具有非光滑核的多线性奇异积分算子交换子的Lipschitz估计(英文)
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摘要
考虑由具有非光滑核的多线性奇异积分算子与Lipschitz函数生成的交换子.建立了这些算子在Lipschitz空间和Triebel-Lizorkin空间中的有界性.
This paper is concerned with certain commutators generated by the Lipschitz functions and multilinear singular integrals with non-smooth kernels.Some mapping properties for these operators on the Lipschitz spaces and Triebel-Lizorkin spaces are obtained.
This paper is concerned with certain commutators generated by the Lipschitz functions and multilinear singular integrals with non-smooth kernels.Some mapping properties for these operators on the Lipschitz spaces and Triebel-Lizorkin spaces are obtained.
引文
[1]Coifman,R.and Meyer,Y.,On commutators of singular integral and bilinear singular integrals,Trans.Amer.Math.Soc,1975,212:315-331.
[2]Coifman,R.and Meyer,Y.,Au-dela Des Operateurs Pseudo-differentiels,Asterisque,Vol.57,Societe Mathematique de Prance,1978(in French).
[3]Coifman,R.and Meyer,Y.,Ondelettes et operateurs III,Paris:Hermann,1990.
[4]Duong,X.T.and Mcintosh,A.,Singular integral operators with non-smooth kernels on irregular domains,Rev.Mat.Iberoamericana,1999,15(2):233-265.
[5]Duong,X.T.and Yan L.X.,Commutators of BMO functions and singular integral operators with non-smooth kernels,Bull.Austral.Math.Soc,2003,67(2):187-200.
[6]Duong,X.T.,Grafakos,L.and Yan L.X.,Multilinear operators with non-smooth kernels and commutators of singular integrals,Trans.Amer.Math.Soc,2010,362(4):2089-2113.
[7]Duong,X.T.,Gong,R.M.,Grafakos,L.,Li J.and Yan L.X.,Maximal operator for multilinear singular integrals with non-smooth kernels,Indiana Univ.Math.J.,2009,58(6):2517-2541.
[8]Gong R.M.and Li J.,Sharp maximal function estimates for multilinear singular integrals with non-smooth kernels,Anal.Theory Appl.,2009,25(4):333-348.
[9]Grafakos,L.and Torres,R.H.,On multilinear singular integrals of Calder6n-Zygmund type,In:Proceedings of the 6th International Conference on Harmonic Analysis and Partial Diferential Equations(El Escorial,2000),Publ.Mat,2002,Extra,57-91.
[10]Grafakos,L.and Torres,R.H.,Multilinear Ca!der6n-Zygmund theory,Adv.Math.,2002,165(1):124-164.
[11]Grafakos,L.,Classical and Modern Fourier Analysis,Prentice Hall:Pearson Education,Inc.,2004.
[12]Kenig,C.and Stein,E.,Multilinear estimates and fractional integration,Math.Res.Lett.,1999,6(1):1-15.
[13]Lian J.L.,Li J.and Wu H.X.,Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels,Appl.Math.J.Chinese Univ.,2011,26(1):109-120.
[14]Lian J.L.,Ma B.and Wu H.X.,Commutators of weighted Lipschitz functions and multilinear singular integrals with non-smooth kernels,Appl.Math.J.Chinese Univ.,2011,26(3):353-367.
[15]Lian J.L.and Wu H.X.,A class of commutators for multilinear fractional integrals in nonhomogeneous spaces,J.Inequal.Appl,2008,Article ID:373050,17 pages.
[16]Paluszynski,M.,Characterization of the Besov spaces via the commutator operator of Coifman Rochberg and Weiss,Indiana Univ.Math.J.,1995,44(1):1-18.
[17]Perez,C,Pradolini,G.,Torres,R.and Trujillo-Gonzalez,R.,End-point estimates for iterated commutators of multilinear singular integrals,arXiv:1004.4976vl[math.CA],2010.
[18]Stein,E.M.,Harmonic Analysis:Real Variable Methods,Orthogonality and Oscillatory Integrals,Princeton,N.J.:Princeton Univ.Press,1993.
[19]Torchinsky,A.,Real-variable Methods in Harmonic Analysis,New York:Academic Press,1986.
[20]Wang W.and Xu J.S.,Commutators of multilinear singular integrals with Lipschitz functions on products of variable exponent Lebesgue spaces,Advances in Mathematics(China),2009,38(6):669-677.
