Analysis on disconnected sets

Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in Rn and that E is a closed set contained in the boundary of U such that for every x ∈ U and r > 0 there is a connected component of U contained in the ball with center x and radius r. For instance, E might be the boundary of U . As a uniform version of the condition, one might ask that there be a constant C > 0 such that the aforementioned connected component of U contains a ball of radius C r. The connected components of U might be quite regular, even if there are infinitely many of them. As a basic example, E could be a Cantor set in the real line, and U could be the complement of E or the union of the bounded complementary components of E. This does not work in higher dimensions, where the complement of a Cantor set is connected. Sierpinski gaskets and carpets in the plane are very interesting cases where E is connected. One could also consider more abstract versions of this, in metric spaces, for instance. The open set U could be discrete, and the limiting set E could be arbitrary. Let us restrict our attention to the case where the closure of U is compact and E = ∂U . Consider Bergman spaces of locally constant functions on U , which can be identified with l spaces. Toeplitz operators associated to continuous functions on U can be defined in the usual way, by multiplying and projecting. The Toeplitz operators associated to functions vanishing on the boundary are compact. Thus, modulo compact operators, it is really the functions on the boundary that are important. Invertible functions on the boundary correspond to Fredholm operators, but their indices are automatically equal to 0. Of course, these are well-known themes.