SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE

Abstract

We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation

$$ \begin{align*} w(i)=W_{\beta,\gamma}(w^q)(i), \quad i \in \mathbb{Z}^n. \end{align*} $$

Here, $n \geq 1$, $\min \{q,\beta \}>0$, $1<\gamma \leq 2$ and $\beta \gamma <n$. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of $w(i)$ when $|i| \to \infty $.

1. Introduction

The Wolff potential of a nonnegative function $f \in L_{loc}^1(\mathbb {R}^n)$ is defined as in [10] by

$$ \begin{align*} W_{\beta,\gamma}(f)(x)=\int_0^{\infty} \bigg[\int_{B_t(x)} \frac{f(y)}{t^{n-\beta\gamma}} \,dy\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t}, \end{align*} $$

where $1<\gamma <\infty $ , $\beta>0$ , $\beta \gamma <n$ and $B_t(x)$ is a ball of radius t centred at x. It is not difficult to see that $W_{1,2}(f)$ is the Newton potential and $W_{{\alpha }/{2},2}(f)$ is the Reisz potential.

The Wolff potentials are helpful for understanding nonlinear partial differential equations (see [15, 16, 23]). For example, if $\mu $ is a positive Borel measure, $W_{1,\gamma }(\mu )$ can be used to estimate positive solutions of the $\gamma $ -Laplace equation

(1.1) $$ \begin{align} {-}\mbox{div}(|\nabla u|^{\gamma-2}\nabla u)=\mu. \end{align} $$

If $\inf _{R^n}u=0$ , then there exist positive constants $C_1$ and $C_2$ such that (see [15, 23])

(1.2) $$ \begin{align} C_1W_{1,\gamma}(\mu)(x) \leq u(x) \leq C_2W_{1,\gamma}(\mu)(x), \quad x \in \mathbb{R}^n. \end{align} $$

Set $R(x)=u(x)[W_{1,\gamma }(\mu )(x)]^{-1}$ . Then u solves the integral equation

(1.3) $$ \begin{align} u(x)=R(x)W_{\beta,\gamma}(\mu)(x) \quad\mbox{and}\quad u>0, \quad x\in\mathbb{R}^n, \end{align} $$

with $\beta =1$ , where $R(x)$ is a double-bounded function (in view of (1.2)). When ${\mu =u^q}$ , the qualitative properties of positive solutions of (1.3) are well studied. Existence results can be seen in [17] and the radial symmetry of positive solutions can be seen in [4, 21]. See also [22] for the integrability of finite energy solutions and [25] for the asymptotic behaviour of those solutions at infinity which shows that the decay rate of those positive solutions of (1.1) with $\mu =u^q$ is the same as the fast decay rate in [8, 14].

Equation (1.3) with $R(x) \equiv 1$ and $\mu \!=\!u^q$ is a generalisation of the Hardy–Littlewood– Sobolev integral equation. Namely, when $\gamma =2$ and $\beta =\alpha /2$ , (1.3) reduces to

(1.4) $$ \begin{align} u(x) = \int_{\mathbb{R}^{n}} \frac{u^{q}(y)}{|x-y|^{n-\alpha}}\,dy \quad\mbox{and}\quad u>0, \quad x\in \mathbb{R}^n. \end{align} $$

This equation is associated with the extremal functions of the Hardy–Littlewood– Sobolev inequality (see [20]):

(1.5) $$ \begin{align} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{f(x)g(y)}{|x-y|^{n-\alpha}}\,dx\,dy \leq C(n,s,\alpha)\|f\|_{L^r(\mathbb{R}^n)}\|g\|_{L^s(\mathbb{R}^n)}, \end{align} $$

where $\min \{s,r\}>1$ , ${1}/{r}+{1}/{s}={(n+\alpha )}/{n}$ , $f \in L^r(\mathbb {R}^n)$ , $g \in L^s(\mathbb {R}^n)$ . Positive solutions of (1.4) and the corresponding partial differential equations of the Lane–Emden type are well studied. In particular, see [2, 13] for the integrability of positive finite energy solutions of the corresponding system. Based on this result, [18] gives the asymptotic behaviour of those positive solutions.

