Hausdorff算子的双权范数估计.pdf

第40卷第4期2023年7月新疆大学学报（自然科学版）（中英文）Journal of Xinjiang University(Natural Science Edition in Chinese and English)Vol.40,No.4Jul.,2023Two-Weight Norm Estimates for HausdorffOperatorsYANG Shuai,LI Baode(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)Abstract:Two-weight inequalities for Hausdorffoperators have been established in weighted Lebesgue spaces under certainassumptions of kernel function.We study the Hausdorffoperators of special kind on the real line R+and also establish corre-sponding two-weight inequalities under weaker assumptions of kernel function.Then we give some specific examples that satisfythese conditions of kernel function.Key words:Hausdorffoperators;weighted Lebesgue spaces;weight functionDOI：10.13568/ki.651094.651316.2022.11.12.0001CLC number：O177.6Document Code：AArticle ID：2096-7675(2023)04-0398-07引文格式：杨帅,李宝德.Hausdorff算子的双权范数估计J.新疆大学学报(自然科学版)(中英文),2023,40(4):398-404.英文引文格式：YANG Shuai,LI Baode.Two-weight norm estimates for HausdorffoperatorsJ.Journal of Xinjiang Univer-sity(Natural Science Edition in Chinese and English),2023,40(4):398-404.Hausdorff算子的双权范数估计杨 帅，李宝德(新疆大学 数学与系统科学学院，新疆 乌鲁木齐830017)摘要：Hausdorff算子的核函数在满足一定条件时,其在加权 Lebesgue 空间上的双权不等式已经被建立 研究了定义在 R+上的 Hausdorff算子,当 Hausdorff算子的核函数在更弱的条件下仍能建立相应的双权不等式,并给出了满足这种条件的具体核函数的例子关键词：Hausdorff算子；加权 Lebesgue 空间；权函数0Introduction and Main ResultsFix a locally integrable function on the interval(0,).The one-dimensional Hausdorffoperator His defined byHf(x)=Z0(xy)yf(y)dy.Forthesakeofsimplicity,weinitiallyassumethatfunctions f areintheclassofSchwartzfunctions.TheHausdorffoperatorisdeeply rooted in the study of one-dimensional Fourier analysis and has become an essential part of modern harmonic analysis.In particular,it is closely related to the summability of the classical Fourier series1.Additionally,many classical operators inanalysis are special cases of the Hausdorffoperator if one chooses suitable kernel functions 25.For example,if one choosea suitable kernel function:(1,)(t)/t,(0,1)(t),max1,t,(1t)1(0,1)(t),respectively,the Hausdorffoperator is reducedto Hardy operator,adjoint Hardy operator6,Hardy-Littlewood-P olya operator7and Ces aro operator89,respectively.Thus,the Hausdorffoperator has been received a lot of attention in harmonic analysis.Two-weight norm inequalities estimates arealso an important part of harmonic analysis1011.We recall the definition of the weighted Lebesgue spaces.Received Date：2022-11-12Foundation Item：This project was supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China“Function spacesadapted to multi-level anisotropic ellipsoid covers and the boundedness of related operators”(2020D01C048);the National Natural Science Foundation