Weak type $(1,1)$ bounds for Riesz transforms for elliptic operators in non-divergence form
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916630565486592 |
|---|---|
| author | Song, Liang Zhang, Huohao |
| author_facet | Song, Liang Zhang, Huohao |
| contents | Let $L=-\sum_{i,j=1}^n a_{ij}D_iD_j$ be the elliptic operator in non-divergence form with smooth real coefficients satisfying uniformly elliptic condition. Let $W$ be the global
nonnegative adjoint solution. If $W\in A_2$, we prove that the Riesz transforms $\nabla L^{-\frac{1}{2}}$ is of weak type $(1,1)$ with respect to the measure $W(x)dx$. This, together with $L^2_W$ boundedness of Riesz transforms \cite{EHH}, implies that the Riesz transforms are bounded in $L^p_W$ for $1<p<2$. Our results are applicable to the case of real coefficients having sufficiently small BMO norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18711 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak type $(1,1)$ bounds for Riesz transforms for elliptic operators in non-divergence form Song, Liang Zhang, Huohao Classical Analysis and ODEs Analysis of PDEs Let $L=-\sum_{i,j=1}^n a_{ij}D_iD_j$ be the elliptic operator in non-divergence form with smooth real coefficients satisfying uniformly elliptic condition. Let $W$ be the global nonnegative adjoint solution. If $W\in A_2$, we prove that the Riesz transforms $\nabla L^{-\frac{1}{2}}$ is of weak type $(1,1)$ with respect to the measure $W(x)dx$. This, together with $L^2_W$ boundedness of Riesz transforms \cite{EHH}, implies that the Riesz transforms are bounded in $L^p_W$ for $1<p<2$. Our results are applicable to the case of real coefficients having sufficiently small BMO norm. |
| title | Weak type $(1,1)$ bounds for Riesz transforms for elliptic operators in non-divergence form |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2502.18711 |