圆盘上在二面体群下不变测度的调和分析
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摘要
从H.Pollard的研究开始，Jacobi级数理论及与此相关的更深入的函数论问题有
了很大的发展。B.Muckenhoupt和E.M.Stein在1965年系统研究了超球级数（单参
Jacobi级数）的共轭级数，并建立了相应的H~p空间的基本理论。这些工作被李中凯教
授在1996年推广到一般形式（双参数）的Jacobi级数，但他们所讨论的广义解析函数
F=u+iv中的u和v分别是单位圆盘上的全偶函数和全奇函数，而且u和v满足不
同的广义Laplace方程。如何把他们的结果推广到单位圆盘上的一般函数，一直困扰着
我们。1988年以来，C.F.Dunkl等研究了关于在有限反射群下不变测度的球面调和函
数（称为h-调和函数），特别是找到了单位圆盘上相应于二面体群D_2的正交基，该正交
基由四组参数不同的Jacobi多项式系组成，这为解决上述问题带来了希望。
本文就是在Dunkl理论的基础上研究单位圆盘上相应于二面体群D_2的h-调和
函数，共分五章。第一章主要介绍Jacobi级数和Dunkl理论；第二章介绍对于（0,2π）
上的函数f相应于D_2的h-调和级数展式及其Poisson积分u(x,y)，u(x,y)在区域
D={(x,y)：x~2+y~2＜1}内是h-调和的，即△_hu(x,y)=0，此处
借助Dunkl定义的一阶微分-差分算子T_1，T_2，在第二章中还引入了广义Cauchy-
Riemann方程组：
得到u(x,y)的共轭h-调和函数v(x,y).v(x,y)在D内也是h-调和的。由此定义了h-
共轭函数（广义的Hilbert变换）f→f；第三章得到了相应于D_2的h-调和函数的极大值
原理、平均值定理、Poisson积分的特征等；第四章主要研究共轭Poisson核Q(r,θ)，
给出了Q的准确估计，证明了共轭Poisson积分是(p,p)(1＜p＜∞)有界，弱(1,1)
有界的；第五章引进了h-解析函数F=u+iv（u,v在D内满足(★)）并研究了相应的
H~p空间。当u是全偶函数时，v一定是全奇函数，这时△_h和(★)中的差分项消失，恰
好回到李中凯教授所研究的通常Jacobi级数的情况，从而解决了前面所提出的问题。
Based on Dunkl theory, we investigate h-harmonic functions in the unit disc with
respect to D2 group in this paper and divide it into five chapters.
The first chapter includes some preliminaries about Jacobi series and Dunkl theory.
We study the function f defined in (0, 2ir) thoroughout the second chapter. Firstly,
we prove that the Poisson integral uQr, y) of f is h-harmonic in the unit disc, i.e. ~hu(X, y)
0, where
a2 a2 2~38 2&? ~ ~rix
?9x2 +~2 + +uy ttkx,y)j ?憉~x ~
y
Then by Dunkl differential-difference operator T1 and T? of order one , we introduce the
generalized Cauchy-Riemann equations:
{ ~ 梪(梮,y) _ v(x,y) 梫(x,--y) =
2; y
u(x,y)梪(x,梱) v(x,y)梫(梮,y) _ (*)
梠
y 2;
and the conjugate h-harmonic function vQr, y) related to uQr, y). And v is also h-harmonic
for (x, y) in the unit disc. This leads to the generalized Hubert transform f ?f.
In chapter III we obtain the maximum principle, mean value property and charac-
terization of Poisson integral of h-harmonic function with respect to D2 group.
In chapter IV we give the precise estimate of the conjugate kernel Q(r, 0, so) and
verify that the conjugacy mapping f ?f is a operator of weak (1.1) and strong (p,p).