In 2015, Huang et al. [12] used (1.5) to prove a discrete Hardy–Littlewood–Sobolev inequality and deduce the Euler–Lagrange system satisfied by the extremal sequences. When $f \equiv g$ , such a system can be seen as the discrete form of (1.4). Chen and Zheng [6] obtained the summability of some positive solutions and their asymptotic behaviour is obtained in [19].

The discrete Wolff potential is used to study some nonlinear problems in [1, 10, 23]. Let $f=f(i)$ , $i \in \mathbb {Z}^n$ , be a nonnegative sequence. Define a discrete form of the Wolff potential by

$$ \begin{align*} W_{\beta,\gamma}(f)(i)=\int_0^{\infty} \bigg[\sum_{j \in Z^n,|j-i|<t} \frac{f(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t}. \end{align*} $$

The discrete form of (1.3) is

(1.6) $$ \begin{align} w(i)=R(i)W_{\beta,\gamma}(w^{q})(i), \quad i\in \mathbb{Z}^n, \end{align} $$

where

(1.7) $$ \begin{align} n \geq 1, \ \beta>0, \ \gamma \in (1,2], \ \beta\gamma<n, \ q>0, \ C^{-1} \leq R(i) \leq C. \end{align} $$

Here, $C>1$ is an absolute constant and such an $R(i)$ is called a double-bounded sequence. This equation appears in the study of crystal physics, neural networks and other nonlinear problems (see [7, 9, 11]). We investigate the summability and the asymptotic behaviour of positive solutions of (1.6). We use a regularity lifting lemma (Lemma 2.2) to obtain an initial summability interval of positive solutions. We then extend the interval to an optimal one by means of a Wolff-type inequality (Lemma 2.1).

We now state the main results in this paper. Write

$$ \begin{align*} s_0:=\frac{n(q-\gamma+1)}{\beta\gamma}. \end{align*} $$

Theorem 1.1. Assume $w \in l^{s_0}(\mathbb {Z}^n)$ solves (1.6) with (1.7). If

(1.8) $$ \begin{align} q \geq \frac{n(\gamma-1)}{n-\beta\gamma}, \end{align} $$

we have the following summability and asymptotic behaviour results:

1. (i) w belongs to $l^s(\mathbb {Z}^n)$ for any $s \in ({n(\gamma -1)}/{(n-\beta \gamma )},\infty ]$ and the left end point ${n(\gamma -1)}/{(n-\beta \gamma )}$ is optimal;

2. (ii) if $w(i)$ belongs to a radially symmetric and decreasing surface, then the sequence $w(i) |i|^{{(n-\beta \gamma )}/{(\gamma -1)}}$ is double-bounded.

Remark 1.2. Condition (1.8) in Theorem 1.1 is not essential. In fact, by a similar argument to that in [17], (1.6) has no positive solution when $0<q \leq {n(\gamma -1)}/{(n-\beta \gamma )}$ . An analogous nonexistence result for another discrete Wolff-type equation can be seen in [23].

Remark 1.3. The assumption ‘the deceasing surface’ in Theorem 1.1(ii) is not essential. In fact, for bounded i, it is easy to see that $w(i) |i|^{{(n-\beta \gamma )}/{(\gamma -1)}}$ is double-bounded. Therefore, we only consider the case when $|i|$ is sufficiently large. By the ideas in Step 1 of the proof of [4, Theorem 1], and by the same argument in [19, Section 2.1], we also deduce that $w(i)$ is decreasing about some $i_0 \in \mathbb {Z}^n$ when $|i|$ is large.

Remark 1.4. When $\beta =\alpha /2$ , $\gamma =2$ , Theorem 1.1(i) and (ii) are consistent with the results in [6, 19], respectively. The initial summability condition $w \in l^{s_0}(\mathbb {Z}^n)$ appears in [5], but is different to the initial integrability condition in [22, 25]. In fact, $s_0$ can take all the values mentioned in these papers for the different critical exponents q which determine the existence of positive solutions (see [24]).

