$\theta$-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

1.   Introduction

In 1931, Orlicz first obtained the definition of Lebesgue space with variable exponent $L^{p(\cdot)}(\Omega)$ (see ^[15]), i.e., for any measurable functions $f$ and sets $\Omega\subset\mathbb{R}^{n}$, if there exists a positive constant $\eta$ such that,

where $p$ is a function on $\Omega$ satisfying $1 < p(x) < \infty$. Respectively, the norm of Luxemburg-Nakano is defined by

Since then, many papers focus on the variable exponent spaces and their applications. For example, Kováčik and Rákosník ^[9] systematically researched variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^{n})$ and Sobolev spaces $W^{k, p(\cdot)}(\mathbb{R}^{n})$. In ^[16], Radulescu and Repovs studied the Lebesgue and Morrey spaces with variable exponent on $\mathbb{R}^{n}$, and also obtained some applications in partial differential equations. In ^[17], Ragusa and Tachikawa established the $C^{1, \gamma}_{\mathrm{loc}}(\Omega)$-regularity result for $W^{1, 1}$-local minimizers $\mu$ of the double phase functional with $x$-dependent exponents. In 2021, with the nonstandard growth conditions, Mingione and R$\breve{a}$dulescu provide an overview of recent results concerning elliptic variational problems (see ^[12]). The more development and research on the variable exponents, we refer readers to see ^{[3,4,8,11,13,21,22,23]} and reference therein.

On the other hand, fractional integrals, which regard as an important class of operators in harmonic analysis, have played a key role in the fields of harmonic analysis, applied probability and physics communities. For example, Sawano and Tanaka in ^[18] proved that fractional integral is bounded on Morrey space over non-doubling measures. Based on this work, the boundedness of fractional integral on Morrey space over non-homogeneous metric measure space is obtained by Cao and Zhou in ^[1]. Shen et.al used the generalization of a parameterized inexact Uzawa method to solve such a kind of saddle point problem for fractional diffusion equations (see ^[19]). However, in this paper, we will mainly consider the boundedness of $\theta$-type generalized fractional integrals, which are slightly modified in ^[5], on Lebesgue and Morrey spaces with variable exponents over non-homogeneous spaces. What's more, the results of this paper extend the contents of fractional integral on variable exponent spaces over $\mathbb{R}^{n}$ and non-homogeneous spaces.

Let $X: = (X, d, \mu)$ be a quasimetric measure space, if $\mu$ is a complete measure, and there exists a non-negative real-valued function $d$ on $X\times X$ satisfying the following conditions:

$\mathrm{(1)} \; d(x, x) = 0$ for all $x$ in $X$; $\mathrm{(2)} \; d(x, y) > 0$ for all $x\neq y$, $x, \ y\in X$;

$\mathrm{(3)}$ for all $x, \ y, \ z\in X$, there exists a constant $a_{1} > 0$, such that $d(x, y)\leq a_{1}(d(x, z)+d(y, z))$;

$\mathrm{(4)}$ there exists a constant $a_{0} > 0$, such that $d(x, y)\leq a_{0}d(y, x)$ for all $x, y\in X$.

Moreover, we always assume that balls $B(x, r): = \{y\in X: d(x, y) < r\}$ are measurable, $0\leq \mu(B(x, r)) < \infty$, $\mu(X) < \infty$ and $\mu(\{x\}) = 0$ for all $x\in X$ and $r > 0$ in this paper.

A measure $\mu$ on $X$ is said to satisfy the following growth condition, if there exists a constant $C > 0$ such that, for all $x\in X$ and $r > 0$,

Then the space $(X, d, \mu)$ with measure $\mu$ satisfying (1.1) is called a non-homogeneous space. In this setting, Kokilashvili and Meskhi obtained the boundedness of Maximal function and Riesez potential on variable Morrey spaces(see ^[7]). In ^[10], Lu proved that parameter Marcinkiewicz integral and its commutator are bounded on Morrey spaces with variable exponent and so on.

