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Littlewood-Paley theory on Gaussian spaces

Published online by Cambridge University Press:  22 January 2016

Jürgen Potthoff
Jürgen Potthoff*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya 464, Japan, and Department of Mathematics, MA7-1, Technical University Berlin, Strasse d. 17. Juni 135, D-1000, Berlin, 12
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In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).

In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 109 , March 1988 , pp. 47 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Burkholder, D. L., A geometrical characterization of Banach spaces in which martingale differences are unconditional, Ann. Probab., 9 (1981), 997–1011.CrossRefGoogle Scholar
[1] Cook, J., The mathematics of second quantization, Trans. Amer. Math. Soc, 74 (1953), 222–245.Google Scholar
[1] Gel’fand, I. M., Vilenkin, N. Y., Generalized functions, vol. 4. New York, London: Academic Press 1964.Google Scholar
[1] Gröbner, W., Hofreiter, N., Integraltafel, Zweiter Teil. Wien und Innsbruck : Springer-Verlag 1958.Google Scholar
[1] Hida, T., Brownian motion, Berlin, Heidelberg, New York: Springer 1980.Google Scholar
[1] Meyer, P. A., Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Séminaire de probabilités X, ed. Azéma, J., Yor, M. Berlin, Heidelberg, New York: Springer 1976.Google Scholar
[1] Meyer, P. A., Quelques resultats analytiques sur le semigroupe d’Ornstein-Uhlenbeck en dimension infinie, In: Theory and application of random fields (ed. by Kallianpur, G.), Berlin, Heidelberg, New York: Springer 1983.Google Scholar
[1] Nelson, E., The free Markoff field, J. Funct. Anal., 12 (1973), 211–227.Google Scholar
[1] Nelson, E., Probability theory and Euclidean quantum field theory, In Constructive quantum field theory (ed. by Velo, G. and Wightman, A.), Berlin, Heidelberg, New York: Springer 1973.Google Scholar
[10] Reed, M., Simon, B., Methods in mathematical physics I, New York, London: Academic Press 1975.Google Scholar
[11] Simon, B., The P(ø)2 Euclidean (quantum) field theory, Princeton: Princeton University Press 1970.Google Scholar
[12] Stein, E. M., Singular integrals and differentiability properties of functions, Princeton: Princeton University Press 1970.Google Scholar
[13] Stein, E. M., Topics in harmonic analysis, Princeton: Princeton University Press 1970.Google Scholar