It is shown that the only pseudoconvex sets with smooth boundary in $\mathbf{C}^{n}$ on which $\bar{\partial}$ satisfies Lipschitz smoothing estimates of order $1/2$ 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 $\bar{\partial}$ can hold on a pseudoconvex set in $\mathbf{C}^{n}$ only if the domain is of finite type in the sense of Kohn.

Let $X$ be an $H$-space of the homotopy type of a connected finite CW complex. Suppose the generators of the rational cohomology of $X$ all have dimension $\leq m$. Theorem. If $p$ is a prime satisfying $2p-1\geq m$, then $X$ is mod $p$ equivalent to a product of odd dimensional spheres and generalized Lens spaces $L(p,1,\ldots,1)$ obtained as the orbit space of an action of $Z_{\mathrm{p}}$ on $S^{2\mathrm{p}-1}$.