In this paper, we set that $p$ is a $\mu$-measurable function on $X$, and respectively define

where $E\subset X$ is a $\mu$-measurable. Moreover, we also denote $p_{-} = p_{-}(X)$ and $p_{+} = p_{+}(X)$.

We now recall the following definitions introduced in ^[7].

Definition 1.1. Let $N\geq 1$ be a constant. Suppose that $p$ is a function on $X$ such that $0 < p_{-} < p_{+} < \infty$. We say that $p\in\mathcal{P}(N)$ if there exists a constant $C > 0$ such that,

for all $x\in X$ and $r > 0$.

Definition 1.2. Let $0 < p_{-}\leq p_{+} < \infty$. We say that a function $p$ on $X$ satisfies the Log-Hölder continuity condition $p\in \mathrm{LH}(X)$ if

where constant $A > 0$ does not depend on $x, \ y\in X$.

For any ball $B$, we respectively denote its center and radius by $c_{B}$ and $r_{B}$ (or $r(B)$). Let $\eta > 1$ and $\beta > \eta$, a ball $B$ is said to be an $(\eta, \beta)$-doubling ball if $\mu(\eta B)\leq\beta\mu(B)$, where $\eta B$ denotes the ball with the same center as $B$ and $r(\eta B) = \eta r(B)$. Especially, for any given ball $B$, we denote by $\widetilde{B}$ the smallest doubling ball which contains $B$ and has the same center as $B$. Given two balls $B\subset S$ in $X$, set

where $N_{B, S}$ is denoted by the smallest integer $k$ such that $r(2^{k}B)\geq r(S)$.

The following notion of regular bounded mean oscillation $\mathrm{(RBMO)}$ space is from ^[20].

Definition 1.3. Let $\tau > 1$. A function $f\in L^{1}_{\mathrm{loc}}(\mu)$ is said to be in the space $\mathrm{RBMO}(\mu)$ if there exists a constant $C > 0$ such that for any ball $B$ centered at some point of $\mathrm{supp}(\mu)$,

and

for any two doubling balls $B\subset S$, where $m_{B}(f)$ represents the mean value of function $f$ over ball $B$, that is,

Moreover, the minimal constant $C$ satisfying $\mathrm{(1.5)}$ and $\mathrm{(1.6)}$ is defined to be the norm of $f$ in the space $\mathrm{RBMO}(\mu)$ and denoted by $\|f\|_{\mathrm{RBMO}(\mu)}$.

Now we state the definition of $\theta$-type generalized fractional integral kernel as follows.

Definition 1.4. Let $\alpha\in(0, 1)$, and $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ satisfying

A function $K_{\alpha}\in L^{1}_{\mathrm{loc}}(X\times X\setminus\{(x, x):\ x\in X\})$ is called an $\theta$-type generalized fractional integral kernel if there exists a positive constant $C_{K_{\alpha}}$ depending on $K_{\alpha}$, such that

$\mathrm{(1)}$ for all $x, \ y\in X$ with $x\neq y$,

$\mathrm{(2)}$ there exists a constant $c_{K_{\alpha}}\in(0, \infty)$ such that, for all $x, \tilde{x}, y\in X$ with $d(x, y)\geq c_{K_{\alpha}}d(x, \tilde{x})$,

Remark 1.1. If we take the function $\theta(t)\equiv t^{\delta}$ with $\delta\in(0, 1]$, then the $\theta$-type generalized fractional integral kernel $K_{\alpha}$ is just the fractional kernel of order 1 (see ^[7]).

Let $L^{\infty}_{b}(\mu)$ be the space of all $L^{\infty}(\mu)$ functions with bounded support. A linear $T_{\alpha}$ is called an $\theta$-type generalized fractional integral with $K_{\alpha}$ satisfying (1.8) and (1.9) if, for all $f\in L^{\infty}_{b}(\mu)$ and $x\notin \mathrm{supp}(f)$,

Given a function $b\in \mathrm{RBMO}(\mu)$, the commutator $[b, T_{\alpha}]$ which is generated by $T_{\alpha}$ and $b$ is defined by

The following definition of variable exponent Morrey space $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ is from ^[7].

