具有非卷积型核的多线性Littlewood-Paley算子在Campanato空间上的新估计
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摘要
考虑具有非卷积型核的多线性Littlewood-Paley算子在Campanato空间上的有界性,其中包括多线性g-函数,多线性Lusin面积积分S和多线性g_λ*-函数.证明了如果f=(f_1,…,f_n),f_i∈ε~(α_i,p_i)(R~n),i=1,…,m,那么g(f),S(f),g_λ*(f)几乎处处等于无穷或几乎处处有限,且在后一种情形下,算子[g(f)]~2,[S(f)]~2,[g_λ*(f)]~2从ε~(α_1,p_1)(R~n)×…×ε~(α_m,p_m)(R~n)到ε_*~(2_(α,p)/2)(R~n)是有界的.
This paper considers the boundedness of multilinear Littlewood-Paley operators with nonconvolution type on Campanato spaces,including the multilinear g-function,multilinear Lusin's area integral Sand multilinear g_λ*-function.If f=(f_1,…,f_n),f_i∈ε~(α_i,p_i)(R~n),i=1,…,m,then g(f),S(f),g_λ*(f)are either infinite everywhere or finite almost everywhere,and in the latter case,[g(f)]~2,[S(f)]~2,[g_λ*(f)]~2 are bounded from ε~(α_1,p_1)(R~n)×…×ε~(α_m,p_m)(Rn)to ε_*~(2_(α,p)/2)(R~n).
This paper considers the boundedness of multilinear Littlewood-Paley operators with nonconvolution type on Campanato spaces,including the multilinear g-function,multilinear Lusin's area integral Sand multilinear g_λ*-function.If f=(f_1,…,f_n),f_i∈ε~(α_i,p_i)(R~n),i=1,…,m,then g(f),S(f),g_λ*(f)are either infinite everywhere or finite almost everywhere,and in the latter case,[g(f)]~2,[S(f)]~2,[g_λ*(f)]~2 are bounded from ε~(α_1,p_1)(R~n)×…×ε~(α_m,p_m)(Rn)to ε_*~(2_(α,p)/2)(R~n).
引文
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[13]YABUTA K.Existence and boundedness of g*λ-function and Marcinkiewicz functions on Campanato spaces[J].Sci Math Jpn,2004,59(1):93.
[14]HU Guo-en,MENG Yan,YANG Da-chun.Estimate for Marcinkiewic integrals in BMO and Campanato spaces[J].Glasgow Math J,2007,49(2):167.
[15]HE Sha,XUE Qing-ying,MEI Ting,et al.Existence and boundedness of multilinear Littlewood-Paley operators on Campanato spaces[J].J Math Anal Appl,2015,432(1):86.
[16]CAMPANATOS.Propriet1di h9lderianit1di alcune classi di funzioni[J].Ann Scuola Norm sup Pisa,1963,17(3):175.
[17]COIFMAN R R,ROCHBERG R.Another characterization of BMO[J].Proc Amer Math Soc,1980,79(2):2494.
[18]STEIN E M.On the functions of Littlewood-Paley,Lusin and Marcinkiewicz[J].Trans Amer Math Soc,1958,88(2):430.
[19]XUE Qing-ying,YAN Jing-quan.On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels[J].J Math Anal Appl,2015,422(2):1342.
[20]FABES E B,JOHNSON R L,NERI U.Spaces of harmonic functions representable by Poisson integrals of functions in BMO and[J].Indiana Univ Math J,1976,25(2):159.
[2]LUSIN H.Sur une propriete des fonctions a carre sommable[J].Bull Calcutta Math Soc,1930,20:139.
[3]COIFMAN R R,MEYER Y.On commutators of singular integrals and bilinear singular integrals[J].Trans Amer Math Soc,1975,212:315.
[4]GRAFAKOS L,TORRES R H.Multilinear Caldero′n-Zygmund theory[J].Adv Math,2002,165(1):124.
[5]GRAFAKOS L,TORRES R H.Maximal operator and weighted norm inequalities for multilinear singular integrals[J].Indiana Univ Math J,2002,51(5):1261.
[6]COIFMAN R R,MEYER Y.Au delàdes Ope′rateurs Pseudo-diffe′rentiels[M].Aste′risque:Socie′te′Mathe′matique de France,1978:57.
[7]YABUTA K.A multilinearization of Littlewood-Paley's g-function and Carleson measures[J].Tohoku Math J,1982,34(2):251.
[8]SATO S,YABUTA K.Multilinearized Littlewood-Paley operators[J].Sci Math Jpn,2002,55(3):447.
[9]HART J.Bilinear square functions and vector-valued Calderón-Zygmund operators[J].J Fourier Anal Appl,2012,18(6):1291.
[10]SHI Shao-guang,XUE Qing-ying,YABUTA K.On the boundedness of multilinear Littlewood-Paley g*λ-function[J].J Math Pures Appl,2014,101(3):394.
[11]王斯雷,陈杰诚.关于平方函数的几点注记[J].数学年刊A辑,1990,11(5):630.
[12]SUN Yong-zhong.On the existence and boundedness of square function operators on Campanato spaces[J].Nagoya Math J,2004,173:139.
[13]YABUTA K.Existence and boundedness of g*λ-function and Marcinkiewicz functions on Campanato spaces[J].Sci Math Jpn,2004,59(1):93.
[14]HU Guo-en,MENG Yan,YANG Da-chun.Estimate for Marcinkiewic integrals in BMO and Campanato spaces[J].Glasgow Math J,2007,49(2):167.
[15]HE Sha,XUE Qing-ying,MEI Ting,et al.Existence and boundedness of multilinear Littlewood-Paley operators on Campanato spaces[J].J Math Anal Appl,2015,432(1):86.
[16]CAMPANATOS.Propriet1di h9lderianit1di alcune classi di funzioni[J].Ann Scuola Norm sup Pisa,1963,17(3):175.
[17]COIFMAN R R,ROCHBERG R.Another characterization of BMO[J].Proc Amer Math Soc,1980,79(2):2494.
[18]STEIN E M.On the functions of Littlewood-Paley,Lusin and Marcinkiewicz[J].Trans Amer Math Soc,1958,88(2):430.
[19]XUE Qing-ying,YAN Jing-quan.On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels[J].J Math Anal Appl,2015,422(2):1342.
[20]FABES E B,JOHNSON R L,NERI U.Spaces of harmonic functions representable by Poisson integrals of functions in BMO and[J].Indiana Univ Math J,1976,25(2):159.