Research article

Sharp bounds for multilinear Hardy operators on central Morrey spaces with power weights

  • Received: 09 April 2025 Revised: 23 May 2025 Accepted: 05 June 2025 Published: 19 June 2025
  • MSC : 42B25, 40A30

  • In this paper, we studied the precise norm of the multilinear Hardy operators Pm and Qm on central Morrey spaces with power weights. Furthermore, the precise norm of the multilinear Hardy operator Qm on Lebesgue spaces with power weights was also obtained.

    Citation: Meichuan Lv, Wenming Li. Sharp bounds for multilinear Hardy operators on central Morrey spaces with power weights[J]. AIMS Mathematics, 2025, 10(6): 14183-14195. doi: 10.3934/math.2025639

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  • In this paper, we studied the precise norm of the multilinear Hardy operators Pm and Qm on central Morrey spaces with power weights. Furthermore, the precise norm of the multilinear Hardy operator Qm on Lebesgue spaces with power weights was also obtained.



    Let f be a locally integrable function on R+=(0,), and the classical Hardy operator H and its adjoint H are defined by

    Hf(x)=1xx0f(y)dy,  Hf(x)=xf(y)ydy.

    Hardy [9,10] proved that

    HfLp(R+)pfLp(R+),   HfLp(R+)pfLp(R+),

    where 1<p<, 1/p+1/p=1, and the constants p and p are best possible, i.e.,

    HLp(R+)Lp(R+)=p,HLp(R+)Lp(R+)=p.

    Faris [6] introduced the following n-dimensional Hardy operator:

    Pf(x)=1Ωn|x|n|y|<|x|f(y)dy,  xRn{0}, (1.1)

    for nonnegative functions on Rn, where Ωn=πn2Γ(1+n2) is the volume of the unit ball in Rn and Γ(1+n2)=0tn2etdt. Christ and Grafakos [3] obtained that for 1<p<,

    PLp(Rn)Lp(Rn)=p.

    Zhao, Fu, and Lu [20] obtained the sharp bound for the weak-type (p,p) inequality,

    PLp(Rn)Lp,(Rn)=1,  1p.

    We use the following notation: dy=dy1dym, B(0,R) denotes a ball of radius R centered at the origin, for 1im, yi=(y1i,,yni) denotes elements of Rn, and the Euclidean norm of each yi is |yi|=(nj=1|yji|2)1/2 and of the m-tuple (y1,,ym) is |(y1,,ym)|=(mi=1|yi|2)1/2.

    The Hardy operator was extended to the multilinear setting by Fu, Grafakos, Lu, and Zhao [8]. Let mN, f1,,fm be locally integrable functions on Rn, and the multilinear Hardy operator Pm is defined by

    Pm(f1,,fm)(x)=1Ωmn|x|mn|(y1,,ym)|<|x|f1(y1)fm(ym)dy, (1.2)

    where xRn{0}. They obtained the sharp bounds for the multilinear Hardy operator Pm mapping from product Lebesgue spaces to Lebesgue spaces (both equipped with power weights) and from product central Morrey spaces to central Morrey spaces. After that, many researchers studied sharp estimates for the multilinear Hardy operators and their variants on Lebesgue spaces and Morrey-type spaces (see e.g., [13,14,16] and the references therein). Wei and Yan [18] gave the sharp bounds for the multilinear Hardy operators Pm on mixed radial-angular central Morrey spaces. Readers are referred to [7,11,17] for more details. Two other variants of multilinear Hardy operators were introduced and studied by Bényi and Oh [2].

    The adjoint operator of the n-dimensional Hardy operator P is

    Qf(x)=1Ωn|y||x|f(y)|y|ndy,xRn, (1.3)

    defined for locally integrable functions on Rn. Usually, we refer to both P and Q as n-dimensional Hardy operators. Using the fact that

    PLp(Rn)Lp(Rn)=p,  1<p<,

    and by duality, we obtain QLp(Rn)Lp(Rn)p. Consider f(x)=|x|np+εχB(0,1)(x), and it is easy to get that

    QLp(Rn)Lp(Rn)=p,  1<p<.