In charter V h-harmonic analytic function F u ?iv (where u and v satisfy (*) for
(x, y) in the unit disc) and the corresponding H~ space are considered. We obtain that
4o~ 43
the functions in H~ space have nontangential limit as p max( ) and find
4a+l?~3眑
that if u is even in x and y, then v must be odd in x and y. Under this condition, the
differential-difference terms of 1~h and (*) dissaper, so we return to the Jacobi series case
which Zh.-K. Li has studied
了很大的发展。B.Muckenhoupt和E.M.Stein在1965年系统研究了超球级数（单参
Jacobi级数）的共轭级数，并建立了相应的H~p空间的基本理论。这些工作被李中凯教
授在1996年推广到一般形式（双参数）的Jacobi级数，但他们所讨论的广义解析函数
F=u+iv中的u和v分别是单位圆盘上的全偶函数和全奇函数，而且u和v满足不
同的广义Laplace方程。如何把他们的结果推广到单位圆盘上的一般函数，一直困扰着
我们。1988年以来，C.F.Dunkl等研究了关于在有限反射群下不变测度的球面调和函
数（称为h-调和函数），特别是找到了单位圆盘上相应于二面体群D_2的正交基，该正交
基由四组参数不同的Jacobi多项式系组成，这为解决上述问题带来了希望。
本文就是在Dunkl理论的基础上研究单位圆盘上相应于二面体群D_2的h-调和
函数，共分五章。第一章主要介绍Jacobi级数和Dunkl理论；第二章介绍对于（0,2π）
上的函数f相应于D_2的h-调和级数展式及其Poisson积分u(x,y)，u(x,y)在区域
D={(x,y)：x~2+y~2＜1}内是h-调和的，即△_hu(x,y)=0，此处
借助Dunkl定义的一阶微分-差分算子T_1，T_2，在第二章中还引入了广义Cauchy-
Riemann方程组：
得到u(x,y)的共轭h-调和函数v(x,y).v(x,y)在D内也是h-调和的。由此定义了h-
共轭函数（广义的Hilbert变换）f→f；第三章得到了相应于D_2的h-调和函数的极大值
原理、平均值定理、Poisson积分的特征等；第四章主要研究共轭Poisson核Q(r,θ)，
给出了Q的准确估计，证明了共轭Poisson积分是(p,p)(1＜p＜∞)有界，弱(1,1)
有界的；第五章引进了h-解析函数F=u+iv（u,v在D内满足(★)）并研究了相应的
H~p空间。当u是全偶函数时，v一定是全奇函数，这时△_h和(★)中的差分项消失，恰
好回到李中凯教授所研究的通常Jacobi级数的情况，从而解决了前面所提出的问题。
Based on Dunkl theory, we investigate h-harmonic functions in the unit disc with
respect to D2 group in this paper and divide it into five chapters.
The first chapter includes some preliminaries about Jacobi series and Dunkl theory.
We study the function f defined in (0, 2ir) thoroughout the second chapter. Firstly,
we prove that the Poisson integral uQr, y) of f is h-harmonic in the unit disc, i.e. ~hu(X, y)
0, where
a2 a2 2~38 2&? ~ ~rix
?9x2 +~2 + +uy ttkx,y)j ?憉~x ~
y
Then by Dunkl differential-difference operator T1 and T? of order one , we introduce the
generalized Cauchy-Riemann equations:
{ ~ 梪(梮,y) _ v(x,y) 梫(x,--y) =
2; y
u(x,y)梪(x,梱) v(x,y)梫(梮,y) _ (*)
梠
y 2;
and the conjugate h-harmonic function vQr, y) related to uQr, y). And v is also h-harmonic
for (x, y) in the unit disc. This leads to the generalized Hubert transform f ?f.
In chapter III we obtain the maximum principle, mean value property and charac-
terization of Poisson integral of h-harmonic function with respect to D2 group.
In chapter IV we give the precise estimate of the conjugate kernel Q(r, 0, so) and
verify that the conjugacy mapping f ?f is a operator of weak (1.1) and strong (p,p).
In charter V h-harmonic analytic function F u ?iv (where u and v satisfy (*) for
(x, y) in the unit disc) and the corresponding H~ space are considered. We obtain that
4o~ 43
the functions in H~ space have nontangential limit as p max( ) and find
4a+l?~3眑
that if u is even in x and y, then v must be odd in x and y. Under this condition, the
differential-difference terms of 1~h and (*) dissaper, so we return to the Jacobi series case
which Zh.-K. Li has studied
引文
[1] R.Askey, Mean convergence of orthogonal series and Lagrange interpolation,Acta. Math.Acad.Sci.Hung.23(1-2) ,(1972) ,p71-85．
[2] R.Askey, Summability of Jacobi series, Trans. Amer. Math. Soc. Vol.179 (1973) , 71-84．
[3] W.N.Bailey, Generalized hypergeometric series,Cambridge Univ. Press, Cambridge, (1935) .
[4] C.T.Benson and L.C.Grove, Finite reflection groups.2nd edition.Berlin Heidelberg New York Springer ,(1985) .
