Solutions of d and d-bar with small support.

Abstract

Two related problems are studied in this thesis. We refer to them as the real and the complex case. In the real case, we are interested in hulls K of compact sets K in ${\bf R}\sp{n}.$ A hull K is a minimal compact set in ${\bf R}\sp{n}$ that contains K and has the property that the equation $df = \phi$ has a solution f whose support is arbitrarily close to K, for any given q-form $\phi$ on ${\bf R}\sp{n}$ whose support is contained in K. In the complex case, the analogous problem for the equation $\overline{\partial}f = \phi$ on $\doubc\sp{n}$ is studied (here $\phi$ is a (0,q)-form). In both cases, we prove that hulls exist but are not unique, unless K = K or $q = 1.$ In the real case, we use de Rham theory to characterize hulls K as the minimal compact sets containing K and satisfying the condition $i\sb{\*} = 0,$ where $i\sb{\*}:\tilde H\sb{n-q}(K)\longrightarrow\tilde H\sb{n-q} (\ K)$ is the natural map on the real homology groups induced by the inclusion $i:K\rightarrow\ K.$ As a consequence, we observe that K = K precisely when $\tilde H\sb{n-q}(K) = 0.$ Next, we turn to the study of polyhedral hulls. We prove that they always exist and are obtained by adding to K a finite number of simplicial $(n - q + 1)$-chains that lie in the complement of K and have their boundary in K. Finally, we apply these results to prove a theorem about the topology of hyperplane sections of K in ${\bf R}\sp{n}.$ The study of the complex case is complicated by the fact that geometric and analytic dualities for the Dolbeault cohomology hold only under certain conditions. We use Andreotti-Grauert theory to find some conditions that imply K = K and, in case K is polyhedral, prove some estimates on the dimension of $\ K\\ K$. In particular, we show that there exist polyhedral hulls that satisfy dim($\ K\\ K)\le 2n - q + 1$ and that arbitrary hulls can be approximated by such polyhedral hulls. Next, fibered hulls are considered. They turn out to be related to polynomial hulls and provide some interesting examples. As an application of the theory developed, we prove that $\overline{\partial}$-cohomology classes of open subsets of $\doubc\sp{n}$ can be represented by (0,q)-forms supported arbitrarily close to a closed set of Hausdorff dimension at most $2n - q.$