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

In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by (-∞, -1] ∪ [1,∞) equipped with the measure dμ = |r|^d₁-1dr for r ≤ -1 and dμ = r^d₂-1dr for r≥ 1, where d₁, d₂ >1. For the Riesz transform, we show that the range of its Lᵖ boundedness depends solely on the smaller dimension, d₁ d₂. Furthermore, we establish a Lorentz type estimate at the endpoint. In our subsequent investigation, we consider the reverse Riesz inequality by rigorously verifying the Lᵖ lower bounds for the Riesz transform for almost every p∈ (1,∞). Notably, unlike most previous studies, we do not assume the doubling condition or the Poincar\'e inequality. Our approach is based on careful estimates of the Riesz kernel and a method known as harmonic annihilation.