On exponential decay properties of solutions of the (3 + 1)-dimensional modified Zakharov-Kuznetsov equation

1.   Introduction

In this paper, we would like to investigate the initial value problem (IVP) associated with the (3+1)-dimensional modified Zakharov-Kuznetsov (mZK) equation

where $u = u(x, y, z, t)$ is a real-valued function, $u_{0} = u_{0}(x, y, z)$, and $\gamma$ is a nonzero constant. Furthermore, it is proved that its solutions $u(x, y, z, t)$ have the properties of exponential decay above the plane $x+y+z = 0$.

Equation (1.1) was proposed by Zakharov and Kuznetsov ^[1] as a three-dimensional generalization of the Korteweg-de Vries (KdV) equation, which was derived from the Euler-Poisson system with magnetic field by Lannes et al. in ^[2]. This equation describes the unidirectional propagation of ionic-acoustic waves in magnetized plasma.

It is easy to see that the ZK equation can be regarded as a multidimensional generalization of the one-dimensional KdV equation

It is worth mentioning that two dimensional versions of the KdV equation and modified Korteweg-de Vries (mKdV) equation are the ZK equation and the mZK equation, respectively. Up to now, to the best of our knowledge, for the two-dimensional ZK equation, the well-posedness and uniqueness results have been studied extensively. For some related works, refer to ^{[3,4,5,6,7,8]} and references therein. At the same time, the Cauchy problem for the 2D mZK equation has also been discussed. The local well-posedness in $H^{1}(\mathbb{R}^{2})$ was obtained by Biagioni and Linares in ^[9]. The local result was generalized to the data in $H^{s}(\mathbb{R}^{2})$, $s > 3/4$, by Linares and Pastor in ^[10]. In terms of a smallness assumption on the $L^{2}$-norm of the data ^[11], they proved the global well-posedness in $H^{s}(\mathbb{R}^{2})$, $s > 53/63$. In ^[12], Ribaud and Vento investigated the local well-posedness in $H^{s}(\mathbb{R}^{2})$, $s > 1/4$.

On the other hand, for the three-dimensional ZK equation, many interesting results have been obtained. For initial data in $H^{s}(\mathbb{R}^{3})$ with $s > 9/8$, Linares and Saut ^[13] showed the local well-posedness of this initial problem. The local well-posedness theory of the Benjamin-Ono equation was established in ^[14] by utilizing similar techniques in ^[13]. For more discussions of the 3D ZK equation, see ^[15,16,17] and reference therein. Moreover, for the 3D mZK equation, in ^[18], Grünrock proved the local well-posedness of the Cauchy problem (1.1) for initial data in $H^{s}(\mathbb{R}^{3})$ with $s > 1/2$. Kinoshita ^[19] established the well-posedness in the critical space $H^{1/2}(\mathbb{R}^{3})$ for the Cauchy problem of the mZK equation. Ali et al. ^[20] developed the propagation of dispersive wave solutions for (3+1)-dimensional nonlinear mZK equation in plasma physics. Further analysis results can be also found in ^[21] and references therein.

The properties of decay preservation are of great interest. In an innovative paper, Isaza and León ^[22] studied the optimal exponential decay properties of solutions to the KdV equation. Larkin and Tronco ^[23] derived the decay properties of small solutions for the ZK equation posed on a half-strip. In ^[24], Larkin further established the exponential decay of the $H^{1}$-norm for the 2D ZK equation. Recently, the decay properties for solutions of the ZK equation were also obtained in ^[25].

It is obvious that the decay properties are closely related to the aspect of unique continuation. It is noted that Bustamante et al. ^[15] derived the unique continuation property of the solutions of the 3D Zakharov-Kuznetsov equation. In recent years, the unique continuation principles of several models arising in nonlinear dispersive equations were investigated, see references ^{[26,27,28,29,30]} for example.

The well-posedness for the two dimensional generalized ZK equation in anisotropic weighted Sobolev spaces was discussed in ^[31]. In ^[32], Bustamante et al. established the well-posedness of the IVP for the 2D ZK equation in weighted Sobolev spaces $H^{s}(\mathbb{R}^{2})\cap L^{2}((1+x^{2}+y^{2})^{r}dxdy)$ for $s, r\in\mathbb{R}$. Furthermore, they also showed in ^[15] that, for some small $\varepsilon > 0$,

are solutions of the IVP for the three-dimensional ZK equation. Then, there exists a constant $a_{0} > 0$ such that if for some $a > a_{0}$,

then $u_{1}\equiv u_{2}$.