ofthe Peoples Republic of China“Real-variable theory of variable exponential functions on anisotropic Euclidean spaces and its applications”(12261083).Biography：YANG Shuai(1996-),male,postgraduate,research fields:harmonic analysis,E-mail:.Corresponding author：LI Baode(1981-),male,professor,research fields:harmonic analysis,E-mail:.No.4YANG Shuai,et al:Two-Weight Norm Estimates for HausdorffOperators399Definition 1 Let 1 p and be a weight on R+(i.e.is an integrable and-a.e.positive function on R+).Theweighted Lebesgue space,denoted by Lp(R+),is the class of all Lebesgue measurable functions f on R+for which the normkfkLp(R+):=Z0|f(x)|p(x)dx!1p.In 2021,via adding certain assumptions on,Bandaliyev and Safarova established necessary and sufficient conditionson monotone weight functions for the boundedness of Hausdorffoperators Hin weighted Lebesgue spaces12.Their resultsare stated as follows.Theorem 1 Let 1 p ,and let u and v be increasing weight functions defined on R+.Let be a positive functionon R+satisfying the following conditions:(i)Z10(t)t11pdt0,i=1,2 such thatC1t(t)C2t,t1.Then the inequality?Hf?Lpu(R+)CkfkLpv(R+)holds if and only ifsupt0 Ztu(x)xpdx!1p Zt0v(x)1pdx!1p.Theorem 2 Let 1 p ,and let u and v be increasing weight functions defined on R+.Let be a positive functionon R+satisfying the following conditions:(i)Z1(t)t11pdt0,i=1,2 such thatC1(t)C2,t1.Then the inequality?Hf?Lpu(R+)CkfkLpv(R+)holds if and only ifsupt0 Zt0u(x)dx!1p Ztv(x)1pxpdx!1p.In this paper,by extending the assumptions of,in Theorem 3 and Theorem 4,we obtain some new boundedness resultsof Hon weighted Lebesgue spaces as follows.Theorem 3 Let 1 p and be a positive function on R+satisfying the following conditions:(i)Z10(t)t11pdt0 such that,for all t1,(t)Cta(1)Increasing weight functions u and v defined on R+satisfyB:=supt0 Ztu(x)minxp,xapdx!1p Zt0v(x)1pdx!1p1,b p1and a(p1,1.(t)=1/tbt(0,11/tat(1,)is an example of of Theorem 3.Theorem 4 Let 1 p and be a positive function on R+satisfying the following conditions:(i)Z1(t)t11pdt0 such that,for all 00 Zt0u(x)max1,xapdx!1p Ztv(x)1pxpdx!1p1,b p1and a(p1,0.(x)=tat(0,11/tbt(1,)is an example of of Theorem 4.Remark 3Under the assumptions of Theorem 3 or Theorem 4,we can choose(t)=(1,)(t)/t,(0,1)(t),(1 t)1(0,1)(t)(1),then we obtain the boundedness of corresponding Hardy operator,adjoint Hardy operator,Ces aro operatoron weighted Lebesgue spaces.1Proofs of Theorem 3 and Theorem 4We need the following Lemmas to prove the main results.Lemma 113Let 1 p and be a positive function on R+,ifK,p:=Z0(t)t11pdt(5)then?Hf?Lp(R+)K,pkfkLp(R+).Lemma 214Let 1 p0 Ztu1(x)xpdx!1p Zt0v1(x)1pdx!1pholds,where the constant C satisfies B1C p1/p(p)1/pB1.Lemma 314Let 1 p0 