一类粗糙核奇异积分算子的端点估计
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摘要
设T_Ω是带粗糙核的Calderón-Zygmund奇异积分算子,I为任意真包含在单位圆周S~1上的闭圆弧.本文证明,若Ω支在I上并在I上单调,那么T_Ω是从Hardy空间H~1(R~2)到L~1(R~2)的有界算子当且仅当‖Ω‖_(LlogL(S~1))<∞.
Assume T_Ω is the Calderón-Zygmund singular integral operator with rough kernel and I is the closed arc of unit circle that I(?) S~1. In this paper, we prove that if Ω is supported in I and monotonous on I, then T_Ω is bounded from Hardy space H~1(R~2) to L~1(R~2) if and only if ‖Ω‖_(L log L)<∞.
Assume T_Ω is the Calderón-Zygmund singular integral operator with rough kernel and I is the closed arc of unit circle that I(?) S~1. In this paper, we prove that if Ω is supported in I and monotonous on I, then T_Ω is bounded from Hardy space H~1(R~2) to L~1(R~2) if and only if ‖Ω‖_(L log L)<∞.
引文
[1]Bownik M.,Seeger A.,Boundedness of operators on Hardy spaces via atomic decompositions,Proceedings of the American Mathematical Society,2005,133:3535-3542.
[2]Calderón A.P.,Zygmund A.,On singular integrals,Amer.J.Math.,1956,78:289-309.
[3]Chen J.,Zhu X.,A note on weak type L~1 boundness of CZOs with rough kernels,J.Math.Anal.Appl.,2008,339:438-453.
[4]Christ M.,Weak type(1,1)bounds for rough kernels,Ann.Math.,1988,128:19-42.
[5]Christ M.,Rubio de Francia J.L.,Weak type(1,1)bounds for rough operators II,Invent.Math.,1988,93:229-237.
[6]Coifman R.R.,Weiss G.,,Extensions of Hardy spaces and their use in analysis,Bull.Amer.Math.Soc.,1997,83:569-645.
[7]Connett W.C.,Singular integrals near L~1,Proc.Sympos.Pure Math.Amer.Math.Soc.(S.Wainger and G.Weiss,eds.),1979:163-169.
[8]Grafakos L.,Classical Fourier Analysis,Springer,New York,2004.
[9]Grafakos L.,Endpoint bounded for an analytic family of Hilbert transforms,Duke Math.,1991,62:23-59.
[10]Grafakos L.,Modern Fourier Analysis,Springer,New York,2004.
[11]Grafakos L.,Stefanov A.,Convolution Calderón-Zygmund singular integral operators with rough kernels,Analysis of Divergence Applied and Numerical Harmonic Analysis,1999:119-143.
[12]Hofmann S.,Weak(1,1)boundedness of singular integrals with nonsmooth kernels,Proc.Amer.Math.Soc.,1988,103:260-264.
[13]Ricci F.,Weiss G.,A characterization of H~1(∑_(n-1)),in:S.Wainger,G.Weiss(Eds.),Proc.Sympos.Pure Math.Amer.Math.Soc.,1979:289-294.
[14]Seeger A.,Singular integral operator with rough convolution kernels,J.Amer.Math.Soc.,1996,9:99-109.
[15]Seeger A.,Tao T.,Sharp Lorentz space estimates for rough operators,Math.Ann.,2001,320:381-415.
[16]Stefanov A.,Weak type estimates for certain Calderón-Zygmund singular integral operators,Studia Math.,2001,147(1):1-13.
[17]Stein E.M.,Singular Integral and Differentiability Properties of Functions,Princeton Univ.Press,Princeton,1970.
[18]Tao T.,The weak-type(1,1)of L log L homogeneous convolution operators,Indiana Univ.Math.,1999,48:1547-1584.
[2]Calderón A.P.,Zygmund A.,On singular integrals,Amer.J.Math.,1956,78:289-309.
[3]Chen J.,Zhu X.,A note on weak type L~1 boundness of CZOs with rough kernels,J.Math.Anal.Appl.,2008,339:438-453.
[4]Christ M.,Weak type(1,1)bounds for rough kernels,Ann.Math.,1988,128:19-42.
[5]Christ M.,Rubio de Francia J.L.,Weak type(1,1)bounds for rough operators II,Invent.Math.,1988,93:229-237.
[6]Coifman R.R.,Weiss G.,,Extensions of Hardy spaces and their use in analysis,Bull.Amer.Math.Soc.,1997,83:569-645.
[7]Connett W.C.,Singular integrals near L~1,Proc.Sympos.Pure Math.Amer.Math.Soc.(S.Wainger and G.Weiss,eds.),1979:163-169.
[8]Grafakos L.,Classical Fourier Analysis,Springer,New York,2004.
[9]Grafakos L.,Endpoint bounded for an analytic family of Hilbert transforms,Duke Math.,1991,62:23-59.
[10]Grafakos L.,Modern Fourier Analysis,Springer,New York,2004.
[11]Grafakos L.,Stefanov A.,Convolution Calderón-Zygmund singular integral operators with rough kernels,Analysis of Divergence Applied and Numerical Harmonic Analysis,1999:119-143.
[12]Hofmann S.,Weak(1,1)boundedness of singular integrals with nonsmooth kernels,Proc.Amer.Math.Soc.,1988,103:260-264.
[13]Ricci F.,Weiss G.,A characterization of H~1(∑_(n-1)),in:S.Wainger,G.Weiss(Eds.),Proc.Sympos.Pure Math.Amer.Math.Soc.,1979:289-294.
[14]Seeger A.,Singular integral operator with rough convolution kernels,J.Amer.Math.Soc.,1996,9:99-109.
[15]Seeger A.,Tao T.,Sharp Lorentz space estimates for rough operators,Math.Ann.,2001,320:381-415.
[16]Stefanov A.,Weak type estimates for certain Calderón-Zygmund singular integral operators,Studia Math.,2001,147(1):1-13.
[17]Stein E.M.,Singular Integral and Differentiability Properties of Functions,Princeton Univ.Press,Princeton,1970.
[18]Tao T.,The weak-type(1,1)of L log L homogeneous convolution operators,Indiana Univ.Math.,1999,48:1547-1584.