    Duoandikoetxea, Martín-Reyes, and Ombrosi [4] gave weighted inequalities for the n-dimensional Hardy operators P and Q.

    Let mN, f1,,fm be locally integrable functions on Rn, and we define the multilinear Hardy operator Qm as

    Qm(f1,,fm)(x)=1Ωmn|(y1,,ym)||x|f1(y1)fm(ym)|(y1,,ym)|mndy, (1.4)

    and obviously, Q1=Q.

    Let 1p<, 1/pλ<0, αR, and f be a measurable function on Rn, and we define the central (also known as local) Morrey spaces with power weights ˙Bp,λ(|x|αdx) by

    ˙Bp,λ(|x|αdx)={fLploc(|x|αdx):f˙Bp,λ(|x|αdx)<},

    where

    f˙Bp,λ(|x|αdx)=supR>0(1|B(0,R)|1+λpB(0,R)|f(x)|p|x|αdx)1p.

    Yee and Ho [19] obtained the boundedness of the Hardy operators on weighted local Morrey spaces. When α=0, ˙Bp,λ(|x|αdx) is the central homogeneous Morrey space ˙Bp,λ(Rn) introduced by Alvarez, Guzmán-Partida, and Lakey [1]. The classical Morrey space was introduced by Morrey [12], and the Morrey spaces with general weights were introduced in [15]. The case λ=1/p corresponds to the Lebesgue spaces with power weights.

    Let mN, 1<pi<, 1p<, αi<pn(11/pi), i=1,,m, 1p=1p1++1pm, and α=mj=1αj. Fu, Grafakos, Lu, and Zhao [8] obtained the multilinear Hardy operator Pm maps Lp1(|x|α1p1pdx)××Lpm(|x|αmpmpdx) to Lp(|x|αdx) with norm equal to the constant

    pmnωmn2m1ωmn(pmnnα)mi=1Γ(n2(11piαipn))Γ(n2(m1pαpn)),

    where ωn=nΩn. Let mN, 1<pi<, 1/pi<λi<0, 1p<, 1p=mj=11pj, pλ=piλi, i=1,,m, and λ=mj=1λj. They also obtained that Pm maps ˙Bp1,λ1(Rn)××˙Bpm,λm(Rn) to ˙Bp,λ(Rn) with norm

    Pm˙Bp1,λ1(Rn)××˙Bpm,λm(Rn)˙Bp,λ(Rn)=mωmn2m1ωmn(m+λ)mi=1Γ(n2(1+λi))Γ(n2(m+λ)).

    In this paper, we establish the sharp bounds for the multilinear Hardy operators Pm and Qm mapping from product central Morrey spaces to central Morrey spaces (both equipped with power weights) in Sections 2 and 3, respectively. We also obtain the precise norm of the multilinear Hardy operator Qm on Lebesgue spaces with power weights.

    We recall the definitions of the beta function B(z,w)=10tz1(1t)w1dt and the gamma function Γ(z)=0tz1etdt, where z and w are complex numbers with positive real parts. These functions satisfy the identity B(z,w)Γ(z+w)=Γ(z)Γ(w).

    In this section, we obtain sharp bounds for the multilinear Hardy operator Pm on the central Morrey spaces with power weights. Our results include, as special cases, the sharp bounds for the central Morrey spaces.

    Theorem 2.1. Let mN, 1<pi<, 1/pi<λi<0, 1p<, 1p=1p1++1pm, αi<pnλi+pn, i=1,,m, α=mj=1αj, and λ=mj=1λj. Then the multilinear Hardy operator Pm defined in (1.2) maps the product of central Morrey spaces with power weights ˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx) to ˙Bp,λ(|x|αdx) and

    Pm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)C1,

    where

    C1=pmnωmn2m1ωmn(pmn+pnλα)mi=1Γ(n2(1+λiαipn))Γ(n2(m+λαpn)).

    If for i=1,,m, pλ=piλi, then

    Pm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)=C1.

    Remark 2.2. For 1<p< and 1/p<λ<0, the operator norm from ˙Bp,λ(Rn) to itself of the n-dimensional Hardy operator P defined in (1.1) was evaluated in [8]. It was found to be independent of n.

