It is shown that the only pseudoconvex sets with smooth boundary in $$ on which $$ satisfies Lipschitz smoothing estimates of order $$ are the strongly pseudoconvex ones. Various extensions of this result are made to weakly pseudoconvex domains of finite type and in various norms. It is proved that subelliptic estimates for $$ can hold on a pseudoconvex set in $$ only if the domain is of finite type in the sense of Kohn.

Let $$ be an $$-space of the homotopy type of a connected finite CW complex. Suppose the generators of the rational cohomology of $$ all have dimension $$. Theorem. If $$ is a prime satisfying $$, then $$ is mod $$ equivalent to a product of odd dimensional spheres and generalized Lens spaces $$ obtained as the orbit space of an action of $$ on $$.