Hardy型不等式与NA群上可积函数的性质
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摘要
受到欧氏空间与H-型群上类似问题的启发,首先,我们将关于具有紧支撑的光滑函数的Hardy-Sobolev不等式推广到径向导数的情形,并指出不等式中常数是最佳的.其次,我们将Heisenberg群上的Hardy-Rellich不等式推广到Canrot群上,也得到相应的最佳常数,并指出有关Grushin算子的类似不等式也是成立.
NA群包含了一些秩为一的黎曼对称空间,如复双曲空间,四元数双曲空间与Cayley双曲空间.我们仿照秩为一的黎曼对称空间的情形,用球面变换和带状域上解析函数的谱综合的方法,证明了NA群上满足平均值性质可积函数为调和函数的一种充分条件.
H-型群是重要的幂零黎曼流形,NA群是典型的可解流形；我们按照黎曼流形和乐群的Berger分类,分别给出了H-型群,NA群的和乐群.运用球面变换,我们证明了NA群上径向群代数的半单性.利用Banach代数的上同调和顺从性等概念,我们得到了H-型群,NA群上群代数,径向群代数的顺从性及上同调群的性质.
Having in mind similar problems in the Euclidean and H-type groups setting, first we generalized Hardy-Sobolev inequalities from smooth functions with compact support to radial derivatives in Euclidean spaces, and point out that the constants in such Hardy-Sobolev inequalities are the best. Secondly, we prove Hardy-Rellich inequalities on Canrot groups as the same as on Heisenberg groups, and the constants in these inequalities are sharp. We show the analogue inequalities for Grushin operators also.
NA groups include some Riemannian symmetric spaces with rank one as com-plex hyperbolic spaces, quaternionic hyperbolic spaces and Cayley hyperbolic space. We give a sufficient condition for integrable functions satisfy mean value property on NA groups to be harmonic with spherical transformations and spectral synthe-sis on analytic functions in strip.The method comes from the similar result about Riemannian symmetric spaces with rank one.
H-type groups as nilpotent manifolds and NA groups as solvable manifolds are important Riemann manifolds. We obtain the holonomy groups of H-type groups and NA groups according to Berger classification of Riemannian holonomy groups. We use the spherical transformation to prove the semisimplity of radial group alge-bras on NA groups, and show these algebras are amenable.
NA群包含了一些秩为一的黎曼对称空间,如复双曲空间,四元数双曲空间与Cayley双曲空间.我们仿照秩为一的黎曼对称空间的情形,用球面变换和带状域上解析函数的谱综合的方法,证明了NA群上满足平均值性质可积函数为调和函数的一种充分条件.
H-型群是重要的幂零黎曼流形,NA群是典型的可解流形；我们按照黎曼流形和乐群的Berger分类,分别给出了H-型群,NA群的和乐群.运用球面变换,我们证明了NA群上径向群代数的半单性.利用Banach代数的上同调和顺从性等概念,我们得到了H-型群,NA群上群代数,径向群代数的顺从性及上同调群的性质.
Having in mind similar problems in the Euclidean and H-type groups setting, first we generalized Hardy-Sobolev inequalities from smooth functions with compact support to radial derivatives in Euclidean spaces, and point out that the constants in such Hardy-Sobolev inequalities are the best. Secondly, we prove Hardy-Rellich inequalities on Canrot groups as the same as on Heisenberg groups, and the constants in these inequalities are sharp. We show the analogue inequalities for Grushin operators also.
NA groups include some Riemannian symmetric spaces with rank one as com-plex hyperbolic spaces, quaternionic hyperbolic spaces and Cayley hyperbolic space. We give a sufficient condition for integrable functions satisfy mean value property on NA groups to be harmonic with spherical transformations and spectral synthe-sis on analytic functions in strip.The method comes from the similar result about Riemannian symmetric spaces with rank one.
H-type groups as nilpotent manifolds and NA groups as solvable manifolds are important Riemann manifolds. We obtain the holonomy groups of H-type groups and NA groups according to Berger classification of Riemannian holonomy groups. We use the spherical transformation to prove the semisimplity of radial group alge-bras on NA groups, and show these algebras are amenable.
