Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

• Published in: Potential analysis Vol. 33; no. 1; pp. 85 - 105
• Main Authors: Carbonaro, Andrea, Mauceri, Giancarlo, Meda, Stefano
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2010
• Subjects: Functional Analysis; Geometry; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; Gaussian measure; Homogeneous tree; 42B30; 58C99; 42B20; Hardy spaces; Singular integrals; Imaginary powers; Riemannian manifold; Ornstein–Uhlenbeck operator
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-009-9160-6

Denote by γ the Gauss measure on ℝ n and by the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space of Goldberg type and show that for each u in ℝ ∖ {0} and r  > 0 the operator is unbounded from to L 1 γ. This result is in sharp contrast both with the fact that is bounded from H 1 γ to L 1 γ, where H 1 γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007 ), and with the fact that in the Euclidean case is bounded from the Goldberg space to L 1 ℝ n . We consider also the case of Riemannian manifolds M with Riemannian measure μ . We prove that, under certain geometric assumptions on M , an operator , bounded on L 2 μ , and with a kernel satisfying certain analytic assumptions, is bounded from H 1 μ to L 1 μ if and only if it is bounded from to L 1 μ . Here H 1 μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009 ), and is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009 ). The case of translation invariant operators on homogeneous trees is also considered.