    Remark 2.3. Assume that 1p=mj=11pj and λ=mj=1λj. Then for all i=1,,m, the inequality pλpiλi holds if and only if pλ=piλi.

    Proof of Theorem 2.1. As in the proof of [8], the operator Pm and its restriction to radial functions have the same operator norm on the spaces ˙Bp,λ(|x|αdx). Taking radial functions fi˙Bpi,λi(|x|αipipdx),i=1,,m, by the Minkowski integral inequality and the Hölder inequality, we have

    (1|B(0,R)|1+λpB(0,R)|Pm(f1,,fm)(x)|p|x|αdx)1p=(1|B(0,R)|1+λpB(0,R)|1Ωmn|x|mn|(y1,,ym)|<|x|f1(y1)fm(ym)dy|p|x|αdx)1p=1Ωmn(1|B(0,R)|1+λpB(0,R)||(y1,,ym)|<1mi=1fi(|x|yi)dy|p|x|αdx)1p1Ωmn|(y1,,ym)|<1(1|B(0,R)|1+λpB(0,R)|mi=1fi(x|yi|)|p|x|αdx)1pdy1Ωmn|(y1,,ym)|<1mi=1(1|B(0,R)|1+λipiB(0,R)|fi(x|yi|)|pi|x|αipipdx)1pidy=1Ωmn|(y1,,ym)|<1mi=1(1|B(0,R)|1+λipiB(0,|yi|R)|fi(t)|pi|t|αipipdt1|yi|n+αipip)1pidy1Ωmn|(y1,,ym)|<1mi=1|yi|nλiαipdymi=1fi˙Bpi,λi(|x|αipipdx),

    and

    1Ωmn|(y1,,ym)|<1mi=1|yi|nλiαipdy=mnωmnωmnπ20π2010rmn+nλαp1m1i=1(cosθi)n+nλiαip1(sinθi)mi1+mj=i+1(n+nλjαjp1)drdθ1dθm1.

    By an elementary calculation, we obtain that

    π20(cosθ)2z1(sinθ)2w1dθ=10(1t2)z1t2w1dt(t=sinθ)=1210(1x)z1xw1dx(x=t2)=12B(w,z). (2.1)

    Therefore, we have

    1Ωmn|(y1,,ym)|<1mi=1|yi|nλiαipdy=pmnωmn2m1ωmn(pmn+pnλα)B(mi=2(n+nλiαip)2,n+nλ1α1p2)B(mi=3(n+nλiαip)2,n+nλ2α2p2)B(n+nλmαmp2,n+nλm1αm1p2)=C1.

    Thus

    Pm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)C1.

    If for i=1,,m, pλ=piλi, let ˜fi(x)=|x|nλiαip, xRn, and then

    ˜fi˙Bpi,λi(|x|αipipdx)=supR>0(1|B(0,R)|1+λipiB(0,R)|˜fi(x)|pi|x|αipipdx)1pi=nλiωλin(1+piλi)1pi.

    We have

    Pm(˜f1,,˜fm)(x)=1Ωmn|(y1,,ym)|<1mi=1fi(|x|yi)dy=1Ωmn|x|nλαp|(y1,,ym)|<1mi=1|yi|nλiαipdy=C1mi=1˜fi(x),

    and

    Pm(˜f1,,˜fm)˙Bp,λ(|x|αdx)=supR>0(1|B(0,R)|1+λpB(0,R)|Pm(˜f1,,˜fm)(x)|p|x|αdx)1p=C1supR>0(1|B(0,R)|1+λpB(0,R)mi=1|˜fi(x)|p|x|αdx)1p=C1supR>0(1|B(0,R)|1+λpB(0,R)|x|pnλdx)1p=C1nλωλn(1+pλ)1p=C1mi=1nλiωλin(1+piλi)1pi=C1mi=1˜fi˙Bpi,λi(|x|αipipdx).

    This ends the proof.

    When αi=0 for all i=1,,m, the result above was proved in [8].

    In this section, we obtain sharp bounds for the multilinear Hardy operator Qm on the central Morrey spaces with power weights. Additionally, the sharp bounds for central Morrey spaces are derived.

