Some Remarks on the Riesz and reverse Riesz transforms on Broken Line
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| Format: | Preprint |
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2025
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| _version_ | 1866909542365790208 |
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| author | He, Dangyang |
| author_facet | He, Dangyang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Remarks on the Riesz and reverse Riesz transforms on Broken Line He, Dangyang Classical Analysis and ODEs Analysis of PDEs Functional Analysis 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. |
| title | Some Remarks on the Riesz and reverse Riesz transforms on Broken Line |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2503.14885 |