Zt0u2(x)dx!1p Ztv2(x)1pxpdx!1pNo.4YANG Shuai,et al:Two-Weight Norm Estimates for HausdorffOperators401holds,where the constant C satisfies B1C p1/p(p)1/pB1.Proof of Theorem 3By(1)and p1a1,we haveZ0(t)tt1pdt=Z10(t)tt1pdt+Z1(t)tt1pdtZ10(t)tt1pdt+CZ1dtt1+a1pZ10(t)tt1pdt+Cpap10 Ztu(x)xpdx!1p Zt0v(x)1pdx!1psupt0u(t)v(t)#1p Ztdxxp!1p Zt0dx!1p=1(p1)1psupt0u(t)v(t)#1p.Thus,for every t0,condition(2)implies thatu(t)(p1)Bpv(t)(7)Now we estimate E1and E2separately.By(6)and(7),we haveE1=Z0?Hf(x)?pu(0)dx!1p=(u(0)1p Z0?Hf(x)?pdx!1pK,p(u(0)1p Z0|f(x)|pdx!1pK,p Z0|f(x)|pu(x)dx!1p B(p1)1pK,pkfkLpv(R+)B(p1)1pK,pkfkLpv0(R+),andE2=Z0?Hf(x)?p Zx0(t)dt!dx#1p=Z0?Hf(x)?p Z0(t)xt(x)dt!dx#1p=Z0Z0?Hf(x)?p(t)xt(x)dtdx!1p=Z0(t)Zt?Hf(x)?pdx!dt#1p21pZ0(t)Zt?Zt0(xy)yf(y)dy?pdxdt1p+21pZ0(t)Zt?Zt(xy)yf(y)dy?pdxdt1p=:E21+E22.402Journal of Xinjiang University(Natural Science Edition in Chinese and English)2023Then we estimate E21and E22separately.Notice that if xt,yt,then x/y1.From(1)and 1/pa1,it follows thatE21=21pZ0(t)Zt?Zt0?xy?yf(y)dy?pdxdt1p21pZ0(t)ZtZt0?xy?y|f(y)|dypdxdt1p21pCZ0(t)Ztdxxap!Zt0ya1|f(y)|dy!pdt#1p=21pC p1ap1!1pZ0(t)t1+papp1 1tZt0ya1|f(y)|dy!pdt#1p.Moreover,we have1p1Zt(s)s1+papspds=Zt(s)spap Zsdxxp!ds=Z0(s)spap(t,)(s)Z0(s,)(x)xpdx!ds=Z0Z0(s)spapxp(t,)(s)(s,)(x)dxds=Ztxp Zxt(s)spapds!dxZtxpxpap Zx0(s)ds!dxZtu(x)xpapxpdx.From this and(2),it follows that 1p1Zt(s)s1+papspds!1p Zt0v(x)1pdx!1pt(y)dy?pxt(x)dxdt1p21pZ0(t)Z0?Z0?xy?yf(y)yt(y)dy?pdxdt1p21pK,pZ0(t)Z0|f(x)|pxt(x)dx!dt#1p=21pK,pZ0|f(x)|p Zx0(t)dt!dx#1p21pK,p Z0|f(x)|pu(x)dx!1p21pK,pB(p1)1pkfkLpv(R+)21pK,pB(p1)1pkfkLpv0(R+).The proof is completed.No.4YANG Shuai,et al:Two-Weight Norm Estimates for HausdorffOperators403Proof of Theorem 4By(3)and p1a0,we haveZ0(t)tt1pdt=Z10(t)tt1pdt+Z1(t)tt1pdtCZ101t1a1pdt+Z1(t)tt1pdt=Cpap+1+Z1(t)tt1pdt0 Zt0u(x)dx!1p Ztv(x)1pxpdx!1psupt0u(t)v(t)#1p Ztdxxp!1p Zt0dx!1p=1(p1)1psupt0u(t)v(t)#1p.Thus,for every t0,condition(4)implies thatu(t)(p1)p1(B)pv(t)(8)Now we estimate E1and E2separately.By(6)and(8),we haveE1=Z0?Hf(x)?pu()dx!1p=(u()1p Z0?Hf(x)?pdx!1pK,p(u()1p Z0|f(x)|pdx!1pK,p Z0|f(x)|pu(x)dx!1pK,p(p1)1pBkfkLpv(R+)K,p(p1)1pBkfkLpv0(R+),andE2=Z0?Hf(x)?p Zx(t)dt!dx#1p=Z0?Hf(x)?p Z0(t)xt(x)dt!dx#1p=Z0Z0?Hf(x)?p(t)xt(x)dtdx!1p=Z0(t)Zt0?Hf(x)?pdx!dt#1p21pZ0(t)Zt0?Zt0(xy)yf(y)dy?pdxdt1p+21pZ0(t)Zt0?Zt(xy)yf(y)dy?pdxdt1p=:E21+E22.Then we estimate E21and E22separately.By(6)and(8),we haveE21=21pZ0(t)Z0?Z0?xy?yf(y)yt(y)dy?pxt(x)dxdt1p21pZ0(t)Z0?Z0?xy?yf(y)yt(y)dy?pdxdt1p21pK,pZ0(t)Z0|f(x)|pxt(x)dx!dt#1p=21pK,pZ0|f(x)|p Zx(t)dt!dx#1p21pK,p Z0|f(x)|pu(x)dx!1p21pK,p(p1)1pBkfkLpv(R+)21pK,p(p1)1pBkfkLpv0(R+).404Journal of Xinjiang University(Natural Science Edition in Chinese and English)2023Next we estimate