2. Preliminaries

2.1. Wolff-type inequality

In 2008, Phuc and Verbitsky [23] gave the following relation between the Wolff potential and the Riesz potential. If $q>\gamma -1>0$ , there exists ${C>1}$ such that

$$ \begin{align*} C^{-1}\|W_{\beta,\gamma}(\mu)\|_{L^q(\mathbb{R}^n)}^q \leq \|I_{\beta\gamma}(\mu)\|_{L^{q/(\gamma-1)}(\mathbb{R}^n)}^{q/(\gamma-1)} \leq C\|W_{\beta,\gamma}(\mu)\|_{L^q(\mathbb{R}^n)}^q. \end{align*} $$

Combining this with the Hardy–Littlewood–Sobolev inequality, one can obtain the Wolff-type inequality (see [22, Corollary 2.1]):

(2.1) $$ \begin{align} \|W_{\beta,\gamma}(f)\|_{L^q(\mathbb{R}^n)} \leq C \|f\|_{L^{qn/[n(\gamma-1)+q\beta\gamma]}(\mathbb{R}^n)}^{1/(\gamma-1)}. \end{align} $$

For the discrete sequence $f(i)$ , we have the same kind of inequality.

Lemma 2.1. When $q>\gamma -1>0$ ,

(2.2) $$ \begin{align} \|W_{\beta,\gamma}(f)(i)\|_{l^q(\mathbb{Z}^n)} \leq C \|f(i)\|_{l^{qn/[n(\gamma-1)+q\beta\gamma]}(\mathbb{Z}^n)}^{1/(\gamma-1)}. \end{align} $$

Proof. Let $\epsilon \in (0, 1/3)$ be sufficiently small and in (2.1), take

$$ \begin{align*}f(x) = \begin{cases} f(i) &\mbox{for}\ |x-i|<\epsilon \mbox{ and all } i \in \mathbb{Z}^n, \\ 0 &\mbox{otherwise}. \end{cases} \end{align*} $$

We claim that

(2.3) $$ \begin{align} \|f\|_{L^{qn/[n(\gamma-1)+q\beta\gamma]}(\mathbb{R}^n)} \leq C \|f(i)\|_{l^{qn/[n(\gamma-1)+q\beta\gamma]}(\mathbb{Z}^n)}. \end{align} $$

In fact, since $B_{\epsilon }(i) \cap B_{\epsilon }(j) =\emptyset $ for $i \neq j$ ,

$$ \begin{align*}\begin{aligned} \int_{\mathbb{R}^n} f^{qn/[n(\gamma-1)+q\beta\gamma]}(y)\,dy & =\sum_{i \in \mathbb{Z}^n} \int_{B_\epsilon(i)}f^{qn/[n(\gamma-1)+q\beta\gamma]}(i)\,dy \\ & =|B_1(0)|\epsilon^n \sum_{i \in \mathbb{Z}^n} f^{qn/[n(\gamma-1)+q\beta\gamma]}(i). \end{aligned} \end{align*} $$

This implies (2.3).

However, when $t>2\epsilon $ , we have $\bigcup _{|j-i|<t-2\epsilon }B_{\epsilon }(j) \subset B_{t-\epsilon }(i) \subset B_t(x)$ . Therefore,

(2.4) $$ \begin{align} \int_{B_t(x)}f(y)\,dy \geq c\sum_{j \in \mathbb{Z}^n,|j-i|<t-2\epsilon}f(j)\int_{B_\epsilon(j)}dy \geq c|B_1(0)|\epsilon^{n}\sum_{j \in \mathbb{Z}^n,|j-i|<t-2\epsilon}f(j). \end{align} $$