引文
[1]Adimurthi, A. Sekar, Role of the fundamental solution in Hardy-Sobolev-type in-equalities, Proccedings of the Royal Society of Edinburgh[J],136A, (2006),1111-1130.
[2]Anker,J.P.,E.Damek, C.Yacob, Spherical Analysis on Harmonic AN Groups[J], Ann.Scuola.Norm.Sup.Pisa, (4)23(1996), no.4643-679.
[3]Ahern, P.,M. Flores and W. Rudin, An Invariant Volume-mean-value Prop-erty [J],J.Functional Analysis,111,380-397,1993.
[4]Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities, Nonlinear Analysis,70(2009),2826-2833.
[5]T. Aubin, Probleme isoperimetric et espace de Sobolev[J], J. Differential Geom.,11 (1976),573-598.
[6]A. Balinsky, W. D. Evans, D. Hundertmark, R. T. Lewis, On inequalities of Hardy-Sobolev type[J], Banach J. Math. Anal.2 (2008), no.2,94-106.
[7]W. Beckner, On Hardy-Sobolev embedding[J], arXiv:0907.3932v1[math.AP].
[8]Z. Balogh, J. Tyson, Polar coordinates on Carnot groups [J], Math. Z.241(4)(2002), 697-730.
[9]Z. Balogh, J. Tyson, Polar coordinates on Carnot groups [J], Math. Z.241(4)(2002), 697-730.
[10]Berndt J., Tricerri F., Vanhecke L.:Generalized Heisenberg groups and Damek-Ricci Spaces[M], LNM 1589,1995
[11]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical exponents[J], Comm. Pure Appl. Math.36 (1983) 437-477.
[12]Bonsall,F.,F.,J.Dancall, Complet normed algebras[M], Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag,1970,171-185
[13]Birindelli,I., Cutrl.A., A similinear problem for the Heisenberg Laplacian[J], Rend. Mat. Univ. Padova 94(1995),137-153
[14]Bony J.M., Principle du Maximum, Inegalite de Harnack et unnicite du probleme de Cauchy pour les operaeurs elliptiques degeneres [J], Ann. Inst. Fourier Grenobles, 19,277-304 (1969)
[15]Caffarelli L., Garofalo N., Segala F., A gradient bound for entire solutions of quasi-linear equations and its consequences [J], Comm. Pure Appl. Math.47 1457-1473 (1994).
[16]Corwin L.J., Greenleaf F.P., Representations of nilpotent Lie groups and their ap-plications.1:Basic theory and examples[M], Cambridge Studies in Advanced Math-ematics 18, Cambridge University Press,1990
[17]Conway, J.,B., A Course in Functional Analysis[M], GTM96,1st edition.377,Springer-Verlag, New York.1985.
[18]Chen,W., Li, C., Classification of solution of some nonlinear elliptic equations [J], Duke Math. J.63(1991),615-622
[19]Damek, E., F.Ricci, Harmonic analysis on Solvable Extension of H-type Groups[J], J.Geometic Analysis,2(1992).213-248.
[20]D'Ambrosio L., Lucente S., Nonlinear Liouville theorems for Grushin and Tricomi operators[J], J. Diff. Equa.193 (2003),511-541
[21]D'Ambrosio L., Some Hardy inequalities on the Heisenberg group[J], Diff. Equa. 40(4) (2004),552-564.
[22]D'Ambrosio L., Hardy inequalities related to Grushin type operators [J], Proc. Amer. Math. Soc.132 (2004),725-734.
[23]Deng Hongjun, Amenable Banach Algebras[R], Procedins in Asia Mathematics,Hongkong,1991,178-182.
[24]Dales et.al. Introduction to Banach Algebras,Operators,and Harmonic Analy-sis [M], Cambridge University Press:2003,75-109.
[25]Folland G.B., A fundamental solution for a subelliptic operator [J], Bull. Amer. Math. Soc.,79(2) (1973),373-376.
