction, when
$|i| \leq m$
, for the above
$\varepsilon $
, we can find
$N_m \geq N_{m-1}$
such that
$$ \begin{align} a_i^\ast \geq (1-\varepsilon)a(N_m)_i. \end{align} $$
This and (3.7) imply that when
$m \to \infty $
,
$(1-\varepsilon )a(N_m)_i-a_i^\ast \to 0^-$
for each
$|i| \leq m$
. Therefore, we can find
$\bar {m}>0$
such that
$a(N_{\bar {m}})_i \geq a(\bar {m})_i$
when
$|i| \leq \bar {m}$
.
In view of (3.18), applying the reversed Minkowski inequality to
$a^*=[a^*- (1-\varepsilon )a(N_{\bar{m}})]+(1-\varepsilon )a(N_{\bar{m}})$
, by (3.6) we get
$$ \begin{align} \begin{aligned} \gamma=\|a^*\|_{l^{r}(\mathbb{Z}^n)} \geq & \|a^*-(1-\varepsilon)a(N_{\bar{m}})\|_{l^{r}(\mathbb{Z}^n)} +(1-\varepsilon)\|a(N_{\bar{m}})\|_{l^{r}(\mathbb{Z}^n)}\\ \geq & (1-\varepsilon)\|a(\bar{m})\|_{l^{r}(\mathbb{Z}_{\bar{m}}^n)} =1-\varepsilon. \end{aligned} \end{align} $$
Letting
$\varepsilon \to 0$
, we can see
$\gamma \geq 1$
. Combining this result with (3.16), we immediately obtain (3.14).
Step 4. Complete the proof.
Multiply both sides of (3.12) by
$(a_{i+i_1}^*)^{r}$
and sum for
$i \in \mathbb {Z}^n$
. Using (3.14), we derive that
$$ \begin{align*}Q= \sum_{i,j \in \mathbb{Z}^n,i \neq j} \frac{a_{i+i_1}^*b_{j+j_2}^*}{|i-j|^\lambda} +\sum_{j \in \mathbb{Z}^n} {a_{j+j_1}^*b_{j+j_2}^*}. \end{align*} $$
This result, together with (3.8), (3.9), (3.14), and (3.15), shows that Q is attainable and
$(\bar {a}^*,\bar {b}^*)$
is an extremal sequence. Here
$\bar {a}_i^*=a_{i+i_1}^*$
and
$\bar {b}_i^*=b_{i+i_2}^*$
.
Remark 3.1 There seems to be no direct correlation between extremal sequences of (1.6) and (1.8). However, the proof of Theorem 1.4 shows some convergence relation of their translations. In fact, noticing (3.6) and (3.14), we have
$$ \begin{align} \lim_{N\to \infty}\|a(N)\|_{l^{r}(\mathbb{Z}^n)}=\|a^*\|_{l^{r}(\mathbb{Z}^n)}. \end{align} $$
Similarly,
$$ \begin{align} \lim_{N\to \infty}\|b(N)\|_{l^{s}(\mathbb{Z}_N^n)} =\|b^*\|_{l^{s}(\mathbb{Z}^n)}. \end{align} $$
$$ \begin{align} \lim_{N\to \infty}\|a^*-(1-\varepsilon_N)a(N)\|_{l^{r}(\mathbb{Z}^n)}=0. \end{align} $$
Similarly,
$$ \begin{align} \lim_{N\to \infty}\|b^*-(1-\varepsilon_N)b(N)\|_{l^{s}(\mathbb{Z}^n)}=0. \end{align} $$
Acknowledgements
The authors thank the unknown referees very much for their useful suggestions.
