带权的球面调和乘子
|
摘要
根据近年来C．Dunkl等发展的关于在有限反射群下不变测度的球面调和理论框架
（简称h-调和理论），研究了R~n中单位球面S~(n-1)上带权调和展开的乘子问题，内容包
括Littlewood-Paley理论和乘子有界的充分条件。这里我们考虑的权函数是
其中α=（α_1，α_2，…，α_n），αi≥0。
本文的第一章介绍乘子理论的发展背景和h-调和理论的知识；第二章引入相应于
权函数（1）的球面h-调和展开的辅助函数，并研究了其性
质，得到一系列范数不等式（Littlewood-Paley理论）；第三章给出带权球面h-调和乘子
有界的Horrnander型条件。在证明过程中，关键之一是利用Y.Xu得到的关于h-调和
的缠绕算子V的积分变换公式，从而使多个变量的积分简化为一个变量的积分，再利用
Parseval等式，使问题处理起来简捷。
本文结合Fourier级数，Jacobi级数和经典球面调和展开，首次研究了带权的球面
调和展开Littlewood-Paley理论和乘子有界性。它们之间的关系如下图：
归纳起来，本文的创新点在于如下两方面：
（i）当所有的参数α_i=0时，带权的球面h-调和展开问题就退化为经典球面调和展
开，但是研究带权的球面h-调和展开困难之处在于它不能用旋转交换，必须采用其它
方法，计算显得相对复杂。
（ii）当n=2时，带权的球面调和展开就退化为Jacobi展开，其中参数为α=α_1-
1/2，β=α_2-1/2，但在Jacobi展开的Littlewood-Paley理论和乘子理论中可以利用所谓的
Jacboi卷积结构，而在高维时不能定义相应的卷积。
在研究带权的球面h-调和展开的Littlewood-Paley理论时，我们得到了与经典球
面调和展开和Jacobi展开完全相应的结论，但在研究乘子的有界性时，本文仅得到在
这一特殊情况下的结果（相对于经典球面调和展开和Jacobi展开而言），
对于一般的参数α_i≥0，目前尚未得到理想的结果，这是一个非常值得研究的问题。
Based on the theor of spherical harmonics for measures invariant under a finite reflection group developed by C. Diinkl recentl. the Littlewood-Pale~?theor is studied and an application is given to riiultipliers for ~plirioal li梙larllo(jlncs in IJ?h~: .Sr~).
The paper is divided as follows: In Chapter Oiie I lie developrtieiit background of ;iitilti?pliers theory arid spherical h-harmonics theory are introduced. In Chapter T~vo the auxiliary functions g6(J. x?, S(f, x? and g~(f, x? related to spherical h-harmonics with respect t.o the weight function F~=1 Ix~I~?o~ > 0) are defined. We study the properties of those functionals and establish Littlewood-Palev theory. In Chapter Three the g~(f. x?. S(J. x? and g(f. x? functionals are used to prove a iuiiultiphier tiieoreuiu for spherical u梙armonics with respect to the weight function fl~ .cJ~ (o~ > 0) on the unit sphere S~ in R~. 慖L proof of this theorem uses a number of the results which are proved in Chapter Two. In particular. we use the integral transformation formula of the intertwining operator V discovered by Y. Xu with respect to spherical h-harmonics. so that integral of several variables involved V turns to integral of a single variable. Finally using Parseval equality of ultraspherical series l
et us solve the problem in an easy way.
Combined Fourier multiplier. Jacobi multiplier arid multipliers for classical spherical cxpanlsions, we study multipliers for spherical lu梙armonics for the first time. If all parameters o~ =0. then the multipliers for spherical h-harmonics reduce to multipliers for classical spherical expansions. But, since multipliers for spherical h-harmonics cani commute with tile action of the rotation group SO(n) on we meet a lot of difficulties in studying them. We use other method which involves complicated calculation. For the case tu~2, then the multipliers for spluerical h-hartnomiics reduce to multipliers for Jacobi expansions, where o o~ ?1/2.1 02?1/2. The convolution structure is used in studying the Littlewood桺alev theory and multipliers for Jacobi expansions, but there isn抰 corresponding convolution in higher dimensions.
In this thesis, we obtain for spherical h-harmonics the coml)let.e analogue of the LittlewoodPaley theory to classical spherical expansions and Jacobi expansions. But in studying the bound conditions for multipliers of spherical h-harmonics, tile result in this paper is true only in the special case mint<~<~ o~ = 0. For general parameters a, > 0 we don抰 obtain an ideal result at present. This problem is well worth studying further.