Since $i,j \in \mathbb {Z}^n$ , when $\epsilon $ is sufficiently small, $|j-i| \geq 2\epsilon $ for $i \neq j,$ and when $t>2\epsilon $ ,

$$ \begin{align*} \{j \in \mathbb{Z}^n : |j-i|<t-2\epsilon\}=\{j \in \mathbb{Z}^n : |j-i|<t\} \quad \mbox{for all } i \in \mathbb{Z}^n. \end{align*} $$

Therefore,

$$ \begin{align*} \sum_{j \in \mathbb{Z}^n,|j-i|<2\epsilon}f(j)=0 \quad \mbox{and} \quad \sum_{j \in \mathbb{Z}^n,|j-i|<t-2\epsilon}f(j)=\sum_{j \in \mathbb{Z}^n,|j-i|<t}f(j), \end{align*} $$

when $\epsilon $ is sufficiently small. Thus, from (2.4),

$$ \begin{align*} W_{\beta,\gamma}(f)(i) =0+\int_{2\epsilon}^\infty \bigg[\sum_{j \in \mathbb{Z}^n,|j-i|<t}f(j)\bigg]^{{1}/{(\gamma-1)}} t^{{(\beta\gamma-n)}{(\gamma-1)}}\,\frac{dt}{t} \leq C W_{\beta,\gamma}(f)(x). \end{align*} $$

Inserting this result and (2.3) into (2.1), we obtain (2.2).

2.2. Regularity lifting lemma

Let V be a topological vector space. Suppose there are two extended norms, $\|\cdot \|_X, \|\cdot \|_Y :V \to [0,\infty ]$ , defined on V (that is, allowing that the norm of an element in V might be infinity). Let

$$ \begin{align*} X := \{ v \in V : \|v\|_X < \infty\} \quad \mbox{and} \quad Y := \{ v \in V : \|v\|_Y < \infty\}. \end{align*} $$

Lemma 2.2 [3, Theorem 3.3.1].

Let T be a contraction map from X into itself and from Y into itself. Assume that $f \in X$ and that there exists a function $g \in Z := X \cap Y$ such that $f = Tf + g$ in X. Then f also belongs to Z.

This regularity lifting lemma can be used to study integral equations involving the Riesz potentials and the Wolff potentials. It was used in [2, 5, 6, 13, 22] to obtain better integrability results for positive solutions of Hardy–Littlewood–Sobolev-type equations, Stein–Weiss-type equations, discrete Stein–Weiss-type equations and Wolff-type equations.

3. Summability

Theorem 3.1. Assume $w \in l^{s_0}(\mathbb {Z}^n)$ solves (1.6) with (1.7) and (1.8). Then,

(3.1) $$ \begin{align} w \in l^s(\mathbb{Z}^n) \quad \mbox{for all } {s} \in \bigg(\frac{n(\gamma-1)}{n-\beta\gamma},\infty \bigg]. \end{align} $$

Proof. By [6, Lemma 2.2] or [12, Lemma 2.1], $w \in l^{s_0}(\mathbb {Z}^n)$ implies $w \in l^{\infty }(\mathbb {Z}^n)$ , and hence

(3.2) $$ \begin{align} w \in l^{s}(\mathbb{Z}^n) \quad \mbox{for all } s \in [s_0,\infty]. \end{align} $$

From (1.8), it follows that $s_0 \geq {n((\gamma -1))}/{(n-\beta \gamma )}$ . We lift the summability from (3.2) to (3.1).

Step 1. Establish an operator equation. For $A>0$ , set

$$ \begin{align*}w_A(i) = \begin{cases} w(i) & \mbox{if } |i|>A, \\ 0 & \mbox{otherwise}, \end{cases} \end{align*} $$

and $w_B(i)=w(i)-w_A(i)$ . Let $\sigma $ satisfy

(3.3) $$ \begin{align} \frac{2-\gamma}{s_0}<\frac{1}{\sigma}<\frac{2-\gamma}{s_0} +\frac{n-\beta\gamma}{n}. \end{align} $$