    Theorem 3.1. Let mN, 1<pi<, 1/pi<λi<0, 1p<, 1p=mj=11pj, αi<pnλi+pn, i=1,,m, α=mj=1αj, λ=mj=1λj, and pnλ<α. Then the multilinear Hardy operator Qm defined in (1.4) maps ˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx) to ˙Bp,λ(|x|αdx) and

    Qm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)C2,

    where

    C2=pmnωmn2m1ωmn(αpnλ)mi=1Γ(n2(1+λiαipn))Γ(n2(m+λαpn)).

    If pλ=piλi,i=1,,m, then

    Qm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)=C2.

    Remark 3.2. For 1<p< and 1/p<λ<0, the operator norm from ˙Bp,λ(Rn) to itself of the n-dimensional Hardy operator Q defined in (1.3) is independent of n.

    Proof of Theorem 3.1. As before, we note that the operator Qm and its restriction to radial functions have the same operator norm in ˙Bp,λ(|x|αdx), and taking radial functions fi˙Bpi,λi(|x|αipipdx),i=1,m, then

    Qm(f1,,fm)(x)=1Ωmn|(y1,,ym)|>1f1(|x|y1)fm(|x|ym)|(y1,,ym)|mndy.

    By the Minkowski integral inequality and the Hölder inequality, we have

    (1|B(0,R)|1+λpB(0,R)|Qm(f1,,fm)(x)|p|x|αdx)1p=(1|B(0,R)|1+λpB(0,R)|1Ωmn|(y1,,ym)|>1f1(x|y1|)fm(x|ym|)|(y1,,ym)|mndy|p|x|αdx)1p1Ωmn|(y1,,ym)|>1(1|B(0,R)|1+λpB(0,R)|f1(x|y1|)fm(x|ym|)|(y1,,ym)|mn|p|x|αdx)1pdy1Ωmn|(y1,,ym)|>11|(y1,,ym)|mnmi=1(1|B(0,R)|1+λipiB(0,R)|fi(x|yi|)|pi|x|αipipdx)1pidy=1Ωmn|(y1,,ym)|>1mi=1|yi|nλiαip|(y1,,ym)|mnmi=1(1|B(0,|yi|R)|1+λipiB(0,|yi|R)|fi(t)|pi|t|αipipdt)1pidy1Ωmn|(y1,,ym)|>1mi=1|yi|nλiαip|(y1,,ym)|mndymi=1fi˙Bpi,λi(|x|αipipdx),

    and by Eq (2.1), we obtain that

    1Ωmn|(y1,,ym)|>1mi=1|yi|nλiαip|(y1,,ym)|mndy=mnωmnωmnπ20π201rnλαp1m1i=1(cosθi)n+nλiαip1(sinθi)mi1+mj=i+1(n+nλjαjp1)drdθ1dθm1=pmnωmn2m1ωmn(αpnλ)B(mi=2(n+nλiαip)2,n+nλ1α1p2)B(mi=3(n+nλiαip)2,n+nλ2α2p2)B(n+nλmαmp2,n+nλm1αm1p2)=C2.

    Thus

    Qm˙Bp1,λ1(|x|α1p1pdx)××˙Bpm,λm(|x|αmpmpdx)˙Bp,λ(|x|αdx)C2.

    If for i=1,,m, pλ=piλi, as in the proof of Theorem 3.1, let ˜fi(x)=|x|nλiαip, xRn, and we have

    Qm(˜f1,,˜fm)(x)=1Ωmn|(y1,,ym)|>1˜f1(x|y1|)˜fm(x|ym|)|(y1,,ym)|mndy=1Ωmn|x|nλαp|(y1,,ym)|>1mi=1|yi|nλiαip|(y1,,ym)|mndy=C2mi=1˜fi(x),

    and

    Qm(˜f1,,˜fm)˙Bp,λ(|x|αdx)=supR>0(1|B(0,R)|1+λpB(0,R)|Qm(˜f1,,˜fm)(x)|p|x|αdx)1p=C2supR>0(1|B(0,R)|1+λpB(0,R)mi=1|˜fi(x)|p|x|αdx)1p=C2mi=1nλiωλin(1+piλi)1pi=C2mi=1˜fi˙Bpi,λi(|x|αipipdx).