The main goal of the present paper is to formally derive the decay properties of exponential type solutions $u(x, y, z, t)$ to the IVP (1.1). In order to achieve this goal, we shall utilize Kato's approach to prove Kato's estimation in three-dimensional form.

Now, we are in the position to state our main results.

Theorem 1.1. Let $a_{0}$ be a positive constant. For any given data

the unique solution $u(\cdot, \cdot, \cdot)$ of the IVP (1.1) provided in ^[18]

satisfies

where

with

Let us consider weighted spaces with symmetric weight, which take the form

Regardless of whether the time direction is forward $t > 0$ or backward $t < 0$, its persistent properties should hold.

Theorem 1.2. Let $a_{0}$ be a positive constant. Let $u_{1}$, $u_{2}$ be solutions of the IVP (1.1) such that

If

then

where $c^{**} = c^{**}\left(a_{0}; \|u_{0, 1}\|_{H^{4}(\mathbb{R}^{3})}; \|u_{0, 2}\|_{H^{4}(\mathbb{R}^{3})}; \|x^{2}u_{0, 1}\|_{L^{2}(\mathbb{R}^{3})}; \|xu_{0, 2}\|_{L^{2}(\mathbb{R}^{3})}; \Lambda; T\right)$ and

Let us denote the norm of the functional space $X^{s}$ by

for $s > 1/2$, where $\widehat{J_{x}^{s}}(\xi, \eta, \gamma) = (1+\xi^{2})^{s/2}\widehat{f}(\xi, \eta, \gamma)$.

In terms of the context, the previous result shows that it is necessary to have a similar property in a suitable Sobolev space $H^{s}(\mathbb{R}^{3})$ for a solution of the IVP (1.1) to satisfy the persistent property in $L^{2}(\langle x+y+z\rangle^{b}dxdydz)$.

Theorem 1.3. Let $u_{0}\in X^{s}$, $s > 9/8$. There exists $T = T(\|u_{0}\|_{X^{s}})$ and a unique solution of the IVP (1.1) such that $u\in C([0, T];X^{s})$ provided by Lemma 2.1 below. If there exist $\alpha > 0$ and two different instants of time $t_{0}, t_{1}\in[0, T]$ such that

then for any $t\in[0, T]$,

The rest of this paper is organized as follows. In Section 2, some details on known results of the three-dimensional mZK equation will be introduced. In Section 3, the weights will be constructed to put forward the theory. Section 4 is devoted to proving Theorems 1.1 and 1.2. Finally, in Section 5, we demonstrate Theorem 1.3.

2.   Preliminaries

Attention in this section is now turned to prove some preliminary estimates which we often use in our analysis. We first give the following result.

Lemma 2.1. Given $u_{0}\in X^{s}$, $s > 9/8$, there exists $T = T(\|u_{0}\|_{X^{s}})$ and a unique solution of the IVP (1.1) such that $u\in C([0, T];X^{s})$, $u, \partial_{x}u\in L_{T}^{1}L_{xyz}^{\infty}$. Moreover, the map $u_{0}\mapsto u$ is continuous from a neighborhood of $u_{0}\in X^{s}$ into $C([0, T];X^{s})$.

The proof is similar to Theorem 3.9 in ^[13]. Meanwhile, using the assumptions of Theorem 1.3, we deduce that

where $c_{T}$ is a constant.

Lemma 2.2. Let $u\in C([0, T];H^{2}(\mathbb{R}^{3}))$ be a solution of the IVP (1.1), corresponding to data $u_{0}\in H^{2}(\mathbb{R}^{3})\cap L^{2}(e^{\beta(x+y+z)}dxdydz)$, $\beta > 0$. Then,

and

Proof. Applying Kato's approach in ^[33], let us now prove this lemma. First, we consider the equation

Next, multiplying by $u\varphi_{\delta}$ on both sides of Eq (2.2) and integrating by parts, a direct computation gives rise to

For $\beta > 0$, we define

Thus, we see that

and then

hence

Therefore,

Moreover,

and

We apply properties (2.3) and (2.4) to obtain the estimate

In the case of $u\in C([0, T];H^{2}(\mathbb{R}^{3}))$, there exists a positive constant $c$ such that

Next, we consider the last term of (2.5). We write

Inserting this estimate into (2.5), one has

Using Gronwall's lemma and integrating (2.6) in $t\in[0, T]$, we deduce that

where $c$ is a constant.