For $g \in l^\sigma (\mathbb {Z}^n)$ , define operators T and S by

$$ \begin{align*} Tg:=R(i)\int_0^\infty \bigg(\frac{\sum_{|j-i|<t}w^{q}(j)} {t^{n-\beta\gamma}}\bigg)^{{(2-\gamma)}/{(\gamma-1)}} \frac{\sum_{|j-i|<t}w_A^{q-1}(j)g(j)}{t^{n-\beta\gamma}} \,\frac{dt}{t}, \end{align*} $$

$$ \begin{align*} Sg:=\int_0^\infty \bigg(\frac{\sum_{|j-i|<t}w_A^{q-1}(j)g(j)} {t^{n-\beta\gamma}}\bigg)^{{1}/{(\gamma-1)}} \,\frac{dt}{t} \end{align*} $$

and write

$$ \begin{align*} F:=R(x)\int_0^\infty \bigg(\frac{\sum_{|j-i|<t}w^{q}(j)} {t^{n-\beta\gamma}}\bigg)^{{(2-\gamma)}/{(\gamma-1)}} \frac{\sum_{|j-i|<t}w_B^{q}(j)}{t^{n-\beta\gamma}} \,\frac{dt}{t}. \end{align*} $$

Clearly, w is a solution of the operator equation $g=Tg+F$ .

Step 2. T is a contraction map from $l^\sigma (\mathbb {Z}^n)$ into itself. In fact, the Hölder inequality and (1.7) imply $|Tg| \leq Cw^{2-\gamma } |Sg|^{\gamma -1}$ . Using the Hölder inequality again yields

(3.4) $$ \begin{align} \|Tg\|_\sigma \leq C\|w\|_{s_0}^{2-\gamma} \|Sg\|_t^{\gamma-1}, \end{align} $$

where $t>0$ satisfies

(3.5) $$ \begin{align} \frac{1}{\sigma}=\frac{2-\gamma}{s_0}+\frac{\gamma-1}{t}. \end{align} $$

Hereafter, for simplicity, we denote $\|\cdot \|_{l^s(\mathbb {Z}^n)}$ by $\|\cdot \|_s$ .

By (3.3) and (3.5),

(3.6) $$ \begin{align} 0<\frac{\gamma-1}{t}<1-\frac{\beta\gamma}{n}. \end{align} $$

Therefore, we can use Lemma 2.1 to obtain

(3.7) $$ \begin{align} \|Sg\|_t \leq C\|w_A^{q-1} g\|_{{nt}/{(n(\gamma-1)+t\beta\gamma)}}^{{1}/{(\gamma-1)}}. \end{align} $$

Since (3.5) leads to

$$ \begin{align*} \frac{\gamma-1}{t}-\frac{1}{\sigma} =\frac{q-1}{s_0}-\frac{\beta\gamma}{n}, \end{align*} $$

it follows from (3.7) and the Hölder inequality that $\|Sg\|_t^{\gamma -1} \leq C\|w_A\|_{s_0}^{q-1} \|g\|_\sigma $ . Inserting this into (3.4) yields

$$ \begin{align*} \|Tg\|_\sigma \leq C\|w\|_{s_0}^{2-\gamma} \|w_A\|_{s_0}^{q-1} \|g\|_\sigma. \end{align*} $$

Since $w \in l^{s_0}(\mathbb {Z}^n)$ ,

$$ \begin{align*} C\|w\|_{s_0}^{2-\gamma} \|w_A\|_{s_0}^{q-1} \leq \tfrac{1}{2} \end{align*} $$

when A is sufficiently large. Thus, T is a shrinking operator. Since T is linear, it follows that T is a contraction map from $l^{\sigma }(\mathbb {Z}^n)$ to itself as long as $\sigma $ satisfies (3.3).

