Distributions in spaces with thick submanifolds

We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on $\mathbb{R}^n$ whose domain contains a smooth submanifold on which the test functions may be singular. We define several operations, including ``thick partial derivatives'', and clarify their connection with their classical counterparts in Schwartz distribution theory. We also introduce and study a number of special thick distributions, including new thick delta functions, or more generally thick multilayer distributions along a submanifold.