Littlewood-Paley-Stein inequality for a symmetric diffusion

Ichiro SHIGEKAWA; Nobuo YOSHIDA

doi:10.2969/jmsj/04420251

Ichiro SHIGEKAWA, Nobuo YOSHIDA

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1992 Volume 44 Issue 2 Pages 251-280

DOI https://doi.org/10.2969/jmsj/04420251

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Published: 1992 Received: November 07, 1990 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces-the construction of the associated diffusion process, Probab. Th. Rel. Fields, 83 (1989), 405-434.
2) T. Aubin, Espace de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 100 (1976), 149-173.
3) D. Bakry and M. Emery, Diffusions hypercontractives, Séminaire de Prob. XIX, Lecture Notes in Math., 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
4) D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Séminaire de Prob. XIX, Lecture Notes in Math., 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985, pp. 179-206,
5) D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète, Comptes Rendes Acad. Sc. Paris, 303, Série I (1986), 23-26.
6) D. Bakry, Etude des transformations de Riesz dans les variétés riemaniennes à courbure de Ricci minorée, Séminaire de Prob. XXI, Lecture Notes in Math., 1247, Springer-Verlag, Berlin-Heidelberg-New York, 1987, pp. 137-172.
7) E. B. Davies, One-parameter semigroups, Academic Press, New York, 1980.
8) E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.
9) G.-M. de Rham, Differentiable manifolds, Springer-Verlag, 1984.
10) R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier theory, Springer-Verlag, 1977.
11) M. Fukushima, Dirichlet forms and Markov Processes, North Holland/Kodansha Amsterdam/Tokyo, 1980.
12) N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North Holland/Kodansha, Amsterdam/Tokyo, 1981.
13) S. Kobayashi and K. Nomizu, Foundations of differential geometry, I, Interscience Publishers, New York-London, 1963.
14) S. Kusuoka, Dirichlet forms and diffusion processes on Banach space, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 29 (1982), 79-95.
15) E. Lenglart, D. Lépingle and M. Pratelli, Présentation unifiée de certaines inégalités de théorie des martingales, Séminaire de Prob. XIV, Lecture Notes in Math., 784, Springer-Verlag, Berlin-Heidelberg-New York, 1980, pp. 26-48.
16) J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series (I), J. London Math. Soc., 6 (1931), 230-233.
17) N. Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemanienne à courbure non positive, J. Funct. Anal., 61 (1985), 164-201.
18) P. A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Séminaire de Prob. X, Lecture Notes in Math., 511, Springer-Verlag, Berlin-Heidelberg-New York, 1976, pp. 125-183.
19) P. A. Meyer, Retour sur la théorie de Littlewood-Paley, Séminaire de Prob. XV, Lecture Notes in Math., 850, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 151-166.
20) J. Potthoff, Littlewood-Paley theory on Gaussian spaces, Nagoya Math. J., 109 (1988), 47-61.
21) G. C. Rota, An “Alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc., 68 (1962), 95-102.
22) B. Schmuland, An alternative compactification for classical Dirichlet forms on topological vector spaces, preprint.
23) I. Shigekawa, Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator, preprint.
24) E. M. Stein, Singular integrals and differentiable properties of functions, Princeton University Press, Princeton, New Jersey, 1970.
25) E. M. Stein, Topics in harmonic analysis related to Littlewood-Paley theory, Annals of Math. Study, no. 63, Princeton, 1970.
26) R. S. Strichartz, Analysis of Laplacian on the complete Riemannian manifold, J. Funct. Anal., 52 (1983), 48-79.
27) N. T. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal., 38 (1980), 25-60.
28) S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Tata Institute of Fundamental Research, Springer-Verlag, Berlin-Heidelberg-New York, 1984.
29) K. Yosida, Functional analysis, 6th ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980.
30) N. Yoshida, On the equivalence of Sobolev spaces over the Riemannian manifold, preprint.

Right : [1] S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces-the construction of the associated diffusion process, Probab. Th. Rel. Fields, 83 (1989), 405-434.
[2] T. Aubin, Espace de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 100 (1976), 149-173.
[3] D. Bakry and M. Emery, Diffusions hypercontractives, Séminaire de Prob. XIX, Lecture Notes in Math., 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
[4] D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Séminaire de Prob. XIX, Lecture Notes in Math., 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985, pp. 179-206,
[5] D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète, Comptes Rendes Acad. Sc. Paris, 303, Série I (1986), 23-26.
[6] D. Bakry, Etude des transformations de Riesz dans les variétés riemaniennes à courbure de Ricci minorée, Séminaire de Prob. XXI, Lecture Notes in Math., 1247, Springer-Verlag, Berlin-Heidelberg-New York, 1987, pp. 137-172.
[7] E. B. Davies, One-parameter semigroups, Academic Press, New York, 1980.
[8] E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.
[9] G. -M. de Rham, Differentiable manifolds, Springer-Verlag, 1984.
[10] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier theory, Springer-Verlag, 1977.
[11] M. Fukushima, Dirichlet forms and Markov Processes, North Holland/Kodansha Amsterdam/Tokyo, 1980.
[12] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North Holland/Kodansha, Amsterdam/Tokyo, 1981.
[13] S. Kobayashi and K. Nomizu, Foundations of differential geometry, I, Interscience Publishers, New York-London, 1963.
[14] S. Kusuoka, Dirichlet forms and diffusion processes on Banach space, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 29 (1982), 79-95.
[15] E. Lenglart, D. Lépingle and M. Pratelli, Présentation unifiée de certaines inégalités de théorie des martingales, Séminaire de Prob. XIV, Lecture Notes in Math., 784, Springer-Verlag, Berlin-Heidelberg-New York, 1980, pp. 26-48.
[16] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series (I), J. London Math. Soc., 6 (1931), 230-233.
[17] N. Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemanienne à courbure non positive, J. Funct. Anal., 61 (1985), 164-201.
[18] P. A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Séminaire de Prob. X, Lecture Notes in Math., 511, Springer-Verlag, Berlin-Heidelberg-New York, 1976, pp. 125-183.
[19] P. A. Meyer, Retour sur la théorie de Littlewood-Paley, Séminaire de Prob. XV, Lecture Notes in Math., 850, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 151-166.
[20] J. Potthoff, Littlewood-Paley theory on Gaussian spaces, Nagoya Math. J., 109 (1988), 47-61.
[21] G. C. Rota, An “Alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc., 68 (1962), 95-102.
[22] B. Schmuland, An alternative compactification for classical Dirichlet forms on topological vector spaces, preprint.
[23] I. Shigekawa, Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator, preprint.
[24] E. M. Stein, Singular integrals and differentiable properties of functions, Princeton University Press, Princeton, New Jersey, 1970.
[25] E. M. Stein, Topics in harmonic analysis related to Littlewood-Paley theory, Annals of Math. Study, no. 63, Princeton, 1970.
[26] R. S. Strichartz, Analysis of Laplacian on the complete Riemannian manifold, J. Funct. Anal., 52 (1983), 48-79.
[27] N. T. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal., 38 (1980), 25-60.
[28] S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Tata Institute of Fundamental Research, Springer-Verlag, Berlin-Heidelberg-New York, 1984.
[29] K. Yosida, Functional analysis, 6th ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980.
[30] N. Yoshida, On the equivalence of Sobolev spaces over the Riemannian manifold, preprint.

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