[26]Folland G.B., Subelliptic estimates and function spaces on nilpotent Lie group[J], Ark. Mat.,13.161-220 (1975)
[27]Folland G.B., Stein E.M., Estimates for the (?)h complex and analysis on the Heisen-berg group[J], Comm. Pure Apll. Math..27,492-522 (1974)
[28]Gaveau B.:Principle de moindre action, propogation de la chaleur er estimees sous-elliptiques sur certains groups nilpotents[J], Acta Math.,139,95-153 (1977)
[29]Gidas B.. Ni W.M., Nirenberg L.,:Symmetry and related properties via the maxi-mum principle [J], Comm. Math. Phys.,68,209-243 (1979)
[30]Gidas B., Sprugk J.,Global and local behavior of positive solutions of nonlinear elliptic equations [J], Comm. Pure App. Math.,35,525-598 (1981)
[31]J. A. Goldstein, I. Kombe, The Hardy inequality and nonlinear parabolic equations on Carnot groups[J]. Nonlinear Analysis (2007), doi:10.1016/j.na.2007.
[32]Hardy G., Note on a theorem of Hilbert [J], Math.Zeitschr.6 (1920),314-317.
[33]Hierro M.D.:Dispersive and strichartz estimates on H-type groups[J], Stud. Math., 169(1),1-20 (2005)
[34]Hormander L.,Hypoelliptic second order differential equation [J], Acta Math., Upp-sala,119,147-171 (1967)
[35]Helgason.S.,Groups and Geometric Analysis[M], Academic Press,381-382,1984.
[36]Helgason.S.,Differential Geometry,Lie Groups and Symmetric Spaces[M], Academic Press,515-518,1978.
[37]Helemskii.A. Ya.,The Homology of Banach and Topological Algebras[M],Kluwer,183-195,1989.
[38]Hewitt, E., K.A.Ross, Abstract Harmonic Analysis[M], vol.1,291-292, Springer-Verlag, Berlin,1963.
[39]Hulanicki A., The distribution of energy of the Brownian motion in Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group [J]. Studia Math.,56 (1976),165-173.
[40]Johnson B.E., Cohomology in Banach Algebras [M]. Memoirs of the AMS,25, 81-85 (1972)
[41]Jerison D.S., Lee J.M., Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem[J], J. Amer. Math. Soc.,1,1-13 (1998)
[42]Kaplan A.:Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms [J], Trans. Amer. Math. Soc.,258(1),147-153 (1980)
[43]Kombe I., Sharp Hardy-type inequalities on Carnot groups[J]. preprint,(2005)
[44]Kombe I., The Hardy-type inequalities and nonliear parabolic equations on Carnot groups [J], preprint, (2005)
[45]Koranyi,A., On a Mean Value Property for Hyperbolic Spaces[J], Contemporary Mathematics, vol.191.107-116.1995.
[46]Lust-Piquard F., A sinple-minded computation of heat kernels on Heisenberg groups[J]. Coll. Math.,2003.97(2):233-249.
[47]Marakami,S.,Introduction to Homogenous Manifolds(in Chinese)[M], Shanghai Sci-ence and Technology Press,51-52,1984.
[48]Naimak.M.A., Normed Algebras[M], Wolters-Northoff,1970,381-385.
[49]Ni W.M., On a singular elliptic equation[J], Proc. Amer. Math. Soc.,88(1983), 614-616.
[50]Niu P., Zhang H., Wang Y., Hardy type and Rellich type inequalities on the Heisenberg group [J], Proc. Amer. Math. Soc.129(2001),3623-3630.
[51]R. Monti, D. Morbidelli, Kelvin transform for Grushin operators and critical semi-linear equations[J], Duke Math. J.,131(2006),167-202.
[52]Naten,B.,Y., Y.Weit,, Integrable Harmonic Functions on Symmetric Spaces of Rank one[J], J. Functional Analysis, vol.160,141-149,1998.
[53]P. Niu, H. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisen-berg group [J]. Proc. Amer. Math. Soc.,129 (2001),3623-3630.
[54]Opic B., Kufner A., Hardy-type Inequalities[M], Pitman Research Notes in Math., 219, Long-man 1990.
