L Boundedness of Commutators of Riesz Transforms Associated to Schrödinger Operator
نویسندگان
چکیده
Abstract: In this paper we consider Lp boundedness of some commutators of Riesz transforms associated to Schrödinger operator P = −∆+ V (x) on Rn, n ≥ 3. We assume that V (x) is non-zero, nonnegative, and belongs to Bq for some q ≥ n/2. Let T1 = (−∆ + V ) −1V, T2 = (−∆ + V )−1/2V 1/2 and T3 = (−∆+ V ) −1/2∇. We obtain that [b, Tj ] (j = 1, 2, 3) are bounded operators on Lp(Rn) when p ranges in a interval, where b ∈ BMO(Rn). Note that the kernel of Tj (j = 1, 2, 3) has no smoothness.
منابع مشابه
Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators
In this paper, we discuss the H1 L -boundedness of commutators of Riesz transforms associated with the Schrödinger operator L =−4+ V , where H1 L (R n) is the Hardy space associated with L . We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q > n/2. Let T1 = V (x)(−4+ V )−1, T2 = V 1/2(−4+ V )−1/2 and T3 =∇(−4+ V )−1/2. We prove that, for b ∈ BMO(Rn), the com...
متن کاملCommutators with Lipschitz Functions and Nonintegral Operators
Let T be a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators with T and Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elli...
متن کاملOn Fundamental Solutions of Generalized Schrödinger Operators
We consider the generalized Schrr odinger operator ? + where is a nonnegative Radon measure in R n , n 3. Assuming that satisses certain scale-invariant Kato condition and doubling condition, we establish the following bounds for the fundamental solution of ? + in R n : where d(x; y;) is the distance function for the modiied Agmon metric m(x;)dx 2 associated with. We also study the boundedness ...
متن کاملBoundedness for Riesz transform associated with Schrödinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials
*Correspondence: [email protected] School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China Abstract Let L = – + V be a Schrödinger operator, where is the Laplacian on Rn and the nonnegative potential V belongs to the reverse Hölder class Bq for q≥ n/2. The Riesz transform associated with the operator L is denoted by T =∇(– + V)– 2 and the dual Ri...
متن کاملRiesz transforms through reverse Hölder and Poincaré inequalities
We study the boundedness of Riesz transforms in L for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L for p in some interval [2, 2 + ε), and that L gradient estimates ...
متن کامل