Sobolev–Orlicz Imbeddings, Weak Compactness, and Spectrum

The aim of this wok is to show how the weak compactness in the L1(X, m) space may be used to relate the existence of a Sobolev–Orlicz imbedding to the L2(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 L2-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 Perssonʹs characterization of the bottom of the essential spectrum holds true, a Sobolev–Orlicz imbedding always implies the discreteness of the L2-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 L2-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.