E22.Notice that if xt,yt,then x/y1.From(3)and p1a0,it follows thatE22=21pZ0(t)Zt0?Zt?xy?yf(y)dy?pdxdt1p21pZ0(t)Zt0Zt?xy?y|f(y)|dypdxdt1p21pCZ0(t)Zt0 xapdx!Ztya|f(y)|ydy!pdt#1p=21pC(11+ap)1pZ0(t)t1+ap Ztya|f(y)|ydy!pdt#1p.Moreover,we haveZt0(s)s1+apds=Zt0(s)sap Zs0dx!ds=Z0(s)sap(0,t)(s)Z0(0,s)(x)dx!ds=Z0Z0(s)sap(0,t)(s)(0,s)(x)dxds=Zt0 Ztx(s)sapds!dxZt0 xap Zx(s)ds!dxZt0u(x)xapdx.From this and(4),it follows that Zt0(s)s1+apds!1p Ztv(x)1pxpdx!1p B.By this and Lemma 3 with u2(t)=(t)t1+apand v2(t)=v(t),we haveE2221pC(11+ap)1pZ0(t)t1+ap Ztya|f(y)|ydy!pdt#1p21pC(11+ap)1pp1p(p)1pBkyaf(y)kLpv(R+)21pC(11+ap)1pp1p(p)1pBkf(y)kLpv0(R+).The proof is completed.References：1LAL S N,RAM S.On the absolute Hausdorffsummability of a Fourier seriesJ.Pacific Journal of Mathematics,1972,42(2):439-451.2CHEN J C,FAN D S,LI J.Hausdorffoperators on function spacesJ.Chinese Annals of Mathematics,Series B,2012,33(4):537-556.3CHEN J C,FAN D S,ZHANG C J.Multilinear Hausdorffoperators and their best constantsJ.Acta Mathematica Sinica,2012,28(8):1521-1530.4LIFLYAND E,MORICZ F.The Hausdorffoperator is bounded on the real Hardy space H1(R)J.Proceedings of the American Mathematical Society,2000,128(5):1391-1396.5FAN D S,LIN X Y.Hausdorffoperator on real Hardy spacesJ.Analysis,2014,34(4):319-337.6FU Z W,LIU Z G,LU S Z,et al.Characterization for commutators of n-dimensional fractional Hardy operatorsJ.Science in China(Series A),2007,50(10):1418-1426.7BENYIA,OH C.Best constants for certain multilinear integral operatorsJ.Journal of Inequalities and Applications,2006,2006:28582.8GIANG D V,MORICZ F.The Ces aro operator is bounded on the Hardy spaces H1J.Acta Mathematica Scientia,1995,61(4):533-544.9KANJIN Y.The Hausdorffoperator on the real Hardy spaces Hp(R)J.Studia Mathematica,2001,148(1):37-45.10 WELLAND G V.Weighted norm inequalities for fractional integralsJ.Proceedings of the American Mathematical Society,1975,51(1):143-148.11 ZHANG H,QI C Y,LI B D.A weighted estimates of anisotropic fractional integral operatorsJ.Journal of Xinjiang University(Natural ScienceEdition),2017,34(1):35-39.12 BANDALIYEV R,SAFAROVA K.On two-weight inequalities for Hausdorffoperators of special kind in Lebesgue spacesJ.Hacettepe Jouenal ofMathematics and Statistics,2021,50(5):1334-1346.13 BROWN G,MORICZ F.The Hausdorffoperator and the quasi Hausdorffoperators on the space Lp,1 p J.Mathematical Inequalities andApplications,2000,3(1):105-115.14 KUFNER A,MALIGRANDA L,PERSSON L E.The hardy inequality:about its history and some related resultsM.Plzen:Vydavatelsk y Servis,2007.15 BANDALIEV R A,OMAROVA K K.Two-weight norm inequalities for certain singular integralsJ.Taiwanese Journal of Mathematics,2012,16(2):713-732.责任编辑：张自强刘 敏