Sharp Inequalities for Functional Integrals and Traces of Conformally Invariant Operators

The intertwining operators Ad = Ad(g) on the round sphere (Sn, g) are the conformal analogues of the power Laplacians 1d/2 on the flat Rn . To each metric ρg, conformally equivalent to g, we can naturally associate an operator Ad(ρg), which is compact, elliptic, pseudodifferential of order d, and which has eigenvalues λ j (ρ); the special case d = 2 gives precisely the conformal Laplacian in the metric ρg. In this paper we derive sharp inequalities for a class of trace functionals associated to such operators, including their zeta function ∑ j λ j (ρ) −s , and its regularization between the first two poles. These inequalities are expressed analytically as sharp, conformally invariant Sobolev-type (or log Sobolev type) inequalities that involve either multilinear integrals or functional integrals with respect to d-symmetric stable processes. New strict rearrangement inequalities are derived for a general class of path integrals.