On the continuity of the product of distributions in local Sobolev spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917174087516160 |
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| author | Tutić, Stefan |
| author_facet | Tutić, Stefan |
| contents | 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 Hörmander's idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ of the tensor product of two distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the continuity of the product of distributions in local Sobolev spaces Tutić, Stefan Functional Analysis 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 Hörmander's idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ of the tensor product of two distributions. |
| title | On the continuity of the product of distributions in local Sobolev spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2512.23521 |