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

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外文摘要：Let n ≥ 2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in Rn.In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L1/2 satisfies the weighted Lp estimates ||L1/2(f)||Lpω(Rn)≤C||▽f||Lpω(Rn;Rn)for any p ∈(1,∞)and ω∈ Ap(Rn)(the class of Muckenhoupt weights),and that ||▽f||Lpω(Rn;Rn)≤ C||L1/2(f)||Lpω(Rn)for any p ∈(1,2+ε)and ω ∈ Ap(Rn)∩RH(2+ε/p)'(Rn)(the class of reverse Hölder weights),where e ∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q ∈(2,∞),we give a sufficient condition to obtain that||▽f||Lpω(Rn;Rn)≤ C||L1/2(f)||Lpω(Rn)for any p ∈(1,q)and ω ∈ Ap(Rn)∩ RH(q/p)'(Rn).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform ▽L-1/2 is bounded on Lpω(Rn)for any given p ∈(1,∞)and ω ∈ Ap(Rn).Furthermore,applications to the weighted L2-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

外文关键词：

elliptic operatorKato square root problemMuckenhoupt weightRiesz trans-formreverse Hölder inequality

作者：

马文贤、杨四辈

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作者单位：

School of Mathematics and Statistics,Gansu Key Laboratory of Applied Mathematics and Complex Systems,Lanzhou University,Lanzhou 730000,China

基金：

Key Project of Gansu Provincial National Science FoundationNational Natural Science Foundation of ChinaFundamental Research Funds for the Central UniversitiesInnovative Groups of Basic Research in Gansu Province

项目编号：

23JRRA102212071431lzujbky-2021-ey1822JR5RA391

出版年：

2024

DOI：

10.1007/s10473-024-0209-9

数学物理学报(英文版)

中科院武汉物理与数学研究所

数学物理学报(英文版)

CSTPCD

影响因子：0.256

ISSN：0252-9602

年,卷(期)：2024.44(2)

参考文献量32