Step 3. Estimating F to lift the regularity. Similar to the derivation of (3.4) and (3.7), for all $\sigma $ satisfying (3.3), we also deduce that

$$ \begin{align*} \|F\|_\sigma \leq C\|w\|_{s_0}^{2-\gamma} \|w_B^{q}\|_{{nt}/{(n(\gamma-1)+t\beta\gamma)}}, \end{align*} $$

where t satisfies (3.6). Noting $w\in l^{s_0}(\mathbb {Z}^n)$ and the definition of $w_B$ , we have $F \in l^{\sigma }(\mathbb {Z}^n)$ as long as $\sigma $ satisfies (3.3). Taking $X=l^{s_0}(\mathbb {Z}^n)$ , $Y=l^{\sigma }(\mathbb {Z}^n)$ and $Z=l^{s_0}(\mathbb {Z}^n) \cap l^{\sigma }(\mathbb {Z}^n)$ in Lemma 2.2, yields $w \in l^{\sigma }(\mathbb {Z}^n)$ for all $\sigma $ satisfying (3.3).

Step 4. Extend the interval from (3.3) to that in (3.1). Let

(3.8) $$ \begin{align} \frac{1}{s} \in \bigg(0,\frac{n-\beta\gamma}{n(\gamma-1)}\bigg). \end{align} $$

Then we can use Lemma 2.1 to deduce that

(3.9) $$ \begin{align} \|w\|_s \leq C\|w^{q}\|_{{ns}/{(n(\gamma-1)+s\beta\gamma)}}^{{1}/{(\gamma-1)}} = C\|w\|_{{nsq}/{(n(\gamma-1)+s\beta\gamma)}}^{{q}/{(\gamma-1)}}. \end{align} $$

Noting (3.3), from (3.9), we see that $\|w\|_s <\infty $ as long as s satisfies

(3.10) $$ \begin{align} \frac{2-\gamma}{s_0}< \frac{n(\gamma-1)+s\beta\gamma}{nsq} <\frac{2-\gamma}{s_0}+\frac{n-\beta\gamma}{n}. \end{align} $$

Next, we will prove that (3.10) is true as long as (3.8) holds. We only need to verify

(3.11) $$ \begin{align} \frac{n-\beta\gamma}{n(\gamma-1)} \leq \frac{q}{\gamma-1}\bigg[\frac{2-\gamma}{s_0}+\frac{n-\beta\gamma}{n}\bigg] -\frac{\beta\gamma}{n(\gamma-1)}. \end{align} $$

In fact,

$$ \begin{align*}\begin{aligned} \frac{n-\beta\gamma}{n(\gamma-1)} &\leq \frac{q}{\gamma-1}\bigg[\frac{\beta\gamma(2-\gamma)}{n(q-\gamma+1)} +\frac{n-\beta\gamma}{n}\bigg]-\frac{\beta\gamma}{n(\gamma-1)}\\ &\Longleftrightarrow 1 \leq q\bigg[\frac{\beta\gamma}{n}\bigg(\frac{2-\gamma}{q-\gamma+1}-1\bigg)+1\bigg] =q\bigg[\frac{\beta\gamma}{n}\frac{1-q}{q-\gamma+1}+1\bigg]\\ &\Longleftrightarrow \beta\gamma q(q-1) \leq n(q-\gamma+1)(q-1)\\ &\Longleftrightarrow q \geq \frac{n(\gamma-1)}{n-\beta\gamma}. \end{aligned} \end{align*} $$

Thus, (3.11) is true, and hence $w \in l^s(\mathbb {Z}^n)$ for all s satisfying (3.8). Combining this with (3.2), we see that (3.1) is true.

Step 5. If

(3.12) $$ \begin{align} 1/s \geq (n-\beta\gamma)/[n(\gamma-1)], \end{align} $$

we claim $w \not \in l^s(\mathbb {Z}^n)$ . In fact, when $|i|$ is suitably large,

$$ \begin{align*} w(i) \geq c\int_{2|i|}^\infty \bigg[\frac{\sum_{|j| \leq 1}w^q(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{\gamma}} \geq c|i|^{{(\beta\gamma-n)}/{(\gamma-1)}}. \end{align*} $$

Therefore, by (3.12), for suitably large $M>0$ ,

$$ \begin{align*} \sum_{|i| \geq M} w^s(i) \geq c\sum_{|i| \geq M} |i|^{{s(\beta\gamma-n)}/{(\gamma-1)}}=\infty. \end{align*} $$

The claim is proved.