[5] L.Bers, Function-theoretical properties of solutions of partial differential equations of elliptic type, Contributions to the theory of partial differential equations, pp. 69-94,Annals of Math.Studies No.33,Princeton Univ. Press, Princeton, N. J. (1954) .
[6] A.P.Calderon and A. Zygmund, On higher gradients of harmonic functions,Studia Math.,24(1964) ,211-216．
[7] W.C. Connett and A.L.Schwartz, A multiplier theorem for Jacobi expansions ,Studia Math . 52(1975) ,243-261． See also A correction to the paper " A multiplier theorem for Jacobi expansions ", Studia Math. No.54(1975) ,197．
[8] W.C.Connett and A.L.Schwartz, The Littlewood-Paley theory for Jacobi expansions, Trans. Amer. Math. Soc. Vol.251(1979) , 219-234．
[9] C.F.Dunkl, Differential-difference operators associated to reflection groups,Trans. Amer. Math. Soc. Vol.311(1989) , 167-183．
[10] C.F.Dunkl, Integral kernels with reflection group invariance , Can. J. Math. Vol.43 (1991) ,1213-1227．
[11] C.F.Dunkl, Intertwinging operators associated to the group S3, Trans. Amer. Math. Soc. Vol.347(1995) , 3347-3374．
[12] C.F.Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with Dihedral Symmerty, J. Math. Anal. Appl. 143(1989) , 459-470．
[13] C.F.Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988) , 33-60．
[14] J.Garcia-Cuerva and Rubio de J.L.Francia, Weighted norm inequalities and relate topics ,North-Holland Math. Studies 116, Amsterdam, (1985) .
[15] G.Gasper, Positivity and the convolution structure for Jacobi series , Ann. Math. Vol.93 (1971) ,112-118．
[16] Zh.-K.Li, Conjugate Jacobi series and conjugate functions, J, Approx. Theory. Vol. 86, No.2． (1996) , 179-196．
[17] Zh.-K.Li, Hardy spaces for Jacobi expansions, Analysis 16(1996) , 27-49．
[18] Zh.-K.Li, and Y.Xu, Summability of h-harmonic and orthogonal polynomials on balls and simplices, Trans. Amer. Math. Soc., to appear.
[19] B.Muckenhoupt, Mean convergence of Jacobi series,Proc.Amer.Math.Soc.,23(1969) , 306-310．
[20] B.Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. math. Soc.356(1986) .
[21] B.Muckenhoupt and E.M.Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. ,Vol.118(1965) , 17-92．
[22] H.Pollard,The mean convergence of orthogonal series I,Trans. Amer.Math.Soc.62 (1947) ,387-403．
[23] H.Pollard,The mean convergence of orthogonal series II,Trans. Amer.Math.Soc.63 (1948) ,355-367．
[24] H.Pollard,The mean convergence of orthogonal series III,Duke Math.J., 16(1949) ,189-191．
[25] G.Szego, Orthogonal Polynomials, Amer.Math.Soc. Colloq. Publ. 23,4th ed., Amer. Math. Soc. ,Providence, RI. ,1975．
[26] A.Torchinsky, Real-variable methode in harmonic analysis,Academic Press, orlando, Florida, (1986) .
[27]Y.Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups,Proc. Amer. Math. Soc. Vol.125(1997), 2963-2973.
[28]Y.Xu, Intertwining operator and h-harmonic associated with reflection groups, Can.J. Math. Vol.50(1),(1998),193-209.
[29]Y.Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. Vol. 49(1)(1997), 175-192.
[30]Y.Xu, Summability of Fourier orthogonal series for Jacobi polynomials on a ball in R_d, Trans. Amer. Math. Soc. Vol. 351 No.6(1999), 2439-2458.
[31]Y.Xu, Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in R_d,Proc. Amer. Math. Soc. Vol.126(1998), 3027-3036.
[32]A.Zygmund, Trigonometric Series,Vols,I and pi,2nd ed.,Cambridge Univ. Press.New York,(1959).
[33]黄国宣，李群基础，复旦大学出版社，(1995)．
[34]李中凯，Fourier-Jacobi级数与逼近论，博士论文，大连理工大学，(1992)．
[35]孙永生，函数逼近论，北京师范大学出版社，(1989)．
[2] R.Askey, Summability of Jacobi series, Trans. Amer. Math. Soc. Vol.179 (1973) , 71-84．
[3] W.N.Bailey, Generalized hypergeometric series,Cambridge Univ. Press, Cambridge, (1935) .
