Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finner-like inequalities in the Heisenberg group

Kaiyi Huang

Abstract

We completely characterize the range of $L^p$-boundedness of certain multilinear Radon-like transforms involving vertical projections in the Heisenberg group.

Finner-like inequalities in the Heisenberg group

Abstract

We completely characterize the range of -boundedness of certain multilinear Radon-like transforms involving vertical projections in the Heisenberg group.
Paper Structure (23 sections, 42 theorems, 284 equations, 4 figures, 4 tables)

This paper contains 23 sections, 42 theorems, 284 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Prior results and innovation
  3. A historical overview
  4. Innovation of the paper
  5. Proof overview
  6. Examples
  7. Proof sketch for Example \ref{['Ex:isoperimetric']}
  8. Proof sketch for Example \ref{['Ex:aniso']}
  9. Reduce Theorem \ref{['T:main']} to a base case
  10. Auxiliary maps with nested kernels
  11. Finner's inequality and generalization thereof
  12. Refining Omega
  13. An inflation map
  14. Approximation by ellipsoids
  15. Semiregularity, regularity, and quasiextremality
  16. ...and 8 more sections

Key Result

Theorem 1.1

For $\pi_j$ of the form (E:pi_j vp), there exists $0<C<\infty$ depending only on $n,\pi_j,p_j$ such that holds uniformly over continuous functions $f_1,\dots,f_M$ with compact support if and only if $p_j\in[1,\infty]$ for all $1\leq j\leq M$, and for arbitrary coordinate subspaces $V$ of $\mathbb{R}^n$.

Figures (4)

  • Figure 1:
  • Figure 3: These are the forms of $s^{j,l}$ and $s^{j+1,l}$ when $\tau=()$. Each $\xi^{j,l}$ is distinctly colored. $\Xi_{j+1}$ includes $\xi^{r,l}$ for $r\leq j$ or $r=j+1,l=1$.
  • Figure 4:
  • Figure 5:

Theorems & Definitions (99)

  • Theorem 1.1
  • Remark
  • Theorem 3.1
  • Definition 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Definition 4.1
  • Definition 4.2
  • ...and 89 more