Let G be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure gamma on R-d. We prove a sharp estimate of the operator norm of the imaginary powers of L on L-P(gamma), 1 < p < infinity. Then we use this estimate to prove that if b is in [0, infinity] and M is a bounded holomorphic function in the sector {z is an element of C : arg(z - b) < arcsin /p - 1} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on L-P(gamma). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea
Mauceri, G., Meda, S., Sjogren, P. (2004). Sharp estimates for the Ornstein-Uhlenbeck operator. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 3(3), 447-480 [10.2422/2036-2145.2004.3.01].
Sharp estimates for the Ornstein-Uhlenbeck operator
MEDA, STEFANO;
2004
Abstract
Let G be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure gamma on R-d. We prove a sharp estimate of the operator norm of the imaginary powers of L on L-P(gamma), 1 < p < infinity. Then we use this estimate to prove that if b is in [0, infinity] and M is a bounded holomorphic function in the sector {z is an element of C : arg(z - b) < arcsin /p - 1} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on L-P(gamma). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea
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