各向异性分数次积分算子的加权范数不等式（英文）
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摘要
设A是一个扩张矩阵,α∈[0,1),p:=1/α且函数v满足各向异性Muckenhoupt Ap,∞(A)权条件.本文研究了各向异性分数次积分算子的有界性的问题.利用L(p,∞)空间的Holder不等式和范数‖·‖p,1的σ-次可加性得到了各向异性分数次积分算子关于权vp的一些加权范数不等式.这些结果是Muckenhoupt和Wheeden的结果[6]在各向异性情形下的推广.
Let A be an expansive dilation, α ∈(0, 1), p := 1/α and function v satisfy the anisotropic Muckenhoupt condition A_(p, ∞)(A). In this paper, we study the boundedness of anisotropic fractional integral operators. By L(p, ∞) H¨older's inequality and the σ-subaddictive property of ‖·‖ p, 1, we obtain some weighted norm inequalities for anisotropic fractional integral operators associated with the weight vp, which are anisotropic extension of Muckenhoupt and Wheeden [6].
Let A be an expansive dilation, α ∈(0, 1), p := 1/α and function v satisfy the anisotropic Muckenhoupt condition A_(p, ∞)(A). In this paper, we study the boundedness of anisotropic fractional integral operators. By L(p, ∞) H¨older's inequality and the σ-subaddictive property of ‖·‖ p, 1, we obtain some weighted norm inequalities for anisotropic fractional integral operators associated with the weight vp, which are anisotropic extension of Muckenhoupt and Wheeden [6].
引文
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[5]Calder′on A-P,Torchinsky A.Parabolic maximal functions associated with a distribution II[J].Adv.Math.,1977,24(2):101-171.
[6]Muckenhoupt B,Wheeden R.Weighted norm inequalities for fractional integrals[J].Trans.Amer.Math.Soc.,1974,92(1):261-274.
[7]Welland G.Weighted norm inequalities for fractional integrals[J].Proc.Amer.Math.Soc.,1975,51(1):143-148.
[8]Zhang Chaonan,Zhou Jiang,Cao Yonghui.The boundedness of generalized fractional integral operators on the weighted homogeneous Morrey-Herz spaces[J].J.Math.,2016,36(1):199-206.
[9]Eridani A,Kokilashvili V,Meskhi A.Morrey spaces and fractional integral operators[J].Expos.Math.,2008,27(3):227-239.
[10]Lan Jiacheng.The boundedness of multilinear fractional integral opeators on weak type Hardy spaces[J].J.Math.,2006,26(3):343-348.
[11]Stein E M.Singular integrals and differentiability properties of functions[M].Princeton:Princeton Univ.Press,1970.
[12]Hardy G H,Littlewood J E.Some properties of fractional integrals[J].I.Math.Z.,1928,27(1):565-606.
[13]Sobolev S.On a theorem in functional analysis[J].Mat.Sob.,1938,46(4):471-497.
[14]Ding Yong,Lan Senhua.Fractional integral operators on anisotropic Hardy spaces[J].Int.Equ.Oper.The.,2008,60(3):329-356.
[15]Lan Senhua,Lee Mingyi,Lin Chincheng.Fractional integral operators on weighted anisotropic Hardy spaces[J].J.Oper.The.,2012,68(1):3-17.
[16]Coifman R R,Weiss G.Analyse harmonique non-commutative sur certains espaces homogenes[M].Lect.Notes Math.,Vol.242,Berlin:Springer,1971.
[17]Coifman R R,Weiss G.Extensions of Hardy spaces and their use in analysis[J].Bull.Amer.Math.Soc.,1977,83(4):569-645.
[18]V′agi S,Gatto A,Fern′andez C.On fractional differentiation and integration on spaces of homogeneous type[J].Rev.Mat.Iberoamericana,1996,12(1):111-146.
[19]Hunt R.On L(p,q)spaces[J].Enseignement Math.,1966,12(2):249-276.
[2]Bownik M,Li Baode,Yang Dachun,Zhou Yuan.Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[J].Indiana Univ.Math.J.,2008,57(7):3065-3100.
[3]Fefferman C,Stein E M.Hpspaces of several variables[J].Acta Math.,1972,129(1):137-193.
[4]Calder′on A-P,Torchinsky A.Parabolic maximal functions associated with a distribution[J].Adv.Math.,1975,16(16):1-64.
[5]Calder′on A-P,Torchinsky A.Parabolic maximal functions associated with a distribution II[J].Adv.Math.,1977,24(2):101-171.
[6]Muckenhoupt B,Wheeden R.Weighted norm inequalities for fractional integrals[J].Trans.Amer.Math.Soc.,1974,92(1):261-274.
[7]Welland G.Weighted norm inequalities for fractional integrals[J].Proc.Amer.Math.Soc.,1975,51(1):143-148.
[8]Zhang Chaonan,Zhou Jiang,Cao Yonghui.The boundedness of generalized fractional integral operators on the weighted homogeneous Morrey-Herz spaces[J].J.Math.,2016,36(1):199-206.
[9]Eridani A,Kokilashvili V,Meskhi A.Morrey spaces and fractional integral operators[J].Expos.Math.,2008,27(3):227-239.
[10]Lan Jiacheng.The boundedness of multilinear fractional integral opeators on weak type Hardy spaces[J].J.Math.,2006,26(3):343-348.
[11]Stein E M.Singular integrals and differentiability properties of functions[M].Princeton:Princeton Univ.Press,1970.
[12]Hardy G H,Littlewood J E.Some properties of fractional integrals[J].I.Math.Z.,1928,27(1):565-606.
[13]Sobolev S.On a theorem in functional analysis[J].Mat.Sob.,1938,46(4):471-497.
[14]Ding Yong,Lan Senhua.Fractional integral operators on anisotropic Hardy spaces[J].Int.Equ.Oper.The.,2008,60(3):329-356.
[15]Lan Senhua,Lee Mingyi,Lin Chincheng.Fractional integral operators on weighted anisotropic Hardy spaces[J].J.Oper.The.,2012,68(1):3-17.
[16]Coifman R R,Weiss G.Analyse harmonique non-commutative sur certains espaces homogenes[M].Lect.Notes Math.,Vol.242,Berlin:Springer,1971.
[17]Coifman R R,Weiss G.Extensions of Hardy spaces and their use in analysis[J].Bull.Amer.Math.Soc.,1977,83(4):569-645.
[18]V′agi S,Gatto A,Fern′andez C.On fractional differentiation and integration on spaces of homogeneous type[J].Rev.Mat.Iberoamericana,1996,12(1):111-146.
[19]Hunt R.On L(p,q)spaces[J].Enseignement Math.,1966,12(2):249-276.