Definition 1.5. Let $N\geq 1$ be a constant and $1 < q_{-}\leq q(x)\leq p(x)\leq p_{+} < \infty$. Then, the variable exponent Morrey space $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ is defined by

where

Remark 1.2. If we take $p(x) = q(x)$ in (1.12), then, the variable exponent Morrey space $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ is just variable exponent Lebesgue space $L^{p(\cdot)}(X)$ (see ^[7]), namely, for any $\mu$-measurable subset $E\subset X$ and $1\leq p_{-}(E)\leq p_{+}(E) < \infty$, then variable exponent Lebesgue space $L^{p(\cdot)}(E)$ is defined by

where

The organization of this paper is stated as follows. In section 2, via some known results, we prove that $\theta$-type fractional integral $T_{\alpha}$ is bounded from variable exponent Lebesgue spaces $L^{p(\cdot)}(X)$ into spaces $L^{q(\cdot)}(X)$ for $\alpha\in(0, 1)$ and $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}-\alpha$, and bounded from variable exponent Morrey spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ into spaces $M^{s(\cdot)}_{t(\cdot)}(X)_{N\bar{a}}$, where $\frac{t(\cdot)}{s(\cdot)} = \frac{q(\cdot)}{p(\cdot)}$, $\frac{1}{s(\cdot)} = \frac{1}{p(\cdot)}-\alpha$, $\bar{a}: = a_{1}(a_{1}(a_{0}+1)+1)$ and $N$ is a constant with $N\geq 1$. By establishing the sharp maximal function for commutator $[b, T_{\alpha}]$ generated by $T_{\alpha}$ and $b\in\mathrm{RBMO}(\mu)$, the boundedness of the $[b, T_{\alpha}]$ on spaces $L^{p(\cdot)}(X)$ and on spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ is also obtained in sections 3 and 4.

Finally, we make some conventions on notation. Throughout the whole paper, $C$ represents a positive constant being independent of the main parameters. For any subset $E$ of $X$, we use $\chi_{E}$ to denote its characteristic function.

2.   Estimate for $T_{\alpha}$ on variable exponent spaces

In this section, by applying some known results, the boundedness of $\theta$-type generalized fractional integral $T_{\alpha}$ on variable Lebsgue spaces $L^{p(\cdot)}(X)$ and on variable exponent Morrey spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ is obtained. Now we state the main theorems as follows.

Theorem 2.1. Let $N\geq 1$ be a constant, $K_{\alpha}$ satisfy $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $1 < p_{-}\leq p(x)\leq q(x)\leq q_{+} < \infty$, $\frac{1}{q(x)} = \frac{1}{p(x)}-\alpha$ and $0 < p_{+} < \frac{1}{\alpha}$. Suppose that $p\in\mathcal{P}(N)$, $q\in\mathcal{P}(1)$ and $\mu$ satisfies $\mathrm{(1.1)}$. Then $T_{\alpha}$ defined as in $\mathrm{(1.10)}$ is bounded from variable Lebesgue spaces $L^{p(\cdot)}(X)$ into spaces $L^{q(\cdot)}(X)$.

Theorem 2.2. Let $N\geq 1$ be a constant, $K_{\alpha}$ satisfy $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $1 < t_{-}\leq t(x)\leq s(x)\leq s_{+} < \infty$, $\frac{s(x)}{t(x)} = \frac{p(x)}{q(x)}$ and $\frac{1}{s(x)} = \frac{1}{p(x)}-\alpha$ with $0 < p_{+} < \frac{1}{\alpha}$. Suppose that $\mu$ satisfies $\mathrm{(1.1)}$, $p\in\mathcal{P}(N)$ and $q\in\mathcal{P}(1)$. Then $T_{\alpha}$ defined as in $\mathrm{(1.10)}$ is bounded from variable Morrey spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ into spaces $M^{s(\cdot)}_{t(\cdot)}(X)_{N\bar{a}}$.

Remark 2.1. By Remark 1.7, once Theorem 2.2 is proved, it is easy to see that Theorem 2.1 holds. Thus, we only prove Theorem 2.2 in this section.