    This ends the proof.

    Let αi=0, i=1,,m, and we have the following result.

    Corollary 3.4. Let mN, 1<pi<, 1/pi<λi<0, 1p<, 1p=mj=11pj, pλ=piλi, i=1,,m, and λ=mj=1λj. Then the multilinear Hardy operator Qm maps ˙Bp1,λ1(Rn)××˙Bpm,λm(Rn) to ˙Bp,λ(Rn) with norm

    Qm˙Bp1,λ1(Rn)××˙Bpm,λm(Rn)˙Bp,λ(Rn)=mωmn2m1ωmnλmi=1Γ(n2(1+λi))Γ(n2(m+λ)).

    In this section, we establish sharp bounds for the multilinear Hardy operator Qm on Lebesgue spaces with power weights.

    Theorem 4.1. Let mN, 1<pi<, 1p<, 1p=mj=11pj, αi<pn(11/pi), i=1,,m, α=mj=1αj, and n+α>0. Then the multilinear Hardy operator Qm maps the product of weighted Lebesgue spaces Lp1(|x|α1p1pdx)××Lpm(|x|αmpmpdx) to Lp(|x|αdx) with norm equal to the constant

    pmnωmn2m1ωmn(n+α)mi=1Γ(n2(11piαipn))Γ(n2(m1pαpn)).

    Proof. As before, we observe that the operator Qm and its restriction to radial functions have the same operator norm on Lp(|x|αdx). Let fiLpi(|x|αipipdx), i=1,,m, be radial functions, and by Minkowski's integral inequality and Hölder's inequality, we have

    Qm(f1,,fm)Lp(|x|αdx)=1Ωmn(Rn||(y1,,ym)|>1f1(|x|y1)fm(|x|ym)|(y1,,ym)|mndy|p|x|αdx)1p1Ωmn|(y1,,ym)|>1(Rn|f1(|x|y1)fm(|x|ym)|(y1,,ym)|mn|p|x|αdx)1pdy=1Ωmn|(y1,,ym)|>1(Rn|f1(x|y1|)fm(x|ym|)|(y1,,ym)|mn|p|x|αdx)1pdy=1Ωmn|(y1,,ym)|>11|(y1,,ym)|mnmi=1(Rn|fi(x|yi|)|pi|x|αipipdx)1pidy1Ωmn|(y1,,ym)|>11|(y1,,ym)|mnmi=1|yi|npi+αipdymi=1fiLpi(|x|αipipdx)=C3mi=1fiLpi(|x|αipipdx),

    and by Eq (2.1), we obtain that

    C3=mnωmnωmnπ20π201rnpαp1m1i=1(cosθi)nnpiαip1(sinθi)mi1+mj=i+1(nnpjαjp1)drdθ1dθm1=pmnωmn2m1ωmn(n+α)B(mi=2(nnpiαip)2,nnp1α1p2)B(mi=3(nnpiαip)2,nnp2α2p2)B(nnpmαmp2,nnpm1αm1p2)=pmnωmn2m1ωmn(n+α)mi=1Γ(n2(11piαipn))Γ(n2(m1pαpn)).

    To show that C3 is the best possible constant, we should obtain that

    QmLp1(|x|α1p1pdx)××Lpm(|x|αmpmpdx)Lp(|x|αdx)C3.

    For a sufficiently small ε, 0<ε<min{1,n+αpm,1m}, and we define

    fεi(x)={|x|npiαip+pmεpi,|x|1m,0,|x|>1m,

    where i=1,,m. We have that

    fε1p1Lp1(|x|α1p1pdx)==fεmpmLpm(|x|αmpmpdx)=ωnpmε(1m)pmε.

    Qm(fε1,,fεm)(x)=0 when |x|1, and that

    Qm(fε1,,fεm)(x)=|x|npαp+pmεpΩmn|(y1,,ym)|>1;|y1|<mm|x|;;|ym|<mm|x|mi=1|yi|npiαip+pmεpi|(y1,,ym)|mndy

    when |x|<1.

