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 $d\mu = |r|^{ d_{ 1}-1}dr$ for $r \le -1$ and $d\mu = r^{ d_{ 2}-1}dr$ 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 Poincar\'e inequality. Our approach is based on careful estimates of the Riesz kernel and a method known as harmonic annihilation.

Categories: math.CA math.AP math.FA

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