[55]Ricci, F., Spherial Functions on Certain Non-symmetric Harmonic Manifolds[R], Lecture Notes in the Workshop on Representation Theory of Lie Groups, ICTP, 1993.
[56]Rothschild. L.P.. Stein, E.M., Hypoelliptic differential operators and nilpotent groups[J], Acta Math.,137(1976) 247-320
[57]Staubach W.:Wiener path integrals and the fundamental solution for the Heisen-berg Laplacian[J], Journal D:analyse Math..91,389-400 (2003)
[58]Stein E., Harmonic Analysis, Real-Variable Methods, Orthgonality, and Oscillatory Integrals[M], Princeton University Press, Princeton, NJ.1993.
[59]J. Stubbe, Bounds on the number of bound states for potentials with critical decay at infinity [J], J. Math. Phys.31(1990), no.5,1177-1180.
[60]G. Talenti, Best constant in Sobolev inequality [J], Ann. Mat. Pura Appl.,110(4) (1976),353-372.
[61]Thangavelu S., Harmonic Analysis on the Heisenberg Group[M], Progress in Math-ematics 159, Birkhauser, Boston,1998
[62]A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements[J], Adv. Math.209 (2007),407-459.
[63]Wolf,J.A., Harmonic Analysis on Commutative Spaces[M], Mathematical Surveys and Monographs,Vol.142,2007,185-188.
[64]Warner,G., Harmonic Analysis on Senisimple Lie Groups[M], vol.2,335-337, Springer-Verlag, Berlin,1972.
[65]Zhang H., Niu. P., Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications[J], Nonlinear Anal.54(1) (2003), 165-186.
[66]Q. Yang. Best constants in the Hardy-Rellich type inequalities on the Heisenberg group[J], J. Math. Anal. Appl.,342(2008).423-431.
[67]Zhu Fuliu:The heat kernel and the Riesz transform on the quaternionic Heisenberg groups[J], Pacific J. Math.209(1),175-199 (2003)
[68]Wu,H.,Topics in Riemann Geometry[M],Peking Uni.Press,1993,51-88.
[69]Besse,A.,Einstein Aanifolds[M].Springer-Verlag.1980,185-188.
[2]Anker,J.P.,E.Damek, C.Yacob, Spherical Analysis on Harmonic AN Groups[J], Ann.Scuola.Norm.Sup.Pisa, (4)23(1996), no.4643-679.
[3]Ahern, P.,M. Flores and W. Rudin, An Invariant Volume-mean-value Prop-erty [J],J.Functional Analysis,111,380-397,1993.
[4]Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities, Nonlinear Analysis,70(2009),2826-2833.
[5]T. Aubin, Probleme isoperimetric et espace de Sobolev[J], J. Differential Geom.,11 (1976),573-598.
[6]A. Balinsky, W. D. Evans, D. Hundertmark, R. T. Lewis, On inequalities of Hardy-Sobolev type[J], Banach J. Math. Anal.2 (2008), no.2,94-106.
[7]W. Beckner, On Hardy-Sobolev embedding[J], arXiv:0907.3932v1[math.AP].
[8]Z. Balogh, J. Tyson, Polar coordinates on Carnot groups [J], Math. Z.241(4)(2002), 697-730.
[9]Z. Balogh, J. Tyson, Polar coordinates on Carnot groups [J], Math. Z.241(4)(2002), 697-730.
[10]Berndt J., Tricerri F., Vanhecke L.:Generalized Heisenberg groups and Damek-Ricci Spaces[M], LNM 1589,1995
[11]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical exponents[J], Comm. Pure Appl. Math.36 (1983) 437-477.
[12]Bonsall,F.,F.,J.Dancall, Complet normed algebras[M], Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag,1970,171-185
[13]Birindelli,I., Cutrl.A., A similinear problem for the Heisenberg Laplacian[J], Rend. Mat. Univ. Padova 94(1995),137-153
[14]Bony J.M., Principle du Maximum, Inegalite de Harnack et unnicite du probleme de Cauchy pour les operaeurs elliptiques degeneres [J], Ann. Inst. Fourier Grenobles, 19,277-304 (1969)
[15]Caffarelli L., Garofalo N., Segala F., A gradient bound for entire solutions of quasi-linear equations and its consequences [J], Comm. Pure Appl. Math.47 1457-1473 (1994).