Letting $\delta\downarrow 0$, this completes the proof of Lemma 2.2.

□

3.   Construction of weights

We multiply $u\phi_{N}$ on both sides of (2.2). Then, for a fixed $t\in[0, T]$, integrating over $\mathbb{R}^{3}$ with $x$, $y$, and $z$, and making use of integration by parts yields

A sequence of the weights $\{\phi_{N}\}_{N = 1}^{\infty}$ will be constructed, which plays an important role in the proof of our main theorems.

Theorem 3.1. Given $a_{0} > 0$, there exists a sequence $\{\phi_{N}\}_{N = 1}^{\infty}$ of functions with

satisfying for any $N\in\mathbb{Z}^{+}$:

(i) $\phi_{N}\in C^{2}(\mathbb{R}^{3}\times[0, \infty))$ with $\partial_{x}^{3}\phi_{N}(\cdot, \cdot, \cdot, t)$, $\partial_{xyy}\phi_{N}(\cdot, \cdot, \cdot, t)$, $\partial_{xzz}\phi_{N}(\cdot, \cdot, \cdot, t)$ having a jump discontinuity at $x+y+z = N$.

(ii) $\phi_{N}(x, y, z, t) > 0$ for all $(x, y, z, t)\in\mathbb{R}^{3}\times[0, \infty)$.

(iii) $\partial_{x}\phi_{N}(x, y, z, t) = \partial_{y}\phi_{N}(x, y, z, t) = \partial_{z}\phi_{N}(x, y, z, t) > 0$ for all $(x, y, z, t)\in\mathbb{R}^{3}\times[0, \infty)$.

(iv) There exist constants $c_{N} = c(N) > 0$ and $c_{0} = c_{0}(a_{0}) > 0$ such that

with

(v) For $T > 0$, there is $N_{0}\in\mathbb{Z}^{+}$ such that

Also,

for any $t > 0$ and $x+y+z\in(-\infty, 0)\cap(1, \infty)$, where

(vi) There exists a constant $c_{0} = c_{0}(a_{0}) > 0$ such that

for any $(x, y, z, t)\in\mathbb{R}^{3}\times[0, \infty)$.

(vii) There exists a constant $c_{1} = c_{1}(a_{0}) > 0$ such that

for any $(x, y, z, t)\in\mathbb{R}^{3}\times[0, \infty)$.

Proof. For $N\in\mathbb{Z}^{+}$, given $a_{0} > 0$, let us first define

where

$a_{0}$ being the initial parameter.

for $x+y+z\in(-\infty, 1]$ where $\eta\in C^{\infty}(\mathbb{R}^{3})$, $\eta_{x} = \eta_{y} = \eta_{z}\geq0$, and

i.e., for each $x+y+z\in[0, 1]$, $\varphi(x, y, z)$ is a convex combination of $(x+y+z)^{3}$ and $(x+y+z)^{3/2}$. $P_{N}(x, y, z, t)$ is a polynomial of order 2 in $(x+y+z)$, which matches the value of $e^{a(t)(x+y+z)^{3/2}}$ and its partial derivatives up to order 2 at $x+y+z = N$:

with $a = a(t)$ as in $(3.4)$.

Thus, to prove Theorem 3.1, let us consider the regions $x+y+z\in(-\infty, 0]$, $[0, 1]$, $[1, N]$, and $[N, \infty)$, respectively.

In the first region $x+y+z\leq0$, we get

which clearly satisfies Theorem 3.1.

In the region $x+y+z\in [0, 1]$, we deduce that

with

with $\eta$ as in (3.5). Since in this region $(x+y+z)^{3/2}\geq (x+y+z)^{3}$, it follows that

likewise

with

and there exists $c > 0$ such that

Note that for $a'(t)\leq0$, it is found that

and it is deduced that

Next, let us prove that there exists $c_{0} = c_{0}(a_{0}) > 0$ in this region such that

Since

From (3.6), we derive that

thus, for $x+y+z\thicksim0$ $(x+y+z\geq0)$,

Hence, for $x+y+z\thicksim0$ $(0\leq x+y+z\leq1)$,

Using (3.7) (i.e., $a(t)\leq a_{0}$ for $t\geq0$), it is easy to see that there exist $\delta > 0$ and a universal constant $c > 0$ such that

In the region $x+y+z\in [1, \delta]$, we conclude that (3.8) still holds (with a possible large $c > 0$). Applying the above estimates, it then follows that Theorem 3.1 holds in this region.

