Some Remarks on the Riesz and reverse Riesz transforms on Broken Line

TL;DR

This work analyzes the Riesz transform on a broken-line model with ends carrying different homogeneous dimensions, proving that $L^p$-boundedness of $\nabla\Delta^{-1/2}$ depends only on the smaller end-dimension $d_*$. The authors develop explicit kernel decompositions into $kk$ and $kl$ parts, derive precise resolvent asymptotics, and establish Hardy–Hilbert type inequalities to control the kernels, yielding a Lorentz endpoint estimate and a complete $L^p$ range with a sharp endpoint. They also prove a reverse Riesz inequality in this non-doubling setting using harmonic annihilation, Hardy-type estimates, and a resolvent-based bilinear form, covering all dimensional configurations. The results extend non-doubling harmonic analysis for Riesz-type operators and illuminate how end-geometry governs transform boundedness and reverse inequalities without relying on doubling or Poincaré conditions.

Abstract

In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by $(-\infty, -1] \cup [1,\infty)$ equipped with the measure $dμ= |r|^{d_{1}-1}dr$ for $r \le -1$ and $dμ= r^{d_{2}-1}dr$ for $r\ge 1$, where $d_{1}, d_{2} >1$. For the Riesz transform, we show that the range of its $L^{p}$ boundedness depends solely on the smaller dimension, $d_{1} \wedge d_{2}$. Furthermore, we establish a Lorentz type estimate at the endpoint. In our subsequent investigation, we consider the reverse Riesz inequality by rigorously verifying the $L^{p}$ lower bounds for the Riesz transform for almost every $p\in (1,\infty)$. Notably, unlike most previous studies, we do not assume the doubling condition or the Poincaré inequality. Our approach is based on careful estimates of the Riesz kernel and a method known as harmonic annihilation.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof