Dimension results for mappings of jet space Carnot groups

TL;DR

This paper introduces new projection-like mappings in jet space Carnot groups, proves Marstrand-type theorems for them, and determines the Hausdorff dimensions of their images, advancing geometric measure theory in these groups.

Contribution

It develops analogues of projections in jet space Carnot groups and establishes dimension results, extending classical geometric measure theory to these structures.

Findings

Proved Marstrand-type theorems for the new mappings.

Determined possible Hausdorff dimensions of images.

Established a framework for projections in jet space Carnot groups.

Abstract

We propose analogues of horizontal and vertical projections for model filiform jet space Carnot groups. Every pair consisting of the jet of a smooth function on $\mathbb{R}$ and a vertical hyperplane with first coordinate fixed provides a splitting of a model filiform group, which induces mappings of the group. We prove Marstrand-type theorems for these mappings and determine the possible Hausdorff dimensions of images of sets under these mappings. Analogues of projections for general jet space Carnot groups could be defined similarly.