In the domain $x+y+z\in [1, N]$, we observe that

Then, a direct computation gives rise to

Hence, $\phi_{N} > 0$ and

Taking advantage of

we eliminate the terms with power $3/2$ on the right-hand side of (3.10). Therefore,

We show that

with $c_{0} = c_{0}(a_{0}) > 0$, and it is easy to find that for $1\leq x+y+z\leq N$,

since $a(t) = a\leq a_{0}$, $-\frac{9}{8}a(x+y+z)^{-\frac{3}{2}}\leq0$.

Next, it follows from (3.9) that

Lastly, we remark that

which completes the proof of Theorem 3.1 in this region.

Finally, let us consider the last region $x+y+z\in [N, \infty]$. In this domain,

with $a = a(t)$ as in (3.11). Hence, we obtain that there exists $c > 0$ such that for $x+y+z\geq N$,

which proves (ⅱ) in Theorem 3.1 in this region. Furthermore, one gets

which proves (ⅲ) in Theorem 3.1 in this domain, and

where

Next, we shall prove that if $x+y+z\geq N$,

Note that

Combining the above estimates completes the proof of (3.14). Then, (3.13) yields (3.3) in this region $x+y+z\geq N$.

In order to complete the proof, it is necessary to prove $(v)$ in the region $x+y+z\in [N, \infty)$. Taking advantage of (3.9) with $t = 0$, we need only prove that for $x+y+z\geq N$,

Let $x+y+z = \omega$ to prove (3.15). We need to prove that

Considering $P_{N}(\omega, 0)$ and $e^{a_{0}\omega_{+}^{3/2}}$, and their derivatives up to the second order, which coincide at $\omega = N$, to prove (3.16), it is sufficient to prove that

For this purpose, we deduce from (3.12) that the constant value of $\partial_{\omega}^{2}P_{N}(\omega, 0)$ is given by

and coincides at $\omega = N$ with

Let us observe that

if $\omega > N$ and $N > (9a_{0}^{2})^{-1/3}\equiv N_{0}(a_{0})$. Therefore, if $N\geq N_{0}$, $\frac{d^{2}}{d\omega^{2}}e^{a_{0}\omega^{3/2}}$ is an increasing function with regard to the variable $\omega$ for $\omega\geq N$. According to this fact, we obtain (3.17). Hence, (3.16) and (3.15) hold, which gives the proof of (v) in this region.

Thus, the proof of Theorem 3.1 has been completed. □

4.   Decay of solutions

4.1.   Proof of Theorem 1.1

By virtue of Lemma 2.2, we deduce that the solution $u$ of IVP (1.1) satisfies

In general, if for some $\beta > 0$, $e^{\beta(x+y+z)}f, \partial_{x}^{2}f\in L^{2}(\mathbb{R}^{3})$, then $e^{\beta(x+y+z)/2}\partial_{x}f\in L^{2}(\mathbb{R}^{3})$ since

To prove (4.2), one initially assumes that $f\in H^{2}(\mathbb{R}^{3})$ with compact support to obtain (4.2) by integration by parts, and then the density of this class is employed to achieve the desired result.

Therefore, applying the last argument and (4.1), it follows that

In particular, for any $k$, $u\in C([0, T];L^{2}(\langle x+y+z\rangle^{k} dxdydz))$, we assume that $u$ is sufficiently regular, that is, $u\in C([0, T];H^{3}(\mathbb{R}^{3}))$. Then, we derive energy estimates on $u$ applying the weights $\{\phi_{N}\}$ (since $\phi_{N}\leq c\langle x+y+z\rangle^{2}$). Thus, multiplying by $u\phi_{N}$ on both sides of (2.2), and integrating the result in $\mathbb{R}^{3}$ with $x$, $y$, $z$, we obtain

Applying (3.1) and property (ⅲ) in Theorem 3.1, it is inferred that

From (3.2), it follows that

with $c_{0} = c_{0}(a_{0})$.