Proof of Theorem $\mathrm{2.2.}$ For any $x\in X$, by (1.9), we can deduce that

where $I_{\alpha}$ represents the homogeneous fractional integral operator (see ^[7]), namely, for any $x\in X$, set

Further, by applying the $(M^{p(\cdot)}_{q(\cdot)}(X)_{N}, M^{s(\cdot)}_{t(\cdot)}(X)_{N\bar{a}})$-boundedness of $I_{\alpha}$ in ^[7], we have

3.   Estimate for $[b, T_{\alpha}]$ on $L^{p(\cdot)}(X)$

In this section, by establishing the sharp maximal function for commutator $[b, T_{\alpha}]$, which is generated by $T_{\alpha}$ and $b\in\mathrm{RBMO}(\mu)$, we prove that the $[b, T_{\alpha}]$ is bounded from space $L^{p(\cdot)}(X)$ into space $L^{q(\cdot)}(X)$. The main theorem of this section is as follows.

Theorem 3.1. Let $N\geq 1$ be a constant, $b\in\mathrm{RBMO}(\mu)$, $K_{\alpha}$ satisfy $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $1 < p_{-}\leq p(x) < p_{+} < \frac{1}{\alpha}$ and $\frac{1}{q(x)} = \frac{1}{p(x)}-\alpha$ with $0 < \alpha < 1$. Suppose that $\mu$ satisfies condition $\mathrm{(1.1)}$. Then $[b, T_{\alpha}]$ defined as in $\mathrm{(1.11)}$ is bounded from $L^{p(\cdot)}(X)$ into $L^{q(\cdot)}(X)$.

To prove the above theorem, we need to recall and establish the following corollary and lemmas, see ^[6,7], respectively.

Corollary 3.1. If $f\in\mathrm{RBMO}(\mu)$, then there exists a constant $C > 0$ such that, for any balls $B$, $\rho\in(1, \infty)$ and $r\in[1, \infty)$,

Lemma 3.1. Let $\mu(X) < \infty$, $N\geq 1$ be a constant, $1 < p_{-}\leq p(x)\leq p_{+} < \infty$ and $s\in(1, p_{-})$. If there exists a positive constant $C$ such that for all $x\in X$ and $r > 0$, the following inequality

holds, then $M_{s, N}$ is bounded on $L^{p(\cdot)}(X)$, where maximal operator $M_{s, N}$ is defined by, for any $f\in L^{1}_{\mathrm{loc}}(X)$,

Moreover, if $s = 1$ in $\mathrm{(3.2)}$, we simply denote $M_{N}: = M_{1, N}$.

Lemma 3.2. Let $\mu(B) < \infty$, $N\geq 1$ be a constant, $\tau\in(0, 1)$, $s\in(1, \frac{1}{\tau})$, $s < p_{-}\leq p(\cdot) < \frac{1}{\tau}$ and $\frac{1}{q(x)} = \frac{1}{p(x)}-\tau$. Then there exists a constant $C > 0$ such that, for all $f\in L^{p(\cdot)}(X)$,

where

and the supremum is taken over all balls $B\ni x$.

Remark 3.1. With a way similar to that used in the proof of Theorem 1.3 in ^[2], it is easy to show that Lemma 3.4 hold on $(X, d, \mu)$.

Also, by applying Theorem 1.13 in ^[5], we have the following result on $(X, d, \mu)$.

Lemma 3.3. Let $K_{\alpha}$ satisfying $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $\alpha\in(0, 1)$ and $\frac{1}{q} = \frac{1}{p}-\alpha$. Suppose that $T_{\alpha}$ defined as in $\mathrm{(1.10)}$ is bounded on $L^{2}(\mu)$. Then $T_{\alpha}$ is bounded from $L^{p}(\mu)$ into $L^{q}(\mu)$.

