Let 0 〈 p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and yang[11] established that the Riesz transforms R j, j = 1,2,..., n, ...Let 0 〈 p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and yang[11] established that the Riesz transforms R j, j = 1,2,..., n, are bounded on Hwp (Rn). In this note we extend this to the general case of weight w in the Muckenhoupt class A.. through molecular characterization. One difficulty, which has not been taken care in [11] consists in passing from atoms to all functions in HwP(Rn). Furthermore, the HwP-boundedness of θ- Calderon-Zygmund operators are also given through molecular characterization and atomic decomposition.展开更多
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...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.展开更多
Let L = -△+V be a Schrodinger operator acting on L^2(R^n), n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp (w) associated wit...Let L = -△+V be a Schrodinger operator acting on L^2(R^n), n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp (w) associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform △↓L^-1/2 associated with L is bounded from our new space HP(w) to the classical weighted Hardy sp ace HP ( w ) when n / (n + 1 ) 〈 p 〈 1 and w ∈ A 1 ∩ R H( 2 / p )'.展开更多
In this paper,we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator■=-Δ+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spa...In this paper,we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator■=-Δ+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces.The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder's inequality.Our results are new and general in many cases of problems.As an application of the boundedness property of these singular integral operators,we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.展开更多
Let T be a strongly singular Calderon-Zygmund operator and b∈L_(loc)(R^(n)).This article finds out a class of non-trivial subspaces BMO_(ω,p,u)(R^(n))of BMO(R^(n))for certain ω∈A_(1),0<p≤1 and 1<u≤∞,such ...Let T be a strongly singular Calderon-Zygmund operator and b∈L_(loc)(R^(n)).This article finds out a class of non-trivial subspaces BMO_(ω,p,u)(R^(n))of BMO(R^(n))for certain ω∈A_(1),0<p≤1 and 1<u≤∞,such that the commutator[b,T]is bounded from weighted Hardy space H_(ω)^(p)(R^(n))to weighted Lebesgue space L_(ω)^(p)(R^(n))if b∈BMO_(ω,p,∞)(R^(n)),and is bounded from weighted Hardy space H_(ω)^(p)(R^(n)) to itself if T^(∗)1=0 and b∈BMO_(ω,p,u)(R^(n))for 1<u<2.展开更多
In this paper, we study integral operators of the formappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.
文摘Let 0 〈 p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and yang[11] established that the Riesz transforms R j, j = 1,2,..., n, are bounded on Hwp (Rn). In this note we extend this to the general case of weight w in the Muckenhoupt class A.. through molecular characterization. One difficulty, which has not been taken care in [11] consists in passing from atoms to all functions in HwP(Rn). Furthermore, the HwP-boundedness of θ- Calderon-Zygmund operators are also given through molecular characterization and atomic decomposition.
基金supported by the Key Project of Gansu Provincial National Science Foundation(23JRRA1022)the National Natural Science Foundation of China(12071431)+1 种基金the Fundamental Research Funds for the Central Universities(lzujbky-2021-ey18)the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).
文摘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.
文摘Let L = -△+V be a Schrodinger operator acting on L^2(R^n), n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp (w) associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform △↓L^-1/2 associated with L is bounded from our new space HP(w) to the classical weighted Hardy sp ace HP ( w ) when n / (n + 1 ) 〈 p 〈 1 and w ∈ A 1 ∩ R H( 2 / p )'.
文摘In this paper,we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator■=-Δ+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces.The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder's inequality.Our results are new and general in many cases of problems.As an application of the boundedness property of these singular integral operators,we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.
基金Supported by the NNSF of China(Grant Nos.11771358,11871101,12171399)。
文摘Let T be a strongly singular Calderon-Zygmund operator and b∈L_(loc)(R^(n)).This article finds out a class of non-trivial subspaces BMO_(ω,p,u)(R^(n))of BMO(R^(n))for certain ω∈A_(1),0<p≤1 and 1<u≤∞,such that the commutator[b,T]is bounded from weighted Hardy space H_(ω)^(p)(R^(n))to weighted Lebesgue space L_(ω)^(p)(R^(n))if b∈BMO_(ω,p,∞)(R^(n)),and is bounded from weighted Hardy space H_(ω)^(p)(R^(n)) to itself if T^(∗)1=0 and b∈BMO_(ω,p,u)(R^(n))for 1<u<2.
基金Supported by Consejo Nacional de Investigaciones Científica y Tcnicas (CONICET)Secretaría de Ciencia y Tecnología de la Universidad Nacional de Córdoba (SECyT-UNC)
文摘In this paper, we study integral operators of the formappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.