Let us estimate the second term on the right-hand side of (4.4). To bound the contribution of the second term using (3.3), we write

hence

Combining the Sobolev embedding theorem and (4.3), one has

Inserting the above estimates into (4.4), for any $N\in\mathbb{Z}^{+}$, we deduce that

with $L(t)\in L^{\infty}([0, T])$, where $L(\cdot) = L(a_{0};\|e^{\frac{1}{2}(x+y+z)}u_{0}\|_{L^{2}(\mathbb{R}^{3})}; \|u_{0}\|_{H^{1}(\mathbb{R}^{3})})$. In view of property $(v)$ in Theorem 3.1, and using Gronwall's lemma, it follows that for $t\in[0, T]$,

We are now in a position to establish (4.5) for our less regular solution $u\in C([0, T];H^{2}(\mathbb{R}^{3}))$. For this purpose, let us consider IVP (1.1) with regularized initial data $u_{0, \delta}: = \rho_{\delta}\ast u(\cdot+\delta, \cdot+\delta, \cdot+\delta, 0)$, where $\delta > 0$, $\rho_{\delta} = \frac{1}{\delta^{3}}\rho(\frac{\cdot}{\delta}, \frac{\cdot}{\delta}, \frac{\cdot}{\delta})$, $\rho\in C^{\infty}(\mathbb{R}^{3})$ is supported in $(-1, 1)\times(-1, 1)\times(-1, 1)$, and $\int_{\mathbb{R}^{3}}\rho d\xi d\eta d\zeta = 1$. Since

for IVP (1.1) in $H^{2}(\mathbb{R}^{3})$, according to the well-posedness result in ^[18], the corresponding solutions $u_{\delta}$ satisfy $u_{\delta}(t)\rightarrow u(t)$ in $H^{2}(\mathbb{R}^{3})$ uniformly for $t\in[0, T]$ as $\delta\rightarrow0$. Furthermore, in terms of the Sobolev embedding theorem, for fixed $t$,

Meanwhile, by applying Minkowski's integral inequality, we prove that

Note that $u_{\delta}$ is sufficiently regular, and we obtain (4.5) with $u_{\delta}$ and $u_{0, \delta}$ instead of $u$ and $u_{0}$. For fixed $t$, taking account of (4.6), (4.7) and using Fatou's lemma, one can deduce that

Taking $N\uparrow\infty$, and making use of Fatou's lemma and the property $(v)$ in Theorem 3.1, it follows that

which completes the proof of Theorem 1.1.

4.2.   Proof of Theorem 1.2

Let us consider the difference of the two solutions to the equation

that is,

Taking advantage of the argument developed in Theorem 1.1, we multiply (4.8) by $w\phi_{N}$, integrate the result in $\mathbb{R}^{3}$ with $x, y, z$, and formally use integration by parts to deduce that

where

Using (3.3), it follows that

Altogether,

Thus, combining the equality

and the above estimates, one has

i.e.,

where

with $G(t)\in L^{\infty}([0, T])$. Therefore,

which completes the proof of Theorem 1.2.

5.   Proof of Theorem 1.3

Proof. We suppose $t_{0} = 0$ and $0 < t_{1} < T$. First, let us consider $\alpha\in(0, 1/2]$. For $x+y+z\geq0, N\in\mathbb{Z}^{+}$, and $\alpha > 0$, we are in a position to define

with $\varphi_{N, \alpha}(x, y, z)\in C^{3}(x+y+z\geq0)$, $\varphi_{N, \alpha}(x, y, z)\geq0$, and $\partial_{x}\varphi_{N, \alpha} = \partial_{y}\varphi_{N, \alpha} = \partial_{z}\varphi_{N, \alpha}\geq0$, and for $\alpha\in(0, 1/2]$,

where $C$ is independent of $N$.

Let $\theta_{N, \alpha}$ be defined as the following:

Note that

Then, let $(u_{0, m})_{m\in\mathbb{Z^{+}}}$ be a sequence in $C_{0}^{\infty}(\mathbb{R}^{3})$ such that

and let $u_{m}\in C([0, T];H^{\infty}(\mathbb{R}^{3}))$ be the solution of Eq (1.1) corresponding to the initial data $u_{0, m}$. We have

From the continuous dependence of the solution upon the data (see Lemma 2.1), (5.3), and (5.4), there exists $T > 0$ such that