（简称h-调和理论），研究了R~n中单位球面S~(n-1)上带权调和展开的乘子问题，内容包
括Littlewood-Paley理论和乘子有界的充分条件。这里我们考虑的权函数是
其中α=（α_1，α_2，…，α_n），αi≥0。
本文的第一章介绍乘子理论的发展背景和h-调和理论的知识；第二章引入相应于
权函数（1）的球面h-调和展开的辅助函数，并研究了其性
质，得到一系列范数不等式（Littlewood-Paley理论）；第三章给出带权球面h-调和乘子
有界的Horrnander型条件。在证明过程中，关键之一是利用Y.Xu得到的关于h-调和
的缠绕算子V的积分变换公式，从而使多个变量的积分简化为一个变量的积分，再利用
Parseval等式，使问题处理起来简捷。
本文结合Fourier级数，Jacobi级数和经典球面调和展开，首次研究了带权的球面
调和展开Littlewood-Paley理论和乘子有界性。它们之间的关系如下图：
归纳起来，本文的创新点在于如下两方面：
（i）当所有的参数α_i=0时，带权的球面h-调和展开问题就退化为经典球面调和展
开，但是研究带权的球面h-调和展开困难之处在于它不能用旋转交换，必须采用其它
方法，计算显得相对复杂。
（ii）当n=2时，带权的球面调和展开就退化为Jacobi展开，其中参数为α=α_1-
1/2，β=α_2-1/2，但在Jacobi展开的Littlewood-Paley理论和乘子理论中可以利用所谓的
Jacboi卷积结构，而在高维时不能定义相应的卷积。
在研究带权的球面h-调和展开的Littlewood-Paley理论时，我们得到了与经典球
面调和展开和Jacobi展开完全相应的结论，但在研究乘子的有界性时，本文仅得到在
这一特殊情况下的结果（相对于经典球面调和展开和Jacobi展开而言），
对于一般的参数α_i≥0，目前尚未得到理想的结果，这是一个非常值得研究的问题。
Based on the theor of spherical harmonics for measures invariant under a finite reflection group developed by C. Diinkl recentl. the Littlewood-Pale~?theor is studied and an application is given to riiultipliers for ~plirioal li梙larllo(jlncs in IJ?h~: .Sr~).
The paper is divided as follows: In Chapter Oiie I lie developrtieiit background of ;iitilti?pliers theory arid spherical h-harmonics theory are introduced. In Chapter T~vo the auxiliary functions g6(J. x?, S(f, x? and g~(f, x? related to spherical h-harmonics with respect t.o the weight function F~=1 Ix~I~?o~ > 0) are defined. We study the properties of those functionals and establish Littlewood-Palev theory. In Chapter Three the g~(f. x?. S(J. x? and g(f. x? functionals are used to prove a iuiiultiphier tiieoreuiu for spherical u梙armonics with respect to the weight function fl~ .cJ~ (o~ > 0) on the unit sphere S~ in R~. 慖L proof of this theorem uses a number of the results which are proved in Chapter Two. In particular. we use the integral transformation formula of the intertwining operator V discovered by Y. Xu with respect to spherical h-harmonics. so that integral of several variables involved V turns to integral of a single variable. Finally using Parseval equality of ultraspherical series l
et us solve the problem in an easy way.
Combined Fourier multiplier. Jacobi multiplier arid multipliers for classical spherical cxpanlsions, we study multipliers for spherical lu梙armonics for the first time. If all parameters o~ =0. then the multipliers for spherical h-harmonics reduce to multipliers for classical spherical expansions. But, since multipliers for spherical h-harmonics cani commute with tile action of the rotation group SO(n) on we meet a lot of difficulties in studying them. We use other method which involves complicated calculation. For the case tu~2, then the multipliers for spluerical h-hartnomiics reduce to multipliers for Jacobi expansions, where o o~ ?1/2.1 02?1/2. The convolution structure is used in studying the Littlewood桺alev theory and multipliers for Jacobi expansions, but there isn抰 corresponding convolution in higher dimensions.
In this thesis, we obtain for spherical h-harmonics the coml)let.e analogue of the LittlewoodPaley theory to classical spherical expansions and Jacobi expansions. But in studying the bound conditions for multipliers of spherical h-harmonics, tile result in this paper is true only in the special case mint<~<~ o~ = 0. For general parameters a, > 0 we don抰 obtain an ideal result at present. This problem is well worth studying further.
