On Energy Inequalities

✍ Scribed by Vesna Perišić

• Book ID: 102597619
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 118 KB
• Volume: 244
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.6711

✦ Synopsis

We consider the Schrodinger operator H s y q q in a bounded domain

Ž . in R , d G 3, with q in the Kato class K K D and finite gauge g x s w Ž . x Ž . E exp H q X ds , where X is a Brownian motion and is the first exit time of

the Brownian motion X from the domain D. Let K and G denote the Green t ⌬ operators in D for H and , respectively. We prove that there is a positive

where f is a real, measurable function for which G f , f is finite. As a direct consequence of this double inequality, we have that the potential Gf is in the 1 Ž . 1 Ž . Sobolev space H D if and only if Kf g H D .