Sobolev–Orlicz Imbeddings, Weak Compactness, and Spectrum

✍ Scribed by Fabio Cipriani

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 183 KB
• Volume: 177
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3633

✦ Synopsis

The aim of this wok is to show how the weak compactness in the L 1 (X, m) space may be used to relate the existence of a Sobolev Orlicz imbedding to the L 2 (X, m)spectral properties of an operator H. In the first part we show that a Sobolev Orlicz imbedding implies that the bottom of the L 2 -spectrum of H is an eigenvalue (i.e. the existence of the ground state) with finite multiplicity, provided m is finite. In the second part we prove that for a large class of operators, namely those for which Persson's characterization of the bottom of the essential spectrum holds true, a Sobolev Orlicz imbedding always implies the discreteness of the L 2 -spectrum of H, provided m is finite. In the third part we show a certain converse of this last result in the sense that the discreteness of the L 2 -spectrum of H always implies the existence of an Orlicz space for which a Sobolev Orlicz imbedding holds true for H. The case of logarithmic Sobolev inequalities is considered and provides the original motivations for this research.