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Sharp estimates for the Ornstein-Uhlenbeck operator. (English) Zbl 1116.47036
The purpose of this paper is to obtain a sharp functional calculus for the Ornstein-Uhlenbeck operator \(T\) acting on the \(L^p\) spaces with respect to the Gaussian measure \(\gamma\) on \(\mathbb R^d\). The authors prove a sharp estimate of the operator norm of the imaginary powers of \(T\) on \(L^p(\gamma), \quad 1<p<+\infty\). This result improves earlier results of the authors with J. Garcia-Cuerva and J. L. Torrea.

MSC:
47D06 One-parameter semigroups and linear evolution equations
47A60 Functional calculus for linear operators
47N30 Applications of operator theory in probability theory and statistics
60G15 Gaussian processes
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References:
[1] M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267-283. Zbl0528.42006 MR690846 · Zbl 0528.42006
[2] M. Cowling - I. Doust - A. McIntosh - A. Yagi, Banach space operators with a bounded \(H^\infty \) functional calculus, J. Aust. Math. Soc. 60 (1996), 51-89. Zbl0853.47010 MR1364554 · Zbl 0853.47010
[3] M. Cowling - S. Meda, Harmonic analysis and ultracontractivity, Trans. Amer. Math. Soc. 340 (1993), 733-752. Zbl0798.47032 MR1127154 · Zbl 0798.47032
[4] E. B. Davies, “Heat Kernels and Spectral Theory”, Cambridge Tract. in Math. 92, Cambridge University Press, Cambridge, 1989. Zbl0699.35006 MR990239 · Zbl 0699.35006
[5] J. B. Epperson, The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel, J. Funct. Anal. 87 (1989), 1-30. Zbl0696.47028 MR1025881 · Zbl 0696.47028
[6] J. Garcia-Cuerva - G. Mauceri - P. Sjögren - J.L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999), 281-305. Zbl0939.42007 MR1714425 · Zbl 0939.42007
[7] J. García-Cuerva - G. Mauceri - S. Meda - P. Sjögren - J. L. Torrea, Functional Calculus for the Ornstein-Uhlenbeck Operator, J. Funct. Anal. 183 (2001), 413-450. Zbl0995.47010 MR1844213 · Zbl 0995.47010
[8] W. Hebisch - G. Mauceri - S. Meda, Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator, J. Funct. Anal. 210 (2004), 101-124. Zbl1069.47017 MR2052115 · Zbl 1069.47017
[9] L. Hörmander, Estimates for translation invariant operators in \(L^p\) spaces, Acta Math. 104 (1960), 93-140. Zbl0093.11402 MR121655 · Zbl 0093.11402
[10] L. Hörmander, “The Analysis of Linear Partial Differential Operators”, Vol. 1 Springer Verlag, Berlin, 1983.
[11] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110 (1990), 639-647. Zbl0760.42007 MR1028046 · Zbl 0760.42007
[12] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973), 211-227. Zbl0273.60079 MR343816 · Zbl 0273.60079
[13] E. M. Stein, “Topics in Harmonic Analysis Related to the Littlewood-Paley Theory”, Annals of Math. Studies, No. 63, Princeton N. J., 1970. Zbl0193.10502 MR252961 · Zbl 0193.10502
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