[16]Corwin L.J., Greenleaf F.P., Representations of nilpotent Lie groups and their ap-plications.1:Basic theory and examples[M], Cambridge Studies in Advanced Math-ematics 18, Cambridge University Press,1990
[17]Conway, J.,B., A Course in Functional Analysis[M], GTM96,1st edition.377,Springer-Verlag, New York.1985.
[18]Chen,W., Li, C., Classification of solution of some nonlinear elliptic equations [J], Duke Math. J.63(1991),615-622
[19]Damek, E., F.Ricci, Harmonic analysis on Solvable Extension of H-type Groups[J], J.Geometic Analysis,2(1992).213-248.
[20]D'Ambrosio L., Lucente S., Nonlinear Liouville theorems for Grushin and Tricomi operators[J], J. Diff. Equa.193 (2003),511-541
[21]D'Ambrosio L., Some Hardy inequalities on the Heisenberg group[J], Diff. Equa. 40(4) (2004),552-564.
[22]D'Ambrosio L., Hardy inequalities related to Grushin type operators [J], Proc. Amer. Math. Soc.132 (2004),725-734.
[23]Deng Hongjun, Amenable Banach Algebras[R], Procedins in Asia Mathematics,Hongkong,1991,178-182.
[24]Dales et.al. Introduction to Banach Algebras,Operators,and Harmonic Analy-sis [M], Cambridge University Press:2003,75-109.
[25]Folland G.B., A fundamental solution for a subelliptic operator [J], Bull. Amer. Math. Soc.,79(2) (1973),373-376.
[26]Folland G.B., Subelliptic estimates and function spaces on nilpotent Lie group[J], Ark. Mat.,13.161-220 (1975)
[27]Folland G.B., Stein E.M., Estimates for the (?)h complex and analysis on the Heisen-berg group[J], Comm. Pure Apll. Math..27,492-522 (1974)
[28]Gaveau B.:Principle de moindre action, propogation de la chaleur er estimees sous-elliptiques sur certains groups nilpotents[J], Acta Math.,139,95-153 (1977)
[29]Gidas B.. Ni W.M., Nirenberg L.,:Symmetry and related properties via the maxi-mum principle [J], Comm. Math. Phys.,68,209-243 (1979)
[30]Gidas B., Sprugk J.,Global and local behavior of positive solutions of nonlinear elliptic equations [J], Comm. Pure App. Math.,35,525-598 (1981)
[31]J. A. Goldstein, I. Kombe, The Hardy inequality and nonlinear parabolic equations on Carnot groups[J]. Nonlinear Analysis (2007), doi:10.1016/j.na.2007.
[32]Hardy G., Note on a theorem of Hilbert [J], Math.Zeitschr.6 (1920),314-317.
[33]Hierro M.D.:Dispersive and strichartz estimates on H-type groups[J], Stud. Math., 169(1),1-20 (2005)
[34]Hormander L.,Hypoelliptic second order differential equation [J], Acta Math., Upp-sala,119,147-171 (1967)
[35]Helgason.S.,Groups and Geometric Analysis[M], Academic Press,381-382,1984.
[36]Helgason.S.,Differential Geometry,Lie Groups and Symmetric Spaces[M], Academic Press,515-518,1978.
[37]Helemskii.A. Ya.,The Homology of Banach and Topological Algebras[M],Kluwer,183-195,1989.
[38]Hewitt, E., K.A.Ross, Abstract Harmonic Analysis[M], vol.1,291-292, Springer-Verlag, Berlin,1963.
[39]Hulanicki A., The distribution of energy of the Brownian motion in Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group [J]. Studia Math.,56 (1976),165-173.
