多线性分数次积分算子在加权变指数Herz乘积空间上的有界性
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摘要
利用加权变指数Lebesgue空间的特征和多线性分数次积分算子的L~p有界性,基于加权变指数Herz空间的定义,运用调和分析实方法进行不等式的估计,证明了多线性分数次积分算子在加权变指数Herz乘积空间的有界性.
The definitions and some basic properties of the variable exponent Lebesgue space are mentioned.Then, by the properties of weighted Lebesgue spaces with variable exponents and the boundedness of the multilinear fractional integral operator on L~p, based on the definition of the weighted Herz spaces with variable exponent, using the real methods in harmonic analysis, the boundedness of the multilinear fractional integral operators on the product weighted Herz spaces with variable exponents is obtained.
The definitions and some basic properties of the variable exponent Lebesgue space are mentioned.Then, by the properties of weighted Lebesgue spaces with variable exponents and the boundedness of the multilinear fractional integral operator on L~p, based on the definition of the weighted Herz spaces with variable exponent, using the real methods in harmonic analysis, the boundedness of the multilinear fractional integral operators on the product weighted Herz spaces with variable exponents is obtained.
引文
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[7]Grafakos L,Torres R H.Multilinear Calderòn-Zygmund theary[J].Advances in Mathematics,2002,165(1):124-164.DOI:10.1006/aima.2001.2028.
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[9]Cruz-Uribe D,Wang L A.Extrapolation and weighted norm inequalities in the variable Lebesgue Spaces[J].Transactions of the American Mathematical Society,2014,369(2):1 205-1 235.
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[11]张立平,赵凯,耿素丽.与微分算子相关的分数次积分算子交换子的有界性[J].云南大学学报:自然科学版,2014,36(1):1-5.Zhang L P,Zhao K,Geng S L.Boundedness of the commutators of the fractional integral operators related to differential operators[J].Journal of Yunnan University:Natural Sciences Edition,2014,36(1):1-5.
[12]Capone C,Cruz-Uribe D,Fiorenza A.The fractional maximal operator and fractional integrals on variable Lp spaces[J].Revista Matematica Iberoamericana,2007,23(23):743-770.
[13]Ho K P.The fractional integral operators on Morrey spaces with variable exponent on unbounded domains[J].Mathematical Inequalities&Applications,2013,16(2):363-373.
[14]Izuki M.Fractional integrals on Herz-Morrey spaces with variable exponent[J].Hiroshima Mathematical Journal,2010,40(3):343-355.DOI:10.32917/hmj/1291818849.
[15]Kenig C E,Stein E M.Multilinear estimates and fractional integration[J].Mathematical Research Letters,1999,6(1):1-15.DOI:10.4310/MRL.1999.v6.n1.a1.
[16]Shi Y L,Tao X X.Boundedness for multilinear fractional integral operators on Herz type spaces[J].Applied Mathematics AJournal of Chinese Universities,2008,23(4):437-446.DOI:10.1007/s11766-008-1995-x.
[17]Shi Y L,Tao X X.Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces[J].Hokkaido Mathematical Journal,2009,38(4):635-662.DOI:10.14492/hokmj/1258554238.
[18]董楠楠,赵凯.内蕴平方函数在加权变指数Herz-Morrey空间上的有界性[J].云南大学学报:自然科学版,2017,39(6):924-929.Dong N N,Zhao K.Boundedness of intrinsic square functions on weighted Herz-Morrey spaces with variable exponent[J].Journal of Yunnan University:Natural Sciences Edition,2017,39(6):924-929.
[19]Izuki M,Noi T.An intrinsic square function on weighted Herz spaces with variable exponent[J].Journal of Mathematical Inequalities,2017,11(3):799-816.
[20]吴青.变指数空间中的多线性算子[D].大连:大连海事大学,2015.Wu Q.Multilinear operators in variable space[D].Dalian:Dalian Maritime University,2015.
[2]Tang C Q,Yang D C,Zhang P.Boundedness of generalized fractional operators[J].Approximation Theory and its Applications,2002,18(4):38-54.
[3]Ková?ik O,Rákosnik J.On spaces Lp(x)and Wk,p(x)[J].Czechoslovak Mathematical Journal,1994,41(4):592-618.
