$\ell^{p}$ improving estimates for multilinear forms motivated by distance graphs

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, $\Lambda_{G}$, 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 $G$.