[40]Johnson B.E., Cohomology in Banach Algebras [M]. Memoirs of the AMS,25, 81-85 (1972)
[41]Jerison D.S., Lee J.M., Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem[J], J. Amer. Math. Soc.,1,1-13 (1998)
[42]Kaplan A.:Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms [J], Trans. Amer. Math. Soc.,258(1),147-153 (1980)
[43]Kombe I., Sharp Hardy-type inequalities on Carnot groups[J]. preprint,(2005)
[44]Kombe I., The Hardy-type inequalities and nonliear parabolic equations on Carnot groups [J], preprint, (2005)
[45]Koranyi,A., On a Mean Value Property for Hyperbolic Spaces[J], Contemporary Mathematics, vol.191.107-116.1995.
[46]Lust-Piquard F., A sinple-minded computation of heat kernels on Heisenberg groups[J]. Coll. Math.,2003.97(2):233-249.
[47]Marakami,S.,Introduction to Homogenous Manifolds(in Chinese)[M], Shanghai Sci-ence and Technology Press,51-52,1984.
[48]Naimak.M.A., Normed Algebras[M], Wolters-Northoff,1970,381-385.
[49]Ni W.M., On a singular elliptic equation[J], Proc. Amer. Math. Soc.,88(1983), 614-616.
[50]Niu P., Zhang H., Wang Y., Hardy type and Rellich type inequalities on the Heisenberg group [J], Proc. Amer. Math. Soc.129(2001),3623-3630.
[51]R. Monti, D. Morbidelli, Kelvin transform for Grushin operators and critical semi-linear equations[J], Duke Math. J.,131(2006),167-202.
[52]Naten,B.,Y., Y.Weit,, Integrable Harmonic Functions on Symmetric Spaces of Rank one[J], J. Functional Analysis, vol.160,141-149,1998.
[53]P. Niu, H. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisen-berg group [J]. Proc. Amer. Math. Soc.,129 (2001),3623-3630.
[54]Opic B., Kufner A., Hardy-type Inequalities[M], Pitman Research Notes in Math., 219, Long-man 1990.
[55]Ricci, F., Spherial Functions on Certain Non-symmetric Harmonic Manifolds[R], Lecture Notes in the Workshop on Representation Theory of Lie Groups, ICTP, 1993.
[56]Rothschild. L.P.. Stein, E.M., Hypoelliptic differential operators and nilpotent groups[J], Acta Math.,137(1976) 247-320
[57]Staubach W.:Wiener path integrals and the fundamental solution for the Heisen-berg Laplacian[J], Journal D:analyse Math..91,389-400 (2003)
[58]Stein E., Harmonic Analysis, Real-Variable Methods, Orthgonality, and Oscillatory Integrals[M], Princeton University Press, Princeton, NJ.1993.
[59]J. Stubbe, Bounds on the number of bound states for potentials with critical decay at infinity [J], J. Math. Phys.31(1990), no.5,1177-1180.
[60]G. Talenti, Best constant in Sobolev inequality [J], Ann. Mat. Pura Appl.,110(4) (1976),353-372.
[61]Thangavelu S., Harmonic Analysis on the Heisenberg Group[M], Progress in Math-ematics 159, Birkhauser, Boston,1998
[62]A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements[J], Adv. Math.209 (2007),407-459.
[63]Wolf,J.A., Harmonic Analysis on Commutative Spaces[M], Mathematical Surveys and Monographs,Vol.142,2007,185-188.
[64]Warner,G., Harmonic Analysis on Senisimple Lie Groups[M], vol.2,335-337, Springer-Verlag, Berlin,1972.
[65]Zhang H., Niu. P., Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications[J], Nonlinear Anal.54(1) (2003), 165-186.
[66]Q. Yang. Best constants in the Hardy-Rellich type inequalities on the Heisenberg group[J], J. Math. Anal. Appl.,342(2008).423-431.
[67]Zhu Fuliu:The heat kernel and the Riesz transform on the quaternionic Heisenberg groups[J], Pacific J. Math.209(1),175-199 (2003)
[68]Wu,H.,Topics in Riemann Geometry[M],Peking Uni.Press,1993,51-88.
[69]Besse,A.,Einstein Aanifolds[M].Springer-Verlag.1980,185-188.