Steps 4 and 5 show that the integrability interval is the one in (3.1). The proof of Theorem 3.1 is complete.

4. Decay rates

Proposition 4.1. Assume $w \in l^{s_0}(\mathbb {Z}^n)$ solves (1.6) with (1.7) and (1.8). If $w(i)$ belongs to a radially symmetric and decreasing surface, we can find $C>1$ such that for large  $|i|$ ,

$$ \begin{align*} C^{-1} |i|^{{(\beta\gamma-n)}/{(\gamma-1)}} \leq w(i) \leq C |i|^{{(\beta\gamma-n)}/{(\gamma-1)}}. \end{align*} $$

Proof. Without loss of generality, we assume $w(i)$ is radially symmetric about $i_0$ . Write

$$ \begin{align*} \omega(r)=w(i) \quad with \quad r=|i-i_0|. \end{align*} $$

Step 1. For any $s>{(n-\beta \gamma )}/{n(\gamma -1)}$ , we can find $C>0$ such that for large $R>0$ ,

(4.1) $$ \begin{align} \omega(R) \leq CR^{-n/s}. \end{align} $$

In fact, from Theorem 3.1, $w \in l^s(\mathbb {Z}^n)$ when s belongs to the interval of (3.1). Thus, we can denote $\sum _{i \in \mathbb {Z}^n}w^s(i)$ by a constant $C_s$ . In view of the monotonicity of $\omega (R)$ , we deduce that

$$ \begin{align*} c \ \omega^s(R) R^n \leq \sum_{R/2<|j-i_0|<R} w^s(j) \leq C_s. \end{align*} $$

Thus, (4.1) is verified.

Step 2. There exists $c>0$ such that $w(i) \geq c|i|^{-{(n-\beta \gamma )}/{(\gamma -1)}}$ for large $|i|$ . In fact, if ${|j-i_0|<2}$ , then for large $|i|$ and $t \in (2|i|,4|i|)$ ,

$$ \begin{align*} |j-i| \leq |j-i_0|+|i-i_0| <2+|i_0|+|i|<t. \end{align*} $$

This means $\{j : |j-i_0|<2\} \subset \{j : |j-i|<t\}$ . Therefore, by the monotonicity of $\omega (r)$ ,

$$ \begin{align*} \sum_{|j-i|<t}w^{q}(j) \geq \sum_{|j-i_0|<2}w^{q}(j) \geq \omega^q(2)\bigg(\sum_{|j-i_0|<2}1\bigg)=c, \end{align*} $$

when $t \in (2|i|,4|i|)$ . Thus,

$$ \begin{align*} w(i) \geq \int_{2|i|}^{4|i|} \bigg[\frac{\sum_{|j-i|<t}w^{q}(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t} \geq c \int_{2|i|}^{4|i|} \frac{1}{t^{{(n-\beta\gamma)}/({\gamma-1)}}} \,\frac{dt}{t} \geq \frac{c}{|i|^{{(n-\beta\gamma)}/{(\gamma-1)}}}. \end{align*} $$

Step 3. Estimate the upper bound of $w(i)$ for large $|i|$ . Clearly,

$$ \begin{align*}\begin{array}{ll} w(i) & \leq C\displaystyle\int_0^{{|i|}/{2}} \bigg[\frac{\sum_{|j-i|<t}w^{q}(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t} +C\displaystyle\int_{{|i|}/{2}}^{\infty} \bigg[\frac{\sum_{|j-i|<t}w^{q}(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t} \\ &:=C(I_1+I_2). \end{array} \end{align*} $$