    By Eq (2.1), we have

    Qm(fε1,,fεm)Lp(|x|αdx)=1Ωmn(|x|<1||x|npαp+pmεp|(y1,,ym)|>1;|y1|,,|ym|<mm|x|mi=1|yi|npiαip+pmεpi|(y1,,ym)|mndy|p|x|αdx)1p1Ωmn(|x|<ε||x|npαp+pmεp|(y1,,ym)|>1;|y1|,,|ym|<mmεmi=1|yi|npiαip+pmεpi|(y1,,ym)|mndy|p|x|αdx)1p(ωnεpmεpmε)1pmnωmnωmnπ20π20mmε1rnpαp+pmεp1m1i=1(cosθi)nnpiαip+pmεpi1(sinθi)mi1+mj=i+1(nnpjαjp+pmεpj1)drdθ1dθm1=mnωmnmpmεpεpmεp(1(mε)np+αppmεp)2m1ωmn(np+αppmεp)B(mi=2(nnpiαip+pmεpi)2,nnp1α1p+pmεp12)B(mi=3(nnpiαip+pmεpi)2,nnp2α2p+pmεp22)B(nnpmαmp+ε2,nnpm1αm1p+pmεpm12)mi=1fεiLpi(|x|αipipdx).

    Finally, let ε0, and we get

    QmLp1(|x|α1p1pdx)××Lpm(|x|αmpmpdx)Lp(|x|αdx)C3.

    This ends the proof.

    Duoandikoetxea, Martín-Reyes, and Ombrosi [4] introduced the n-dimensional maximal operator

    Nf(x)=supr>|x|1Ωnrn|y|<r|f(y)|dy

    for locally integrable functions on Rn. The operator N plays a crucial role in proving the boundedness of both the Hardy operator P and the Calderón operator S (defined as S=P+Q) on weighted Lebesgue spaces. In [5], the operator N was further employed to build weights that yield the boundedness of the fractional hardy operator. Let f be the nonnegative integrable on Rn, and we obtain

    Pf(x)Nf(x)Pf(x)+Qf(x).

    For 1<p<, 1p<λ<0, and αR, by Minkowski's integral inequality, we have

    P˙Bp,λ(|x|αdx)˙Bp,λ(|x|αdx)N˙Bp,λ(|x|αdx)˙Bp,λ(|x|αdx)P˙Bp,λ(|x|αdx)˙Bp,λ(|x|αdx)+Q˙Bp,λ(|x|αdx)˙Bp,λ(|x|αdx),

    and

    PLp(|x|αdx)Lp(|x|αdx)NLp(|x|αdx)Lp(|x|αdx)PLp(|x|αdx)Lp(|x|αdx)+QLp(|x|αdx)Lp(|x|αdx).

    An immediate consequence of Theorems 2.1 and 3.1 is that for all 1<p<, 1p<λ<0, pnλ<α<pnλ+pn,

    pnpn+pnλαN˙Bp,λ(|x|αdx)˙Bp,λ(|x|αdx)pnpn+pnλα+pnαpnλ.

    By Theorems 4.1 and 1 in [8], we obtain that for 1<p< and n<α<pnn,

    pnpnnαNLp(|x|αdx)Lp(|x|αdx)pnpnnα+pnn+α.

    We establish the precise norm of the multilinear Hardy operators Pm and Qm on central Morrey spaces with power weights. Following the method developed in the proof of [8], we also obtain the exact operator norms of the multilinear Hardy operator Qm on Lebesgue spaces with power weights. This approach may be adapted to study the multilinear Hardy operators Pm and Qm on Herz spaces and Herz-Morrey spaces.

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

    Meichuan Lv: Conceptualization, writing-original draft, methodology, writing-review and editing; Wenming Li: Conceptualization, methodology, writing-review, supervision, language editing and funding acquisition. All the authors have read and agreed to the published version of the manuscript.

    The work was supported by the Natural Science Foundation of Hebei Province (No. A2021205013) and the Innovation Fund of the School of Mathematical Sciences, Hebei Normal University (No. ycxzzbs202502).

    The authors state that there are no conflicts of interest in this paper.



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