From ^[6], the sharp maximal function $\widetilde{M}^{\sharp, \alpha}$ is defined by, for all $x\in X$, $\alpha\in[0, 1)$ and $f\in L^{1}_{\mathrm{loc}}(\mu)$,

where $\Delta_{x} = \{x\in B\subset S \hbox{ and } B, \ S \hbox{ are doubling balls}\}$ and coefficient $\widetilde{K}^{(\alpha)}_{B, S}$ is defined by

For $0 < r < \infty$ and $x\in X$, set $M^{\sharp, \alpha}_{r}f(x) = [M^{\sharp, \alpha}(|f|^{r})(x)]^{\frac{1}{r}}$. A simple computation shows that if $0 < r < 1$, we have

Lemma 3.4. Let $\tau\in(0, 1)$, $g\in L^{1}_{\mathrm{loc}}(X)$ and $\mu$-measurable function $f$ satisfy the following condition

then

Lemma 3.5. Let $K_{\alpha}$ satisfy the conditions $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $s\in(1, \infty)$ and $p_{0}\in(1, \infty)$. If $T_{\alpha}$ is bounded on $L^{2}(\mu)$, then there exists a positive constant $C$ such that, for all $f\in L^{\infty}(\mu)\cap L^{p_{0}}(\mu)$,

Proof. By applying the definition of sharp maximal function $M^{\sharp, \alpha}$ defined as in (3.4), for any ball $B$, it suffices to show that, for all $x$ and $B$ with $B\ni x$,

and, for all balls $B, \ S$ with $B\subset S$ and $B\ni x$,

where

To prove (3.8), decompose $f$ as $f: = f_{1}+f_{2}: = f\chi_{\frac{3}{2}B}+f\chi_{X\setminus(\frac{3}{2}B)}$, then write,

From Hölder inequality, Corollary 3.2 and (3.2), it follows that

To estimate $\mathrm{D}_{2}$, take $t = \sqrt{s}$ and $\frac{1}{r} = \frac{1}{t}-\alpha$. By applying Hölder inequality, Corollary 3.2 and Lemma 3.6, we obtain that

Since

thus, we only estimate the difference $|T_{\alpha}([b-b_{B}]f_{2})(y)-T_{\alpha}([b-b_{B}]f_{2})(z)|$. For all $y, \ z\in B$, by applying (1.7), (1.9), Corollary 3.2 and Hölder inequality, we have

where we have used the following fact that

Thus,

which, together with $\mathrm{D}_{1}$ and $\mathrm{D}_{2}$, implies (3.8).

Now let us estimate (3.9). Consider two balls $B\subset S$ with $x\in B$ and let $N: = N_{B, S}+1$. Write

With arguments similar to that used in the estimate of $\mathrm{D}_{3}$ and Theorem 1 in ^[22], it is not difficult to obtain that

and

For any $y\in B$, by applying Hölder inequality, Corollary 3.2 and (3.2), we obtain that

Taking the mean over ball $B$, we get $\mathrm{E}_{3}\leq C\widetilde{K}_{B, S}\|b\|_{\mathrm{RBMO}(\mu)}M^{(\alpha)}_{s, \frac{3}{2}}(f)(x)$. Similarly, we have

Which, combining the estimates $\mathrm{E}_{1}$, $\mathrm{E}_{2}$ and $\mathrm{E}_{3}$, implies (3.9).

Proof of Theorem $\mathrm{3.1.}$ By applying Lemmas 3.3 and 3.4, Lemmas 3.6-3.8 and Hölder inequality, we can deduce that

4.   Estimate for $[b, T_{\alpha}]$ on spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$

The main theorem of this section is stated as follows.

Theorem 4.1. Let $b\in\mathrm{RBMO}(\mu)$, $K_{\alpha}$ satisfy $\mathrm{(1.8)}$ and $\mathrm{(1.9)}$, $\mu(X) < \infty$, $N: = a_{1}(1+2a_{0})$, $1 < p_{-}\leq p(x)\leq q(x)\leq q_{+} < \infty$, $1 < t_{-}\leq t(x)\leq s(x)\leq s_{+} < \infty$, $\frac{t(x)}{s(x)} = \frac{q(x)}{p(x)}$, $\frac{1}{s(x)} = \frac{1}{p(x)}-\alpha$ and $0 < p(\cdot) < \frac{1}{\alpha}$ and $p\in\mathcal{P}(N)$. Suppose that $\mu$ satisfies $\mathrm{(1.1)}$. Then $[b, T_{\alpha}]$ defined as in $\mathrm{(1.11)}$ is bounded from spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ into spaces $M^{s(\cdot)}_{t(\cdot)}(X)_{N\bar{a}}$.

