On the continuity of the product of distributions in local Sobolev spaces

TL;DR

This paper extends the product operation of smooth functions to local Sobolev distributions with controlled wave front sets, establishing conditions for continuity in a new locally convex topology.

Contribution

It introduces a topology on local Sobolev distributions and proves the continuous extension of the product under specific conditions on wave front sets.

Findings

The product of distributions extends continuously in the new topology.

Conditions on wave front sets ensure the product's well-definedness.

The approach uses H"ormander's pullback technique on tensor products.

Abstract

We consider the space $\mathscr{H}_L ^{s,r} (O)$ consisting of all local Sobolev distributions of order $s$ on an open set $O$ whose Sobolev wave front set of order $r$ is contained in the closed conic set $L\subseteq O\times(\mathbb{R}^m\backslash\{0\})$. We introduce a locally convex topology on $\mathscr{H}_L ^{s,r} (O)$ and show that the ordinary product of smooth functions uniquely extends to a continuous bilinear mapping $\mathscr{H}_{L_1} ^{r_1,r'} (O) \times \mathscr{H}_{L_2} ^{r_2,r''} (O) \to \mathscr{H}_{L} ^{s,r} (O)$, for appropriate $s$ and $r$ when $L_1$ and $L_2$ are in a favorable position. The key ingredient in our proof is to employ H\"ormander's idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ of the tensor product of two distributions.