Functional inequalities for empty essential spectrum

In terms of the equivalence of Poincare inequality and the existence of spe ctral gap, the super-Poincare inequality is suggested in the paper fur the study of essential spectrum. It is proved for symmetric diffusions that, su ch an inequality is equivalent to empty essential spectrum of the correspon ding diffusion operator. This inequality recovers known Sobolev and Nash ty pe ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also prov ed for a more general setting including symmetric jump processes. Moreover. estimates of inequality constants are also presented, which lead to a proo f of a result on ultracontractivity suggested recently by D. Stroock. Final ly, concentration of reference measures for super-Poincare inequalities is studied. the resulting estimates extend previous ones for Poincare and log- Sobolev inequalities. (C) 2000 Academic Press.