$\ell^{p}$ improving estimates for multilinear forms motivated by distance graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918497885356032 |
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| author | Palsson, Eyvindur Smucker, Jennifer |
| author_facet | Palsson, Eyvindur Smucker, Jennifer |
| contents | We undertake a systematic study of the mapping properties of forms based on distance graphs in $\mathbb{Z}^{d}$ to see how the structure of a graph, $G$, affects the $\ell^{p}$ improving estimates of the form, $Λ_{G}$, based on $G$. This extends previous work on $\ell^{p}$ improving properties for the spherical averaging operator, which corresponds to a distance graph of a single distance. We obtain $\ell^{p}$ improving estimates for the collection of forms based on all graphs with 2, 3, and 4 vertices, as well as chains and simplexes of any size in $\mathbb{Z}^{d}$. Surprisingly, certain mapping properties only seem to depend on the number of vertices in the graph, not its structure, and forms based on subgraphs of a graph, $G$, do not necessarily inherit all mapping properties from $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12439 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\ell^{p}$ improving estimates for multilinear forms motivated by distance graphs Palsson, Eyvindur Smucker, Jennifer Classical Analysis and ODEs Number Theory We undertake a systematic study of the mapping properties of forms based on distance graphs in $\mathbb{Z}^{d}$ to see how the structure of a graph, $G$, affects the $\ell^{p}$ improving estimates of the form, $Λ_{G}$, based on $G$. This extends previous work on $\ell^{p}$ improving properties for the spherical averaging operator, which corresponds to a distance graph of a single distance. We obtain $\ell^{p}$ improving estimates for the collection of forms based on all graphs with 2, 3, and 4 vertices, as well as chains and simplexes of any size in $\mathbb{Z}^{d}$. Surprisingly, certain mapping properties only seem to depend on the number of vertices in the graph, not its structure, and forms based on subgraphs of a graph, $G$, do not necessarily inherit all mapping properties from $G$. |
| title | $\ell^{p}$ improving estimates for multilinear forms motivated by distance graphs |
| topic | Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2605.12439 |