Distributions in spaces with thick submanifolds

TL;DR

This paper develops a new theory of distributions in Euclidean space that incorporates thick submanifolds, allowing test functions to be singular along these submanifolds and introducing new types of thick delta functions.

Contribution

It introduces a comprehensive framework for thick distributions on ^n with singularities along submanifolds, including new operations and special distributions.

Findings

Defined thick partial derivatives and connected them to classical derivatives.

Constructed new thick delta functions and multilayer distributions.

Established the theoretical foundation for distributions with singularities along submanifolds.

Abstract

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.