引文
[1] R. Askey, A transplantation for Jacobi series, Illinois J. Math. 13(1969) , 583-590．
[2] A. Bonami and J. L. Clerc, Sommes de Cesaro et multiplicateurs des developments en harmoniques spheriques, Transactions Amer. Math. Soc. 183(1973) , 223-263．
[3] W. C. Connett and A. L. Schwartz, The Little wood-Paley theory for Jacobi expansions, Trans. Amer. Math. Soc 251(1979) , 219-234．
[4] 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 . 54 (1975) , 197． )
[5] W. C. Connett and A. L. Schwartz, The theory of ultraspherical multipliers, Mem. Amer. Math. Soc. No. 183, 1977．
[6] C. Dunkl, Differntial-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) , 167-183．
[7] C. Dunkl, Integral kernels with reflection group invariance , Can. J. Math. 43(1991) , 1213-1227．
[8] G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. 95 (1972) , 261-280．
[9] G. Gasper and W. Trebels, Multiplier criteria of Hormander type for Jacobi expansions, Studia Math. T. LXVIII(1980) , 187-197．
[10] G. Gasper , Positivity and the convolution structure for Jacobi series , Ann. of Math. 93 (1971) , 112-118．
[11] Zh.-K. Li, Conjugate Jacobi series and conjugate functions, J. Approx. Theory 86(1996) , 179-196．
[12] Zh.-K. Li, Hardy spaces for Jacobi expansions, Analysis 16． 27-49(1996) .
[13] Zh.-K. Li and Y. Xu. Summability of orthogonal expansions on spheres, balls and simplices, Trans. Amer. Soc. , to appear.
[14] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118(1965) , 17-92．
[15] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J. , 1971．
[16] E. M. Stein, Singular integrals and differentiability properties of functions , Princeton University Press, Princeton, N. J. , 1970．
[17] E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Studies, no. 63, Princeton Univ. Press, Princeton, N. J. ; Univ. of Tokyo Press, Tokyo, 1970．
[18] R. S. Strichartz, Multipliers for spherical harmonic expansions, Trans. Amer. Math. Soc 167(1972) , 115-124．
[19] G. Szego, Orthogonal polynomials. 4th ed. , Amer. Math. Soc. , Colloq. Publ. 23, Providence, Rhode Island, 1975．
[20] Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125(1997) , 2963-2973．
[21] Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. Vol. 49(1) (1997) , 175-192．
[2] A. Bonami and J. L. Clerc, Sommes de Cesaro et multiplicateurs des developments en harmoniques spheriques, Transactions Amer. Math. Soc. 183(1973) , 223-263．
[3] W. C. Connett and A. L. Schwartz, The Little wood-Paley theory for Jacobi expansions, Trans. Amer. Math. Soc 251(1979) , 219-234．
[4] 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 . 54 (1975) , 197． )
[5] W. C. Connett and A. L. Schwartz, The theory of ultraspherical multipliers, Mem. Amer. Math. Soc. No. 183, 1977．
[6] C. Dunkl, Differntial-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) , 167-183．
[7] C. Dunkl, Integral kernels with reflection group invariance , Can. J. Math. 43(1991) , 1213-1227．
[8] G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. 95 (1972) , 261-280．
[9] G. Gasper and W. Trebels, Multiplier criteria of Hormander type for Jacobi expansions, Studia Math. T. LXVIII(1980) , 187-197．
[10] G. Gasper , Positivity and the convolution structure for Jacobi series , Ann. of Math. 93 (1971) , 112-118．
[11] Zh.-K. Li, Conjugate Jacobi series and conjugate functions, J. Approx. Theory 86(1996) , 179-196．
[12] Zh.-K. Li, Hardy spaces for Jacobi expansions, Analysis 16． 27-49(1996) .
[13] Zh.-K. Li and Y. Xu. Summability of orthogonal expansions on spheres, balls and simplices, Trans. Amer. Soc. , to appear.
[14] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118(1965) , 17-92．
[15] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J. , 1971．
[16] E. M. Stein, Singular integrals and differentiability properties of functions , Princeton University Press, Princeton, N. J. , 1970．
[17] E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Studies, no. 63, Princeton Univ. Press, Princeton, N. J. ; Univ. of Tokyo Press, Tokyo, 1970．
[18] R. S. Strichartz, Multipliers for spherical harmonic expansions, Trans. Amer. Math. Soc 167(1972) , 115-124．
[19] G. Szego, Orthogonal polynomials. 4th ed. , Amer. Math. Soc. , Colloq. Publ. 23, Providence, Rhode Island, 1975．
[20] Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125(1997) , 2963-2973．
[21] Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. Vol. 49(1) (1997) , 175-192．