The intertwining operators $A\sb d=A\sb d(g)$ on the round sphere
$(S\sp n,g)$ are the conformal analogues of the power Laplacians
$\Delta\sp {d/2}$ on the flat $\mathbf {R}\sp n$. To each metric $\rho
g$, conformally equivalent to $g$, we can naturally associate an
operator $A\sb d(\rho g)$, which is compact, elliptic,
pseudodifferential of order $d$, and which has eigenvalues $\lambda\sb
j(\rho)$; the special case $d=2$ gives precisely the conformal
Laplacian in the metric $\rho g$. In this paper we derive sharp
inequalities for a class of trace functionals associated to such
operators, including their zeta function $\sum\sp j\lambda\sp
j(\rho)\sp {-s}$, 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.