[4]Cruz-Uribe D,Fiorenza A,Neugebauer C.The maximal function on variable Lp spaces[J].Annales Academiae Scientiarum Fennicae-Mathematica,2003,28(1):223-238.
[5]Diening L.Riesz potential and Sobolev embeddings on generalized Lebesgue spaces Sobolev spaces L·)p(and Wk,p(·)[J].Mathematische Nachrichten,2004,268(1):31-43.DOI:10.1002/(ISSN)1522-2616.
[6]Diening L,Ru?i?k M.Non-Newtonian fluids and function spaces[J].Nonlinear Analysis Function Spaces&Applications,2007(8):95-143.
[7]Grafakos L,Torres R H.Multilinear Calderòn-Zygmund theary[J].Advances in Mathematics,2002,165(1):124-164.DOI:10.1006/aima.2001.2028.
[8]Huang A W,Xu J S.Multilinear singular integrals and commutators in variable Lebesgue spaces[J].Applied Mathematics AJournal of Chinese Universities,2010,25(1):69-77.DOI:10.1007/s11766-010-2167-3.
[9]Cruz-Uribe D,Wang L A.Extrapolation and weighted norm inequalities in the variable Lebesgue Spaces[J].Transactions of the American Mathematical Society,2014,369(2):1 205-1 235.
[10]赵凯,董彭娟,任晓芳.带粗糙核的分数次积分算子交换子在Morrey-Herz空间的加权有界性[J].云南大学学报:自然科学版,2010,32(6):621-627.Zhao K,Dong P J,Ren X F.Weighted boundedness of commutors of fractional integral operators with rough kernel on Morrey-herz spaces[J].Journal of Yunnan University:Natural Sciences Edition,2010,32(6):621-627.
[11]张立平,赵凯,耿素丽.与微分算子相关的分数次积分算子交换子的有界性[J].云南大学学报:自然科学版,2014,36(1):1-5.Zhang L P,Zhao K,Geng S L.Boundedness of the commutators of the fractional integral operators related to differential operators[J].Journal of Yunnan University:Natural Sciences Edition,2014,36(1):1-5.
[12]Capone C,Cruz-Uribe D,Fiorenza A.The fractional maximal operator and fractional integrals on variable Lp spaces[J].Revista Matematica Iberoamericana,2007,23(23):743-770.
[13]Ho K P.The fractional integral operators on Morrey spaces with variable exponent on unbounded domains[J].Mathematical Inequalities&Applications,2013,16(2):363-373.
[14]Izuki M.Fractional integrals on Herz-Morrey spaces with variable exponent[J].Hiroshima Mathematical Journal,2010,40(3):343-355.DOI:10.32917/hmj/1291818849.
[15]Kenig C E,Stein E M.Multilinear estimates and fractional integration[J].Mathematical Research Letters,1999,6(1):1-15.DOI:10.4310/MRL.1999.v6.n1.a1.
[16]Shi Y L,Tao X X.Boundedness for multilinear fractional integral operators on Herz type spaces[J].Applied Mathematics AJournal of Chinese Universities,2008,23(4):437-446.DOI:10.1007/s11766-008-1995-x.
[17]Shi Y L,Tao X X.Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces[J].Hokkaido Mathematical Journal,2009,38(4):635-662.DOI:10.14492/hokmj/1258554238.
[18]董楠楠,赵凯.内蕴平方函数在加权变指数Herz-Morrey空间上的有界性[J].云南大学学报:自然科学版,2017,39(6):924-929.Dong N N,Zhao K.Boundedness of intrinsic square functions on weighted Herz-Morrey spaces with variable exponent[J].Journal of Yunnan University:Natural Sciences Edition,2017,39(6):924-929.
[19]Izuki M,Noi T.An intrinsic square function on weighted Herz spaces with variable exponent[J].Journal of Mathematical Inequalities,2017,11(3):799-816.
[20]吴青.变指数空间中的多线性算子[D].大连:大连海事大学,2015.Wu Q.Multilinear operators in variable space[D].Dalian:Dalian Maritime University,2015.
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