Functional Inequalities and Spectrum Estimates: The Infinite Measure Case

✍ Scribed by Feng-Yu Wang

• Book ID: 102588977
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 218 KB
• Volume: 194
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2002.3968

✦ Synopsis

As a continuation of [13] where a Poincare´-type inequality was introduced to study the essential spectrum on the L 2 -space of a probability measure, this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite. High-order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality. Criteria of the inequality and estimates of the inequality constants are presented. Finally, some concrete examples are considered to illustrate the main results. In particular, estimates of high-order eigenvalues obtained in this paper are sharp as checked by two examples on the Euclidean space.