Claim 1. There exists $C>0$ such that $ I_1 \leq C|i|^{-{(n-\beta \gamma )}/{(\gamma -1)}} $ for large $|i|$ . In fact, we have ${|i|}/{2} \leq |j| \leq {3|i|}/{2}$ when $|j-i|<t$ and $t<{|i|}/{2}$ . By virtue of the monotonicity of $\omega (r)$ , for large $|i|$ ,

$$ \begin{align*} w(j)=\omega(|j-i_0|) \leq C \omega\bigg(\frac{|j|}{3}\bigg) \leq C \omega\bigg(\frac{|i|}{6}\bigg) \end{align*} $$

when $|j-i|<t$ and $t<{|i|}/{2}$ . Therefore, by (4.1),

(4.2) $$ \begin{align} |i|^{{(n-\beta\gamma)}/{(\gamma-1)}}I_1 & \leq C|i|^{{(n-\beta\gamma)}/{(\gamma-1)}} \displaystyle\int_0^{{|i|}/{2}} \bigg[\frac{\omega^q(|i|/6)t^n }{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}}\,\frac{dt}{t} \notag\\ & \leq C|i|^{{(n-\beta\gamma)}/{(\gamma-1)} -{nq}/{s(\gamma-1)}} \displaystyle\int_0^{{|i|}/{2}} t^{{\beta\gamma}/{(\gamma-1)}} \,\frac{dt}{t} \leq C|i|^{{n}/{(\gamma-1)} -{nq}/{s(\gamma-1)}}. \end{align} $$

We choose s approaching the left end point of the interval of (3.1) such that

(4.3) $$ \begin{align} 1-\frac{q}{s}<0. \end{align} $$

Inserting this in (4.2) proves the claim.

Claim 2. There exists $C>0$ such that $ I_2 \leq C|i|^{-{(n-\beta \gamma )}/{(\gamma -1)}} $ for large $|i|$ . In fact, since $(\gamma -1)^{-1} \geq 1$ , Jensen’s inequality gives

$$ \begin{align*} I_2 &\leq C\displaystyle\int_{{|i|}/{2}}^{\infty} \bigg[\frac{\sum_{|j| \leq 2|i_0|,|j-i|<t}w^q(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t} +C\displaystyle\int_{{|x|}/{2}}^{\infty} \bigg[\frac{\sum_{|j|>2|i_0|,|j-i|<t}w^q(j)}{t^{n-\beta\gamma}}\bigg]^{{1}/{(\gamma-1)}} \,\frac{dt}{t}\\ &:=C(I_{21}+I_{22}). \end{align*} $$

Since $\sum _{|j| \leq 2|i_0|}w^q(j) \leq C$ ,

$$ \begin{align*} |i|^{{(n-\beta\gamma)}/{(\gamma-1)}}I_{21} \leq C|i|^{{(n-\beta\gamma)}/{(\gamma-1)}} \int_{{|i|}/{2}}^{\infty} t^{{(\beta\gamma-n)}/{(\gamma-1)}} \,\frac{dt}{t} \leq C. \end{align*} $$

However, when $|j|> 2|i_0|$ , we have $|j-i_0| \geq |j|-|i_0|> {|j|}/{2}$ . Applying the monotonicity of $\omega (r)$ and (4.1),

$$ \begin{align*} w^q(j)=\omega^q(|j-i_0|) \leq \omega^q\bigg(\frac{|j|}{2}\bigg) \leq C\bigg(\frac{|j|}{2}\bigg)^{-nq/s}. \end{align*} $$

By this result and (4.3),

$$ \begin{align*} \sum_{|j|> 2|i_0|,|j-i|<t}w^q(j) \leq C\sum_{2|i_0|<|j|<|i|+t} |j|^{-{nq}/{s}} \leq C. \end{align*} $$

Inserting this into $I_{22}$ yields

$$ \begin{align*} |i|^{{(n-\beta\gamma)}/{(\gamma-1)}}I_{22} \leq C|i|^{{(n-\beta\gamma)}/{(\gamma-1)}} \int_{{|i|}/{2}}^{\infty} t^{-{(n-\beta\gamma)}/{(\gamma-1)}} \,\frac{dt}{t} \leq C. \end{align*} $$

Claim 2 is proved and the proof of Proposition 4.1 is complete.