[4] C.T.Benson and L.C.Grove, Finite reflection groups.2nd edition.Berlin Heidelberg New York Springer ,(1985) .
[5] L.Bers, Function-theoretical properties of solutions of partial differential equations of elliptic type, Contributions to the theory of partial differential equations, pp. 69-94,Annals of Math.Studies No.33,Princeton Univ. Press, Princeton, N. J. (1954) .
[6] A.P.Calderon and A. Zygmund, On higher gradients of harmonic functions,Studia Math.,24(1964) ,211-216．
[7] W.C. Connett and A.L.Schwartz, A multiplier theorem for Jacobi expansions ,Studia Math . 52(1975) ,243-261． See also A correction to the paper " A multiplier theorem for Jacobi expansions ", Studia Math. No.54(1975) ,197．
[8] W.C.Connett and A.L.Schwartz, The Littlewood-Paley theory for Jacobi expansions, Trans. Amer. Math. Soc. Vol.251(1979) , 219-234．
[9] C.F.Dunkl, Differential-difference operators associated to reflection groups,Trans. Amer. Math. Soc. Vol.311(1989) , 167-183．
[10] C.F.Dunkl, Integral kernels with reflection group invariance , Can. J. Math. Vol.43 (1991) ,1213-1227．
[11] C.F.Dunkl, Intertwinging operators associated to the group S3, Trans. Amer. Math. Soc. Vol.347(1995) , 3347-3374．
[12] C.F.Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with Dihedral Symmerty, J. Math. Anal. Appl. 143(1989) , 459-470．
[13] C.F.Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988) , 33-60．
[14] J.Garcia-Cuerva and Rubio de J.L.Francia, Weighted norm inequalities and relate topics ,North-Holland Math. Studies 116, Amsterdam, (1985) .
[15] G.Gasper, Positivity and the convolution structure for Jacobi series , Ann. Math. Vol.93 (1971) ,112-118．
[16] Zh.-K.Li, Conjugate Jacobi series and conjugate functions, J, Approx. Theory. Vol. 86, No.2． (1996) , 179-196．
[17] Zh.-K.Li, Hardy spaces for Jacobi expansions, Analysis 16(1996) , 27-49．
[18] Zh.-K.Li, and Y.Xu, Summability of h-harmonic and orthogonal polynomials on balls and simplices, Trans. Amer. Math. Soc., to appear.
[19] B.Muckenhoupt, Mean convergence of Jacobi series,Proc.Amer.Math.Soc.,23(1969) , 306-310．
[20] B.Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. math. Soc.356(1986) .
[21] B.Muckenhoupt and E.M.Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. ,Vol.118(1965) , 17-92．
[22] H.Pollard,The mean convergence of orthogonal series I,Trans. Amer.Math.Soc.62 (1947) ,387-403．
[23] H.Pollard,The mean convergence of orthogonal series II,Trans. Amer.Math.Soc.63 (1948) ,355-367．
[24] H.Pollard,The mean convergence of orthogonal series III,Duke Math.J., 16(1949) ,189-191．
[25] G.Szego, Orthogonal Polynomials, Amer.Math.Soc. Colloq. Publ. 23,4th ed., Amer. Math. Soc. ,Providence, RI. ,1975．
[26] A.Torchinsky, Real-variable methode in harmonic analysis,Academic Press, orlando, Florida, (1986) .
[27]Y.Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups,Proc. Amer. Math. Soc. Vol.125(1997), 2963-2973.
[28]Y.Xu, Intertwining operator and h-harmonic associated with reflection groups, Can.J. Math. Vol.50(1),(1998),193-209.
[29]Y.Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. Vol. 49(1)(1997), 175-192.
[30]Y.Xu, Summability of Fourier orthogonal series for Jacobi polynomials on a ball in R_d, Trans. Amer. Math. Soc. Vol. 351 No.6(1999), 2439-2458.
[31]Y.Xu, Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in R_d,Proc. Amer. Math. Soc. Vol.126(1998), 3027-3036.
[32]A.Zygmund, Trigonometric Series,Vols,I and pi,2nd ed.,Cambridge Univ. Press.New York,(1959).
[33]黄国宣，李群基础，复旦大学出版社，(1995)．
[34]李中凯，Fourier-Jacobi级数与逼近论，博士论文，大连理工大学，(1992)．
[35]孙永生，函数逼近论，北京师范大学出版社，(1989)．
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