齊次 Morrey-Herz 空間上廣義 Riesz 變換及其交換子的有界性

楊明華，許明，楊曉轉

（暨南大學數(shù)學系，廣東 廣州 510632

齊次 Morrey-Herz 空間上廣義 Riesz 變換及其交換子的有界性

楊明華，許明，楊曉轉

（暨南大學數(shù)學系，廣東 廣州 510632

1 引言

2 預備知識

3 主要結果及其證明

4 補充說明

[1] Calderˊon A P, Zygmund A. On the existence of certain singular integral[J]. Acta. Math., 1952,88:85-139.

[2] Auscher P, Tchamitchian P. Square Root Problem for Divergence Operators and Related Topics[M]. Astˊerisque: Spring-Verlag, 1998.

[3] Hofmann S,MartellM J.Lpbounds for Riesz transforms and square roots associated to second order elliptic operator [J]. Pub. Math., 2003,47:497-515.

[4]Duong X T,Mcintosh A.The Lpbonudedness of Riesz transform associated with divergence form operators operators [J]. Center for Math. and Appl. Australian National Univ., 1999,37:15-26.

[5] Mcintosh A. Operators which have H∞-calculus, Miniconference on operators theory and partial differential equations[J]. Proc. Center Math. Analysis, ANU., Canberra, 1986,14:210-231.

[6] Auscher P. On necessary and sufficient conditions for Lp estimates of Riesz transform associated to elliptic operators on and related estimates[J]. Mem. Amer. Math. Soc., 2007,186:871.

[7] Fan Dashan, Lu Shanzhen, Yang Dachun. Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients[J]. Georgian. Math. J., 1998,5:425-440.

[8] Fan Dashan, Lu Shanzhen, Yang Dachun. Boundedness of operators in Morrey spaces on homogenous spaces and its applications[J]. Acta. Math. Sinica. Suppl., 1998,14:625-634.

[9] Lu Shanzhen, Xu Lifang. Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces[J]. Hokkaido Math. J., 2005,34(2):299-314.

[10] Lu Shanzhen, Yang Dachun, Zhou Zusheng. Sublinear operators with rough kernel on genenalized Morrey space[J]. Hokkaido Math. J., 1998,27(1):219-232.

[11] Lu Shanzhen, Tang Lin, Yang Dachun. Boundedness of commutators on homogeneous Herz spaces[J]. Sci. China. Ser. A., 1998,41(10):1023-1033

[12] Auscher P, Martell M J. Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators[J]. Funct. Anal. J., 2006,241:703-746.

[13] Stein E M. Singular Integrals and Differentiability Properties of Functions[M]. Princeton New Jersey: Princeton University Press, 1970.

[14] Lu Shanzhen, Yang Dachun. The continuity of commutators on Herz-type Spaces[J]. Michigan Math., 1997, 44(2):255-280.

Boundedness of generalized Riesz transform and its commutator on the homogeneous Morrey-Herz spaces

Yang Minghua,Xu Ming,Yang Xiaozhuan

(Department of Mathematics,Ji′nan University,Guangzhou 510632,China)

In this paper,we study the generalized Riesz transform?L-1/2associated with divergence form elliptic operator and its commutator[b,?L-1/2]generated by generalized Riesz transform and BMO(Rn)functions.By them ethods of studying ring decom position of function and thier corresponding truncated operators, their boundedness of the results form space MKα,λp1,q(Rn)to space MKα,λp2,q(?n)were established.The well-known results gotten by before scholars are extended.

ellip tic operator,commutator,homogeneous Morrey-Herz,Riesz transform,BMO(Rn)

2011-09-17.

國家自然科學基金(10771221)；暨南大學青年自然科學基金(51208036).

楊明華(1986-)，碩士生，研究方向：調和分析.

2010 MSC:42B25