To prove the above theorem, we need to establish the following lemmas.

Lemma 4.1. Let $\mu(X) < \infty$, $1 < p_{-}\leq p(x)\leq q(x)\leq q_{+} < \infty$. Suppose that $N = a_{1}(1+2a_{0})$ and $p\in\mathcal{P}(N)$, $q\in\mathcal{P}(1)$. Then $M_{s, N}$ defined as in $\mathrm{(3.2)}$ is bounded from spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ into spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N\bar{a}}$.

Remark 4.1. With a slight modified argument similar to that use in the proof of Theorem 3.4 in ^[7], it is not difficult to prove that Lemma 4.2 also holds.

Lemma 4.2. Let $N$ be a constant satisfying the condition $N\geq 1$ and $\tau\in(0, 1)$. Suppose that $1\leq s < q_{-}\leq q(x)\leq p(x)\leq p_{+} < \infty$, $s < \frac{1}{\tau}$ and $1 < p(x) < \frac{1}{\tau}$. Suppose that $\mu$ satisfies condition $\mathrm{(1.1)}$. Then

Proof. For any $x\in X$, we set $\ell^{\frac{1}{p(x)}}_{x} = \frac{\|f\|_{M^{p(\cdot)}_{q(\cdot)}(X)_{N}}}{M_{s, N}f(x)}$. Then

For $\mathrm{H}_{1}$, we obtain that

If $\mu(B(x, Nr)) > \ell_{x}$, then there exists a $i\in\mathbb{N}$ such that $2^{i-1}\ell_{x} < \mu(B(x, Nr)) < 2^{i}\ell_{x}$. By applying Hölder inequality and Definition 1.6, we can deduce that

Which, together with estimate $\mathrm{H}_{1}$, the proof of Lemma 4.4 is completed.

By applying Lemmas 4.2 and 4.4, it is easy to get the following result.

Lemma 4.3. Let $\mu(X) < \infty$, $1 < p_{-}\leq p(x)\leq q(x)\leq q_{+} < \infty$, $1 < t_{-}\leq t(x)\leq s(x)\leq s_{+} < \infty$, $\frac{t(x)}{s(x)} = \frac{q(x)}{p(x)}$ and $\frac{1}{s(x)} = \frac{1}{p(x)}-\tau$ satisfying $0 < \tau < \frac{1}{p_{+}}$. Suppose that $N = a_{1}(1+2a_{0})$ and $p\in\mathcal{P}(N)$, $q\in\mathcal{P}(1)$. Then $M^{(\tau)}_{s, N}$ defined as in $\mathrm{(3.3)}$ is bounded from spaces $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$ into spaces $M^{s(\cdot)}_{t(\cdot)}(X)_{N\bar{a}}$.

Proof of Theorem $\mathrm{4.1}$ From Theorem 2.2, Lemmas 3.8, 4.2 and 4.4, it follows that

5.   Conclusions

In this paper, we mainly obtain the boundedness of $\theta$-type generalized fractional integral $T_{\theta}$ and its commutator $[b, T_{\theta}]$ generated by $b$ and $T_{\theta}$ on variable Lebesgue space $L^{p(\cdot)}(X)$ and Morrey space $M^{p(\cdot)}_{q(\cdot)}(X)_{N}$.

Acknowledgments

The authors would like to express sincere thanks to the reviewers for their helpful comments and suggestions. This research is supported by the Innovation Fund Project for Higher Education of Gansu Province (2020A-010) and Young Teachers' Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2020-07).

Conflict of interest

The authors declare that they have no conflict of interest.