臺灣博碩士論文加值系統

在這篇論文，我們證明了weighted atom 有關於Hermite expansions 的不等式。但我們想要把這個不等式套用在一維的加權哈弟空間上，也一樣能得到相同形式的不等式。

In this paper, we prove a weighted atom inequality for Hermite expansions and want to obtain similar type of inequalities for 1-dimensional Hermite expansions of weighted Hardy''s inequalities.

1 Introduction 2
2 Preliminaries 4
2.1 The atomic characterization of Hp(Rn)
2.2 The atomic characterization of Hpw(Rn)
2.3 Hermiteexpansions
3 The main result 15

[1] P. L. Duren, Theory of Hp Spaces, Academic Press, N. Y., (1971).
[2] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, (1959).
[3] L. Colzani and G. Travaglini, Hardy-Lorentz spaces and expansions in eigenfunctions of the Laplace-Beltrami operator on compact manifolds, Colloq. Math. 58 (1990), 305-316. MR 92c:58142.
[4] Y. Kanjin, Hardy’s inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc. 29 (1997), 331-337. MR 98f:42024
[5] M. Statake, Hardy’s inequalities for Laguerre expansions, J. Math. Soc. Japan 52 (2000), 17-24. Mr 2001c:42014
[6] S. THANGAVELU, Lectures on Hermite and Laguerre expansions, Mathematical Notes, 42, Princetion Univ.Press, (1993).
[7] R. RADHA, Hardy type inequalities, Taiwanese J. Math., 4 (2000), 447-456.
[8] R. RADHA and S. THANGAVELU, Hardy’s inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc., 132 (12) (2004) 3525-3536.
[9] C. Fe erman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193.
[10] R.R.Coifman, A real variable characterization of Hp, Studia Math. 51 (1974), 269-274.
[11] R.H.Latter, A characterization of Hp(Rn) in terms of atoms, Studia Math. 62 (1978), 93-101.
[12] J. Garcia-Cuerva, Weighted Hp spaces, Dissertations Math. 162 (1979), 1-63.
[13] J.-O. Str¨omberg and A. Torchinsky, ”Weighted Hardy Spaces,” Lecture Notes in Mathematics, Vol. 1381, Spring-Verlag, Berlin/New York, (1989).
[14] B. Muckenhoupt, Weighted norm inequalities for the Hardy mximal funcions, Trans. Amer. Math. Soc. 165 (1972), 207-226.
[15] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251.
[16] R. R. Coifman and C. Fe erman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.
[17] E. M. Stein, ”Singular Integrals and Di erentiability Properties of Functions,”
Princeton Univ. Press, Princeton, NJ, (1970).
[18] S. THANGAVELU, Lectures on Hermite and Laguerre expansions, Mathematical Notes, 42, Princeton Univ. Press, (1993).
[19] R. BALASUBRAMANIAN AND R. RADHA, Hardy-type inequalities for Hermite expansions, JIPAM 6(1), Article 12, (2005).
[20] GUO BEN-YU, Error estimation of Hermite spectral method for nonlinear partial di erential equations, Math. comp.,V.68, 227 (1999), 1067-1078.
[21] Ming-Yi Lee and Chin-Cheng Lin, The molecular characterization of weighted Hardy spaces, J. Funct. Anal. 188 (2002), 442-460.