THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART  

作　　者：马文贤 杨四辈 Wenxian MA;Sibei YANG(School of Mathematics and Statistics,Gansu Key Laboratory of Applied Mathematics and Complex Systems,Lanzhou University,Lanzhou,730000,China)

机构地区：[1]School of Mathematics and Statistics,Gansu Key Laboratory of Applied Mathematics and Complex Systems,Lanzhou University,Lanzhou,730000,China

出　　处：《Acta Mathematica Scientia》2024年第2期532-550,共19页数学物理学报（B辑英文版）

基　　金：supported by the Key Project of 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);the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).

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 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_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(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||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).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 L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(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 

分 类 号：O177[理学—数学]