Functional Calculus for the Ornstein–Uhlenbeck Operator

✍ Scribed by José Garcı́a-Cuerva; Giancarlo Mauceri; Stefano Meda; Peter Sjögren; José Luis Torrea

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 333 KB
• Volume: 183
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3757

✦ Synopsis

Let # be the Gauss measure on R d and L the Ornstein Uhlenbeck operator, which is self adjoint in L 2 (#). For every p in (1, ), p{2, set , p *=arc sin |2Âp&1| and consider the sector

The main result of this paper is that if M is a bounded holomorphic function on S ,* p whose boundary values on S ,* p satisfy suitable Ho rmander type conditions, then the spectral operator M(L) extends to a bounded operator on L p (#) and hence on L q (#) for all q such that |1Âq&1Â2| |1Âp&1Â2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than S ,* p .

2001