Functional Inequalities for Empty Essential Spectrum

In terms of the equivalence of Poincaré inequality and the existence of spectral gap, the super-Poincaré inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for super-Poincaré inequalities is studied, the resulting estimates extend previous ones for Poincaré and log-Sobolev inequalities.