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

Abstract

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, $\hat{I}{›}_G$, 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 $G$.

41 pages, added a section on the normalization factor