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

TL;DR

This paper systematically studies how the structure of distance graphs in integer lattices influences the - improving estimates of multilinear forms, revealing that some properties depend only on vertex count.

Contribution

It extends previous work by providing - estimates for forms based on various small graphs and chains, highlighting structure-independent properties.

Findings

Certain mapping properties depend only on the number of vertices, not the graph structure.

Forms based on subgraphs do not necessarily inherit all properties from the original graph.

Established - estimates for all graphs with 2, 3, and 4 vertices, as well as chains and simplexes.

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