We study prolongation of the conformal embedding equations. Let $(\mathscr{M}, g)$ be a $C^{\infty}$ Riemannian manifold of dimension $n \geq 3$ and $(\tilde{\mathscr{M}}, \tilde{g})$ be a $C^{\infty}$ Riemannian manifold of dimension $n + d$, $d <\frac{1}{2}n(n-1)$. Suppose that $f : \mathscr{M} \to \tilde{\mathscr{M}}$ is a conformal immersion with conformal factor $v$. If the conformal 1-nullity off at a point $P \in \mathscr{M}$ does not exceed $n - 2$, we prove that the system of conformal embedding equations admits a prolongation to a system of nonlinear partial differential equations of second order which is elliptic at the solution $(f, v)$. In particular, if $(\mathscr{M}, g)$ and $(\tilde{\mathscr{M}}, \tilde{g})$ are analytic and $f$ and $v$ are of differentiability class $C^{2}$ then $f$ and $v$ are analytic on a neighborhood of $P$ in $\mathscr{M}$.

In this paper we derive a formula about the action of $\mathscr{P}^{1}$ on the odd generators of the mod 3 cohomology of a finite simply connected mod 3 $H$-space with associative mod 3 homology. This formula will be used in a subsequent paper to classify all possible cohomology rings that can occur as the mod 3 cohomology of such $H$-spaces.

Let $\mu$ be a probability measure on a Riemannian manifold. It is known that if the semigroup $e^{-t\nabla *\nabla}$ is hypercontractive, then any function $g$ for which $\|\nabla g\|_{\infty}\leq 1$ will satisfy a Herbst inequality, $\int \exp (\alpha g^{2})d\mu < \infty$, for small $\alpha > 0$. If the semigroup is supercontractive, then the above inequality will hold for all $\alpha > 0$. For any $\alpha > 0$ for which $Z = \int \exp(\alpha g^{2})d\mu < \infty$, we define a measure $\mu _{g}$ by $d\mu _{g}=Z^{-1}\exp (\alpha g^{2})d\mu$. We show that if $\mu$ is hyper- or supercontractive, then so is $\mu _{g}$. Moreover, under standard conditions on logarithmic Sobolev inequalities which yield ultracontractivity of the semigroup, Gross and Rothaus have shown that $Z = \int \exp (\alpha g^{2}|\log |g||^{c})d\mu < \infty$ for some constants $\alpha ,c$. We in addition show that the perturbed measure $d\mu _{g} = Z^{-1} \exp (\alpha g^{2}|\log |g||^{c})d\mu$ is ultracontractive.