[21]Xu J.S.,Generalized commutators of multilinear singular integrals,In:Proceedings of A.Razmadze Mathematical Institute,2006,142:109-122.
[22]Xu J.S.,The boundedness of multilinear commutators of singular integrals on Lebesgue spaces with variable exponent,Czech.Mathematical Journal,2007,57(1):13-27.
[23]Xu J.S.,Multilinear commutators of singular integrals operators with non-smooth kernels,Taiwanese J.Math.,2007,11(2):483-496.
[2]Coifman,R.and Meyer,Y.,Au-dela Des Operateurs Pseudo-differentiels,Asterisque,Vol.57,Societe Mathematique de Prance,1978(in French).
[3]Coifman,R.and Meyer,Y.,Ondelettes et operateurs III,Paris:Hermann,1990.
[4]Duong,X.T.and Mcintosh,A.,Singular integral operators with non-smooth kernels on irregular domains,Rev.Mat.Iberoamericana,1999,15(2):233-265.
[5]Duong,X.T.and Yan L.X.,Commutators of BMO functions and singular integral operators with non-smooth kernels,Bull.Austral.Math.Soc,2003,67(2):187-200.
[6]Duong,X.T.,Grafakos,L.and Yan L.X.,Multilinear operators with non-smooth kernels and commutators of singular integrals,Trans.Amer.Math.Soc,2010,362(4):2089-2113.
[7]Duong,X.T.,Gong,R.M.,Grafakos,L.,Li J.and Yan L.X.,Maximal operator for multilinear singular integrals with non-smooth kernels,Indiana Univ.Math.J.,2009,58(6):2517-2541.
[8]Gong R.M.and Li J.,Sharp maximal function estimates for multilinear singular integrals with non-smooth kernels,Anal.Theory Appl.,2009,25(4):333-348.
[9]Grafakos,L.and Torres,R.H.,On multilinear singular integrals of Calder6n-Zygmund type,In:Proceedings of the 6th International Conference on Harmonic Analysis and Partial Diferential Equations(El Escorial,2000),Publ.Mat,2002,Extra,57-91.
[10]Grafakos,L.and Torres,R.H.,Multilinear Ca!der6n-Zygmund theory,Adv.Math.,2002,165(1):124-164.
[11]Grafakos,L.,Classical and Modern Fourier Analysis,Prentice Hall:Pearson Education,Inc.,2004.
[12]Kenig,C.and Stein,E.,Multilinear estimates and fractional integration,Math.Res.Lett.,1999,6(1):1-15.
[13]Lian J.L.,Li J.and Wu H.X.,Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels,Appl.Math.J.Chinese Univ.,2011,26(1):109-120.
[14]Lian J.L.,Ma B.and Wu H.X.,Commutators of weighted Lipschitz functions and multilinear singular integrals with non-smooth kernels,Appl.Math.J.Chinese Univ.,2011,26(3):353-367.
[15]Lian J.L.and Wu H.X.,A class of commutators for multilinear fractional integrals in nonhomogeneous spaces,J.Inequal.Appl,2008,Article ID:373050,17 pages.
[16]Paluszynski,M.,Characterization of the Besov spaces via the commutator operator of Coifman Rochberg and Weiss,Indiana Univ.Math.J.,1995,44(1):1-18.
[17]Perez,C,Pradolini,G.,Torres,R.and Trujillo-Gonzalez,R.,End-point estimates for iterated commutators of multilinear singular integrals,arXiv:1004.4976vl[math.CA],2010.
[18]Stein,E.M.,Harmonic Analysis:Real Variable Methods,Orthogonality and Oscillatory Integrals,Princeton,N.J.:Princeton Univ.Press,1993.
[19]Torchinsky,A.,Real-variable Methods in Harmonic Analysis,New York:Academic Press,1986.
[20]Wang W.and Xu J.S.,Commutators of multilinear singular integrals with Lipschitz functions on products of variable exponent Lebesgue spaces,Advances in Mathematics(China),2009,38(6):669-677.
[21]Xu J.S.,Generalized commutators of multilinear singular integrals,In:Proceedings of A.Razmadze Mathematical Institute,2006,142:109-122.
[22]Xu J.S.,The boundedness of multilinear commutators of singular integrals on Lebesgue spaces with variable exponent,Czech.Mathematical Journal,2007,57(1):13-27.
[23]Xu J.S.,Multilinear commutators of singular integrals operators with non-smooth kernels,Taiwanese J.Math.,2007,11(2):483-496.