数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 532-550.doi: 10.1007/s10473-024-0209-9

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THE WEIGHTED KATO SQUARE ROOT PROBLEM OF ELLIPTIC OPERATORS HAVING A BMO ANTI-SYMMETRIC PART

Wenxian MA, Sibei YANG*   

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
  • 收稿日期:2022-06-22 修回日期:2023-03-07 出版日期:2024-04-25 发布日期:2024-04-16
  • 通讯作者: *Sibei YANG, E-mail: yangsb@lzu.edu.cn
  • 作者简介:Wenxian MA, E-mail: mawx2021@lzu.edu.cn
  • 基金资助:
    Gansu Provincial National Science Foundation (23JRRA1022), the National Natural Science Foundation of China (12071431), the Fundamental Research Funds for the Central Universities (lzujbky-2021-ey18) and the Innovative Groups of Basic Research in Gansu Province (22JR5RA391).

THE WEIGHTED KATO SQUARE ROOT PROBLEM OF ELLIPTIC OPERATORS HAVING A BMO ANTI-SYMMETRIC PART

Wenxian MA, Sibei YANG*   

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
  • Received:2022-06-22 Revised:2023-03-07 Online:2024-04-25 Published:2024-04-16
  • Contact: *Sibei YANG, E-mail: yangsb@lzu.edu.cn
  • About author:Wenxian MA, E-mail: mawx2021@lzu.edu.cn
  • Supported by:
    Gansu Provincial National Science Foundation (23JRRA1022), the National Natural Science Foundation of China (12071431), the Fundamental Research Funds for the Central Universities (lzujbky-2021-ey18) and the Innovative Groups of Basic Research in Gansu Province (22JR5RA391).

摘要: Let $n\ge2$ and let $L$ be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a $\mathrm{BMO}$ anti-symmetric part in $\mathbb{R}^n$. In this article, we consider the weighted Kato square root problem for $L$. More precisely, we prove that the square root $L^{1/2}$ satisfies the weighted $L^p$ estimates $\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}\le C\|\nabla f\|_{L^p_\omega (\mathbb{R}^n;\mathbb{R}^n)}$ for any $p\in(1,\infty)$ and $\omega\in A_p{(\mathbb{R}^n)}$ (the class of Muckenhoupt weights), and that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)}\le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,2+\varepsilon)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{2+\varepsilon}{p})'}(\mathbb{R}^n)$ (the class of reverse Hölder weights), where $\varepsilon\in(0,\infty)$ is a constant depending only on $n$ and the operator $L$, and where $(\frac{2+\varepsilon}{p})'$ denotes the Hölder conjugate exponent of $\frac{2+\varepsilon}{p}$. Moreover, for any given $q\in(2,\infty)$, we give a sufficient condition to obtain that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)} \le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,q)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{q}{p})'}(\mathbb{R}^n)$. As an application, we prove that when the coefficient matrix $A$ that appears in $L$ satisfies the small $\mathrm{BMO}$ condition, the Riesz transform $\nabla L^{-1/2}$ is bounded on $L^p_\omega(\mathbb{R}^n)$ for any given $p\in(1,\infty)$ and $\omega\in A_p(\mathbb{R}^n)$. Furthermore, applications to the weighted $L^2$-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

关键词: elliptic operator, Kato square root problem, Muckenhoupt weight, Riesz transform, reverse Hölder inequality

Abstract: Let $n\ge2$ and let $L$ be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a $\mathrm{BMO}$ anti-symmetric part in $\mathbb{R}^n$. In this article, we consider the weighted Kato square root problem for $L$. More precisely, we prove that the square root $L^{1/2}$ satisfies the weighted $L^p$ estimates $\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}\le C\|\nabla f\|_{L^p_\omega (\mathbb{R}^n;\mathbb{R}^n)}$ for any $p\in(1,\infty)$ and $\omega\in A_p{(\mathbb{R}^n)}$ (the class of Muckenhoupt weights), and that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)}\le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,2+\varepsilon)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{2+\varepsilon}{p})'}(\mathbb{R}^n)$ (the class of reverse Hölder weights), where $\varepsilon\in(0,\infty)$ is a constant depending only on $n$ and the operator $L$, and where $(\frac{2+\varepsilon}{p})'$ denotes the Hölder conjugate exponent of $\frac{2+\varepsilon}{p}$. Moreover, for any given $q\in(2,\infty)$, we give a sufficient condition to obtain that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)} \le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,q)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{q}{p})'}(\mathbb{R}^n)$. As an application, we prove that when the coefficient matrix $A$ that appears in $L$ satisfies the small $\mathrm{BMO}$ condition, the Riesz transform $\nabla L^{-1/2}$ is bounded on $L^p_\omega(\mathbb{R}^n)$ for any given $p\in(1,\infty)$ and $\omega\in A_p(\mathbb{R}^n)$. Furthermore, applications to the weighted $L^2$-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Key words: elliptic operator, Kato square root problem, Muckenhoupt weight, Riesz transform, reverse Hölder inequality

中图分类号: 

  • 35B45