Owing to $u_{m}\in C([0, T], H^{\infty}(\mathbb{R}^{3}))$ satisfying Eq (1.1), we multiply it by $u_{m}\theta_{N}$. Then integrating the result and formally using integration by parts (justified since $\theta_{N}$ is bounded), we obtain

Notice that $\partial_{x}\theta_{N} = \partial_{y}\theta_{N} = \partial_{z}\theta_{N}$, $3\partial_{x}\theta_{N}-\partial_{y}\theta_{N}-\partial_{z}\theta_{N} > 0$, $\partial_{x}\theta_{N}-\partial_{y}\theta_{N} = 0$, and $\partial_{x}\theta_{N}-\partial_{z}\theta_{N} = 0$. We rewrite (5.6) as follows:

For $m$ large enough, thanks to the boundedness of the partial derivatives of $\theta_{N}$, the $L^{2}-$norm conservation law, and the convergence of the sequence $\{u_{0, m}\}$, we deduce that

Integrating (5.7) with regard to $t$ in $[0, t_{1}]$ and using (5.8), it follows that

where $C$ represents a constant, and its value may change from line to line. Meanwhile, it does not depend on the initial parameters of the problem. Setting $m\uparrow\infty$ and making use of (5.5), Lemma 2.1, (2.1), and the assumptions of Theorem 1.3, we deduce that

with $M = M\left(\|\langle x+y+z\rangle^{\alpha}u_{0}\|_{L^{2}(\mathbb{R}^{3})}, \|\langle x+y+z\rangle^{\alpha}u(t_{1})\|_{L^{2}(\mathbb{R}^{3})}\right)$. Next, we use (5.4) and (5.5) to conclude that for any fixed $\bar{N}\in\mathbb{Z}^{+}$ and $\bar{N} > 10N$,

in $L^{2}\left([0, t_{1}]\times\{x+y+z\in[-\overline{N}, \overline{N}]\}\right)$ as $m\uparrow\infty.$

Thanks to $\partial_{y}\theta_{N}$ and $\partial_{z}\theta_{N}$ having compact support, one has

At last, notice that $\partial_{y}\theta_{N} = \partial_{z}\theta_{N}\geq0$, and for $x+y+z > 1$,

Using Fatou's lemma in (5.11), we obtain

From Lemma 2.1 and (2.1), it follows that

We derive that

With (5.13), we reapply the above argument with $\psi_{N, \alpha}(x, y, z) = \psi_{N}(x, y, z)$:

Note that

In Eq (5.6) with $\psi_{N}(x, y, z)$ instead of $\theta_{N}(x, y, z)$, similar computations to that in (5.8) lead us to

and

Collecting all the above estimates and integrating in $[0, t]\subset[0, t_{1}]$ yields

Taking advantage of (5.13), for $t\in[0, t_{1}]$, it is inferred that

Hence, the desired result holds.

Next, let us consider the case $\alpha\in(1/2, 1]$. A direct computation gives rise to

As before, we deduce that

with $u_{m}\in C([0, T];H^{\infty}(\mathbb{R}^{3}))$.

In (5.16), we first utilize that

Next, for the last term on the right-hand side of (5.16), it follows that

with $C$ independent of $N$. Accordingly,

For each fixed $N$, the $\theta_{N, \alpha}$'s are bounded, and

we conclude that

Hence,

with $M = M\left(\|\langle x+y+z\rangle^{1/2} u_{0}\|_{L^{2}(\mathbb{R}^{3})}, \|\langle x+y+z\rangle^{1/2} u(t_{1})\|_{L^{2}(\mathbb{R}^{3})}\right)$ for $m\gg1$.

Plugging the above estimates into (5.16) and applying the same argument in the previous case $\alpha\in(0, 1/2]$, we obtain

for $m\gg1$, and

where $M = M\left(\|\langle x+y+z\rangle^{1/2} u_{0}\|_{L^{2}(\mathbb{R}^{3})}, \|\langle x+y+z\rangle^{1/2} u(t_{1})\|_{L^{2}(\mathbb{R}^{3})}\right)$.

In the following, we use $\psi_{N, \alpha}$ instead of $\theta_{N, \alpha}$. Then we obtain the similar formulation to (5.16). From (5.20) and (5.21), the desired result is valid.

For the case $\alpha\in(1, 3/2]$ and higher $\alpha$, we may apply a similar bootstrap technique to get the desired result. □

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The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

All authors declare no conflicts of interest that may influence the publication of this paper.